mm-1689 === Subject: Force to be calculated It is simple question apparently. But it seems to me not so easy to wor out, high maths are probably needed. Here is the song: One ball of 1 Kg, held 1 meter above the surface of perfect balance. We let the ball fall without any initial speed. Question: what is the value of the force that will be indicated by needle of the balance in the exact moment of the choc, which is suposed to be 100% elastic (no loss of energy)? At the exact time of first contact, the needle will either read infinite force or zero, depending upon the model. If the impact is a mathematically idealised model that takes zero time, then the function of force with time is a delta function, which has an infinite value at zero and zero value everywhere else. If the impact is modelled in a more physically reasonable way, then the force starts at zero, increases to a maximum, and then drops back to zero. In this case, the important feature for your problem is that it starts at zero. This is a physics question; it really belongs in sci.physics. But there's math involved, so... If the ball and scale are completely rigid, the force at the moment of contact will be of infinite force for an infinitessimal amount of time, as it it were a Dirac delta function. Regardless of how rigid it is, the area under the force curve will be enough impulse to accellerate the ball from its downward velocity to an equal upward velocity -- 8.8 kg-m/s. --Keith Lewis klewis {at} mitre.org The above may not (yet) represent the opinions of my employer. === Subject: Re: Force to be calculated It is simple question apparently. But it seems to me not so easy to wor out, high maths are probably needed. Here is the song: One ball of 1 Kg, held 1 meter above the surface of perfect balance. We let the ball fall without any initial speed. Question: what is the value of the force that will be indicated by needle of the balance in the exact moment of the choc, which is suposed to be 100% elastic (no loss of energy)? > This is a physics question; it really belongs in sci.physics. But there's > math involved, so... > If the ball and scale are completely rigid, the force at the moment of > contact will be of infinite force for an infinitessimal amount of time, as > it it were a Dirac delta function. > Regardless of how rigid it is, the area under the force curve will be enough > impulse to accellerate the ball from its downward velocity to an equal > upward velocity -- 8.8 kg-m/s. Yes. As you say, conservation of energy dictates the area under the force curve, but not its shape or its maximum value. To say what the needle reads, you need a more detailed model that can tell you how the transient event translates into a reading. - Randy === Subject: Re: Force to be calculated > But it seems to me not so easy to work out but it is easy .. > high maths are probably needed. No, you need to realize that spring deformation x is to be found from a quadriatic equation given below, from combining potential energies of spring in the balance and mgh. .. > 1 Kg, held 1 meter(h) above the surface of perfect balance.We let > the ball fall without any initial speed. Question: what is the > value of the force that will be indicated by needle of the balance > in the exact moment of the choc, which is suposed to be 100% > elastic (no loss of energy)? HINT: Equate maximum potential energy to spring's potential energy by deformation through x. mg(h+x)=kx^2/2, F= k*x. Note that even if h=0, the balance registers double weight at impact time due to sudden load introduction. === Subject: Re: Force to be calculated Everybody who has done a bungee jump will correctly guess that one will undergo a 2mg apparent force (i.e. twice one's weight) at the deepest point of the jump. The balance will still register zero weight at the moment of impact; == not until the falling body has reached == its deepest point the balance will register double weight. BTW, I never did a bungee jump. In my boyhood and student's times I performed this kind of experiments as well as one can do with common household and university physics stuff. Happy experimenting: Johan E. Mebius >>But it seems to me not so easy to work out >> >but it is easy .. >>high maths are probably needed. >> >No, you need to realize that spring deformation x is to be found from a >quadriatic equation given below, from combining potential energies of >spring in the balance and mgh. >>1 Kg, held 1 meter(h) above the surface of perfect balance.We let >>the ball fall without any initial speed. Question: what is the >>value of the force that will be indicated by needle of the balance >>in the exact moment of the choc, which is suposed to be 100% >>elastic (no loss of energy)? >> >HINT: Equate maximum potential energy to spring's potential energy by >deformation through x. mg(h+x)=kx^2/2, F= k*x. Note that even if h=0, >the balance registers double weight at impact time due to sudden load >introduction. === Subject: Re: Force to be calculated <41C2073E.9010001@xs4all.nl> > Everybody who has done a bungee jump will correctly guess that one will > undergo a 2mg apparent force (i.e. twice one's weight) at the deepest > point of the jump. If so, that must be a property of the bungee cord material, not true in general. Let us consider a completely elastic bungee cord (if it were an ideal elastic, then you would bounce back up to exactly your original height. I don't think this is the case with actual bungee jumping.) The bungee cord has an elastic constant k. Assume you have velocity v when you start stretching it. The cord will continue to stretch until all of your kinetic energy 0.5*m*v^2 has been converted into potential energy 0.5*k*x^2, or x = v*sqrt(m/k). At this point, the force exerted by the bungee cord is kx = k*v*sqrt(m/k) = v*sqrt(mk). The acceleration will be F/m = v*sqrt(k/m). Two things to observe about this: 1. The force and the acceleration depend on the elastic constant. A stiffer cord (higher k) will cause larger force. 2. Even with the same bungee cord, the acceleration depends on mass (a larger mass will end up experiencing less acceleration). The cord may have the right elastic constant that an average adult will experience 2 g's, but above and below average adults will experience different accelerations. The above are valid for any elastic collision, not just bungee jumping. - Randy === Subject: Re: Force to be calculated >> Everybody who has done a bungee jump will correctly guess that one > will >> undergo a 2mg apparent force (i.e. twice one's weight) at the deepest >> point of the jump. > If so, that must be a property of the bungee cord material, not > true in general. > Let us consider a completely elastic bungee cord (if it > were an ideal elastic, then you would bounce back up to > exactly your original height. I don't think this is > the case with actual bungee jumping.) This is not enough to determine the behavior of the cord. One can have a perfectly elastic cord without having it conform to the ideal spring law (force = k * displacement). > The bungee cord has an elastic constant k. Assume you > have velocity v when you start stretching it. The cord > will continue to stretch until all of your kinetic energy > 0.5*m*v^2 has been converted into potential energy 0.5*k*x^2, > or x = v*sqrt(m/k). At this point, the force exerted by > the bungee cord is kx = k*v*sqrt(m/k) = v*sqrt(mk). The > acceleration will be F/m = v*sqrt(k/m). Ok. So you're assuming a cord that obeys the ideal spring law. And you're assuming that we have some non-zero velocity when we start stretching the cord. The latter is indeed a component of real-world bungee jumping -- the cord has significant length, even when not under tension. But your equations are wrong. Gravity doesn't magically stop working when the bungee cord starts to tighten. You have a component of gravitational energy given by mgx. Please redo your calculations accordingly. [Note that bungee cord, like any rope, does not support compressive loads. It will only obey the ideal spring law when under tension. But, since we are focusing our attention on the portion of the jump where all the slack has been taken up and the bungee is under tension, this is no obstacle to a correct analysis] > Two things to observe about this: > 1. The force and the acceleration depend on the elastic > constant. A stiffer cord (higher k) will cause larger > force. Yup. That much is obvious. At least for a non-zero starting velocity. > 2. Even with the same bungee cord, the acceleration depends > on mass (a larger mass will end up experiencing less acceleration). > The cord may have the right elastic constant that an > average adult will experience 2 g's, but above and below > average adults will experience different accelerations. I think you are wrong here. The _minimum_ (over all adult weights) of the maximum (over all points in the jump) acceleration is guaranteed to be at least two g's for an ideal (force = k * displacement) bungee. This minimum is exactly achieved in the case of zero initial velocity > The above are valid for any elastic collision, not just > bungee jumping. Do you know what elasticity means? It is a measure of the degree to which the energy after the rebound matches the energy prior to impact. It is not a measure of the degree to which restoring force is proportional to displacement. It _is_ a measure to the degree to which the restoring force is _any function of_ the displacement, independent of all other factors and, in particular, a measure of the degree to which the restoring force is independent of direction and speed. John Briggs === Subject: Re: help on a recursive function >> I'm interested on the behaviour on a H-length circular queue, empty >> before i=0, equal to P(0)=a and P(i)= b Sum(j=i-1, i-H [P(j)]), that is: >> P(i) = a(1+b)^i i=0, ..., H-1 >> P(i) = (1+b)P(i-1) - b P(i-H) i>= H >> I got just the following results: >> P(i) = a (b(i+1) - 1)/(b - 1) *if H=1* > Oh? If H = 1, I get, for i > 0, > P(i) = (1+b)P(i-1) - bP(i-1) = P(i-1) = ... = a oops... excuse me Rick, I meant with: P(i) = a (b^(i+1) - 1)/(b - 1) (I forgot the ^) > Robert has already given you one answer. Here's another > If k = mH + n, with 0 <= n < H, m >= 0, then > P(k) = > (1+b)^n sum{i=0}^{m} (-1)^i comb{(m-i)H + n + i}{i} b^i (1+b)^{(m-i)H} > Admittedly, that's not in a very closed form. oh, it's perferct :-) I tried H by H with a few results... Could you help me understanding *how* have you got in this? Smoll Est === Subject: Re: help on a recursive function >> If k = mH + n, with 0 <= n < H, m >= 0, then >> P(k) = >> (1+b)^n sum{i=0}^{m} (-1)^i comb{(m-i)H + n + i}{i} b^i (1+b)^{(m-i)H} >> Admittedly, that's not in a very closed form. > oh, it's perferct :-) > I tried H by H with a few results... > Could you help me understanding *how* have you got in this? Pure luck. :-) Actually, all I did was the usual stuff: 1. Write the values for some P(i) 2. Identify a pattern (this is the magic step) 3. Prove that your guess is right (often by induction). But I'm sure that I'm not telling you anything you didn't already know. Rick === Subject: Online equation system solver I need it to solve systems of 3 or more quadratic equations. === Subject: Re: Velocity problem I have seen derivatives, we just never used the ODE acronym to describe them. This course made a lot of assumptions about our previous math experiences, non of which were listed as prerequisites, so hense my stupid questions. See my other post about the answer, and thank you for your help! Elana === Subject: Re: Velocity problem Yep, that's the answer. Since acceleration is constant, velocity is Elana === Subject: Re: Velocity problem course, and calculus was not listed as a pre-requisite. I took calculus as a freshman in college 7 years ago, and, as you can see, hardly remember any. In any case, I solved the problem, and the answer is: v^2 = kH --> v=sqrt(kH) dv/dt (acceleration) = k/2sqrt(kH) dH/dt ----> k/2 because sqrt(kH) is integral (velocity) is a variable to the first power, and thus it is a linear function of time. Looks pretty simple, right? I just started to panic a little when I saw the long-forgotten calculus in the homework, and so I made a post without proper review. I appologize for looking like an idiot :). Elana === Subject: Re: Velocity problem > course, and calculus was not listed as a pre-requisite. Yeah, obviously there's a little disconnect between the course content and the course description. It happens. > I took > calculus as a freshman in college 7 years ago, and, as you can see, > hardly remember any. In any case, I solved the problem, and the answer > is: Congratulations. Feels good, doesn't it? > Looks pretty simple, right? Well no, since I missed the easy approach and was trying to kill a mosquito with a nuclear weapon. By the way, differential equations are usually seen in a course after 4 semesters of calculus, so don't feel bad if you haven't heard of them. They're used a lot in science and engineering. > I just started to panic a little when I > saw the long-forgotten calculus in the homework, and so I made a post > without proper review. I appologize for looking like an idiot :). Not an idiot at all, merely someone in over their head. But obviously I need not have worried, since a few minutes of tutorial got you what you needed. You'll be fine. - Randy === Subject: Re: Earth and Venus - Five Petal Rose ETAuAhUAuEGnVaU4T9fwhgzOA4ghJbTJKPYCFQC8HwGw8gWghcr8zMZhKXO6FNhCvw== The set of rational numbers is dense, so there are rational numbers arbitrarily close to the real ratio of orbital periods. In a word, coincidence. --OL === Subject: Re: Earth and Venus - Five Petal Rose <2166-41C1AAA8-224@storefull-3251.bay.webtv.net> > The set of rational numbers is dense, so there are rational numbers > arbitrarily close to the real ratio of orbital periods. In a word, > coincidence. More like serendipity, 13/8 is made up of Fibonacci numbers 13 and 8. Also the orbits form a rotating pentagram, more 'coincidential' evidence of the Golden Ratio. === Subject: Re: Earth and Venus - Five Petal Rose Originator: richard@cogsci.ed.ac.uk (Richard Tobin) >More like serendipity, 13/8 is made up of Fibonacci numbers 13 and 8. >Also the orbits form a rotating pentagram, more 'coincidential' >evidence of the Golden Ratio. The pentagram is a consequence of the 13 and the 8, so it's not *more* evidence. -- Richard === Subject: Re: Earth and Venus - Five Petal Rose <2166-41C1AAA8-224@storefull-3251.bay.webtv.net> > The pentagram is a consequence of the 13 and the 8, so it's not > *more* evidence. I didn't mean 'more' par se, I only referenced a known fact....geez...are you going to correct all of my politically incorrect and grammatical errors? I ain't stoopid...I speek inglish much gooder than yu... === Subject: Re: Earth and Venus - Five Petal Rose <2166-41C1AAA8-224@storefull-3251.bay.webtv.net> It was talked about rotation around sun. This might be another co-incidence for earth-venus: the rotation around it's axis with reference to earth (or earth-moon): Quote : Venus' rotation is somewhat unusual in that it is both very slow (243 Earth days per Venus day, slightly longer than Venus' year) and retrograde . In addition, the periods of Venus' rotation and of its orbit are synchronized such that it always presents the same face toward Earth when the two planets are at their closest approach. Whether this is a resonance effect or merely a coincidence is not known. http://www.nineplanets.org/venus.html Have fun Hero === Subject: Re: Earth and Venus - Five Petal Rose > Congratulations. You rediscovered something, the old astronomers of > the past observed too : If You mark the points of reappearanve of > Venus between the stars of the zodiac, one will get the > pentagram > a fivepointed star, as a symbol still used in US of A and else. > Of course there's a lot of mystic mist around this. > Reference In German. > Erich Zehren, Das Testament der Sterne > Actually one can tell the future with this knowledge - > but nowadays predicting the date and place of reappearance > of Venus will not be admired, as in the past. > But still, discovering these things can make one happy. > Good luck with more of this > Hero Here's another one that is makes me happy. The boxed diameter of the earth and the circle produced by adding the radius of the earth and the moon shows an instance of 'squaring the circle'. Such finds as these combine math, physics, and philosophy and its primarily responsible for yanking my interest from QM and mathematical physics. At least these mathematical relations have relevance to aesthetics and reason. === Subject: Re: sum of 1/((k+1)*(k+2)) > Telescoping series. Note that 1/((k+1)*(k+2)) = 1/(k+1) - 1/(k+2). have tried this method to no avail....any help please? === Subject: Re: sum of 1/((k+1)*(k+2)) Telescoping series. Note that 1/((k+1)*(k+2)) = 1/(k+1) - 1/(k+2). > have tried this method to no avail....any help please? If you sum 1/(k+1) - 1/(k+2) from k=1 to k=n, all but two of the -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: Deep Thoughts # 17: Liar Paradox is a Formal Metamathematical Theorem <4ujtr05euj9l4q0fcpb8q2sbp141r399op@4ax.com> > You _really_ need to work on the irony detector. Truth > and provability are _different_ things - when you define > one to be the other you exhibit amazing ignorance of the > most basic issues. > ************************ > > David C. Ullrich The definitions of Turing Machines and Lambda Calculus are very different, too, aren't they? > For heaven's sake, at some point you should just accept that > you've made a fool of yourself and drop it. I mean you have > plenty of experience with this phenomenon... (Only at parties.) You are the one who claimed that what Smullyan and I are doing is ignorant. I only agreed with Smullyan. (His 4 texts were the source of many of the theorems that my formalism represents.) Your statement above concerning the ignorance of defining truth and provability to coincide makes sense only in the context of some particular system - the system in which they are being defined. That is exactly what Smullyan is doing. (It sounds to me like you are trying to retract the statement by substituting an alternate explanation of what it means - but haven't come up with a meaningful alternative.) > Ok. To answer your question: Yes, those two definitions > are very different. > I can't imagine what your point is. Two points: 1. Having different definitions doesn't mean that two things are different. 2. Truth and provability need not be defined with different systems. They are defined with different systems only because those who describe them as such just haven't developed a single system for both. The notion of recursive functions is the one system developed to define the functionality of both Turing Machines and Lambda Calculus. Likewise, truth and provability (two sets of sentences) can be formalized within a single system. Both your premise (2) and reasoning (1) are wrong. (I could fault you less for mistake 2 because that, of course, is current conventional wisdom.) > (i) If someone asked for the definition of a Turing Machine > and someone else replied with the definition of the Lambda > Calculus that person would indeed be exhibiting blithering > confusion. Yes! And if they say that any person who is wrong about question A must have property B, in all fairness, does that mean that if they are wrong about A then they themself possess property B? > (ii) On the other hand, it's true that the set of functions > computable by TM's is the same as the set of functions > computable by the LC. _Guessing_ what your point might > be here, I have to point out that truth and provability > are _not_ equivalent. What does it mean to say that truth and provability are or are not equivalent? It depends on the system in which they are being defined. It is meaningless to talk about truth and provability outside of some particular system. They coincide in some systems and they don't coincide in others. (This sounds like more of the ill-formed alternate explanation of your original statement.) > (Validity and provability are equivalent. And truth _in > that one particular model_ is equivalent to provability > _in that one particular formal system_. A very curious > formal system, btw, since there's no procedure to recognize > whether a proof is valid.) How could something be a proof but not be a valid proof? You seem to be trying to say either: 1. We cannot recognize if a given sequence of formulas is a proof. 2. We cannot recognize if a given proof proves a given sentence. 3. We cannot recognize if a given sequence of formulas is a proof of a given sentence. But none of these is definitely true anyway. Given that the set of theorems is not r.e. (system N), any of these can be false if the other two are true. In any case, it is not curious to Smullyan (or me.) Every non-axiomatizable theory has a non-recursive proof predicate, assuming that the sets of (the Godel numbers of) formulas and of proofs are r.e. Is PA very curious because there's no procedure to decide whether a sentence is a theorem? By your reasoning, the user of system N, which is non-axiomatizable and complete, could equally well claim that PA is very curious because its is axiomatizable but not complete. The true lesson, of course, is that there is a trade-off. (The same trade-off that I described in my 2 MetaMathematical Theorems, actually.) It's not that the true and provable sentences can't be the same. It is the fact that there are consequences - both good and bad - if they are the same. After all, determining trade-offs in formal systems is central to MetaMathematics. If a system has property A then it must have property B. Nothing curious about that. C-B We call a formula F(v1,. . .,vn) correct if for every n-tuple (a1,...,an), the sentence F(a1,. . .,an) is true. We let N be the first-order system whose axioms are all the correct formulas (this includes all logically valid formulas.) Thus, the provable formulas of N are nothing more than the axioms of N. Recursion Theory for Metamathematics - Raymond M. Smullyan, 1993 In applications to semantical systems, T will be the truth set, and in syntactical systems, T will be the set of theorems. - Theory of Formal Systems, Chapter III. Incompleteness and Undecidability > ************************ > David C. Ullrich === Subject: Re: pseudo normal random number generator >:-> >I don't really care as much which distriution that I am using, just one >that has the first 4 moments defined, and the higher moments fall >similarly in line with the Normal distribution. Just what do you mean by this statement? If the higher moments agree with those of a normal distribution, the distribution must be normal. Also, unless the first four moments force a two-point distribution, the extreme distributions from the first four moments are three-point distributions, with none of the points roots of the orthogonal polynomial of degree 2. I would be surprised if these three-point distributions are what you desire. Any of the infinite >solutions would be great, I am just not certain how to attack this... >The real motivation is to have a random number generator for a >leptokurtoic, skewed pseudo Normal distribution for use in monticarlo >methods. I suspect you have some kind of reasonable class of distributions of interest. You might try random variables of the form aX-bY, where X and Y are Gamma random variables, easy to generate. But as I said, it is not clear what sort of a distribution is wanted. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Category Theory A set exists and its cause is the categorified set. Meaning the next abstracted set is always in relation to the set of interest. What is debated is the exact relation of set to its abstracted form. And to call the symbol in predicate the next abstraction without debating, is a common means of learning incorrectly, category theory. A foundational relation is postulated to exist, such that all relations may be stated. And in the realm of science, objective relations are used to solve this long standing mystery. Only relations testable in science are to be given to mathematics. IN predicate, set A -> a. And here the relation denoted by, ->, is the correct one. A good teacher will also allow the relations actual cognition, except it is as a trained knowledge, without hardly any symbolic expression to support. Meaning, a good teacher is critical to the good user of the science of category theory. In order to give words to the symbol, ->, here is its definition. ->, a set exists independently of its contents, and the cause to contents is likewise independent of existence. Making the complex relationship of existence to value. And the meaning of dependency and independency identical. A relation of true objective causality exists in nature such that the relation of existence to value is found always independent!!!!! A foundational nonlinear relation is called the, ->. Douglas Eagleson- Gaitherbsurg, MD USA === Subject: Help Needed Hi everyone, I'm currently in the process of forming a web-based security company, that will offer a wide range of services to clients, and I'm searching for people that would be interested into going into partnership with me. I.89m looking for people who specialize, or are interested in any of the following areas: Networking System vulnerability/ penetration testing C/ C++ / C# programming Java Programming Perl Programming Python Programming Electronics/ Hardware design Cryptography Telematics If you.89re interest, please get in touch with me at webmaster@afis.tk Barry *-----------------------* www.GroupSrv.com *-----------------------* >lol - here's the quote that shows how hopeless the problem truly is: >[The American teacher] was especially taken aback by the textbook. By >grades seven and >eight, kids in the Singapore program are doing high-school-level >algebra. I thought, wow, that's complicated -- even for me, says Mr. >Keating. He was eventually won over when he saw how enthusiastic his >own students became about math. >Our own math teachers can't even do 7th grade math. Christ. You may be surprised, but I am not. But it is worse than that; the new math, which succeeded in teaching mathematical concepts, not just how to compute, to children, could not be learned by the teachers, even with major attempts to teach them. The case is no different now. And the schools' emphasis on memorization and routine is greater. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 [The American teacher] was especially taken aback by the textbook. By grades seven and eight, kids in the Singapore program are doing high-school-level algebra. I thought, wow, that's complicated -- even for me, says Mr. Keating. He was eventually won over when he saw how enthusiastic his own students became about math. Our own math teachers can't even do 7th grade math. Christ. > You may be surprised, but I am not. > But it is worse than that; the new math, which succeeded > in teaching mathematical concepts, not just how to compute, > to children, could not be learned by the teachers, even > with major attempts to teach them. The case is no > different now. And the schools' emphasis on memorization > and routine is greater. Whew, just in the nick of time, hand helded graphic calculators that'll do all that math for them. Is this why such emphasis is placed upon graphic calculators in math classes? Is mathematics also, like the Democrats, now on the endangered species list? >[The American teacher] was especially taken aback by the textbook. By >grades seven and >eight, kids in the Singapore program are doing high-school-level >algebra. I thought, wow, that's complicated -- even for me, says Mr. >Keating. He was eventually won over when he saw how enthusiastic his >own students became about math. >Our own math teachers can't even do 7th grade math. Christ. >> You may be surprised, but I am not. >> But it is worse than that; the new math, which succeeded >> in teaching mathematical concepts, not just how to compute, >> to children, could not be learned by the teachers, even >> with major attempts to teach them. The case is no >> different now. And the schools' emphasis on memorization >> and routine is greater. >Whew, just in the nick of time, hand helded graphic calculators >that'll do all that math for them. Is this why such emphasis >is placed upon graphic calculators in math classes? How important is it that people can do arithmetic quickly? If someone understands what addition and multiplication (and the other operations) mean, and the notation, that person can calculate the answer, assuming a sufficiently fast algorithm is used, and no errors are made. Using a calculator is appropriate, if it is understood, and also if the effect of rounding, etc., is understood. >Is mathematics also, like the Democrats, >now on the endangered species list? Mathematics is on the endangered species list, but not because people do not know arithmetic. The current emphasis on getting answers and on applications may well eliminate anyone learning basic concepts. The important parts of mathematics are the concepts and being able to use them, and the language of variables, to formulate problems. I do not care if the engineer knows how to solve a differential equation; it is more important that the formulation be correct, even if he cannot solve it, than that it be formulated so he can. In statistics, that situation is the common one. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 > Whew, just in the nick of time, hand helded graphic calculators > that'll do all that math for them. Is this why such emphasis > is placed upon graphic calculators in math classes? When I teach pre-calc, I have two major uses for graphing calculators. First, students can use them to solve optimization word problems without the use of calculus. Second, students can numerically solve equations of the form Sin(x) = x or 1-x = lnx. A third use, although not as important (imho), is using the graphing capability to verify issues that should be understandable from algebra and analysis -- limiting character of functions, existence of asymptotes, etc. My $0.02. DM >Second, students can numerically solve equations of the form Sin(x) = x Some of us would say that this is an indication of a problem: if you need a graphing calculator to solve this, you don't understand the sine function. dave (Point take, though, about solving 1-x = ln(x) .) Whew, just in the nick of time, hand helded graphic calculators that'll do all that math for them. Is this why such emphasis is placed upon graphic calculators in math classes? > When I teach pre-calc, I have two major uses for graphing calculators. > First, students can use them to solve optimization word problems without > the use of calculus. For example? > Second, students can numerically solve equations of the form Sin(x) = x > or 1-x = lnx. Unless the class is numerical analysis or the topic being taught is Newton's method. > A third use, although not as important (imho), is using the graphing > capability to verify issues that should be understandable from algebra > and analysis -- limiting character of functions, existence of > asymptotes, etc. The graphing capacity is wow, just don't use it for problems. I went thru the graphs with equations in the mathematical handbook. It gave me intuition how formula and graph relate. Using just the graphic calculator would undermine that skill. So have class, homework or graphic calculator contest, dream up wow graphs and give equations. Where intuition is important is look at x^2 cos 1/x Ensued was conservation: It's a parabola. No it's not, zoom in on it. It's still a parabola. No, zoom in on it again. I see something crinkly. Good, zoom in on it again. Oh yea, wow! >> When I teach pre-calc, I have two major uses for graphing calculators. >> First, students can use them to solve optimization word problems without >> the use of calculus. > For example? Suppose that an aluminum can must hold 12 ounces of fluid. It costs $0.03 per square inch to produce/manufacture the top and bottom portion of the can and $0.02 per square inch to produce/manufacture the cylindrical side of the can. Find the dimensions of the can with the least cost. Then consider how your answer depends on the differential cost between top/bottom construction costs and side construction costs. Obviously, there are many other examples of these style (like designing Norman windows). >> Second, students can numerically solve equations of the form Sin(x) = x >> or 1-x = lnx. > Unless the class is numerical analysis or the topic being taught is > Newton's method. I don't cover numerical analysis or Newton's method in pre-calc. I typically use Mathematica or Matlab in numerical analysis courses. > The graphing capacity is wow, just don't use it for problems. > I went thru the graphs with equations in the mathematical handbook. > It gave me intuition how formula and graph relate. Using just > the graphic calculator would undermine that skill. > So have class, homework or graphic calculator contest, dream up wow graphs > and give equations. Where intuition is important is look at > x^2 cos 1/x > Ensued was conservation: > It's a parabola. > No it's not, zoom in on it. > It's still a parabola. > No, zoom in on it again. > I see something crinkly. > Good, zoom in on it again. > Oh yea, wow! And then what? Clearly the issue here is that you are considering f(x)g(x) where one function has a constant limit at 0 and the other at infinity. Don't you think that a student could play these mental games with pencil, paper, and a basic understanding of the graph/periodicity/boundedness of cos(x), x^2, and the concept of composition? I see those fundamental underlying ideas as being lost. That said, I have no problem with students using a calculator and this type of problem to think their way through a rigorous understanding of these concepts. I do this type of laboratory as group work. And interesting question that does not require a calculator at all is to give a fairly asymmetric graph and say it corresponds to a function w(x). Then ask the student to graph f(x) = -2w(4(x+3))+6. DM DM > When I teach pre-calc, I have two major uses for graphing calculators. > First, students can use them to solve optimization word problems without > the use of calculus. For example? > Suppose that an aluminum can must hold 12 ounces of fluid. It costs $0.03 > per square inch to produce/manufacture the top and bottom portion of the can > and $0.02 per square inch to produce/manufacture the cylindrical side of the > can. Find the dimensions of the can with the least cost. Then consider how > your answer depends on the differential cost between top/bottom construction > costs and side construction costs. > Obviously, there are many other examples of these style (like designing > Norman windows). 12 oz * in^3/oz = pi.r^2 h C = 2pi.rh.c_s + 2pi.r^2 c_e You're expecting the graphic calculator to set up the equations??? > Second, students can numerically solve equations of the form Sin(x) = x > or 1-x = lnx. Unless the class is numerical analysis or the topic being taught is Newton's method. > I don't cover numerical analysis or Newton's method in pre-calc. I > typically use Mathematica or Matlab in numerical analysis courses. Practical applications of numerical analysis would benefit from computers, but to teach the basic concepts? So have class, homework or graphic calculator contest, dream up wow graphs and give equations. Where intuition is important is look at x^2 cos 1/x > And then what? Clearly the issue here is that you are considering f(x)g(x) > where one function has a constant limit at 0 and the other at infinity. > Don't you think that a student could play these mental games with pencil, > paper, and a basic understanding of the graph/periodicity/boundedness of > cos(x), x^2, and the concept of composition? I see those fundamental > underlying ideas as being lost. Yes. I do too. > That said, I have no problem with students using a calculator and this > type of problem to think their way through a rigorous understanding of > these concepts. I do this type of laboratory as group work. > And interesting question that does not require a calculator at all is to > give a fairly asymmetric graph and say it corresponds to a function w(x). > Then ask the student to graph f(x) = -2w(4(x+3))+6. ;-) Talk about electronic engineering and wave equations. Does not require? Do you mean, cannot be solved graphic calculator? Any other problems to demostrate the inability of calculators? >You may be surprised, but I am not. >But it is worse than that; the new math, which succeeded >in teaching mathematical concepts, not just how to compute, >to children, could not be learned by the teachers, even >with major attempts to teach them. Solution: Fire those that can't, hire those that can. Simple enough, no? Rich >>You may be surprised, but I am not. >>But it is worse than that; the new math, which succeeded >>in teaching mathematical concepts, not just how to compute, >>to children, could not be learned by the teachers, even >>with major attempts to teach them. >Solution: Fire those that can't, hire those that can. Simple enough, no? This could never have been done, but it could have been effectively done when it was surprisingly found more than 40 years ago. The teachers had tenure, and the schools of education were definitely not going to admit that they had turned out incompetents. If the mathematicians had gone to the public with the clear statement: Your children can learn this; too many of the teachers cannot. Regardless of how we should be teaching mathematics to your children, should these be doing it? a program to remove those who could not learn it from teaching mathematics, and there was an actual movement to have mathematics teacher as an elementary school specialty, I believe it would have succeeded. Instead, there were attempts to reduce it to the level the teachers could understand, and this did not succeed, but by that time, it had been reduced to uselessness. Unlike phonics, it was not possible to produce materials so parents could teach their children, and the public did not even understand the difference, and most did not care whether their children learned anything besides cookbook arithmetic. This is not the only problem. Keeping bright children back is more serious. The educational establishment is oriented to trivial pursuit and memorization, and with age grouping forcing things down, cannot change. Only affordable schools not subject to government control can help at this time. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 > Instead, there were attempts to reduce it to the level > the teachers could understand, and this did not succeed, > but by that time, it had been reduced to uselessness. > Unlike phonics, it was not possible to produce materials > so parents could teach their children, and the public > did not even understand the difference, and most did > not care whether their children learned anything besides > cookbook arithmetic. > This is not the only problem. Keeping bright children > back is more serious. The educational establishment is > oriented to trivial pursuit and memorization, and with > age grouping forcing things down, cannot change. Only > affordable schools not subject to government control > can help at this time. How about 'exact science clubs', or private lessons in some other form, separate from school? Like currently is done in the music and sports world? (How many basketball players could've learned their sport by only playing basketball in high school? And are there any professional violin players who started to play violin in high school and never did anything besides that?) -- Herman Jurjus >> Instead, there were attempts to reduce it to the level >> the teachers could understand, and this did not succeed, >> but by that time, it had been reduced to uselessness. >> Unlike phonics, it was not possible to produce materials >> so parents could teach their children, and the public >> did not even understand the difference, and most did >> not care whether their children learned anything besides >> cookbook arithmetic. >> This is not the only problem. Keeping bright children >> back is more serious. The educational establishment is >> oriented to trivial pursuit and memorization, and with >> age grouping forcing things down, cannot change. Only >> affordable schools not subject to government control >> can help at this time. >How about 'exact science clubs', or private lessons in some other form, >separate from school? You would have to separate the whole science curriculum from the schoolwork, and avoid the schools entirely. Like currently is done in the music and sports >world? (How many basketball players could've learned their sport by only >playing basketball in high school? Most started by playing on the basketball teams in high school, and then went on to get athletic scholarships based on their high school basketball performance, and played basketball in college. They were on the teams, and got special coaching both in high school and college, but had little special coaching outside school. The school athletic coaches may or may not do other teaching. At most universities, they have special contracts, with no tenure. And the athletes do not take the regular physical education, but take instead special classes in their sports. If we had similar coaches for mathematics and science, we could do it. But as these do not involve physical development, as sports does, the groups would be more multi-aged. It is also necessary to start young in academics as distinguished from athletics; most coaches recommend against serious athletics before sufficient physical development has occurred. This is not the case in academics, although those ignorant of the real subject matter seem to think it is. And are there any professional violin >players who started to play violin in high school and never did anything >besides that?) Some may have started in high school, but most started much earlier. Mozart was composing at age six. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 >If we had similar coaches for mathematics and science, >we could do it. I used to think this would be a great idea. Now I'm not so sure I'd want to perform drug tests on the students before they could take the Putnam exam. > Like currently is done in the music and sports >>world? (How many basketball players could've learned their sport by only >>playing basketball in high school? >Most started by playing on the basketball teams in high school, You guys don't have vacant lots over there in W. Lafayette? Take a look some time. I assure you the basketball stars of tomorrow are not learning to play ball in high school; they're shooting hoops as six-year-olds. > And are there any professional violin >>players who started to play violin in high school and never did anything >>besides that?) >Some may have started in high school, but most started >much earlier. Mozart was composing at age six. want to skew this discussion by mentioning the Mozarts (not to mention the Michael Jordans). The question of how to nurture the talents of the most gifted is a difficult one but is distinct from the issues which prompted this thread. There have been two international studies released this week (PISA and TIMSS) which showed disparities from country to country with regard to the AGGREGATE performance of their students, across all skill levels. One can take a Libertarian position and argue that only the success of the top 5% will matter in society anyway, but if one is concerned about the education of the general population, then the extraordinary talents of young Wolfie are not really relevant. Surely most students in the US could do better in math with a better education, but let's remember to focus on the 95% who are _not_ headed towards degrees in mathematics, statistics, etc. dave >>If we had similar coaches for mathematics and science, >>we could do it. >I used to think this would be a great idea. Now I'm not so sure I'd want >to perform drug tests on the students before they could take the Putnam exam. >> Like currently is done in the music and sports >world? (How many basketball players could've learned their sport by only >playing basketball in high school? >>Most started by playing on the basketball teams in high school, >You guys don't have vacant lots over there in W. Lafayette? Take a look >some time. I assure you the basketball stars of tomorrow are not learning >to play ball in high school; they're shooting hoops as six-year-olds. >> And are there any professional violin >players who started to play violin in high school and never did anything >besides that?) >>Some may have started in high school, but most started >>much earlier. Mozart was composing at age six. >want to skew this discussion by mentioning the Mozarts (not to mention >the Michael Jordans). The question of how to nurture the talents of >the most gifted is a difficult one but is distinct from the issues >which prompted this thread. There have been two international studies >released this week (PISA and TIMSS) which showed disparities from >country to country with regard to the AGGREGATE performance of their >students, across all skill levels. One can take a Libertarian position >and argue that only the success of the top 5% will matter in society >anyway, but if one is concerned about the education of the general >population, then the extraordinary talents of young Wolfie are not >really relevant. Surely most students in the US could do better in >math with a better education, but let's remember to focus on the 95% >who are _not_ headed towards degrees in mathematics, statistics, etc. It is not the case that the 95% should all have the same education, either, nor should they go at the same rate in each subject. But even more so, all except the low end can do more and better, but not as they are now being taught. The motivation for the new math was the observation that even bright children taught the mechanics of arithmetic would understand little about the meanings of numbers; I do not know why they only used the cardinal approach, but I guess it was because this was apparently simple. But the ordinal one is the one that is easily axiomatized, and which can easily be shown to be categorical. We CAN teach concepts quickly; it would help to teach the key linguistic idea of variable with beginning reading. Yes, it is linguistic, and really has nothing to do with mathematics except provide the NEEDED linguistic tool to make it easier to communicate. Variables started out as only numbers, but this made them harder to understand. The teachers cannot see that the mathematical concepts are really prior to notation, and that they can be understood without learning arithmetic first. This does not mean that arithmetic should not be taught, but it does mean that it should be taught as a means to do more quickly what would otherwise be quite laborious, and that notation matters. I see the above average child learning this, and also mathematical logic through the restricted predicate calculus, by the age of 10 or so. This of course includes the ability to use symbolism to formulate problems, and thus most of algebra except for practice. We can then go on from here; the program definitely will have to be worked out, as there has been nothing done on this for 40 years, because of the rejection of the ideas. We must not restrict the average child to what the low end can achieve, or the bright to what the average can achieve. At this time, roughly anyone in the upper half can get into college, and at least those should have an understanding of mathematical concepts. Everyone needs to USE probability and statistics, not to compute them themselves, but to formulate. This requires understanding the structure of the integers and real numbers, the use of algebra, moving up to set theory and high school level measure theory, and somewhat more. Machines can do the computing, and facts and formulas can be looked up if they are known. Understanding is needed, and it does not come from computation. If one looks at Landau's little book, I doubt if numbers above 2 (I may be wrong in this) occur except for page numbers, theorem numbers, etc. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 >>If we had similar coaches for mathematics and science, >>we could do it. > I used to think this would be a great idea. Now I'm not so sure I'd want > to perform drug tests on the students before they could take the Putnam exam. :-) Where i live, there are separate music schools, but drug tests aren't necessary there. Why should that have to be any different for math clubs? If it could become an 'ordinary' hobby, like playing baseball, chess, or basoon playing, and accepted by the youth-culture as non-nerdy, that might already be a big improvement. No need to push it to the limits, or to restrict to the top 5%. Thousands of people start playing a musical instrument at some moment in their lives. Most of them find out they're not very talented, pretty soon. Even if they quit, so what? As far as they got, they already have gained something quite valuable. If they don't quit, even better. Math needs something like that. -- Herman Jurjus >Instead, there were attempts to reduce it to the level >the teachers could understand, and this did not succeed, >but by that time, it had been reduced to uselessness. >Unlike phonics, it was not possible to produce materials >so parents could teach their children, and the public >did not even understand the difference, and most did >not care whether their children learned anything besides >cookbook arithmetic. >This is not the only problem. Keeping bright children >back is more serious. The educational establishment is >oriented to trivial pursuit and memorization, and with >age grouping forcing things down, cannot change. Only >affordable schools not subject to government control >can help at this time. >>How about 'exact science clubs', or private lessons in some other form, >>separate from school? > You would have to separate the whole science curriculum > from the schoolwork, and avoid the schools entirely. :-) Well, if those who are interested get real education, the school curricula can be dumbed down without much danger (=without much difference). > Like currently is done in the music and sports >>world? (How many basketball players could've learned their sport by only >>playing basketball in high school? > Most started by playing on the basketball teams in high > school, and then went on to get athletic scholarships based > on their high school basketball performance, and played > basketball in college. They were on the teams, and got > special coaching both in high school and college, but > had little special coaching outside school. Netherlands, i can assure you noone can become a good sports(wo)man by restricting to the school curriculum. They all take lessons and spend many hours as a hobby, outside school. Same for musicians; hardly any music is taught in the official schools. Talents (and also many without) will go to special music schools and private teachers. Perhaps the successes of the high school athletics training in the US obfuscates the importance of all these extra efforts, and how dramatically damaging a simple standard training can be? -- Herman Jurjus >>You may be surprised, but I am not. >>But it is worse than that; the new math, which succeeded >>in teaching mathematical concepts, not just how to compute, >>to children, could not be learned by the teachers, even >>with major attempts to teach them. > Solution: Fire those that can't, hire those that can. Simple enough, no? No. Requires: Pay enough, and provide enough job satisfaction, to attract and retain those who can. Those who cannot are cheaper and easier to find and keep than those who can. >You may be surprised, but I am not. >But it is worse than that; the new math, which succeeded >in teaching mathematical concepts, not just how to compute, >to children, could not be learned by the teachers, even >with major attempts to teach them. >> Solution: Fire those that can't, hire those that can. Simple enough, no? >No. >Requires: Pay enough, and provide enough job satisfaction, to attract and >retain those who can. >Those who cannot are cheaper and easier to find and keep than those who can. This will not work NOW. It would require changing how everything is done. You are not going to get away with paying fresh college graduates twice what incompetents with years of seniority are getting. And the schools of education do not believe that teaching concepts in any case is a good idea. It took 40+ years to get a reasonable approach to teaching reading with phonics back, and it is still being fought. In this case, the parents could take over, and object to their children being taught again to read what they could already read, and this was a LARGE proportion of parents, -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 Paul Murray wites: >You may be surprised, but I am not. >But it is worse than that; the new math, which succeeded >in teaching mathematical concepts, not just how to compute, >to children, could not be learned by the teachers, even >with major attempts to teach them. >> Solution: Fire those that can't, hire those that can. Simple enough, no? >No. >Requires: Pay enough, and provide enough job satisfaction, to attract and >retain those who can. True enough. But how many of the teachers Dr. Rubin refered to above were fired? My guess is: none! Given the lack of real consequences for failing to learn the new math concepts well enough to teach them, is the outcome suprising? >Those who cannot are cheaper and easier to find and keep than those who can. I would only add that they are also, clearly, harder to get rid of. Rich There are further (ridiculus!) obstacles. (1) One often needs a teaching certificate to teach high school. Note that you don't need to prove you know the subject you teach; you just need the certificate. (sarcasm intended) (2) people who really know the subject do not want to be required to follow a forced curriculum. (3) Unmotivated students. This is perhaps the main problem. There are sub-cultures, ethnic groups etc. who simple DO NOT VALUE EDUCATION. And this is what is creating the low test scores. You can lead a horse's ass to knowledge, but you can't make him think. at 07:13 PM, pubkeybreaker@aol.comstuff (Bob Silverman) said: >(3) Unmotivated students. Children start out wanting to learn. The public schools systematically undermine that desire by emphasis on drill and uniformity at the expense of content and challenge. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not > at 07:13 PM, pubkeybreaker@aol.comstuff (Bob Silverman) said: >>(3) Unmotivated students. >Children start out wanting to learn. The public schools systematically >undermine that desire by emphasis on drill and uniformity at the >expense of content and challenge. The kids are bored to death. When they begin twitching in protest, they get drugged to death. /BAH Subtract a hundred and four for e-mail. ----- Original Message ----- > Children start out wanting to learn. The public schools systematically > undermine that desire by emphasis on drill and uniformity at the > expense of content and challenge. And yet I have students who are so weak on this issue that they cannot understand the difference between 2/3 and 2*(1/3). DM > ----- Original Message ----- > Asia >> Children start out wanting to learn. The public schools systematically >> undermine that desire by emphasis on drill and uniformity at the >> expense of content and challenge. > And yet I have students who are so weak on this issue that they cannot > understand the difference between 2/3 and 2*(1/3). Let me rephrase. They are convinced there is a distinction between 2/3 and 2*(1/3). DM ----- Original Message ----- Asia > Children start out wanting to learn. The public schools systematically > undermine that desire by emphasis on drill and uniformity at the > expense of content and challenge. And yet I have students who are so weak on this issue that they cannot understand the difference between 2/3 and 2*(1/3). > Let me rephrase. They are convinced there is a distinction between 2/3 and > 2*(1/3). Imagine a bar one foot long. (2/3) of it is a bar 8 inches long 2*(1/3) of it is two bars, each 4 inches long. They have a point. The trick is to get them to put it aside for the time being, without trying to convince them that their point is eeevul. -- Chris Henrich God just doesn't fit inside a single religion. > Imagine a bar one foot long. > (2/3) of it is a bar 8 inches long > 2*(1/3) of it is two bars, each 4 inches long. > They have a point. The trick is to get them to put it aside for the > time being, without trying to convince them that their point is eeevul. They do not have a point. We are not talking about engineering where they are concerned with the distinct difference between the physical properties of a single 8 inch solid piece of steel and two 4 inch pieces of steel. Further, when the discussion is radians (as was the case here except I left off the pi from the expression), your analogy, to me at least, is more confusing than the issue. Which is the basics of fraction multiplication. I seen a number of HS graduates who cannot add 1/3 and 1/2 without a calculator. And when they do they feel that 0.833 is a better answer than 5/6. DM <41c38e1b$18$fuzhry+tra$mr2ice@news.patriot.net> <9eudnR0UuP-P-1ncRVn-jQ@comcast.com> Translating the example of one bar of 8 inches different from two bars of 4 inches: An equality like 2*(1/3)=2/3 gives equal values for different expressions. In this example an multiplication of two numbers is an operation and that is different from a value, which is not an operation. I do not understand, why You don't understand me. says the teacher. Math ain't that easy. Hero > An equality like > 2*(1/3)=2/3 gives > equal values > for different expressions. > In this example > an multiplication of two numbers > is an operation and that > is different > from a value, which is not an operation. > I do not understand, why You don't understand me. > says the teacher. > Math ain't that easy. Yawn. The multiplication operator * : RxR -> R is a binary map. The student was comparing *(2,1/3), a point in the image space of the map, with 2/3, a different notational representation of the same rational number in R. There is NO distinction. There is a distinction between the binary operation and the image of that operation on a pair of elements in R. But that was not the point of discussion. The class was precalc and not group theory or some discussion growing out of Halmos' Naive Set Theory. DM <41c38e1b$18$fuzhry+tra$mr2ice@news.patriot.net> <9eudnR0UuP-P-1ncRVn-jQ@comcast.com> Yawn. The multiplication operator * : RxR -> R is a binary map. The student was comparing *(2,1/3), a point in the image space of the map, with 2/3, a different notational representation of the same rational number in R. There is NO distinction. What You first posted can be compared to : The students think the mapping of (2,1/3)by * is different from the image-point *(2,1/3) Hero > Yawn. The multiplication operator * : RxR -> R is a binary > map. The student was comparing *(2,1/3), a point in the image > space of the map, with 2/3, a different notational representation of > the same rational number in R. There is NO distinction. > What You first posted can be compared to : > The students think the mapping of (2,1/3)by * > is different from the image-point *(2,1/3) Based on my conversations with many students, it is not this subtlety that is the cause of confusion. It was the fact that, as elements of Q, there is no distinction between 2/3, 2*(1/3), 6/9, 1/3 + 1/3, etc. That is, these are all different representations of the exact same point in Q -- some of which may have been viewed as an image point of the + operator and some an image point of the * operator, and some just viewed as elements of Q, as constructed from N. Another issue I have come across is a basic lack of understanding of how to understand and implement the algebraic rules within Q. It is quite common for students to believe that a/(b+c) = (a/b) + (a/c) -- which tends to hamper their ability to rewrite sums of algebraic expressions over a common denominator, among other useful ideas. DM >Another issue I have come across is a basic lack of >understanding of how to >understand and implement the algebraic rules within Q. >It is quite common >for students to believe that a/(b+c) = (a/b) + (a/c) -- Of course they will. Don't use slanty lines. Straight horizontal lines help convey the terms and where they belong. >which tends to hamper their >ability to rewrite sums of algebraic expressions over a common denominator, >among other useful ideas. My high school algebra teacher counted a correct solution wrong if the equation used slanty lines. It didn't take long for everybody to stop that. It's easier to separate terms and whole divisors if horizontal lines are when drawing fractions on paper. This is especially true when stuff below a line has shorter lines. /BAH Subtract a hundred and four for e-mail. > >>Another issue I have come across is a basic lack of >>understanding of how to >>understand and implement the algebraic rules within Q. >>It is quite common >>for students to believe that a/(b+c) = (a/b) + (a/c) -- > Of course they will. Don't use slanty lines. Straight > horizontal lines help convey the terms and where they > belong. Uh.. I was using the slanty lines (aka forward slashes) to post to this board. Normally I write in LaTeX where the notion is quite clear. I.e., $$frac{a}{b+c} = frac{a}{b} + frac{a}{c}$$ > My high school algebra teacher counted a correct solution > wrong if the equation used slanty lines. It didn't take long > for everybody to stop that. It's easier to separate terms > and whole divisors if horizontal lines are when drawing > fractions on paper. This is especially true when stuff > below a line has shorter lines. See above. DM >> >Another issue I have come across is a basic lack of >understanding of how to >understand and implement the algebraic rules within Q. >It is quite common >for students to believe that a/(b+c) = (a/b) + (a/c) -- >> Of course they will. Don't use slanty lines. Straight >> horizontal lines help convey the terms and where they >> belong. >Uh.. I was using the slanty lines (aka forward slashes) >to post to this board. Normally I write in LaTeX where the >notion is quite clear. I.e., >$$frac{a}{b+c} = frac{a}{b} + frac{a}{c}$$ >> My high school algebra teacher counted a correct solution >> wrong if the equation used slanty lines. It didn't take long >> for everybody to stop that. It's easier to separate terms >> and whole divisors if horizontal lines are when drawing >> fractions on paper. This is especially true when stuff >> below a line has shorter lines. >See above. Do arithmetic and algebra teachers also use LaTex or do the kids see these things ala ASCII? I bought my mother an algebra book and some of it has horizontal lines but not the pages that teach BASIC programming. I haven't seen her workbook but I spent a whole phone call trying to get her to write straight horizontal lines with her pencil. /BAH Subtract a hundred and four for e-mail. There *is* a distinction between 2*(1/3) and 2/3, though of course the two quantities are equal. Admittedly, Mr. Mason's students are probably not arguing that subtlety. Sheesh, if you can't convince them that 2*(1/3) = 2/3, how will they ever understand 0.999... = 1? -- Stephen J. Herschkorn sjherschko@netscape.net ^OX9W/.#XpUmm`>TD2zNE-t}emfPkFR.Z5`flY:3QYT$>dUwN^sm;MBV:F7aL9x*q!` ln!l}>Y6_45$%R|P7DSrBkEph@1-;P*s~F_28vO@e4p/'>}Pc?@rl8cz]d9RXOtSheesh, if you can't convince them > that 2*(1/3) = 2/3, how will they ever understand 0.999... = 1? Oh, is that something that students are supposed to learn? I've always wanted to make this a test question at the end of the calculus sequence, to get exact data on who gets it and who doesn't, but I guess it would only be embarrassing all around. > There *is* a distinction between 2*(1/3) and 2/3, though of course > the two quantities are equal. Admittedly, Mr. Mason's students are > probably not arguing that subtlety. I have frequently run across people who think of something like '2*(1/3)' as an un-evaluated expression that can be simplified to give a number, in this case 2/3. The idea that an expression actually denotes its value is not immediately obvious to everyone. My observation of the local school system -- which sends a large fraction of its graduates to selective and highly-selective universities -- is that this kind of basic abstraction is not taught; it is mentioned and some kids will get it or otherwise puzzle it out on their own. Nobody notices or cares that many don't ever get it since it doesn't hurt their state proficiency test or SAT scores. >> There *is* a distinction between 2*(1/3) and 2/3, though of course >> the two quantities are equal. Admittedly, Mr. Mason's students are >> probably not arguing that subtlety. >I have frequently run across people who think of something like >'2*(1/3)' as an un-evaluated expression that can be simplified >to give a number, in this case 2/3. The idea that an expression >actually denotes its value is not immediately obvious to everyone. >My observation of the local school system -- which sends a large >fraction of its graduates to selective and highly-selective >universities -- is that this kind of basic abstraction is not >taught; it is mentioned and some kids will get it or otherwise >puzzle it out on their own. >Nobody notices or cares that many don't ever get it since it doesn't >hurt their state proficiency test or SAT scores. I'm not so sure that it's such a basic abstraction. As I remember it, the vast majority of my mathematical education, from primary school through graduate school, was of the syntactic variety. (And indeed, subjects like point-set topology had no readily-accessible semantic denotations for many of the concepts. My head hurt trying to visualise what a non-Hausdorff space might look like.) Indeed, from a Representationalist viewpoint, one can consider a number to BE exactly the equivalence class of possible expressions for it. It's certainly debatable whether we should be teaching mathematical religion (Platonism) in schools... [added sci.logic] -- --------------------------- | BBB b Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum viditur. | BBB aa a r bbb | ----------------------------- > I'm not so sure that it's such a basic abstraction. As I remember it, the > vast majority of my mathematical education, from primary school through > graduate school, was of the syntactic variety. (And indeed, subjects > like > point-set topology had no readily-accessible semantic denotations for many > of the concepts. My head hurt trying to visualise what a non-Hausdorff > space might look like.) I always liked the following simple example: If X represents the real numbers (or really any set with at least two distinct elements), and 0 represents the empty set (is there an ASCII convention for the empty set?), then the topology T = {X,0} makes the topological space {X, T} not Hausdorff. All four axioms are clearly satisfied. Then there are some of the simple pictorial examples in the beginnings of Munkre's text. DM > I'm not so sure that it's such a basic abstraction. > [...] > Indeed, from a Representationalist viewpoint, one can consider a number to > BE exactly the equivalence class of possible expressions for it. Even there, the basic abstraction is present: that a number is a conceptual entity distinct from any single representation of it. For example, we were taught that we could count objects with marks like ||||||||||||, and that we could write 1 for |, 2 for || and so on up to 9, and 0 for nothing. If we had more we could make groups of ten and maybe have some left over, like (||||||||||) ||. Then we count the groups the same way. So we had two different ways to write the same count. One was really easy to understand and use, and the other was harder but shorter to write. That's what introduced me to the idea that a number might be written different ways but still be the same number. The more difficult concept for me to grasp (long ago) was that operations on representations could also form equivalence classes. Obviously I didn't think in those terms. I was wondering how multiplication of numbers by forming rectangles of objects with certain side lengths and counting them could have anything to do with the manipulation of numerals by means of a big ugly table and some strange rules. For a while I had a strong distrust of multiplication, thinking that maybe if I multiplied big enough numbers then doing the two different procedures might give me two different results. It had occurred to me that the tables we were being taught to memorise could have been chosen so their answers came out right, and so it worked for small numbers. It was a fair bit longer before it occurred to me that the two sets of rules might really be the same thing in different forms like the numbers were the same thing in different forms. I think almost all of mathematics boils down to trying to take some representation or other, and find another form that represents the same thing but is easier to use. - Tim > Children start out wanting to learn. The public schools systematically > undermine that desire by emphasis on drill and uniformity at the > expense of content and challenge. Square pegs are ground down to fit in round holes. The motto of the Administration is the opposite of that of the Black Knight: All shall pass! Bob Kolker > at 07:13 PM, pubkeybreaker@aol.comstuff (Bob Silverman) said: >>(3) Unmotivated students. >Children start out wanting to learn. The public schools systematically >undermine that desire by emphasis on drill and uniformity at the >expense of content and challenge. Many schools (certainly including many non-public schools) spend much effort on stupefying children, with enormous success. This doesn't mean that there aren't also students who arrive at school either stupid (for genetic or non-human-environmental reasons, e.g., heavy metal poisoning) or stupefied (by their pre-school human environment). There are many paths into stupidity/stupefaction, and very few out. Lee Rudolph > There are further (ridiculus!) obstacles. > (1) One often needs a teaching certificate > to teach high school. Note that you > don't need to prove you know the subject > you teach; you just need the certificate. > (sarcasm intended) Knowing the subject is often a minus. Relating to and managing (controling) students is more important. The subject matter usually is trivial. > (2) people who really know the subject > do not want to be required to follow a > forced curriculum. It's complicated. Walking in both worlds, I can say, without fear of contradiction, that teaching a lesson in high school is much, much, much, much, more difficult than college teaching. Maybe you can teach one class (I doubt it) but how about 5 classes each day? > (3) Unmotivated students. This is perhaps > the main problem. There are sub-cultures, > ethnic groups etc. who simple DO NOT > VALUE EDUCATION. And this is what > is creating the low test scores. That partly is true. Students have come from anti-intellectual, working-class backgrounds before. The main problem is with their readiness to learn anything academic because of their own lack of academic inclination. > (3) Unmotivated students. This is perhaps > the main problem. There are sub-cultures, > ethnic groups etc. who simple DO NOT > VALUE EDUCATION. And this is what > is creating the low test scores. I would not try to teach differential equations to my pet dog. He simply does not have the necessaries to comprehend. Similarly trying to teach any more math to slum dwellers past what they need to score dope deals is a waste of time. Bob Kolker > Similarly trying to teach > any more math to slum dwellers past what they need to score dope deals > is a waste of time. As is trying to teach sanity to Republicans. I predict the slum dwellers get out of the slum before RJK figures out how to help. > As is trying to teach sanity to Republicans. > I predict the slum dwellers get out of the slum > before RJK figures out how to help. No help from me. I did not put them there, so I am not obliged to help them leave. Not my problem, except for the taxes I must pay to deal with this sterling citizens. And I am not a Republican. I am a libertarian. Bob Kolker > There are further (ridiculus!) obstacles. > (3) Unmotivated students. This is perhaps > the main problem. There are sub-cultures, > ethnic groups etc. who simple DO NOT > VALUE EDUCATION. And this is what > is creating the low test scores. > You can lead a horse's ass to knowledge, but you can't make him think. As a laid off (and now apparently careerless engineer) who is working on becoming a high school math teacher perhaps I can provide some practical insight > (1) One often needs a teaching certificate > to teach high school. Note that you > don't need to prove you know the subject you teach; you just need the > certificate. > (sarcasm intended) Well, in Massachusette you need to provide transcripts ( I have math degree) and take the MTEL ( I forget what the acronym means ) test. The test can handled easily by anyone who has finished high school math up through trig and probably has a year of AP calculus. The Mass Department of Education is an amazing example of an out of control government bureaucracy and navigating through the hoops is easily equivalent to a masters level thesis > (2) people who really know the subject > do not want to be required to follow a > forced curriculum. In this state you have standardised state wide tests you have to teach to > (3) Unmotivated students. This is perhaps > the main problem. There are sub-cultures, > ethnic groups etc. who simple DO NOT > VALUE EDUCATION. And this is what > is creating the low test scores. We are talking public education, not expensive private schools and unmotivated students will almost by definition be part of the problem. There is also an image problem for mathematics. How often have you seen interviews of prestigiou people who state, proudly, that they only got up to {1,2} year algebra or trig before giving up on math. Math, according to popular culture, is for social incompetents, for those who communicate only to blackboards or computers and not successfully to any member of the opposite sex. BTW Bob..I believe our paths have crossed at MITRE and Security Dynamics > lol - here's the quote that shows how hopeless the problem truly is: > [The American teacher] was especially taken aback by the textbook. By > grades seven and > eight, kids in the Singapore program are doing high-school-level > algebra. I thought, wow, that's complicated -- even for me, says Mr. > Keating. Poor Mr. Keating and his class. Well you know teaching doesn't pay very much. And it's run by the government. So it doesn't work very well. At a shopping center near where I live, there's one of those commercial places where they teach the kids math. Every Saturday morning the place is full of little kids. Their parents pay taxes to run the schools, and then the ones who can afford to buy their kids a decent education on the weekends. === Subject: Solving g( x+3y ,3x + y) = x*y , g :R * R->R by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGHHIG19584; How do you take this?. Can you generalize to : g(m(x,y),n(x,y))=h(x,y) g,m,n,h R*R->R functions, only g(x,y) unknown ? Explain the necessary conditions... Alain. === Subject: Re: Solving g( x+3y ,3x + y) = x*y , g :R * R->R > How do you take this?. > Can you generalize to : > g(m(x,y),n(x,y))=h(x,y) g,m,n,h R*R->R functions, > only g(x,y) unknown ? > Explain the necessary conditions... > Alain. For the problem given in the subject line, let u = x+3y and v = 3x + y. Then you can write x and y as functions of u and v (I'll let you do this) and g(u, v) = x * y. ________________________________ Eric J. Wingler (wingler@math.ysu.edu) Dept. of Mathematics and Statistics Youngstown State University One University Plaza Youngstown, OH 44555-0001 330-941-1817 === Subject: Re: Solving g( x+3y ,3x + y) = x*y , g :R * R->R >> How do you take this?. >> Can you generalize to : >> g(m(x,y),n(x,y))=h(x,y) g,m,n,h R*R->R functions, >> only g(x,y) unknown ? >For the problem given in the subject line, let u = x+3y and v = 3x + >y. Then you can write x and y as functions of u and v (I'll let you do >this) and g(u, v) = x * y. And of course more generally, solve the equations u = m(x,y), v = n(x,y) to express x, y in terms of u, v. Mike Guy === Subject: Re: Binary field; trace; New(?) Conjecture for fast determination of the trace of an element of a binary field. > I would have put your question as follows: > Suppose f(t) is an irreducible polynomial of degree d over k = F_p. > Let K = k[t]/(f(t)) = F_q where q = p^d. > Then the characteristic equation of c = t mod f(t) is f(t) = 0. > Now you are asking for the trace of c^j > as an element of the extension K/k. > If the eigenvalues of c are lambda_1,...,lambda_d > then the coefficients of > f(t) = t^d - h_1 t^{d-1} + h_2 t^{d-2} + ... > are the basic symmetric functions > h_1 = sum lambda_i, h_2 = sum_{i while the traces you are looking for are the symmetric power sumes > s_j = sum_i lambd_i^j . > So you are asking for formulae for the s_j in terms of the h_i. > Newton gave such formulae, based on the logarithmic differential of f(t). > Although some of the terms would disappear in characteristic 2, > I'm not sure if one could get a fundamentally simpler result. Hi Timothy, I think this is indeed the path that I was looking for. I've been searching on the web - but I cannot find information that relates the Newton Symmetric Polynomials, s_j, where s_j(x_1, x_2, ..., x_n) = sum_{i=1}^{m} x_i^j (see definition 4.2 of http://www.cs.wisc.edu/~jyc/810notes/lecture13.pdf) with the Elementary Symmetric Polynomials (p_k(x_1,...x_n) (that you called h_i instead of p_k)). The given lecture gives a relationship the other way around: p_k = 1/k (p_{k-1} s_1 - p_{k-2} s_2 + ... +/- s_k). If you have an (iterative) expression for arbitrary s_k into p_k, p_i and s_i for i < k, then can you give it to me please? I'd appreciate it when the source was somewhere on the web too, but if you have it in a book then please just mail it to me. Carlo Wood === Subject: weighted arithmetic and geometric means Hello Suppose x_1,....x_n and w_1,...w_n are postive numbers and define a = (Sum(i=1,n)(w_i*x_i))/(Sum(i=1,n)(w_i)) and g = (Product(i=1,n)(x_i)^(w_i))^(1/Sum(i=1,n)(w_i))) I want to prove that there holds an inequalty similar to that related to the arithmetic and geometric means, that is, a>= g, with equality if, and only if, x_1 = ....x_n. Since I already now that the arithmetic/geometric means inequalty is true, I tried to do as follows. First, if all the w_i's are positive integers, then we see readily see that a is the arithmetic mean of numbers x_1,...x_n if if we we take each x_i w_i times. Since a similar conclusion is true of g, we apply the a/g means inequality to conclude that, if all the w_i's are integer then the propostion is true for a and g. If all the w_i's are rational, then, representing each w_i as the ratio between 2 positive integers and doing some elementary algebraic transformations, we see a =a' and g =g', where a' and g' are weighted means similar to a and g corresponding, now, to integer weights. Therefore, we are sent back to the integer case, which shows the proposition still holds if all the w_i's are positive rationals. If the w_i's are real positive integers, then, for a fixed but arbitray (x_1,...x_n) , the functions (w_1,...w_n) -> a(w_1,...w_n) and (w_1,...w_n) -> g(w_1,...w_n,)defined on the subset of R^n composed of their points with positive coordinates, are continuous. Since a>=g in the subset of R^n composed of their points with positive and rational coordinates and since this latter subset is dense in the former, it follows that a(w_1,...w_n) >= g(w_1,...w_n) for every positive w_1,...w_n. Since this holds for arbitray positive x_1,...x_n, we have proved that, in fact, a>=g. But we are not done, because these arguments do not imply that equality occurs if and only if x_1...= x_n. Following my reasoning, can anyone suggest how I can complete the proof? Or, maybe, it's better to start at the very beginning, without supposing the a/g means inequalty is known. Amanda === Subject: Re: weighted arithmetic and geometric means > Hello > Suppose x_1,....x_n and w_1,...w_n are postive numbers and define > a = (Sum(i=1,n)(w_i*x_i))/(Sum(i=1,n)(w_i)) and > g = (Product(i=1,n)(x_i)^(w_i))^(1/Sum(i=1,n)(w_i))) > I want to prove that there holds an inequalty similar to that related > to the arithmetic and geometric means, that is, a>= g, with equality > if, and only if, x_1 = ....x_n. > Since I already now that the arithmetic/geometric means inequalty is > true, I tried to do as follows. > First, if all the w_i's are positive integers, then we see readily see > that a is the arithmetic mean of numbers x_1,...x_n if if we we take > each x_i w_i times. Since a similar conclusion is true of g, we apply > the a/g means inequality to conclude that, if all the w_i's are > integer then the propostion is true for a and g. > If all the w_i's are rational, then, representing each w_i as the > ratio between 2 positive integers and doing some elementary algebraic > transformations, we see a =a' and g =g', where a' and g' are weighted > means similar to a and g corresponding, now, to integer weights. > Therefore, we are sent back to the integer case, which shows the > proposition still holds if all the w_i's are positive rationals. > If the w_i's are real positive integers, then, for a fixed but > arbitray (x_1,...x_n) , the functions (w_1,...w_n) -> a(w_1,...w_n) > and (w_1,...w_n) -> g(w_1,...w_n,)defined on the subset of R^n > composed of their points with positive coordinates, are continuous. > Since a>=g in the subset of R^n composed of their points with > positive and rational coordinates and since this latter subset is > dense in the former, it follows that a(w_1,...w_n) >= g(w_1,...w_n) > for every positive w_1,...w_n. Since this holds for arbitray positive > x_1,...x_n, we have proved that, in fact, a>=g. But we are not done, > because these arguments do not imply that equality occurs if and only > if x_1...= x_n. > Following my reasoning, can anyone suggest how I can complete the > proof? Or, maybe, it's better to start at the very beginning, without > supposing the a/g means inequalty is known. > Amanda An induction proof (I discovered it for myself, but I could hardly be the first) - OK, it uses Calculus: Denote W(n) = w(1) + ... + w(n) A(n) = (w(1)*x(1) + ... + w(n)*x(n)) / W(n) G(n) = (x(1)^w(1) * ... * x(n)^(w(n))^(1/W(n)) and similarly for W(n+1), A(n+1), G(n+1). Then F(t), defined as (G(n+1)/A(n+1))^W(n+1) where x(n+1) is replaced with t (t>0), is a differentiable function of t. Use Calculus to establish that F(t) attains its strict maximum when t=A(n), the maximum being (G(n)/A(n))^W(n). So, since (G(1)/(A(1))^W(1) = 1, you have an induction proof, with necessary and sufficient conditions for equality. === Subject: Re: weighted arithmetic and geometric means Hello Suppose x_1,....x_n and w_1,...w_n are postive numbers and define a = (Sum(i=1,n)(w_i*x_i))/(Sum(i=1,n)(w_i)) and g = (Product(i=1,n)(x_i)^(w_i))^(1/Sum(i=1,n)(w_i))) I want to prove that there holds an inequalty similar to that related to the arithmetic and geometric means, that is, a>= g, with equality if, and only if, x_1 = ....x_n. Since I already now that the arithmetic/geometric means inequalty is true, I tried to do as follows. First, if all the w_i's are positive integers, then we see readily see that a is the arithmetic mean of numbers x_1,...x_n if if we we take each x_i w_i times. Since a similar conclusion is true of g, we apply the a/g means inequality to conclude that, if all the w_i's are integer then the propostion is true for a and g. If all the w_i's are rational, then, representing each w_i as the ratio between 2 positive integers and doing some elementary algebraic transformations, we see a =a' and g =g', where a' and g' are weighted means similar to a and g corresponding, now, to integer weights. Therefore, we are sent back to the integer case, which shows the proposition still holds if all the w_i's are positive rationals. If the w_i's are real positive integers, then, for a fixed but arbitray (x_1,...x_n) , the functions (w_1,...w_n) -> a(w_1,...w_n) and (w_1,...w_n) -> g(w_1,...w_n,)defined on the subset of R^n composed of their points with positive coordinates, are continuous. Since a>=g in the subset of R^n composed of their points with positive and rational coordinates and since this latter subset is dense in the former, it follows that a(w_1,...w_n) >= g(w_1,...w_n) for every positive w_1,...w_n. Since this holds for arbitray positive x_1,...x_n, we have proved that, in fact, a>=g. But we are not done, because these arguments do not imply that equality occurs if and only if x_1...= x_n. Following my reasoning, can anyone suggest how I can complete the proof? Or, maybe, it's better to start at the very beginning, without supposing the a/g means inequalty is known. Amanda > An induction proof (I discovered it for myself, but I could hardly be the > first) - OK, it uses Calculus: > Denote W(n) = w(1) + ... + w(n) > A(n) = (w(1)*x(1) + ... + w(n)*x(n)) / W(n) > G(n) = (x(1)^w(1) * ... * x(n)^(w(n))^(1/W(n)) > and similarly for W(n+1), A(n+1), G(n+1). > Then F(t), defined as > (G(n+1)/A(n+1))^W(n+1) where x(n+1) is replaced with t (t>0), > is a differentiable function of t. Use Calculus to establish that > F(t) attains its strict maximum when t=A(n), the maximum being > (G(n)/A(n))^W(n). > So, since (G(1)/(A(1))^W(1) = 1, you have an induction proof, > with necessary and sufficient conditions for equality. Doesn't this follow from Jensen's theorem directly (see Big Rudin for an overly complicated presentation of same). If I am remembering correctly, if f(x) is convex on an interval, then for any collection of constants w_1, ...,w_n adding up to 1, and x_1 < x_2 < ... < x_n in that interval, then the sum of f(x_i*w_i) < the sum of w_i * f(x_i). This is an easy consequence of applying induction and using the definition of convex for the two entry case. Since f(x) = log(x) is convex, this inequality follows. The strong inequalities really shouldn't make any difference and probably don't, and the condition of equality comes out easily. The theorem for w's that don't add up to 1 should be obtainable directly by applying the theorem to the w's divided by the sum of the w's and making the trivial adjustment. Achava === Subject: review topic suggestions (ODE's)? I'm signed up for a class (College of Engineering) that deals with analytical and numerical methods to solve mechanical engineering problems. my ODE class was a long time ago now. Any advice on what particulars to review? I have an ODE textbook, but don't want to tackle it randomly in the next two weeks. I recall fairly well how to set up IVP problems, use separation of variables, and use the method where you take an integral(e ^tx) and apply it to both sides (don't recall what that method's called), and I recall how to (generally ) deal with non-homogeneous cases, though I'd have to review some particular solution cases. Any advice is welcome. Oh, and I'm not even exactly sure what analytical and numerical methods means specifically. The catalog specifies diff eq's as a prereq, that's about it. k wallace === Subject: Re: review topic suggestions (ODE's)? > I'm signed up for a class (College of Engineering) that deals with > analytical and numerical methods to solve mechanical engineering > problems. my ODE class was a long time ago now. Any advice on what > particulars to review? You need to review your reasons for undertaking this boring and most likely fruitless training. === Subject: Re: review topic suggestions (ODE's)? >>I'm signed up for a class (College of Engineering) that deals with >>analytical and numerical methods to solve mechanical engineering >>problems. my ODE class was a long time ago now. Any advice on what >>particulars to review? > You need to review your reasons for undertaking this boring and > most likely fruitless training. It's a required class for a mechanical engineering degree. I have worked in the field for some time, my company went under last year, I found that the piece of paper is what will get me in the door, rather than my engineering experience. At least, with a company that will pay me better than a TA's salary. Although...I have found some of my classes to be fascinating and have learned a lot in areas outside my expertise, (fluid mechanics and dynamics for example), there are also a lot of things that are totally useless to me. As a designer/product& process analyst, I did not need to spend the last 4 months learning in excruciating detail the solubility limits of metals on the atomic level. If I need an alloy with particular characteristics, I have 4 websites of suppliers I can order it from. My professor, however, wasn't interested in my explanation, so I took the class, memorized a s**tload of stuff, took the final, and have promptly forgotten 90% of it. The structural analysis, and FEA stuff, though- that's very useful to me, and I like it a lot. So there's good and bad about being back in school, 15 years after I thought I was done with it. -k wallace === Subject: Question about Surreal Numbers I understand that the surreal numbers are a proper class which some people find a bit problematic and I would like to ask the following question. If we generate sets of surreal numbers using the usual recursive method (from http://en.wikipedia.org/wiki/Surreal_number) S0 = { 0 } S1 = { -1 < 0 < 1 } S2 = { -2 < -1 < -1/2 < 0 < 1/2 < 1 < 2} S3 = { -3 < -2 < -1 1/2 < -1 < -3/4 < -1/2 < -1/4 < 0 < 1/4 < 1/2 < 3/4 < 1 < 1 1/2 < 2 < 3 } S4 = ... and then transfinitely Sw, S(w+1), etc. Couldn't we define (for any ordinal x) F(x) as {s|s is a field (under the usual surreal operations ) and s is a subset of Sx}. This F(x) is the set of fields generated by ordinal x Then M(x) as {s|s in F(x) |s|>=|t| for every t in F(x)} (so this is a set of maximum sized fields generated with x). Ok one question is: is this set a singleton set? Then we can define N(x) as *some* element of M(x). This will be superfluous if M(x) is a singleton set. So why can't we work with sets of the form N(x) for some ordinal? This way we don't worry about the surreals being a proper class, and won't most interesting fields be captured if we set x right? Or what is wrong with my thinking? === Subject: Re: Question about Surreal Numbers > I understand that the surreal numbers are a proper class which some people > find a bit problematic Why would this upset anyone? But I'm really just jumping on your thread to ask why surreal numbers have not been accepted as the best way of defining rationals, reals and infinitesimals? Is it just that mathematicians are innately conservative? -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Question about Surreal Numbers I understand that the surreal numbers are a proper class which some people find a bit problematic > Why would this upset anyone? > But I'm really just jumping on your thread to ask > why surreal numbers have not been accepted > as the best way of defining rationals, reals and infinitesimals? > Is it just that mathematicians are innately conservative? I think it is too much at once in a non-intuitive way. Moreover, doing analysis on it is hairy. On the other hand, in his book ONAG, Conway shows an algebraically closed field of characteristic 2 with that method. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Question about Surreal Numbers Why are they an especially good way of deifning rationals or reals? Surely the other ways we've got are just fine. As for infinitesimals, there are lots of different ways to extend the field of real numbers to include infinitesimals. There's no reason why the surreal numbers would be the best way. === Subject: Re: Question about Surreal Numbers > Why are they an especially good way of deifning rationals or reals? Because everything is defined at once by a single recursive definition, starting with nothing but the empty set: a number x = < L | R > is defined by two sets of numbers L,R with l < r for all l in L, r in R. Logically, this must be simpler than eg defining rationals as equivalence sets of pairs of integers. Or defining reals by Dedekind sections of rationals. > Surely the other ways we've got are just fine. As for infinitesimals, > there are lots of different ways to extend the field of real numbers to > include infinitesimals. There's no reason why the surreal numbers would > be the best way. With surreal numbers, infintesimals are already part of the field. With other approaches I am aware of, infinitesimals are added to a pre-existing field. Will children in 3000 be taught that 0 = ? After decimals (base 10) are abolished in 2500. -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Question about Surreal Numbers Timothy Murphy says... >> Why are they an especially good way of deifning rationals or reals? >Because everything is defined at once by a single recursive definition, >starting with nothing but the empty set: a number > x = < L | R > >is defined by two sets of numbers L,R with l < r for all l in L, r in R. >Logically, this must be simpler than eg defining rationals >as equivalence sets of pairs of integers. >Or defining reals by Dedekind sections of rationals. That is cute, but I don't think you can really get away from equivalence classes. The problem is that this definition of number doesn't have the property that x <= y and y <= x --> x=y. -- Daryl McCullough Ithaca, NY === Subject: Re: Question about Surreal Numbers > Timothy Murphy says... ... Because everything is defined at once by a single recursive definition, starting with nothing but the empty set: a number x = < L | R > is defined by two sets of numbers L,R with l < r for all l in L, r in R. ... > That is cute, but I don't think you can really get away from equivalence > classes. The problem is that this definition of number doesn't have the > property that x <= y and y <= x --> x=y. Well, acually there is, without equivalence classes, but the definition was incomplete. Let's have x = { Lx | Rx } and y = { Ly | Ry }. Definition: x >= y iff (no x_R in R_x <= y and x <= no y_L in Ly). x <= y iff y >= x. x = y iff (x >= y and y <= x). And the numbers are those x for which: every x in Lx <= every x in Rx (the others are games, and for those the ordering relations do not hold). This defines the basis, although the numbers are not yet labelled. Addition, negation and multiplication can be defined on this basis (the definitions are again recursive). For instance, -x is defined approximately as { - Rx | - Lx }. And the first label is given: 0 = { | }. 0 >= 0 is vacuously true. Next we find 1 = { 0 | }, and so we can go on. Finally it can be shown that the numbers form a field. And also you will find that numbers have different representations: 1 = { 0 | } = { 0 | 2 } (== { 0 | { 0 | }} ) ... and so there are equivalence classes, but they are not needed for the definitions. (And, yes, it looks a bit like a two-sided Peano.) -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Question about Surreal Numbers > Timothy Murphy says... > Why are they an especially good way of deifning rationals or reals? >>Because everything is defined at once by a single recursive definition, >>starting with nothing but the empty set: a number >> x = < L | R > >>is defined by two sets of numbers L,R with l < r for all l in L, r in R. >>Logically, this must be simpler than eg defining rationals >>as equivalence sets of pairs of integers. >>Or defining reals by Dedekind sections of rationals. > That is cute, but I don't think you can really get away from equivalence > classes. The problem is that this definition of number doesn't have the > property that x <= y and y <= x --> x=y. Of course, eg < -1 | 1 > = 0. I didn't express myself well. What I meant is that in the standard approach one first defines Z, then defines Q, then defines R, each in different ways, while surreal numbers define all in one step, in the same way. It's interesting that recursive definitions like this are so rare in mathematics, while they seem quite common in computer science. (Of course computer scientists rarely use transfinite induction!) -- Timothy Murphy e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Question about Surreal Numbers > I understand that the surreal numbers are a proper class which some people > find a bit problematic and I would like to ask the following question. If > we generate sets of surreal numbers using the usual recursive method > (from http://en.wikipedia.org/wiki/Surreal_number) > S0 = { 0 } > S1 = { -1 < 0 < 1 } > S2 = { -2 < -1 < -1/2 < 0 < 1/2 < 1 < 2} S3 = { -3 < -2 < -1 1/2 < -1 > < -3/4 < -1/2 < -1/4 < 0 < 1/4 < 1/2 < 3/4 < 1 < 1 1/2 < 2 < 3 } > S4 = ... > and then transfinitely Sw, S(w+1), etc. > Couldn't we define (for any ordinal x) F(x) as > {s|s is a field (under the usual surreal operations ) and s is a > subset of Sx}. > This F(x) is the set of fields generated by ordinal x > Then M(x) as {s|s in F(x) |s|>=|t| for every t in F(x)} > (so this is a set of maximum sized fields generated with x). > Ok one question is: is this set a singleton set? It's an empty set for x Then we can define N(x) as *some* element of M(x). This will be > superfluous if M(x) is a singleton set. Another approach: In Gonshor's An Introduction to the Theory of Surreal Numbers, he shows how to define surreal numbers as well-ordered sequences of pluses and minuses. Then the surreal numbers of length less than an uncountable cardinal form a field (I think). > So why can't we work with sets of the form N(x) for some ordinal? > This way we don't worry about the surreals being a proper class, and won't > most interesting fields be captured if we set x right? Or what is wrong > with my thinking? Nothing wrong with this, but of course you won't get all the surreal numbers this way. === Subject: Re: Question about Surreal Numbers >> I understand that the surreal numbers are a proper class which some > people >> find a bit problematic and I would like to ask the following > question. If >> we generate sets of surreal numbers using the usual recursive method >> (from http://en.wikipedia.org/wiki/Surreal_number) >> S0 = { 0 } >> S1 = { -1 < 0 < 1 } >> S2 = { -2 < -1 < -1/2 < 0 < 1/2 < 1 < 2} S3 = { -3 < -2 < -1 1/2 > < -1 >> < -3/4 < -1/2 < -1/4 < 0 < 1/4 < 1/2 < 3/4 < 1 < 1 1/2 < 2 < 3 } >> S4 = ... >> and then transfinitely Sw, S(w+1), etc. >> Couldn't we define (for any ordinal x) F(x) as >> {s|s is a field (under the usual surreal operations ) and s is >> subset of Sx}. >> This F(x) is the set of fields generated by ordinal x >> Then M(x) as {s|s in F(x) |s|>=|t| for every t in F(x)} >> (so this is a set of maximum sized fields generated with x). >> Ok one question is: is this set a singleton set? > It's an empty set for x It would be an empty set for some > limit ordinals too, the ones where F(x) has no set field of largest > cardinality. In that case here is a followup question. When M(x) is *not* the empty set is it guaranteed to be a singleton set? >> Then we can define N(x) as *some* element of M(x). >> This will be >> superfluous if M(x) is a singleton set. > Another approach: In Gonshor's An Introduction to the Theory of Surreal > Numbers, he shows how to define surreal numbers as well-ordered > sequences of pluses and minuses. Then the surreal numbers of length less > than an uncountable cardinal form a field (I think). Do you know of any resources availale on the web (including journal have to actually get my hands on the book? Because his approach certainly sounds less hairy than mine. >> So why can't we work with sets of the form N(x) for some ordinal? This >> way we don't worry about the surreals being a proper class, and > won't >> most interesting fields be captured if we set x right? Or what is > wrong >> with my thinking? > Nothing wrong with this, but of course you won't get all the surreal > numbers this way. Well we could say that even the proper class of all surreals isn't all the surreals because....we haven't considered hyperordinals....and this business of saying we haven't got all the surreals could go on for ever. Anyway thank you very much for helping me out. === Subject: Re: Question about Surreal Numbers FunnyGuy > Do you know of any resources availale on the web (including journal > have to actually get my hands on the book? Because his approach certainly > sounds less hairy than mine. Restoring part of what I snipped, the surreals are 1-to-1 with the pairs (S,f) where S is an ordinal and f is a mapping Sto {-1,1}. I think Conway talked about this before Gonshor (somebody did), but you can probably work it all out in less time than it would take to get one of the books. If M=(S,f) and N=(T,g) are two such pairs, then the surreal M was created earlier than N (in Knuth's jargon) if and only if S<=T and the functions f and g agree on S (regarding, as we may, S as a subset of T). The relation M<=N is this: M=N or SFunnyGuy >> Do you know of any resources availale on the web (including journal >> have to actually get my hands on the book? Because his approach certainly >> sounds less hairy than mine. >Restoring part of what I snipped, the surreals are 1-to-1 with the pairs >(S,f) where S is an ordinal and f is a mapping Sto {-1,1}. I think Conway >talked about this before Gonshor (somebody did), but you can probably work >it all out in less time than it would take to get one of the books. If >M=(S,f) and N=(T,g) are two such pairs, then the surreal M was created >earlier than N (in Knuth's jargon) if and only if S<=T and the functions f >and g agree on S (regarding, as we may, S as a subset of T). The relation >M<=N is this: >M=N >ST I understand that the surreal numbers are a proper class which some people > find a bit problematic and I would like to ask the following question. If > we generate sets of surreal numbers using the usual recursive method > (from http://en.wikipedia.org/wiki/Surreal_number) > > S0 = { 0 } > S1 = { -1 < 0 < 1 } > S2 = { -2 < -1 < -1/2 < 0 < 1/2 < 1 < 2} S3 = { -3 < -2 < -1 1/2 < -1 > < -3/4 < -1/2 < -1/4 < 0 < 1/4 < 1/2 < 3/4 < 1 < 1 1/2 < 2 < 3 } > S4 = ... > > and then transfinitely Sw, S(w+1), etc. > > Couldn't we define (for any ordinal x) F(x) as > {s|s is a field (under the usual surreal operations ) and s is a > subset of Sx}. > This F(x) is the set of fields generated by ordinal x > > Then M(x) as {s|s in F(x) |s|>=|t| for every t in F(x)} > (so this is a set of maximum sized fields generated with x). > Ok one question is: is this set a singleton set? > It's an empty set for x Yes, of course. I forgot about that when writing the post. It would be an empty set for some limit ordinals too, the ones where F(x) has no set field of largest cardinality. > In that case here is a followup > question. When M(x) is *not* the empty set is it guaranteed to be a > singleton set? I don't find this very likely, but right now I can't see how to prove it one way or another. > Then we can define N(x) as *some* element of M(x). > This will be > superfluous if M(x) is a singleton set. > > Another approach: In Gonshor's An Introduction to the Theory of Surreal Numbers, he shows how to define surreal numbers as well-ordered sequences of pluses and minuses. Then the surreal numbers of length less than an uncountable cardinal form a field (I think). > Do you know of any resources availale on the web (including journal do I > have to actually get my hands on the book? Because his approach certainly > sounds less hairy than mine. I'm afraid I don't know any web resources, no. > So why can't we work with sets of the form N(x) for some ordinal? This > way we don't worry about the surreals being a proper class, and won't > most interesting fields be captured if we set x right? Or what is wrong > with my thinking? Nothing wrong with this, but of course you won't get all the surreal numbers this way. > Well we could say that even the proper class of all surreals isn't all the > surreals because....we haven't considered hyperordinals....and this > business of saying we haven't got all the surreals could go on for ever. > Anyway thank you very much for helping me out. === Subject: Re: Question about Surreal Numbers > Another approach: In Gonshor's An Introduction to the Theory of > Surreal > Numbers, he shows how to define surreal numbers as well-ordered > sequences of pluses and minuses. Then the surreal numbers of length >less > than an uncountable cardinal form a field (I think). p. 103: ... Finally, as a culmination of the results of this chapter we have shown that the subset of surreal numbers a such that |len(a)| <= d for any fixed infinite cardinal d is a real closed field. ... These are all honest fields since their carriers are *sets*. > I'm afraid I don't know any web resources, no. I did find this: http://www.uwec.edu/andersrn/setsxi.pdf I'm thinking of buying the book because I find the topic exciting and think that it may actually have something to say about the real (physical) world but i dunno. === Subject: Re: Question about Surreal Numbers > I understand that the surreal numbers are a proper class which some people > find a bit problematic and I would like to ask the following question. If > we generate sets of surreal numbers using the usual recursive method > (from http://en.wikipedia.org/wiki/Surreal_number) > > S0 = { 0 } > S1 = { -1 < 0 < 1 } > S2 = { -2 < -1 < -1/2 < 0 < 1/2 < 1 < 2} S3 = { -3 < -2 < -1 1/2 < -1 > < -3/4 < -1/2 < -1/4 < 0 < 1/4 < 1/2 < 3/4 < 1 < 1 1/2 < 2 < 3 } > S4 = ... > > and then transfinitely Sw, S(w+1), etc. > > Couldn't we define (for any ordinal x) F(x) as > {s|s is a field (under the usual surreal operations ) and s is a > subset of Sx}. > This F(x) is the set of fields generated by ordinal x > > Then M(x) as {s|s in F(x) |s|>=|t| for every t in F(x)} > (so this is a set of maximum sized fields generated with x). > Ok one question is: is this set a singleton set? > It's an empty set for x Yes, of course. I forgot about that when writing the post. It would be an empty set for some limit ordinals too, the ones where F(x) has no set field of largest cardinality. > In that case here is a followup > question. When M(x) is *not* the empty set is it guaranteed to be a > singleton set? I don't find this very likely, but right now I can't see how to prove it one way or another. > Then we can define N(x) as *some* element of M(x). > This will be > superfluous if M(x) is a singleton set. > > Another approach: In Gonshor's An Introduction to the Theory of Surreal Numbers, he shows how to define surreal numbers as well-ordered sequences of pluses and minuses. Then the surreal numbers of length less than an uncountable cardinal form a field (I think). > Do you know of any resources availale on the web (including journal do I > have to actually get my hands on the book? Because his approach certainly > sounds less hairy than mine. I'm afraid I don't know any web resources, no. > So why can't we work with sets of the form N(x) for some ordinal? This > way we don't worry about the surreals being a proper class, and won't > most interesting fields be captured if we set x right? Or what is wrong > with my thinking? Nothing wrong with this, but of course you won't get all the surreal numbers this way. > Well we could say that even the proper class of all surreals isn't all the > surreals because....we haven't considered hyperordinals....and this > business of saying we haven't got all the surreals could go on for ever. > Anyway thank you very much for helping me out. === Subject: Re: Question about Surreal Numbers > I understand that the surreal numbers are a proper class which some people > find a bit problematic and I would like to ask the following question. If > we generate sets of surreal numbers using the usual recursive method > (from http://en.wikipedia.org/wiki/Surreal_number) > > S0 = { 0 } > S1 = { -1 < 0 < 1 } > S2 = { -2 < -1 < -1/2 < 0 < 1/2 < 1 < 2} S3 = { -3 < -2 < -1 1/2 < -1 > < -3/4 < -1/2 < -1/4 < 0 < 1/4 < 1/2 < 3/4 < 1 < 1 1/2 < 2 < 3 } > S4 = ... > > and then transfinitely Sw, S(w+1), etc. > > Couldn't we define (for any ordinal x) F(x) as > {s|s is a field (under the usual surreal operations ) and s is a > subset of Sx}. > This F(x) is the set of fields generated by ordinal x > > Then M(x) as {s|s in F(x) |s|>=|t| for every t in F(x)} > (so this is a set of maximum sized fields generated with x). > Ok one question is: is this set a singleton set? > It's an empty set for x Yes, of course. I forgot about that when writing the post. It would be an empty set for some limit ordinals too, the ones where F(x) has no set field of largest cardinality. > In that case here is a followup > question. When M(x) is *not* the empty set is it guaranteed to be a > singleton set? I find this unlikely, but I'm afraid right now I can't see how to prove it one way or the other. > Then we can define N(x) as *some* element of M(x). > This will be > superfluous if M(x) is a singleton set. > > Another approach: In Gonshor's An Introduction to the Theory of Surreal Numbers, he shows how to define surreal numbers as well-ordered sequences of pluses and minuses. Then the surreal numbers of length less than an uncountable cardinal form a field (I think). > Do you know of any resources availale on the web (including journal do I > have to actually get my hands on the book? Because his approach certainly > sounds less hairy than mine. I'm afraid I don't know any web resources, no. > So why can't we work with sets of the form N(x) for some ordinal? This > way we don't worry about the surreals being a proper class, and won't > most interesting fields be captured if we set x right? Or what is wrong > with my thinking? Nothing wrong with this, but of course you won't get all the surreal numbers this way. > Well we could say that even the proper class of all surreals isn't all the > surreals because....we haven't considered hyperordinals....and this > business of saying we haven't got all the surreals could go on for ever. > Anyway thank you very much for helping me out. === Subject: Implicit Function Theorem by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGIRwJ25933; Let (x*,y*) in S be a vector such h(x*,y*) = 0 and the matrix nabla_y h(x*,y*) is nonsingular, the implicit function states that there exist (x,y) in the neigborhood of (x*,y*) such that y = f(x) and h(x,f(x)) = 0. Also, nabla f(x) = - nabla_x h(x,f(x)) * [ nabla_y h(x,f(x)) ]^-1. I know that nabla_y h(x*,y*) is nonsingular, but why this applies to nabla_y h(x,f(x)) for x in the neigborhood of x* as well? === Subject: Re: Implicit Function Theorem > Let (x*,y*) in S be a vector such h(x*,y*) = 0 and the matrix nabla_y > h(x*,y*) is nonsingular, the implicit function states that > there exist (x,y) in the neigborhood of (x*,y*) such that y = f(x) and > h(x,f(x)) = 0. You've badly mangled the IFT, omitting hypotheses, misstating the conclusion, ... >Also, > nabla f(x) = - nabla_x h(x,f(x)) * [ nabla_y h(x,f(x)) ]^-1. > I know that nabla_y h(x*,y*) is nonsingular, but why this applies to nabla_y > h(x,f(x)) for x in the neigborhood of x* as well? Well it's because of hypotheses you've left out in your sloppy statement of the IFT. h is C^1, so because det[dh/dy](x*,y*) is nonzero, det[dh/dy] is nonzero in a neighborhood of (x*,y*) by continuity; your conclusion follows. === Subject: Re: More on the State-of-the-Art in Physics by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGIRsp25873; >1)/.../ calculated that 95% of our universe is dark matter. You probably meen dark energy? As far as I know 20% is dark matter. >2) /.../The discovery of the basic constituent of dark matter, the superstrings. Where can you find this? To my knowledge this question is still highly open to debate and has been set to one of the 25 agendas to be set during the coming 25 years by KITP. joccis === Subject: Re: More on the State-of-the-Art in Physics >>1)/.../ calculated that 95% of our universe is dark matter. >You probably meen dark energy? As far as I know 20% is dark matter. So 115% of our universe is dark? Lee Rudolph === Subject: Re: More on the State-of-the-Art in Physics >MORE ON THE STATE-OF-THE-ART IN PHYSICS I guess you didn't even notice that I did a complete overhaul of physics. What's wrong you don't want to give me credit for anything? You want to rename it something else like the others did to steal my theories on the Helix Spiral Spinning Field Theory showing internal and external structure of the atom and and the universe. Oh, you want ot call it the QM or String Vortex Model? My work is about 10 years ahead of millions of the best minds in physics put together working constantly together and they still don't even know what mass is at the sub-atomic level. What's wrong with you people you need a job that bad? So you keep me out of it. Even the Japanese technology in electronic engineering is just barely catching up with me at the sub-atomic level in physics. Oh just think by the just barely understand where I am at. Smart's Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: More on the State-of-the-Art in Physics >1)/.../ calculated that 95% of our universe is dark matter. >>You probably meen dark energy? As far as I know 20% is dark matter. > So 115% of our universe is dark? > Lee Rudolph It's only 115% dark if you're a mathematician. If you're a physicist it's 100% dark. If you're an engineer it's 99.99% dark +/- 5% dark. -- Replace Roman numerals with digits to reply by email === Subject: Re: JSH: Those Ullrich defenders by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGIRse25855; >> against me, by never telling the full story. >This old stuff James? BORING. More operators and objects please! If you >need inspiration, just get drunk and sing to the walls. It always >worked before, didn't it? All hail Operator Ambiguity. Ad Hominem === Subject: plane curve animations some 80 famous plane curves animations for didactic and esthetic purposes are available here: http://xahlee.org/SpecialPlaneCurves_dir/specialPlaneCurves.html Xah xah@xahlee.org http://xahlee.org/PageTwo_dir/more.html === Subject: Re: Polysigned Numbers > Not getting much feedback here. > Just thought I'd mention that I've got some fresh analysis on my > website. > In particular > http://bandtechnology.com/PolySigned/ProductHistograms/Histograms1000000.htm l > shows an interesting property on four-signed numbers. Both their > products and squares are more dynamic than the other signs. > Squared values in four-signed are showing a clean well if you take the > graphics literally. Perhaps a source of stability? I would love to here > anyone's interpretation of it. No comments, but it's nice to see that you're still working on them. I enjoyed our discussions on them before. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Polysigned Numbers > Not getting much feedback here. > Just thought I'd mention that I've got some fresh analysis on my > website. Well, the first thing would be like this: You Idiot! It's just a histogram. You can't build a potential well out of a histogram. Go study some physics you twit. > In particular http://bandtechnology.com/PolySigned/ProductHistograms/Histograms1000000.htm l > shows an interesting property on four-signed numbers. Both their > products and squares are more dynamic than the other signs. > Squared values in four-signed are showing a clean well if you take the > graphics literally. Perhaps a source of stability? I would love to here > anyone's interpretation of it. > -Tim === Subject: Re: Polysigned Numbers The inequality of the metric with polysigned multiplication may be caused by the normal inequality with vector-addition. But the multiplication in 3-space-D of foursigned-numbers is fascinating. It's different from Quaternion-, dot-, and cross- multiplication. One can embedd the postive real number-ray. Numbers on two of the four coordinate-rays don't have a square-root. I really would like to know, where this will come to ? Have fun Hero === Subject: Re: Polysigned Numbers I've been toying with the square root of minus one in four-signed since you posted this. How do you prove that it does not exist? So Far I don't have an answer, but my method goes like this: (-a+b*c#d)(-a+b*c#d) = -1. As a result: | - 2ad - 2bc | = 1 | + aa + cc + 2bd | = 0 | * 2ab * 2cd | = 0 | # 2ac # bb # dd | = 0 . Or in real values: 2ad + 2bc = 1 aa + cc + 2bd = 0 2ab + 2cd = 0 2ac + bb + dd = 0 Now, we can arbitrarily choose a value for a,b,c,or d since the system is nonorthogonal. So I chose a = 1.0 . Then I boiled down to a large polynomial in c which I couldn't solve. So I graphed it in gnuplot and got some roots. But neither of them actually worked out so I must have made a mistake along the way, or the method is bogus. I think that this method is alright, except for the part where I zoom in on the zero solutions of a complicated polynomial in gnuplot. I'll try to spend a little more time with it. -Tim === Subject: Re: Polysigned Numbers > I've been toying with the square root of minus one in four-signed since > you posted this. > How do you prove that it does not exist? Well I can claim to have verified that sqrt( - 1 ) in four signs does not exist now. That is, assuming I haven't made any more errors on top of my old errors. Corrections to the method below... > So Far I don't have an answer, but my method goes like this: > (-a+b*c#d)(-a+b*c#d) = -1. > As a result: > | - 2ad - 2bc | = 1 > | + aa + cc + 2bd | = 0 > | * 2ab * 2cd | = 0 > | # 2ac # bb # dd | = 0 . This should be: | - 2ad - 2bc | = 1 + x | + aa + cc + 2bd | = x | * 2ab * 2cd | = x | # 2ac # bb # dd | = x . where x is a magnitude > Or in real values: > 2ad + 2bc = 1 > aa + cc + 2bd = 0 > 2ab + 2cd = 0 > 2ac + bb + dd = 0 Or in real values: 2ad + 2bc = 1 + x aa + cc + 2bd = x 2ab + 2cd = x 2ac + bb + dd = x Now choose a value for one of a,b,c,d, or x. So choosing x = 0 disallows a free choice of a = 1. Well it's apparent that some of these must be zero. OK. it shows a = 0, b = 0, c = 0, d = 0. Therefore there is no sqrt( -1 ) since square(0) is zero. But these equations are the general square on the left. Just substitute in something that works to check them... sqrt( # 1 ) should make 2ac + bb + dd = 1 and the rest equate to zero. and so a = 0, c = 0, b = 1, d = 0 is a solution and a = 0, c = 0, b = 0, d = 1 is a solution Yes, that check is good. > Now, we can arbitrarily choose a value for a,b,c,or d since the system > is nonorthogonal. So I chose a = 1.0 . > Then I boiled down to a large polynomial in c which I couldn't solve. This step was bogus. I should not have left the magnitude from when a magnitude is equal to zero. e.g. | + x + y | = 0 does not mean that x = - y. It means x = 0, y = 0. whereas if | + x + y | = z then it is ok to say that x = z - y where all values are magnitudes and leaving the magnitude form steps up to the reals with a guarantee that z > x and z > y. > So I graphed it in gnuplot and got some roots. But neither of them > actually worked out so I must have made a mistake along the way, or the > method is bogus. > I think that this method is alright, except for the part where I zoom > in on the zero solutions of a complicated polynomial in gnuplot. I'll > try to spend a little more time with it. > -Tim -Tim === Subject: Re: Polysigned Numbers I didn't check on Your calculation. > Or in real values: > 2ad + 2bc = 1 > aa + cc + 2bd = 0 > 2ab + 2cd = 0 > 2ac + bb + dd = 0 (4 equations with 4 unknown) i calculated first a=f(d,b,c) then c =g (a,b, d) insert this in the first. Both in the remaining equations and by comparing a=b=c=d=0 I've got a nice rule to memorize the sizes in a tetrahedron. On the way to the center start with one corner. Divide the edge or side line of length s 1:1, or go 1/2 of s. Turn right and facing a corner, divide this line 1:2, or go 1/3 of this line (which is sqrt3 / 2 of s long) . Now turn a right angle and face the last corner, divide this line 1:3, or go 1/4 of it ( which is sqrt2/sqrt3 of s long). Have fun Hero === Subject: Re: Polysigned Numbers > Not getting much feedback here. > Just thought I'd mention that I've got some fresh analysis on my > website. Well, the first thing would be like this: You Idiot! It's just a histogram. You can't build a potential well out of a histogram. Go study some physics you twit. > In particular http://bandtechnology.com/PolySigned/ProductHistograms/Histograms1000000.htm l > shows an interesting property on four-signed numbers. Both their > products and squares are more dynamic than the other signs. > Squared values in four-signed are showing a clean well if you take the > graphics literally. Perhaps a source of stability? I would love to here > anyone's interpretation of it. > -Tim === Subject: Positive functionals Hello. Let M denote the set of nxn-matrices with complex entries, and let f:M->C be a linear functional. It is not hard to prove that there is a matrix h such that f(m)=trace(hm) for all m in M. If f(m^*m) is real and positive for all m in M, why must h be self-adjoint? That is, h^*=h. I don't ask for a complete solution, only a hint :-) -- Michael Knudsen === Subject: nth term of the series Can anybody find the general term of the following set series: (1) 1, 4, 10, 20, 35, 56, 84, ... (2) 4, 15, 36, 70, 120, 189, 280, ... (3) 6, 20, 45, 84, 140, 216, 315, ... satya === Subject: Re: nth term of the series > Can anybody find the general term of the following set series: > (1) 1, 4, 10, 20, 35, 56, 84, ... > (2) 4, 15, 36, 70, 120, 189, 280, ... > (3) 6, 20, 45, 84, 140, 216, 315, ... Form sequences from the difference. For first sequence: 3 6 10 15 21 Now repeat: 2, 4, 5, 6 And again: 1,1,1,1 Sequence is cubic. You can form 4 simultaineous equations, as cubic sequence are all of the form: An^3 + Bn^2 + Cn + D and solve but that is pain. Another approach is to think in algebra. Cubic sequences are of the form An^3 + Bn^2 + Cn + D Use this to generate the nth terms for the sequences of differences you have above: First difference sequence is: [A(n+1)^3+B(n+1)^2+C(n+1)+D] - [An^3+Bn^2+Cn+D] = A(3n^2+3n+1) + B (2n+1) + C Notice, that we've 'lost' D. Repeat this process, using A(3n^2+3n+1) + B (2n+1) + C, to get the nth term for the second difference: A (6n+6) + 2B Notice we've lost C And again: Third difference is the sequence: 6A (The above are good for all cubic sequences.) Now, work back: For our sequence, 6A = 1 , so A =1/6 Sub in A(6n+6) + 2B = 3 (first term in sequence of 2nd differences) B = 1/2 Sub for A and B in A(3n^2+3n+1) + B (2n+1) + C = 3 (first term in sequence of 1st differences) C = 1/3 Now look at original sequence to get D, which is 0. So: T(n) = (n^3)/6 + (n^2)/2 + n/3 Check for n=1, 2, 3 against original sequence. -- Brian Reay www.g8osn.org.uk www.amateurradiotraining.org.uk FP#898 === Subject: Re: nth term of the series > Can anybody find the general term of the following set series: > (1) 1, 4, 10, 20, 35, 56, 84, ... > (2) 4, 15, 36, 70, 120, 189, 280, ... > (3) 6, 20, 45, 84, 140, 216, 315, ... Hint: take first differences, second difference etc until you get a constant. Bob Kolker === Subject: Re: Santa Claus is the god of math !!!! by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGKBUw03032; >>There is NOTHING that Santa cannot prove !!!! >>Wish for a proof for Christmas and you'll get it! >Can he solve the Continuum hypothesis? >[In the sense that Paul Erdos said that Christ's answer to >the question Can we solve the CH? was Paul Cohen gave all you >can know.] Oh. This didn't make sense, I think. I read that Erdos said that maybe, maybe the CH can be solved, our brains being the reason we can't solve it. (i.e., Some creature might have a brain who can say if it's true or not.) So the question is, does He have such an intelligence? delta01211 === Subject: Re: Which do you prefer? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGKBUB03024; >For abb. of physics courses, which of PHSX and PHYX do you prefer? >I personally prefer the former. The latter makes me feel that it's >too soft; the Y makes me feel that way. S, on S, on the other hand, is sharp. I was talking with a friend of mine, and so clicked the Send Message button without thought. delta01211 === Subject: Good intro statistics good? I did a quick search of good statistics books and came across the following: * Problem Solving: A Statistician's Guide by Chatfield, Christopher * Statistics a New Approach [Hardcover] by Wallis, Wilson Allen * Introduction to Probability Theory & Statistical Inference by Harold J. Larson However, I'm not sure if these are at the right level and/or cover the topics I'm interested in. The level needs to be at the AP statistics exam level (which although HS is probably more advanced first year undergrad), and should give a good discussion of sample design and experiments (and also regression and statistical inference). Also, I'm not really looking for an AP exam review book (but of course, suggestions are welcome). What I'm mainly looking for is a book that gives a good, yet accessible discussion of what's going on under the hood in the AP topics (sample design, experiments, regression, statistical inference), If any one knows of any books like this or close to this... === Subject: Re: Good intro statistics good? > I did a quick search of good statistics books and came across the > following: > * Problem Solving: A Statistician's Guide by Chatfield, Christopher Based on Amazon, this may fit your bill. > * Statistics a New Approach [Hardcover] by Wallis, Wilson Allen Amazon offered no info on this. > * Introduction to Probability Theory & Statistical Inference by Harold > J. Larson Calculus. It's probably not what you're looking for. > However, I'm not sure if these are at the right level and/or cover the > topics I'm interested in. The level needs to be at the AP statistics > exam level (which although HS is probably more advanced first year > undergrad), and should give a good discussion of sample design and > experiments (and also regression and statistical inference). Also, I'm > not really looking for an AP exam review book (but of course, > suggestions are welcome). What I'm mainly looking for is a book that > gives a good, yet accessible discussion of what's going on under the > hood in the AP topics (sample design, experiments, regression, > statistical inference), If any one knows of any books like this or > close to this... -- Will Twentyman email: wtwentyman at copper dot net === Subject: Is there at least two primes in the closed interval [N,2N] for all N>1? By Bertrand's Postulate there is at least one prime number in the closed interval [N,2N] for all N>1. Now the following question: Is there at least two prime number in the closed interval [N,2N] for all N>1? Alireza Abdollahi === Subject: Re: Is there at least two primes in the closed interval [N,2N] for all N>1? > By Bertrand's Postulate there is at least one prime number in the > closed interval [N,2N] for all N>1. Now the following question: There is at least one prime in the interval [N, 1.4*N] for large enough N. This is enough to solve your problem for large N. Then check by hand the small cases. === Subject: Re: Is there at least two primes in the closed interval [N,2N] for all N>1? Yes. Because the Bertrand-Chevishev theorem says that there is, at least, one prime in the OPEN interval [N,2N] from N>1 === Subject: Re: Is there at least two primes in the closed interval [N,2N] for all N>1? > Yes. Because the Bertrand-Chevishev theorem says that there is, at > least, one prime in the OPEN interval [N,2N] from N>1 Erdos proved that there exist at least one prime of the form 4k + 1 and at least one prime of the form 4k + 3 between n and 2n for all n > 6. Erdos, P. A Theorem of Sylvester and Schur. J. London Math. Soc. 9, 282-288, 1934. === Subject: Re: Is there at least two primes in the closed interval [N,2N] for all N>1? >Yes. Because the Bertrand-Chevishev theorem says that there is, at >least, one prime in the OPEN interval [N,2N] from N>1 Well, that would do it if N happened to be prime. What if it isn't? Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Is there at least two primes in the closed interval [N,2N] for all N>1? Yes. Because the Bertrand-Chevishev theorem says that there is, at least, one prime in the OPEN interval [N,2N] from N>1 === Subject: Re: Is there at least two primes in the closed interval [N,2N] for all N>1? > By Bertrand's Postulate there is at least one prime number in the > closed interval [N,2N] for all N>1. Now the following question: > Is there at least two prime number in the closed interval [N,2N] for > all N>1? Much more than 2 primes. The two theorems of Dusart(1998) permits to demonstrate it. Calling Pi(x) = Number of primes <= x 1.- Pi(x) > x / (Log(x) - 1) x > 5392 2.- Pi(x) < x / (Log(x) - 1.12) x > 7 Ludovicus === Subject: Re: Is there at least two primes in the closed interval [N,2N] for all N>1? > By Bertrand's Postulate there is at least one prime number in the > closed interval [N,2N] for all N>1. Now the following question: > Is there at least two prime number in the closed interval [N,2N] for > all N>1? Based on the commentary in http://mathworld.wolfram.com/BertrandsPostulate.html the answer would be yes. Assuming I'm reading it right. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Is there at least two primes in the closed interval [N,2N] for all N>1? <41c1fc31$1_2@newsfeed.slurp.net> Sorry, I do not see an answer to my question over there. Alireza Abdollahi By Bertrand's Postulate there is at least one prime number in the closed interval [N,2N] for all N>1. Now the following question: Is there at least two prime number in the closed interval [N,2N] for all N>1? > Based on the commentary in > http://mathworld.wolfram.com/BertrandsPostulate.html the answer would be > yes. Assuming I'm reading it right. > -- > Will Twentyman > email: wtwentyman at copper dot net === Subject: Submitting a paper My last post was probably lost in some server... My simple question is: Is it possible to submit a paper to one journal even if my e-mail address doesn't end with .edu? Is there some of you who have post some any paper and is not a professor or a teacher, but is only a simple math graduated ? Piero Giacomelli === Subject: Re: Submitting a paper > My last post was probably lost in some server... > My simple question is: Is it possible to submit a paper to one journal > even if my e-mail address doesn't end with .edu? Yes, it is: for instance, my e-mail address ends with .fr :-) (Besides, having an e-mail address isn't (yet?) mandatory to submit papers, AFAIK.) LD === Subject: Re: Submitting a paper > My last post was probably lost in some server... > My simple question is: Is it possible to submit a paper to one journal > even if my e-mail address doesn't end with .edu? Yes. I've published lots of papers, and my email doesn't end in edu. Email suffixes are not used to judge mathematical papers, their mathematical content is. -- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html Lacan, Jacques, 79, 91-92; mistakes his penis for a square root, 88-9 Francis Wheen, _How Mumbo-Jumbo Conquered the World_ === Subject: In what space a pentagram... ...would have all lines orthogonal (or nearly so)? The usual pentagram, with circle and all. Our drwng would be a plane representation (shadow) of some object in a higer dimension space. Is there any space where it would be a cube, a pyramid, other regular polyhedra? Can somebody imagine a `translation` where angles are chemical links, like saying it is the flat representation of a protein? What (known) molecules would be equivalent to a pentagram under some transformation (or projection too)? What if we add two circles around? And if we consider different colors as representative of a dimension (not geometrical, maybe interpreted as electromagnetism)? Say, the star one color, the circles other colors... The goal is to find equivalent regular polyhedra in high dim spaces, though it would be interesting to find figures with known mechanical properties in higher spaces equivalent to a pentagram embedded on a 2-plane. Can you think of other religious symbolisms interpreted this way? Like the maltese cross or the swastika. It would also be interesting to try to interpret them as electrical circuits deployed in higher dimensions, again with known properties as a key to do a search... Another idea, ritual movements might spell the vertices of other figures having direct interpretations in higher dimensional spaces. Danilo, the Forgotten, the Forsaken, the Ignored, the Wanderer, the Seeker, the Unnamed, the Misconceived, the Karmik, the Betrayed... === Subject: In what space a pentagram... Got inserted in another thread... > ...would have all lines orthogonal (or nearly so)? The usual pentagram, > with circle and all. Our drwng would be a plane representation (shadow) > of some object in a higer dimension space. Is there any space where it > would be a cube, a pyramid, other regular polyhedra? Can somebody > imagine a `translation` where angles are chemical links, like saying it > is the flat representation of a protein? What (known) molecules would > be equivalent to a pentagram under some transformation (or projection > too)? What if we add two circles around? And if we consider different > colors as representative of a dimension (not geometrical, maybe > interpreted as electromagnetism)? Say, the star one color, the circles > other colors... The goal is to find equivalent regular polyhedra in > high dim spaces, though it would be interesting to find figures with > known mechanical properties in higher spaces equivalent to a pentagram > embedded on a 2-plane. Can you think of other religious symbolisms > interpreted this way? Like the maltese cross or the swastika. It would > also be interesting to try to interpret them as electrical circuits > deployed in higher dimensions, again with known properties as a key to > do a search... > Another idea, ritual movements might spell the vertices of other > figures having direct interpretations in higher dimensional spaces. Danilo, the Forgotten, the Forsaken, the Ignored, the Wanderer, the Seeker, the Unnamed, the Misconceived, the Karmik, the Betrayed... === Subject: Who won this competition? In his Cantorian set theory and limitation of size Michael Hallett quotes this: The usefulness of mathematics, the esteem in which it is held, and the honourable name of 'exact science par excellence' rightly given it are all due to the clarity of its principles, the rigour of its proofs and the precision of its theorems. In order to ensure the perpetuation of these precious merits in so beautiful a part of our knowledge, we seek a clear precise theory of what is called Infinite in mathematics. Competition announcement of the Berlin Academy of Science for 1786 Who won, and with what? === Subject: Existence of countable Hamel basis by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGLYee10428; All normed spaces as vector spaces have basis,so called Hamel basis. Hamel bases in infinite dimensional Banach spaces are uncountable.This follows from Baire category theorem. What if the infinite dimensional space isn't complete? Does it always contain countable Hamel basis? The space d={(x_n):only finitely many x_n =/=0} has e_n=(0,..,0,1,0,...0) as Hamel basis. I would like to see other examples as this is the only one I can think of in the moment. === Subject: Re: Existence of countable Hamel basis at 09:34 PM, anonymous@mathforum.org (Felix) said: >All normed spaces as vector spaces have basis,so called >Hamel basis. >Hamel bases in infinite dimensional Banach spaces >are uncountable.This follows from Baire category theorem. What if the >infinite dimensional space isn't complete? >Does it always contain countable Hamel basis? No. >The space d={(x_n):only finitely many x_n =/=0} has >e_n=(0,..,0,1,0,...0) as Hamel basis. Make a slight change; index the elements on an uncountable set. That is, instead of {x in R^I| all but finitely many X_i are zero} use {x in R^U, U uncountable| all but finitely many X_i are zero}. That gives you an example of an incomplete space with no countable Hamel basis. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: Existence of countable Hamel basis >All normed spaces as vector spaces have basis,so called >Hamel basis. >Hamel bases in infinite dimensional Banach spaces >are uncountable.This follows from Baire category theorem. >What if the infinite dimensional space isn't complete? >Does it always contain countable Hamel basis? Of course not; it might, for instance, contain a complete infinite dimensional subspace even though it itself isn't complete. Lee Rudolph === Subject: Re: nth term of the series by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGLkGk11443; >Can anybody find the general term of the following set series: Use difference tables. >(1) 1, 4, 10, 20, 35, 56, 84, ... (n 3) [n choose 3], n = 3, 4, 5, ... >(2) 4, 15, 36, 70, 120, 189, 280, ... 11, 21, 34, 50, 69, 91, ... 10, 13, 16, 19, 22, ... 3, 3, 3, 3, ... 4 + 11(n 1) + 10(n 2) + 3(n 3) n = 0, 1, 2, ... >(3) 6, 20, 45, 84, 140, 216, 315, ... 14, 25, 39, 56, 76, 99, ... 11, 14, 17, 20, 23, ... 3, 3, 3, 3, ... 6 + 14(n 1) + 11(n 2) + 3(n 3) n = 0, 1, 2, ... Key fact is that for fixed k, the successive differences of (n k) are given by (n k-1). Cf. finite difference calculus (e.g. in Graham, Knuth, Patashnik's _Concrete Mathematics_). Todd Trimble === Subject: Re: Existence of countable Hamel basis by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGMAHx14180; >All normed spaces as vector spaces have basis,so called >Hamel basis. >Hamel bases in infinite dimensional Banach spaces >are uncountable.This follows from Baire category theorem. >What if the infinite dimensional space isn't complete? >Does it always contain countable Hamel basis? Of course not. For instance, given a maximal orthonormal set in l_2 and a line which passes through no member of the set, there exist linear complements of the line containing every member of the set. Obviously the complement can't be closed (hence is not complete under the induced norm), but the complement has uncountable linear dimension. Todd Trimble === Subject: Re: WAR IMMINENT betweeen Mathematicians and Physicists! >Although e, the base of the natural logarithms, and i, the principal >value of the square root of minus one, were to be set in Roman type, >h, Planck's constant, would remain in italics. Just be glad that the EE's aren't running the show, or it would be j instead of i. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: WAR IMMINENT betweeen Mathematicians and Physicists! >Just be glad that the EE's aren't running the show, or it would be j >instead of i. I forgot to drag in that one! Of course, i, j, and k all behave identically in any case, so it doesn't really matter *which* one you use. On that topic... for any function f(z) such that f(z*) = (f(z))* (where the postfix asterisk denotes the complex conjugate), it is reasonable to define the function when applied to quaternions as follows: where f(z), z=x+iy is already defined, f(q), q=r+is+jt+ku, is defined as: where t=0 and u=0, f(q)=f(z) where x=r and y=s; where at least one of s, t, and u is not zero, given that n is defined as sqrt(s^2 + t^2 + u^2), and v is defined as (is+jt+ku)/n, and is basically a unit element showing the direction of the imaginary part of q, f(q) = m + vn, where f(r+in) = m+in. This is essentially how Hamilton derived his ponential function from e^z, but this can be generalized to almost all standard functions defined on the complex plane. So if anyone wants to know the Gamma function of a quaternion... John Savard http://home.ecn.ab.ca/~jsavard/index.html === Subject: Re: WAR IMMINENT betweeen Mathematicians and Physicists! <41c1dff6$4$fuzhry+tra$mr2ice@news.patriot.net> <41c219d8.1752342@news.ecn.ab.ca> >Of course, i, j, and k all behave identically in any case, so it >doesn't really matter *which* one you use. Identically? Not unless you're talking Quaternions, which you hadn't previously[1] mentioned. For Engineers i, j, and k are customery labels for for an orthonormal basis of 3-space. [1] You brought them up after the text I quoted. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: How to visualize limits in category theory I came late to this discussion and, for some reason, Google no longer allows me to post follow-ups, so I have to start a new thread. But let me add my 2c. The original limits were actually colimits along directed sets. In this, they very much resembled limits along nets and were doubtless named by analogy. So suppose D is a directed set (a partially ordered set in which for every x and y, there is a z with x < z and y < z (for our purposes, < will stand for less than or equal)). Assume for each d in D, you are given a group (say, other models of finitary first order theories, such as fields, will do) G_d and whenever d < e a homomorphism f_{ed}: G_d --> G_e such that G_{dd} is the identity and whenever c < d < e, f_{ed}.f_{dc} = f_{ec}. This is called a directed system of groups. Then its (co)limit is constructed by taking the union of the G_d, identifying each element of G_d with its image under f_{ed} in G_e and defining the group operations on that set. It is easy and I leave the details to the reader. The dual notion was developed and it was shown that the Pontrjagin dual of a colimit of abelian groups was the limit of the dual groups (carried out in the category--the concept was very new--of compact abelian groups). Anyway, calling the limits was very natural. Why limits and colimits were named as they were is not clear to me, but I remember Eilenberg saying once that product was rarely misused (but cf. free product). Bourbaki generalized it to filtered limits and colimits and eventually Eilenberg and others realized that even that was unnecessary and that any functor could be said to have a limit or colimit. Actually, you don't even need a functor; it is quite natural to use a graph and a graph morphism to a category. By this time, the original connection to Moore-Smith convergence of nets had been left behind, like the Cheshire cat. When I think about limits, I think about subobjects of products. === Subject: Re: Implicit Function Theorem by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGMqWo17652; >> Let (x*,y*) in S be a vector such h(x*,y*) = 0 and the matrix nabla_y >> h(x*,y*) is nonsingular, the implicit function states that >> there exist (x,y) in the neigborhood of (x*,y*) such that y = f(x) and >> h(x,f(x)) = 0. >You've badly mangled the IFT, omitting hypotheses, misstating the >conclusion, ... sorry. >.... h is C^1, so because det[dh/dy](x*,y*) is nonzero, det[dh/dy] >is nonzero in a neighborhood of (x*,y*) by continuity; your >conclusion follows. Any easy way to show this? === Subject: Re: Existence of countable Hamel basis by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBGNSFm20668; >>All normed spaces as vector spaces have basis,so called >>Hamel basis. >>Hamel bases in infinite dimensional Banach spaces >>are uncountable.This follows from Baire category theorem. >>What if the infinite dimensional space isn't complete? >>Does it always contain countable Hamel basis? >Of course not. For instance, given a maximal orthonormal >set in l_2 and a line which passes through no member of the >set, there exist linear complements of the line containing >every member of the set. Obviously the complement can't be >closed (hence is not complete under the induced norm), but >the complement has uncountable linear dimension. >Todd Trimble Obviously I should have said given... a line not in the linear span of the [maximal orthonormal] set, .... === Subject: Re: How big is the electron? >Jack Sarfatti; >> leads to problems of infinite renormalization parameters. >In quantum informationdynamics says that energy is the same as information. >Because enegy is information then we see trivially, how >energy kati - are you attempting to talk? === Subject: Re: How big is the electron? <41bdfafa$1@news.accesscomm.ca> |The electron is a spiral-shaped cloud extending |outward from the nucleus *in exactly the same |way* that a galaxy's arm is made. |They are the same thing. Suppose you were shooting a 3D stereoptic movie for a classroom setting, would it be sufficient to put a few drops of milk on a vinyl record, spin it a little bit on a phonograph player, and then ray-trace it up and down so that it has the appearance of depth and height? Would that be suitable for your basic hydrogen atom? Using the same setup, would two drops work for a helium atom? === Subject: Re: How big is the electron? >|The electron is a spiral-shaped cloud extending >|outward from the nucleus *in exactly the same >|way* that a galaxy's arm is made. >|They are the same thing. >Suppose you were shooting a 3D stereoptic movie for a classroom >setting, would it be sufficient to put a few drops of milk on a >vinyl record, spin it a little bit on a phonograph player, and >then ray-trace it up and down so that it has the appearance of >depth and height? Would that be suitable for your basic hydrogen >atom? Using the same setup, would two drops work for a helium atom? No === Subject: Re: How to prove that Fibonacci sequence is primitive recursive? What is the definition of primitive recursive? > You can look it up. So what does it mean to say that the above > recursion fits the definition? And what conclusion can we draw > from the observation that it does or that it does not fit the > definition? Look up Tarski on satisfaction. === Subject: Re: How to prove that Fibonacci sequence is primitive recursive? > Look up Tarski on satisfaction. What do you take to be the relevance of Tarski on satisfaction? === Subject: Re: How to prove that Fibonacci sequence is primitive recursive? >Robert Israel says... >>The point is that the OP should put some of his/her own effort into >>doing this perfectly straightforward homework question. >It's not perfectly straightforward. The recursive definition of >F(n) (the nth Fibonacci number) is not a primitive recursive >definition, since F(n) involves both F(n-1) and F(n-2), while >a primitive recursive definition can only give F(n) in terms >of F(n-1). To solve the problem, you have to do something that >is *not* straight-forward, which is show that course-of-values >recursion can be defined in terms of primitive recursion. That >involves showing that a list of numbers can be coded as a single >number, and that both the coding and the decoding is primitive >recursive. That might be obvious once you've seen it >done once, but it isn't obvious otherwise. In the Fibonacci case, you can encode F(n) and F(n-1) as 2^n*F(n)+F(n-1). You do need to know that multiplication and division are primitive recursive. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: How to prove that Fibonacci sequence is primitive recursive? > I think you missed the point. The fact that F(n) (the > nth Fibonacci number) is primitive recursive does *not* > follow by inspection from the definition of primitive > recursive. Whether it does or not depends on the particular definition of primitive recursive that is intended. This is the reason for asking that the definition be provided to us, before we can help further. === Subject: Re: How to prove that Fibonacci sequence is primitive recursive? I think you missed the point. The fact that F(n) (the nth Fibonacci number) is primitive recursive does *not* follow by inspection from the definition of primitive recursive. > Whether it does or not depends on the particular definition > of primitive recursive that is intended. This is the reason > for asking that the definition be provided to us, before > we can help further. Definition of primitive recursive function is: 1)f(0) is primitive recursive; 2)f(x)=f(x)+1 is primitive recursive; 3)fn_m(u1, u2, ... un)=um is primitive recursive; 4)Composition: Given f(x) and g(x) primitive recursive functions, the f(g(x)) function is primitive recursive; 5)Primitive recursion: Given g(x) a primitive recursive function, we form a new primitive recursive function f(x) defined by: f(0)=x0; f(x+1)=g(x, f(x)); By the way, my speciality is computer application and I am only interested in computability theory. === Subject: Re: How to prove that Fibonacci sequence is primitive recursive? Discussion, linux) > I think you missed the point. The fact that F(n) (the > nth Fibonacci number) is primitive recursive does *not* > follow by inspection from the definition of primitive > recursive. >> Whether it does or not depends on the particular definition >> of primitive recursive that is intended. This is the reason >> for asking that the definition be provided to us, before >> we can help further. > Definition of primitive recursive function is: > 1)f(0) is primitive recursive; Huh? > 2)f(x)=f(x)+1 is primitive recursive; Wha? I think something ain't quite right here. -- Jesse F. Hughes Like the ski resort full of girls hunting for husbands and husbands hunting for girls, the situation is not as symmetrical as it might seem. -- Alan MacKay === Subject: Re: How to prove that Fibonacci sequence is primitive recursive? Definition of primitive recursive function is: 1)f(0) is primitive recursive; > Huh? 2)f(x)=f(x)+1 is primitive recursive; > Wha? Maybe something like: f(x) = constant is primitive recursive f(x) = x + 1 is primitive recursive etc was meant. ; === Subject: Re: How to prove that Fibonacci sequence is primitive recursive? <877jni3qjf.fsf@phiwumbda.org> Discussion, linux) > > Definition of primitive recursive function is: > > 1)f(0) is primitive recursive; >> Huh? > 2)f(x)=f(x)+1 is primitive recursive; >> Wha? > Maybe something like: > f(x) = constant is primitive recursive Or just f(x) = 0, since other constants can be had via composition with the successor. > f(x) = x + 1 is primitive recursive > etc > was meant. Maybe so. -- A set having three members is a single thing wholly constituted by its members but distinct from them. After this, the theological doctrine of the Trinity as 'three in one' should be child's play. --Max Black, _Caveats and Critiques_ === Subject: Re: How to prove that Fibonacci sequence is primitive recursive? A N Niel says... >> I think you missed the point. The fact that F(n) (the >> nth Fibonacci number) is primitive recursive does *not* >> follow by inspection from the definition of primitive >> recursive. >Whether it does or not depends on the particular definition >of primitive recursive that is intended. This is the reason >for asking that the definition be provided to us, before >we can help further. I'm sorry. I thought that was pretty standard. The primitive recursive functions is the smallest collection of functions including the primitive functions zero(x) = 0 successor(x) = x+1 proj^i_n(x_1,x_2,...,x_n) = x_i closed under the operations of composition and primitive recursion, where primitive recursion takes a function g(x_1,...,x_n) and a function h(x_1,...,x_n,y) and returns a function f(x_1,...,x_n,y) such that f(x_1,...,x_n,0) = g(x_1,...,x_n) f(g(x_1,...,x_n,y+1) = h(x_1,...,x_n,f(x_1,...,x_n,y)) Now, what you can prove (with a bit of work) is that any total recursive function that is bounded by a primitive recursive function is primitive recursive. So once you've proved that, it is pretty simple to show that something is primitive recursive without getting into the details of how primitive recursion is defined. -- Daryl McCullough Ithaca, NY === Subject: Re: How to prove that Fibonacci sequence is primitive recursive? > Whether it does or not depends on the particular definition > of primitive recursive that is intended. I think you will find that there is no significant variation in the definition of primitive recursive. === Subject: Re: How to prove that Fibonacci sequence is primitive recursive? OK, let's use the NIST definition: primitive recursive Definition: A total function which can be written using only nested conditional (if-then-else) statements and fixed iteration (for) loops. --------- (National Institute of Standards and Technology... http://www.nist.gov/dads/HTML/primitivrecr.html === Subject: Re: How to prove that Fibonacci sequence is primitive recursive? A N Niel says... >OK, let's use the NIST definition: >primitive recursive >Definition: A total function which can be written using only nested >conditional (if-then-else) statements and fixed iteration (for) loops. No, let's not. That's not the usual definition. (Although it gives the same set of functions---but there is a proof involved in showing that it is the same set of functions as the usual definition.) -- Daryl McCullough Ithaca, NY === Subject: Re: How to prove that Fibonacci sequence is primitive recursive? > OK, let's use the NIST definition: > primitive recursive > Definition: A total function which can be written using only nested > conditional (if-then-else) statements and fixed iteration (for) loops. This isn't actually a definition of anything. However, somebody who knows a bit about programming will be able to specify a suitable language and make it into a definition: a function which is definable by a program in language P. === Subject: Re: logic is innate? > >> > > >>The term Paradox of Material Implication refers to the fact >> These aren't paradoxes of material implication. They are problems >> of natural language, relevance, and so on. > They are called Paradoxes of Material Implication for historical > reasons. Note that these aren't problems for natural language > *speakers*, just people who wish to translate a natural language > into the FOL. >>If they're relevant at all in >> the current discussion, then they cast doubt on the claim that logic >> is innate. How does one claim that logic is part of the fundamental >> features of the human brain, when natural language conditionals are so >> obviously not truth-functional? > Because there are different natural language conditionals, some of > which are truth functional. > I still stand by my Eat veggies, get ice cream example, since in my > experience, if the father gives the kid ice cream without him having > eaten his vegetables, people would say that the caved. Of course, > your experience could be different. Assuming that this reaction is > universal is too far out. > Granting this, your example of the non-monotonicity of this > conversational conditional doesn't really show what you think it does. > It shows that the context generated (this may not be the best phrase > for what I mean, but...) by the sentences Eat veggies, get ice cream > and Eat veggies and kill sis, get ice cream are different -- very > different. In short, you've shown the existence of two classes of > conversational conditionals. This isn't new -- we should not forget > that logical english, like what a mathematician would use when > stating a theorem, is a fragment of the whole of english. > For what it's worth, my view is that our natural language abilities > combined with rough, though strong intuitions of truth and falsity > constitute most people's logic abilities. Conversational implicatures > can lead people to stray from classical logic because they cannot > differentiate between a truth functional and a non-truth functional > conditional. > 'cid 'ooh hoo hoo for ooh. Some philosophers believe that symbolic logic can reveal the structure of all possible good inference, and so reveal the common skeletal structure that underlies all reasonable thought processes. Bertrand Russell, Ludwig Wittgenstein, and other philosophers have argued that there is an intimate connection among these three things: predicate logic (which is the main kind of symbolic logic studied in this course), the human mind, and the deep structure of the physical world. SH: Since our brains --> minds receive have perceptions of the physical world, and our brains evolved in the physical world notions of time and motion appear to be inherent; these notions lead to survival strategies. Our minds interpret the universe in terms of causality. So logic is innate to the degree that it matches our perception of present events and then intuits or rehearses future event scenarios, counterfactuals. Natural language represents these abstract ideas, concretely. I found the following musing on the web. http://www.ebtx.com/ntx/ntx14.htm Existence consists of logic embodying itself, interacting with itself, and validating itself by infinite causal demonstration. Conversely, acausality would consist of logic invalidating itself by acausal demonstration. And what of randomness, Stephen === Subject: Re: logic is innate? <30hnelF2vr3i3U1@uni-berlin.de> <310mqkF35dgm5U1@uni-berlin.de> <87u0r73f4i.fsf@phiwumbda.org> <313slaF34ah79U1@uni-berlin.de> <314djvF377aqnU1@individual.net> <871xeasayt.fsf@phiwumbda.org> >> >> >The term Paradox of Material Implication refers to the fact > These aren't paradoxes of material implication. They are problems > of natural language, relevance, and so on. They are called Paradoxes of Material Implication for historical reasons. Note that these aren't problems for natural language *speakers*, just people who wish to translate a natural language into the FOL. >If they're relevant at all in > the current discussion, then they cast doubt on the claim that logic > is innate. How does one claim that logic is part of the fundamental > features of the human brain, when natural language conditionals are so > obviously not truth-functional? Because there are different natural language conditionals, some of which are truth functional. I still stand by my Eat veggies, get ice cream example, since in my experience, if the father gives the kid ice cream without him having eaten his vegetables, people would say that the caved. Of course, your experience could be different. Assuming that this reaction is universal is too far out. Granting this, your example of the non-monotonicity of this conversational conditional doesn't really show what you think it does. It shows that the context generated (this may not be the best phrase for what I mean, but...) by the sentences Eat veggies, get ice cream and Eat veggies and kill sis, get ice cream are different -- very different. In short, you've shown the existence of two classes of conversational conditionals. This isn't new -- we should not forget that logical english, like what a mathematician would use when stating a theorem, is a fragment of the whole of english. For what it's worth, my view is that our natural language abilities combined with rough, though strong intuitions of truth and falsity constitute most people's logic abilities. Conversational implicatures can lead people to stray from classical logic because they cannot differentiate between a truth functional and a non-truth functional conditional. 'cid 'ooh > hoo hoo for ooh. > Some philosophers believe that symbolic logic can reveal the structure of > all possible good inference, and so reveal the common skeletal structure > that underlies all reasonable thought processes. Bertrand Russell, Ludwig > Wittgenstein, and other philosophers have argued that there is an intimate > connection among these three things: predicate logic (which is the main kind > of symbolic logic studied in this course), the human mind, and the deep > structure of the physical world. Wittgenstein eventually came to his senses. > SH: Since our brains --> minds receive have perceptions of the physical > world, and our brains evolved in the physical world notions of time and > motion appear to be inherent; these notions lead to survival strategies. Our > minds interpret the universe in terms of causality. So logic is innate to > the degree that it matches our perception of present events and then intuits > or rehearses future event scenarios, counterfactuals. > Natural language represents these abstract ideas, concretely. I found the > following musing on the web. > http://www.ebtx.com/ntx/ntx14.htm Existence consists of logic embodying > itself, interacting with itself, and validating itself by infinite causal > demonstration. Conversely, acausality would consist of logic invalidating > itself by acausal demonstration. That looks like an indecipherable string of words. (I don't deny the possibility that it could make sense in context -- any string can be given meaning by explication) What sense of consistence, interaction, and validation are meant? Given that I'd like to keep my ontology austere, do I even want to know? 'cid 'ooh === Subject: Re: Wantzel trisection impossibility proof You might want to look in I.N. Herstein's book, Topics in Algebra. I am pretty sure he takes up the topic there. There are many books that have this material. I guess my age is showing here, but I find books to be a better source of mathematics information than the net. As another poster mentions, the idea is that the only points constructible in the plane have have coordinates that lie in a succession of algebraic extensions of Q in which each step is of degree 2, so that the total degree of the extension is a power of 2. Subfields of such extensions must have degrees dividing the degree of the extension containing them. In particular an extension of degree 3 or 6 (the solution of the cubic equation required for the trisection) and so it cannot be done with a straightedge and a compass. Achava === Subject: Re: Wantzel trisection impossibility proof The short version is that ruler and compass constructions involve the equivalent of arithmetic plus square roots, if the Cartesian coordinates of constructible points are considered. This, from the standpoint of Galois theory, means that with such constructions one can characterize the field extension over the rational numbers in which the coordinates lie as being of degree a power of two. Trisection of an arbitrary angle (or for that matter, duplication of a cube) involves finding a real root of an irreducible monic cubic polynomial. Hence the corresponding degree of the field extension would have a factor of three, which powers of two are known not to have. QED Gauss pretty well laid the groundwork for this by showing how to construct by ruler and compass those regular polygons which can be. === Subject: Re: Cantor's diagonal proof wrong? > Here's something all of you should have some fun with. > > Nath is not something I specialize in (and I don't read this group > normally), but I've been looking at a few things lately and I've decided > that some very big mistakes have been made in math because people started > playing around the concept of infinity without realizing the trouble they > were creating for themselves. > > When I was shown Cantor's diagonal proof that the number of reals was not > countable back in college, I thought it was a fascinating proof. It seemed > to uncover some great mystery about the nature of numbers that was not at > first obvious. It sounded very logical and I quickly embraced it as fact. > > Lately however, I've come to see things very differently. I now belief the > proof is totally bogus. And the huge body of work built on top of the > concept is likewise, totally bogus. AS the diagonal proof was Cantor's SECOND proof of the uncountability of the reals, and there have been several subsequent proofs, all of which are totally independent of the diagonal construction, it would not affect the validity of the theorem itself even if the diagonal proof were to be found flawed. For which reason, no sensible mathematician is the least worried that such a flaw would in any way weaken the validity of the theorem itself. > I'm writing to belabor the binary case is sufficient and necessary. > I'm reminded of my request about belaborment, which was about > communication and confusion issues, Virgil. Why do you think the > antidiagonal argument is flawed? > In the binary case, there is one specific anti-diagonal. > Consider an arbitrary base. Any method you use to generate some > antidiagonal will affect more than one location in the expansion as a > binary number. In that way, it might reset one of the previous > locations that would have been different, thus that the antidiagonal > would not be different at that location. That's an implication that a > number represented in a different base is a different number, and > stranger things are known to occur. That is perhaps just an artifact > of the algorithm. > That's similar to the argument that any number is representable in any > radix (base). The point is being that if there is some list, to > generate the list in a base greater than three, where three is as well > shown useless as a base to definitely generate an antidiagonal, and > construct an antidiagonal in some way that it is not rational so it > couldn't be dually represented. > Add a leading zero to each element of the list, then only in a > specific case is the antidiagonal an element in the range. > You refer to other arguments about the naturals and the reals, so do > with EF. Then again, my line of reasoning easily uses what you would > not term standard real numbers. > The rationals are dense in the real numbers. > Curt, you might want to learn about Skolem. Skolem extended the work > of Loewenheim to show that everything is countable. People handwave > about that and they're quite nonsensical in their ludicrous nonsense, > because the extensions are no different than the set. What that means > is they say that they have a set of integers that maps to a powerset > of integers, but in a receding slippery slope type of way that still > claims the opposite true. That's why they call that quandary Skolem's > paradox. > If you accept that the powerset result does not always hold true, > then, both Skolem's and Cantor's paradoxes dissolve, where Cantor's > is that a set of all sets would be its own powerset, and would map to > itself with the identity function. Without transfinite cardinals, for > everyone, measure theory needs some few new foundations, or rather, > just rephrased foundations, with perhaps some meaningful results, and, > that is about it, and all of transfinite cardinal mathematics is its > own little subfield where you axiomatize that so, just so all the work > put into transfinite cardinals was not a total waste of time, like a > pickled three-headed sheep. > Curt, what's the point, man? Do you want to map the reals to the > integers? What good is mapping the reals to the integers? Do you > think calculus is easier to understand if dx is a llittle > infinitesimal coefficient and when you sum the product of the function > and dx over the range that you get the integral? Even if that was too > slippery for general use, the limit being a safety feature of sorts, > and all the calculus was done using limit, wouldn't that be better? > Me, I was just offended that somebody claimed infinite sets weren't > equivalent. Now I feel better about it, because I've proved a few > things about that to people. > Do Zeno's paradoxes prevent Achilles from catching the tortoise? No, > they don't. Does Skolem's paradox prevent there being uncountable > sets? It does. You've probably heard of the paradoxical barber, > there are no paradoxes and so that barber does not exist anywhere, > because everybody in that town is shaved by the barber unless (if and > only if they don't) they shave themselves, everybody shaves, nobody > ever leaves town, and the well-meaning barber, who as an expert > probably shaves himself, also is the barber shaving himself. So, the > barber shaves himself and anybody else who doesn't shave themself. > Take two infinite sets. If there is a way that for each you can > select an element of each set and remove it from that set, do that. > That's a terse constructive proof that infinite sets are equivalent. > Cantor's results have meaning, they in a way force certain conclusions > about the nature of binary logic, because of that one element that is > unmapped, call it the antidiagonal or something, infinity rolls right > back over to zero like an odometer. > That gets into that any set X is an ordinal, and that the order type, > and successor, and X+1, and the powerset, are all the same thing. > When you're talking about mapping the naturals to the reals, there's > probably actually some useful formulas or functions that be used to > derive mathematical results that are not otherwise immediately > apparent. Here's a mapping between the natural numbers (0, 1, 2, ..., > non-negative integers) and the unit interval of the reals ( R[0,1], > every real number between zero and one inclusive): the natural/unit > equivalency function, EF. It's simple, order the reals from least to > greatest or greatest to least, and then map zero from the integer to > least or greatest, and then, in order, pair elements of those sets. > The binary antidiagonal does not exist on the range or is dually > represented, or you can add leading zeroes, and non-standard real > numbers, which are very much real numbers, are used thus that results > about mapping the naturals and reals do not apply. So anyways, > integrate EF and the result is equal to one, where you might think it > would be equal to one half, because you'd figure it would be just like > f(x)=x from zero to one. > That has to do with how points on the real number line are defined in > terms of preceding and following points on that same line, and that > points on a continuous line are in a sense one-sided, where that side > is in the direction of the ray's passage on that line, as the reals > are ordered thus that for two different real numbers one is lesser and > one is greater, or oppositely one is greater and one is lesser. When > the number is by itself then it has two sides and twice the weight, > because two different straight lines can pass through it. > You may as well consider a different method for sweeping through those > points, such as a spiral of sorts or alternatively taking the next > indefinite real element on the lesser and greater side. Again, that > leads to models of non-standard reals, which are real numbers. > Anyways, Curt, some people are very attached to their notions of > cardinality and the uncountable, they think it's very sophisticated > and urbane. A lot of work has been done based upon the simple notion > that f(x) = x+1 doesn't equal x+1. Most don't give a damn either way. > You can say that half of the integers are even, and that half of the > integers are positive, and that a given fraction of the integers are > primes or perfect squares, without the necessity of the transfinite > cardinals. As well, it is shown that a proper subset of a set has > less elements than the superset. There are more rationals than > integers, and more reals than irrationals or rationals. A powerset > has more elements than the set, in a sense, that's not the problem. > Curt, 1+1=2, and 2+2=4. Can't you leave the Cantorians their paradise > and well enough alone? Biblically, Adam and Eve were cast from > Paradise after they partook of the tree of knowledge. If they hadn't, > they'd still be there and that would be the end of the story. > Ross F. === Subject: Re: Cantor's diagonal proof wrong? <41ad4f31$14$fuzhry+tra$mr2ice@news.patriot.net> <41b2432e$17$fuzhry+tra$mr2ice@news.patriot.net> If you accept that the whole set of natural numbers is created by the Peano axioms, > Axioms don't create, they describe. That is a matter of personal opinion. For me, Peano's axioms create the set of natural numbers. but if you simultaneously deny that induction is capable of reaching all natural numbers (all lines of the list), > issue is not any imaginary claims that there is an unreachable natural > number; the issue is Cantor's antidiagonal argument, which neither > makes nor requires any such claim. By induction I have shown that the distance of any natural number from the origin 0 is n - 0 = n. Hence finite. Any natural number has a finite distance from the origin means: The set IN of all natural numbers ist not infinite. (This does not imply that IN was finite. IN is not at all!) === Subject: Re: Cantor's diagonal proof wrong? <41ad4f31$14$fuzhry+tra$mr2ice@news.patriot.net> <41b2432e$17$fuzhry+tra$mr2ice@news.patriot.net> >By induction I have shown that the distance of any natural number >from the origin 0 is n - 0 = n. Irrelevant; omega is not a natural number. >The set IN of all natural numbers ist not infinite. The mapping *2: n->2n does not exist? Because if it exists then by definition the set of naturals is infinite. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: Cantor's diagonal proof wrong? <41ad4f31$14$fuzhry+tra$mr2ice@news.patriot.net> <41b2432e$17$fuzhry+tra$mr2ice@news.patriot.net> <41c22f1d$9$fuzhry+tra$mr2ice@news.patriot.net> The mapping *2: n->2n does not exist? No, it does not exist. It is impossible to store more than 10^100 bits in the universe. If you have used them up , then further numbers can only be created by forgetting existing numbers. The existence of actual infinity is an irrealistic demand. === Subject: Re: Cantor's diagonal proof wrong? part: >That is a matter of personal opinion. For me, Peano's axioms create the >set of natural numbers. How did we ever manage to count anything before Peano? John Savard http://home.ecn.ab.ca/~jsavard/index.html === Subject: Re: Cantor's diagonal proof wrong? <41ad4f31$14$fuzhry+tra$mr2ice@news.patriot.net> <41b2432e$17$fuzhry+tra$mr2ice@news.patriot.net> <41c21d91.2705820@news.ecn.ab.ca> > part: That is a matter of personal opinion. For me, Peano's axioms create the set of natural numbers. > How did we ever manage to count anything before Peano? The principle of creating natural numbers is very old. Peano only formalized it in a very precise way. === Subject: Re: Cantor's diagonal proof wrong? <41ad4f31$14$fuzhry+tra$mr2ice@news.patriot.net> <41b2432e$17$fuzhry+tra$mr2ice@news.patriot.net> <41c21d91.2705820@news.ecn.ab.ca> !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > part: >>That is a matter of personal opinion. For me, Peano's axioms create the >>set of natural numbers. > How did we ever manage to count anything before Peano? That is not the question. How did we ever manage to rigorously prove something about counting before Peano? -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Cantor's diagonal proof wrong? part: >That is a matter of personal opinion. For me, Peano's axioms create the >set of natural numbers. How did we ever manage to count anything before Peano? > That is not the question. How did we ever manage to rigorously prove > something about counting before Peano? We didn't. Peano died in April, 1932, before I ever started counting. === Subject: Re: Cantor's diagonal proof wrong? That is not the question. How did we ever manage to rigorously prove something about counting before Peano? > We didn't. > Peano died in April, 1932, before I ever started counting. So you started counting *after* Peano. -- Alec McKenzie === Subject: Re: Cantor's diagonal proof wrong? > That is a matter of personal opinion. For me, Peano's axioms create the > set of natural numbers. What, if anything, does this statement mean? === Subject: Re: Cantor's diagonal proof wrong? <41aa5d5a$14$fuzhry+tra$mr2ice@news.patriot.net> <41ae4af6$6$fuzhry+tra$mr2ice@news.patriot.net> <41b243a2$20$fuzhry+tra$mr2ice@news.patriot.net> If you think, that then his list would be impossible, we do agree. > No, we don't agree; he proved that no such list is possible. Or, to > put it another way, he proved that for any list of real numbers there > is a real number not on the list. He proved that a complete list of all real numbers IR is impossible. His result is correct, because no complete set IR does exist. But his proof is wrong. You can see this best by considering the following Cantor-list: 0.0 0.1 0.11 0.111 ... If you start to construct the diagonal number 0.111..., you will see that it is always contained in the next line, how far you ever proceed. You can never keep up, except in a nice dream. === Subject: Re: Cantor's diagonal proof wrong? <41aa5d5a$14$fuzhry+tra$mr2ice@news.patriot.net> <41ae4af6$6$fuzhry+tra$mr2ice@news.patriot.net> <41b243a2$20$fuzhry+tra$mr2ice@news.patriot.net> >If you start There is not start. Cantor defines a specific number in terms of the list, not a sequence of numbers. >to construct the diagonal number 0.111..., you will see that it is >always contained in the next line, No. A number totally unrelated to any of Cantor's arguments appears in the list. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: Cantor's diagonal proof wrong? <41aa5d5a$14$fuzhry+tra$mr2ice@news.patriot.net> <41ae4af6$6$fuzhry+tra$mr2ice@news.patriot.net> <41b243a2$20$fuzhry+tra$mr2ice@news.patriot.net> <41c22e94$8$fuzhry+tra$mr2ice@news.patriot.net> If you start > There is not start. Cantor defines a specific number in terms of the > list, not a sequence of numbers. Define a number different from another whithout having seen the other one. Such a claim is typical for set theory but void of any rational meaning. to construct the diagonal number 0.111..., you will see that it is always contained in the next line, > No. A number totally unrelated to any of Cantor's arguments appears in > the list. I know, I cannot convince set theorists by any rational arguing. But I hope that some young mathematicians, not yet spoiled by set theory, may read my arguments and may find their own way. === Subject: Re: Cantor's diagonal proof wrong? <41aa5d5a$14$fuzhry+tra$mr2ice@news.patriot.net> <41ae4af6$6$fuzhry+tra$mr2ice@news.patriot.net> <41b243a2$20$fuzhry+tra$mr2ice@news.patriot.net> <41c22e94$8$fuzhry+tra$mr2ice@news.patriot.net> >Define a number different from another whithout having seen the other >one. Mathematics is not about seeing, it is about logical inferences. But please note that if I accepted your argument then it would immediately lead to Cantor's result without having to resort to an antidiagonal argument; since I haven't seen[1] a list of all real numbers, it doesn't exist. >I know, I cannot convince set theorists by any rational arguing. Indeed ;-) >But I hope that some young mathematicians, not yet spoiled by set >theory, may read my arguments and may find their own way. ITYM find your way. I doubt that you would be pleased should their way include infinite sets. [1] And there aren't enough atoms in the Universe to construct one. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: Cantor's diagonal proof wrong? <41aa5d5a$14$fuzhry+tra$mr2ice@news.patriot.net> <41ae4af6$6$fuzhry+tra$mr2ice@news.patriot.net> <41b243a2$20$fuzhry+tra$mr2ice@news.patriot.net> >If you think, that then his list would be impossible, we do agree. No, we don't agree; he proved that no such list is possible. Or, to put it another way, he proved that for any list of real numbers there is a real number not on the list. > He proved that a complete list of all real numbers IR is impossible. > His result is correct, because no complete set IR does exist. But his > proof is wrong. > You can see this best by considering the following Cantor-list: > 0.0 > 0.1 > 0.11 > 0.111 > ... > If you start to construct the diagonal number 0.111..., you will see > that it is always contained in the next line, Really? Is 0.111... a number? Let us call it x. Do you agree with the following statement: Either x is in the list, or x is not in the list. Both can not be true. What about the following statement: If x is in the list, it occurs at position n where n is a natural number. - Randy === Subject: Re: Cantor's diagonal proof wrong? <41aa5d5a$14$fuzhry+tra$mr2ice@news.patriot.net> <41ae4af6$6$fuzhry+tra$mr2ice@news.patriot.net> <41b243a2$20$fuzhry+tra$mr2ice@news.patriot.net> If you think, that then his list would be impossible, we do agree. > > No, we don't agree; he proved that no such list is possible. Or, to > put it another way, he proved that for any list of real numbers > there > is a real number not on the list. He proved that a complete list of all real numbers IR is impossible. His result is correct, because no complete set IR does exist. But his proof is wrong. You can see this best by considering the following Cantor-list: 0.0 0.1 0.11 0.111 ... If you start to construct the diagonal number 0.111..., you will see that it is always contained in the next line, > Really? > Is 0.111... a number? Let us call it x. It is 1/9. > Do you agree with the following statement: Either x is in > the list, or x is not in the list. Both can not be true. In a finite list, only one of these alternatives could be true. But why do you think that an infinite list obeys the same logic? - a list, which lacks not only the last line, but also the line next to the last one, ... the second half of all lines, 99 % of all lines, and even much more? Arithmetics is very different for infinite sets. Why should logic be the same? > What about the following statement: If x is in the list, > it occurs at position n where n is a natural number. Before 1/9 will be constructed as the antidiagonal number (exchanging 0 by 1) it will appear in a line. It occurs in the line subsequent to the completed 1/9. I cannot name the line number n because I don't know, where 1/9 is completed. But would 1/9 not completely occur as the antidiagonal number, then Cantor's method of constructing transcendental numbers would fail too (and with it the whole proof). === Subject: Re: Cantor's diagonal proof wrong? <41aa5b47$13$fuzhry+tra$mr2ice@news.patriot.net> <41ad5139$15$fuzhry+tra$mr2ice@news.patriot.net> <41b2427f$15$fuzhry+tra$mr2ice@news.patriot.net> WM: Every natural number is an ordinal number and is simultaneously the cardinal number of the sequence of all its successors including itself: 1,2,3,...,n. The sequence up to a natural number n can never have the cardinal number aleph_0. > Cantor's proofs do not involve finite sequences. Omega is not a > natural number. Hence the set of natural numbers cannot be actually infinite. It is potentially infinite, i.e., there is no threshold but there is no infinite number. And the same holds for the cardinal number of each sequence 1,2,3,...,n. You see it best, if you imagine Card{1,2,3,...,n} as a function f(n) of n. The sloop is 1 and that will never change. FOR ALL FINITE NUMBERS n (and there are no others) === Subject: Re: Cantor's diagonal proof wrong? <41aa5b47$13$fuzhry+tra$mr2ice@news.patriot.net> <41ad5139$15$fuzhry+tra$mr2ice@news.patriot.net> <41b2427f$15$fuzhry+tra$mr2ice@news.patriot.net> >Hence the set of natural numbers cannot be actually infinite. Non sequitor. >You see it best, if you imagine Card{1,2,3,...,n} as a function f(n) >of n. You'd have to be hallucinating to see it that way. >(and there are no others) That depends on what is is. It has no relevance to Mathematics, which does not depend on your philosophy. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: Cantor's diagonal proof wrong? <41aa5b47$13$fuzhry+tra$mr2ice@news.patriot.net> <41ad5139$15$fuzhry+tra$mr2ice@news.patriot.net> <41b2427f$15$fuzhry+tra$mr2ice@news.patriot.net> <41c22df7$7$fuzhry+tra$mr2ice@news.patriot.net> Hence the set of natural numbers cannot be actually infinite. > Non sequitor. You see it best, if you imagine Card{1,2,3,...,n} as a function f(n) of n. > You'd have to be hallucinating to see it that way. You could even prove it by using set theory. In the finite realm, ordinal number n is equal to cardinal number n of the sequence 1,2,3,...n. If we have a set of finite numbers only, then all the ordinal numbers are finite, hence giving finite cardinal numbers. There is no chance to form an infinite set with finite numbers. It is so easy to see, but Cantor has spoiled many brains. (and there are no others) > That depends on what is is. It has no relevance to Mathematics, > which does not depend on your philosophy. There are no infinite natural numbers in all mathematics. You'd need no philosophy to see that. > Unsolicited bulk E-mail subject to legal action. I reserve the > right to publicly post or ridicule any abusive E-mail. Reply to > domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: Cantor's diagonal proof wrong? <41aa5b47$13$fuzhry+tra$mr2ice@news.patriot.net> <41ad5139$15$fuzhry+tra$mr2ice@news.patriot.net> <41b2427f$15$fuzhry+tra$mr2ice@news.patriot.net> <41c22df7$7$fuzhry+tra$mr2ice@news.patriot.net> >There are no infinite natural numbers in all mathematics. Well, there are in NSA, but that is not the issue. A set of finite numbers need not be finite any more than a box of red spoons need be red or a spoon. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: Cantor's diagonal proof wrong? > There are no infinite natural numbers in all mathematics. You'd need no > philosophy to see that. No mathematician says that there are infinite naturals, at least in standard models. What they do say is that there is no finite number representing the number of naturals. === Subject: Re: Cantor's diagonal proof wrong? <41aa5b47$13$fuzhry+tra$mr2ice@news.patriot.net> <41ad5139$15$fuzhry+tra$mr2ice@news.patriot.net> <41b2427f$15$fuzhry+tra$mr2ice@news.patriot.net> <41c22df7$7$fuzhry+tra$mr2ice@news.patriot.net> Prof. Dr. Mueckenheim, I disagree with you. First, the universe is infinite. Secondly, if 10^100 is the largest number you can imagine, then it would be infinity. Otherwise, 10^100+1 would be a number, and 10^100 not the largest. The counting numbers are each a finite integer, taken together they're the infinite integer. There are finite integers that as far as we or anyone else are concerned are no different than infinitely large, because no physical process we could ever observe except a highly contrived one would enumerate them all. They are still finite. Breaking through and deductively considering them all at once with a simple tool of mathematical logic called a proof, the set of all of them exists, and has characteristics of being an infinite integer, enough to make it worthwhile to consider that so. That's like the difference between an analytical solution and approximation. Infinite sets are equivalent, because otherwise they wouldn't be infinite. When Cantor has results that appear to show they are not, then, to resolve that, one avenue of rational progression of mathematical logic to reconcile those claims is that there is a shared dual representation of infinity, and zero. That corresponds to the most ancient recorded folklore on the planet of the nature of being. You'd figure they got something correct. It also corresponds to modern philosophical methods, the dualist dichotomy. A consequence of the consideration of the reals and integers due to Cantor and earlier pigeonholing arguments might be that EF, the natural/unit equivalency fucntion, is the necessary building block of functions from the naturals bijectively to and from the real numbers, except those arguments don't apply or else they would apply to rational numbers. There are convenient models of the totally-ordered reals that help define those nonstandard, yet perfectly acceptible when used correctly, contiguous and continuous real numbers on the real number line. That notion also conveniently fits well with other set-theoretical notions attempting to explain the very real nature of mathematical objects, where a set theory needs an ur-element that should be as well a set. It can not be only a set, and the only other things that can not be a set must share a literal value of sorts, because there can only be one proper class, or none. There are a handful of what they call paradoxes or antinomies that have been plaguing set theory since its inception and coagulation upon ZF that are neatly resolved in that way. Consider Planck's constant, h, and Planck's constant, h bar = h / 2 pi. Something measured in a rational number of Planck distances (or Planck lengths) is some irrational number of modified, or corrected, Planck distances, because pi is irrational. That's about that irrational quantities exist. I had this fantastic notion this morning of a matrix of truth values, containing all the numbers and every true statement. Ross F. === Subject: Re: Cantor's diagonal proof wrong? <41aa5b47$13$fuzhry+tra$mr2ice@news.patriot.net> <41ad5139$15$fuzhry+tra$mr2ice@news.patriot.net> <41b2427f$15$fuzhry+tra$mr2ice@news.patriot.net> > ... That depends on what is is. It has no relevance to Mathematics, > which does not depend on your philosophy. Do you think philosophy has any bearing on mathematics? Philosophy has little bearing on mathematics, except the philosophical (philoshopical, philosophical) methods, particularly the rationalist ones, apply. What do you think of Hegel's Being and Nothing dichotomy as model of the ur-element? The set of all sets is its own powerset. An infinite set, in a finitist model, has a powerset that contains a dually represented element, in this way. The set may itself be considered that ur-element, to satisfy some ultrafinitist notions while coincidentally considering the infinite very particularly. Another notion of the philosophical ur-element is Kant's thing-in-itself and noumenon (noumena). What's the class of all classes? Is it the same thing as the set of all sets? Ross F. === Subject: Re: Cantor's diagonal proof wrong? <41aa5b47$13$fuzhry+tra$mr2ice@news.patriot.net> <41ad5139$15$fuzhry+tra$mr2ice@news.patriot.net> <41b2427f$15$fuzhry+tra$mr2ice@news.patriot.net> <41c22df7$7$fuzhry+tra$mr2ice@news.patriot.net> <41C25DB8.EF7C5F2F@tiki-lounge.com> ... That depends on what is is. It has no relevance to Mathematics, which does not depend on your philosophy. > Do you think philosophy has any bearing on mathematics? Philosophy has > little bearing on mathematics, except the philosophical (philoshopical, > philosophical) methods, particularly the rationalist ones, apply. Not only philosophy but, in particular, physics has. Withou matter there is not only no space but there is no means to store any number, not in an abacus, not in a pocket calculator, not in a computer and not in a brain. As there are at most 10^80 protons in the universe and some is 10^10^100. (Though there is not a largest number existing.) Therefore any considering of an actual infinity is nonsense from the scratch. This position isn't finitism, but realism, though not in the euphemistic meaning mathematicians like to use for their utterly unrealistic positions. Actual infinity was also denied by Hegel, by the way. But I don't know much of his his philosophy. > What do you think of Hegel's Being and Nothing dichotomy as model of the > ur-element? > The set of all sets is its own powerset. And, therefore, it cannot exist. The sum of all natural numbers is larger than any natural number. Therefore the set of all natural numbers cannot exist. (Would it actually exist, we could calculate the sum.) That's the same arguing, but mathematicians use to see things different. === Subject: Re: Cantor's diagonal proof wrong? <41aa5b47$13$fuzhry+tra$mr2ice@news.patriot.net> <41ad5139$15$fuzhry+tra$mr2ice@news.patriot.net> <41b2427f$15$fuzhry+tra$mr2ice@news.patriot.net> <41c22df7$7$fuzhry+tra$mr2ice@news.patriot.net> <41C25DB8.EF7C5F2F@tiki-lounge.com> at 01:56 PM, mueckenh@rz.fh-augsburg.de said: >And, therefore, it cannot exist. The sum of all natural numbers is >larger than any natural number. Therefore the set of all natural >numbers cannot exist. That's a non sequitor. >(Would it actually exist, we could calculate the sum.) No. There is no sum of the elements of an infinite set. There may or may not be a limit of a sequence of partial sums; in the case of the integers, there isn't. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: Cantor's diagonal proof wrong? <41aa5b47$13$fuzhry+tra$mr2ice@news.patriot.net> <41ad5139$15$fuzhry+tra$mr2ice@news.patriot.net> <41b2427f$15$fuzhry+tra$mr2ice@news.patriot.net> <41c22df7$7$fuzhry+tra$mr2ice@news.patriot.net> <41C25DB8.EF7C5F2F@tiki-lounge.com> !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw >> Do you think philosophy has any bearing on mathematics? Philosophy >> has little bearing on mathematics, except the philosophical >> (philoshopical, philosophical) methods, particularly the >> rationalist ones, apply. > Not only philosophy but, in particular, physics has. Withou matter > there is not only no space but there is no means to store any > number, not in an abacus, not in a pocket calculator, not in a > computer and not in a brain. But the good thing is that we only need the space for storing the laws that _all_ numbers obey. Like a Shakespearean play: the words of it are spoken by players, but the essence of the play is in the book, and it remains there even when a play is not being performed. And the literary critics can talk about the play without actually seeing a single performance of it. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Cantor's diagonal proof wrong? <41aa5b47$13$fuzhry+tra$mr2ice@news.patriot.net> <41ad5139$15$fuzhry+tra$mr2ice@news.patriot.net> <41b2427f$15$fuzhry+tra$mr2ice@news.patriot.net> <41c22df7$7$fuzhry+tra$mr2ice@news.patriot.net> <41C25DB8.EF7C5F2F@tiki-lounge.com> > Do you think philosophy has any bearing on mathematics? Philosophy > has little bearing on mathematics, except the philosophical > (philoshopical, philosophical) methods, particularly the > rationalist ones, apply. Not only philosophy but, in particular, physics has. Withou matter there is not only no space but there is no means to store any number, not in an abacus, not in a pocket calculator, not in a computer and not in a brain. > But the good thing is that we only need the space for storing the laws > that _all_ numbers obey. Like a Shakespearean play: the words of it > are spoken by players, but the essence of the play is in the book, and > it remains there even when a play is not being performed. > And the literary critics can talk about the play without actually > seeing a single performance of it. What's in a name? that which we call a number by another name would count as well. Could you give me the sum of Integer(pi*10^10^100) and Integer(sqrt(2)*10^10^100), please? Or are your laws insufficient to describe what primarily counts with numbers, namely counting? === Subject: Re: Cantor's diagonal proof wrong? part: >Hence the set of natural numbers cannot be actually infinite. It is >potentially infinite, i.e., there is no threshold but there is no >infinite number. No, what is potentially infinite is the set of natural numbers less than some finite number which I haven't named yet. The set of natural numbers is actually infinite - if one admits all the natural numbers actually exist as a set. One can reject the natural numbers as an object suitable for mathematical study as such, if one wishes to admit only the potentially infinite into mathematics. John Savard http://home.ecn.ab.ca/~jsavard/index.html === Subject: Re: Cantor's diagonal proof wrong? <41aa5b47$13$fuzhry+tra$mr2ice@news.patriot.net> <41ad5139$15$fuzhry+tra$mr2ice@news.patriot.net> <41b2427f$15$fuzhry+tra$mr2ice@news.patriot.net> <41c21cfb.2555895@news.ecn.ab.ca> > part: Hence the set of natural numbers cannot be actually infinite. It is potentially infinite, i.e., there is no threshold but there is no infinite number. > No, what is potentially infinite is the set of natural numbers less > than some finite number which I haven't named yet. The set of natural > numbers is actually infinite - if one admits all the natural numbers > actually exist as a set. The sequence A(n) = {1,2,3,...,n} up to any finite number n is finite too. If there are only finite numbers, then there are only finite sequences. (In the finite realm we have ordinal = cardinal.) We obtain the theorem: n e IN -> Card(A(n)) < aleph_0 which is equipollent to. Card(A(n)) !< aleph_0 -> n !e IN (with ! the symbol of negation). > One can reject the natural numbers as an object suitable for > mathematical study as such, if one wishes to admit only the potentially > infinite into mathematics. If adhering to realism (not what mathematicians call realism, but real realism) then one cannot but reject the set of all natural numbers as an existing entity. === Subject: Re: Cantor's diagonal proof wrong? > If adhering to realism (not what mathematicians call realism, but real > realism) then one cannot but reject the set of all natural numbers as > an existing entity. Plato's realm of Forms is the Real reality. Bob Kolker === Subject: Re: Cantor's diagonal proof wrong? <41aa5b47$13$fuzhry+tra$mr2ice@news.patriot.net> <41ad5139$15$fuzhry+tra$mr2ice@news.patriot.net> <41b2427f$15$fuzhry+tra$mr2ice@news.patriot.net> <41c21cfb.2555895@news.ecn.ab.ca> Plato's kind of philosophy is called idealism in all branches except mathematics, because it has nothing to do with reality. Mathematicians call it realism (cp. the ministery of peace in Orwell's 1984) === Subject: Re: Cantor's diagonal proof wrong? > You see it best, if you imagine Card{1,2,3,...,n} as a function f(n) of > n. The sloop is 1 and that will never change. FOR ALL FINITE NUMBERS n > (and there are no others) The sloop is in fact Sloop John B, and we hope it will never change. Hence the well-known exhortation: So hoist up the John B's sails, see how the main sail sets, Call for the captain ashore, and let me go home. Let me go home, I want to go home, Well I feel so break up, I want to go home. === Subject: Re: Cantor's diagonal proof wrong? >> You see it best, if you imagine Card{1,2,3,...,n} as a function f(n) of >> n. The sloop is 1 and that will never change. FOR ALL FINITE NUMBERS n >> (and there are no others) > The sloop is in fact Sloop John B, and we hope it will never >change. Hence the well-known exhortation: > So hoist up the John B's sails, see how the main sail sets, > Call for the captain ashore, and let me go home. > Let me go home, I want to go home, > Well I feel so break up, I want to go home. Duh, it was a typo. He meant slop. === Subject: re:PROOF that 0.99999... = 1 heres a much simpler proof: statement: .9999999...=1 since 9x=10x-x, 9=9 9=9.9999999...-.9999999... 9(1)=10(.9999999...)-.9999999 it's probably got some blindingly obvious flaw, but what the heck, im a freshman. *-----------------------* www.GroupSrv.com *-----------------------* === Subject: re:PROOF that 0.99999... = 1 >heres a much simpler proof: >statement: .9999999...=1 >since 9x=10x-x, >9=9 >9=9.9999999...-.9999999... >9(1)=10(.9999999...)-.9999999 9(1) =/= 9(.999...) Even if you assume you can do this operation with infinite repeating 9's. 9(.999...) = 8.999... 9 =/= 8.999... >it's probably got some blindingly obvious flaw, but what the heck, im >a freshman. >*-----------------------* > www.GroupSrv.com >*-----------------------* Smart's Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: re:PROOF that 0.99999... = 1 >> heres a much simpler proof: >> statement: .9999999...=1 >> since 9x=10x-x, >> 9=9 >> 9=9.9999999...-.9999999... >> 9(1)=10(.9999999...)-.9999999 > 9(1) =/= 9(.999...) Huh? Nowhere in this proof does he assume that 9(1) = 9(.999...). He assumes 9(1) = 9 (going from the second to last line that you quoted, to the last line). One reason this proof is deficient is because of the assumption that 10(.9999999...) = 9.9999999... (which is true, but needs to be proven). --Mark === Subject: Re: re:PROOF that 0.99999... = 1 > heres a much simpler proof: > statement: .9999999...=1 > since 9x=10x-x, > 9=9 > 9=9.9999999...-.9999999... > 9(1)=10(.9999999...)-.9999999 >> 9(1) =/= 9(.999...) >Huh? Nowhere in this proof does he assume that 9(1) = 9(.999...). He >assumes 9(1) = 9 (going from the second to last line that you quoted, to the >last line). >One reason this proof is deficient is because of the assumption that >10(.9999999...) = 9.9999999... (which is true, but needs to be proven). >--Mark But he assumes .999... = 1 in his equation before it is proven. Smart's Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: re:PROOF that 0.99999... = 1 >> heres a much simpler proof: >> statement: .9999999...=1 >> since 9x=10x-x, >> >> 9=9 >> 9=9.9999999...-.9999999... >> 9(1)=10(.9999999...)-.9999999 > 9(1) =/= 9(.999...) >> Huh? Nowhere in this proof does he assume that 9(1) = 9(.999...). >> He assumes 9(1) = 9 (going from the second to last line that you >> quoted, to the last line). >> One reason this proof is deficient is because of the assumption >> that 10(.9999999...) = 9.9999999... (which is true, but needs to be >> proven). >> --Mark > But he assumes .999... = 1 in his equation before it is proven. Would you point out where he makes this assumption? I repeat the entire proof, expanded a bit, with line numbers added for your convenience: [1] 9=9 [2] 9=9.9999999...-.9999999... [3] 9(1)=10(.9999999...)-.9999999 [4] Let x = .9999999... and substitute in [3] [5] 9(1) = 10x - x [6] 9(1) = 9(x) [7] Therefore x=1 In which line is the assumption .99999... = 1 used? --Mark === Subject: Re: PROOF that 0.99999... = 1 In sci.math, Mark Nudelman : > heres a much simpler proof: > statement: .9999999...=1 > since 9x=10x-x, > > 9=9 > 9=9.9999999...-.9999999... > 9(1)=10(.9999999...)-.9999999 >> >> 9(1) =/= 9(.999...) > Huh? Nowhere in this proof does he assume that 9(1) = 9(.999...). > He assumes 9(1) = 9 (going from the second to last line that you > quoted, to the last line). > One reason this proof is deficient is because of the assumption > that 10(.9999999...) = 9.9999999... (which is true, but needs to be > proven). > --Mark >> But he assumes .999... = 1 in his equation before it is proven. > Would you point out where he makes this assumption? I repeat the entire > proof, expanded a bit, with line numbers added for your convenience: > [1] 9=9 > [2] 9=9.9999999...-.9999999... > [3] 9(1)=10(.9999999...)-.9999999 > [4] Let x = .9999999... and substitute in [3] > [5] 9(1) = 10x - x > [6] 9(1) = 9(x) > [7] Therefore x=1 > In which line is the assumption .99999... = 1 used? [3]. The possibility of an infinite borrow generates headaches. > --Mark -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: PROOF that 0.99999... = 1 > In sci.math, Mark Nudelman >> Would you point out where he makes this assumption? I repeat the >> entire proof, expanded a bit, with line numbers added for your >> convenience: >> [1] 9=9 >> [2] 9=9.9999999...-.9999999... >> [3] 9(1)=10(.9999999...)-.9999999 >> [4] Let x = .9999999... and substitute in [3] >> [5] 9(1) = 10x - x >> [6] 9(1) = 9(x) >> [7] Therefore x=1 >> In which line is the assumption .99999... = 1 used? > [3]. The possibility of an infinite borrow generates headaches. Going from [2] to [3] merely assumes that 10(.99999...) = 9.9999.... This is indeed problematic and needs to be proven, as does the assumption that 9 = 9.99999...- 0.999999 in going from [1] to [2], but I don't see that either of these steps uses the assumption that .99999... = 1. --Mark === Subject: Re: re:PROOF that 0.99999... = 1 >> heres a much simpler proof: >> statement: .9999999...=1 >> since 9x=10x-x, >> >> 9=9 >> 9=9.9999999...-.9999999... >> 9(1)=10(.9999999...)-.9999999 > 9(1) =/= 9(.999...) >>Huh? Nowhere in this proof does he assume that 9(1) = 9(.999...). He >>assumes 9(1) = 9 (going from the second to last line that you quoted, to >>the >>last line). >>One reason this proof is deficient is because of the assumption that >>10(.9999999...) = 9.9999999... (which is true, but needs to be proven). >>--Mark > But he assumes .999... = 1 in his equation before it is proven. > Smart's Alt. Physics News Group > http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 > S. Enterprize (Science Journal) > http://smart1234.s-enterprize.com/ jesus christ! do you know anything about mathematical induction?????????????? let x_n = 9*sum((1/10)^k,k=0..n) = 9*(1 + 1/10 + 1/100 + .. 1/10^n) = 9*(1.11111111...) = 9.999999.. then |10 - x_n| = |10 - 9*sum((1/10^k,k=0..n))| = |10 - ((1/10)^(k+1) - 1)/(1 - 1/10))| = |1/10^n| = 1/10^n < e for all n >= N > -log(e) that means, the difference between the infinitely repeating decimal with period one is the same as 10, i.e. 9.9999999...... = 10 (ofcourse, this work for any number, not just 9) if you don't believe that x_n = 9.9999999999999999999 then thats your fault, you need to learn some simple math.... just try to find me a number sticktly between .999999999999..... and 1! you can do this for all x if you want... x = [x] + {x} = floor(x) + sum((floor((n-x)*10^k) mod 10)/10^k) if x is terminating or repeating in its tail, then the sum has a simple solution and its easy to calculate the answer. if you put x = 1, the {x} = 0 x = .99999...... then sum is just over 9/10^k which is easily to compute again, the only thing that you can have any sorta problem with is how .9999999999 could be reprsented by the sum, but that is your problem... as any halfwit knows that. === Subject: Re: re:PROOF that 0.99999... = 1 > heres a much simpler proof: > statement: .9999999...=1 > since 9x=10x-x, > > 9=9 > 9=9.9999999...-.9999999... > 9(1)=10(.9999999...)-.9999999 >> >> 9(1) =/= 9(.999...) >Huh? Nowhere in this proof does he assume that 9(1) = 9(.999...). He >assumes 9(1) = 9 (going from the second to last line that you quoted, to >the >last line). >One reason this proof is deficient is because of the assumption that >10(.9999999...) = 9.9999999... (which is true, but needs to be proven). >--Mark >> But he assumes .999... = 1 in his equation before it is proven. >> Smart's Alt. Physics News Group >> http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 >> S. Enterprize (Science Journal) >> http://smart1234.s-enterprize.com/ >jesus christ! >do you know anything about mathematical induction?????????????? >let x_n = 9*sum((1/10)^k,k=0..n) = 9*(1 + 1/10 + 1/100 + .. 1/10^n) = >9*(1.11111111...) = 9.999999.. >then |10 - x_n| = |10 - 9*sum((1/10^k,k=0..n))| = |10 - ((1/10)^(k+1) - >1)/(1 - 1/10))| >= |1/10^n| = 1/10^n < e for all n >= N > -log(e) >that means, the difference between the infinitely repeating decimal with >period one is the same as 10, i.e. 9.9999999...... = 10 (ofcourse, this work >for any number, not just 9) >if you don't believe that x_n = 9.9999999999999999999 then thats your fault, >you need to learn some simple math.... just try to find me a number sticktly >between .999999999999..... and 1! >you can do this for all x if you want... >x = [x] + {x} = floor(x) + sum((floor((n-x)*10^k) mod 10)/10^k) >if x is terminating or repeating in its tail, then the sum has a simple >solution and its easy to calculate the answer. >if you put x = 1, the {x} = 0 >x = .99999...... >then sum is just over 9/10^k which is easily to compute >again, the only thing that you can have any sorta problem with is how >.9999999999 could be reprsented by the sum, but that is your problem... as >any halfwit knows that. Hey .999... IS NOT A REAL NUMBER (PERIOD). See math link below: http://mathworld.wolfram.com/HyperrealNumber.html .999... is of the form of a hyper-real number because there is a space between the real numbers between .999... and 1. .999... | | 1 ^ | See space A Hyperreal number is of the form Where n is a real number, x < n x = .999... n = 1 .999... < 1 THEREFORE, .999... =/= 1 Smart's Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: re:PROOF that 0.99999... = 1 >> heres a much simpler proof: >> statement: .9999999...=1 >> since 9x=10x-x, >> >> 9=9 >> 9=9.9999999...-.9999999... >> 9(1)=10(.9999999...)-.9999999 > > 9(1) =/= 9(.999...) >> >>Huh? Nowhere in this proof does he assume that 9(1) = 9(.999...). He >>assumes 9(1) = 9 (going from the second to last line that you quoted, to >>the >>last line). >> >>One reason this proof is deficient is because of the assumption that >>10(.9999999...) = 9.9999999... (which is true, but needs to be proven). >> >>--Mark >> > But he assumes .999... = 1 in his equation before it is proven. > Smart's Alt. Physics News Group > http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 > S. Enterprize (Science Journal) > http://smart1234.s-enterprize.com/ >>jesus christ! >>do you know anything about mathematical induction?????????????? >>let x_n = 9*sum((1/10)^k,k=0..n) = 9*(1 + 1/10 + 1/100 + .. 1/10^n) = >>9*(1.11111111...) = 9.999999.. >>then |10 - x_n| = |10 - 9*sum((1/10^k,k=0..n))| = |10 - ((1/10)^(k+1) - >>1)/(1 - 1/10))| >>= |1/10^n| = 1/10^n < e for all n >= N > -log(e) >>that means, the difference between the infinitely repeating decimal with >>period one is the same as 10, i.e. 9.9999999...... = 10 (ofcourse, this >>work >>for any number, not just 9) >>if you don't believe that x_n = 9.9999999999999999999 then thats your >>fault, >>you need to learn some simple math.... just try to find me a number >>sticktly >>between .999999999999..... and 1! >>you can do this for all x if you want... >>x = [x] + {x} = floor(x) + sum((floor((n-x)*10^k) mod 10)/10^k) >>if x is terminating or repeating in its tail, then the sum has a simple >>solution and its easy to calculate the answer. >>if you put x = 1, the {x} = 0 >>x = .99999...... >>then sum is just over 9/10^k which is easily to compute >>again, the only thing that you can have any sorta problem with is how >>.9999999999 could be reprsented by the sum, but that is your problem... as >>any halfwit knows that. > Hey .999... IS NOT A REAL NUMBER (PERIOD). > See math link below: > http://mathworld.wolfram.com/HyperrealNumber.html > .999... is of the form of a hyper-real number because there is a space > between > the real numbers between .999... and 1. > .999... | | 1 > ^ > | > See space > A Hyperreal number is of the form > Where n is a real number, > x < n > x = .999... > n = 1 > .999... < 1 > THEREFORE, > .999... =/= 1 > Smart's Alt. Physics News Group > http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 > S. Enterprize (Science Journal) > http://smart1234.s-enterprize.com/ your a freaken genius!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! === Subject: Re: PROOF that 0.99999... = 1 In sci.math, Jon Slaughter <10sbusfei7k2lee@corp.supernews.com>: [snipped for sanity] >> .999... =/= 1 >> Smart's Alt. Physics News Group >> http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 >> S. Enterprize (Science Journal) >> http://smart1234.s-enterprize.com/ > your a freaken genius!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! FSVO genius. Most of us use an alternate word with one less letter. :-) -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: re:PROOF that 0.99999... = 1 > heres a much simpler proof: > statement: .9999999...=1 > since 9x=10x-x, > > 9=9 > 9=9.9999999...-.9999999... > 9(1)=10(.9999999...)-.9999999 >> >> 9(1) =/= 9(.999...) > >Huh? Nowhere in this proof does he assume that 9(1) = 9(.999...). He >assumes 9(1) = 9 (going from the second to last line that you quoted, to >the >last line). > >One reason this proof is deficient is because of the assumption that >10(.9999999...) = 9.9999999... (which is true, but needs to be proven). > >--Mark > >> >> But he assumes .999... = 1 in his equation before it is proven. >> >> >> Smart's Alt. Physics News Group >> http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 >> S. Enterprize (Science Journal) >> http://smart1234.s-enterprize.com/ >> >> >jesus christ! >do you know anything about mathematical induction?????????????? >let x_n = 9*sum((1/10)^k,k=0..n) = 9*(1 + 1/10 + 1/100 + .. 1/10^n) = >9*(1.11111111...) = 9.999999.. >then |10 - x_n| = |10 - 9*sum((1/10^k,k=0..n))| = |10 - ((1/10)^(k+1) - >1)/(1 - 1/10))| >= |1/10^n| = 1/10^n < e for all n >= N > -log(e) >that means, the difference between the infinitely repeating decimal with >period one is the same as 10, i.e. 9.9999999...... = 10 (ofcourse, this >work >for any number, not just 9) >if you don't believe that x_n = 9.9999999999999999999 then thats your >fault, >you need to learn some simple math.... just try to find me a number >sticktly >between .999999999999..... and 1! >you can do this for all x if you want... >x = [x] + {x} = floor(x) + sum((floor((n-x)*10^k) mod 10)/10^k) >if x is terminating or repeating in its tail, then the sum has a simple >solution and its easy to calculate the answer. >if you put x = 1, the {x} = 0 >x = .99999...... >then sum is just over 9/10^k which is easily to compute >again, the only thing that you can have any sorta problem with is how >.9999999999 could be reprsented by the sum, but that is your problem... as >any halfwit knows that. >> Hey .999... IS NOT A REAL NUMBER (PERIOD). >> See math link below: >> http://mathworld.wolfram.com/HyperrealNumber.html >> .999... is of the form of a hyper-real number because there is a space >> between >> the real numbers between .999... and 1. >> .999... | | 1 >> ^ >> | >> See space >> A Hyperreal number is of the form >> Where n is a real number, >> x < n >> x = .999... >> n = 1 >> .999... < 1 >> THEREFORE, >> .999... =/= 1 >> Smart's Alt. Physics News Group >> http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 >> S. Enterprize (Science Journal) >> http://smart1234.s-enterprize.com/ >your a freaken genius!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! you're not your Smart's Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: PROOF that 0.99999... = 1 > .999... | | 1 > ^ > | > See space Pure scribble. > A Hyperreal number is of the form You would not know a hyperreal if it bit you. You have not the foggiest notion of how the real number system R is extended to *R. Bob Kolker === Subject: Re: PROOF that 0.99999... = 1 >> .999... | | 1 >> ^ >> | >> See space >Pure scribble. >> A Hyperreal number is of the form >You would not know a hyperreal if it bit you. You have not the foggiest >notion of how the real number system R is extended to *R. >Bob Kolker Hey, I thought you said I didn't know what it was. You are wrong again, and again, again. http://mathworld.wolfram.com/HyperrealNumber.html Smart's Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: PROOF that 0.99999... = 1 In sci.math, S. Enterprize Company > > .999... | | 1 > ^ > | > See space >>Pure scribble. > > A Hyperreal number is of the form >>You would not know a hyperreal if it bit you. You have not the foggiest >>notion of how the real number system R is extended to *R. >>Bob Kolker > Hey, I thought you said I didn't know what it was. You are wrong again, and > again, again. > http://mathworld.wolfram.com/HyperrealNumber.html Like that tells him *anything*. Here's a few Qs for you. [1] If d is such that 0 < d < 1/n for all n in N, what is d^2? d^3? sqrt(d)? [2] Why is 5/5 != 9/9? 5/5 = 1, of course; 0.2 * 5 = 1. 9/9, by contrast, is 0.111... * 9 = 0.999... = 1 - d. In base 12, 1/9 = 0.14(12) but 1/5 = .24972497...(12) ; therefore in this case 9/9 = 1 but 5/5 = 1-d. Does it matter what base one uses for arithmetic? [3] Explain how one computes D_10[.999..., w-1], where w (omega) is the first transfinite ordinal, and D_10[r,n] is r's n'th digit to the right of the decimal point, if n is an integer, then evaluate D_10[(.999... + 9)/10, w-1] and D_10[.999... * 10 - 9, w-1]. (n can be negative but that's not all that important here.) [.sigsnip] -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: PROOF that 0.99999... = 1 > jesus christ! > do you know anything about mathematical induction?????????????? Enterprise does not even know what end comes out of. He is a total mathematical incompetent. He makes JSH look intelligent by comparison. Bob Kolker === Subject: Re: PROOF that 0.99999... = 1 >> jesus christ! >> do you know anything about mathematical induction?????????????? >Enterprise does not even know what end comes out of. He is a total >mathematical incompetent. He makes JSH look intelligent by comparison. >Bob Kolker What's a hyper-real number? Do you even know anything about math? Smart's Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: PROOF that 0.99999... = 1 > What's a hyper-real number? Do you even know anything about math? No. But I do know how the hyperrals are constructed. Bob Kolker === Subject: Re: PROOF that 0.99999... = 1 In sci.math, robert j. kolker : >> What's a hyper-real number? Do you even know anything about math? > No. But I do know how the hyperrals are constructed. > Bob Kolker http://mathworld.wolfram.com/HyperrealNumber.html is extremely bare-bones (is there one hyperreal? more than one? arithmetic operations? proofs?) but at least it's a start. A reference link http://members.tripod.com/PhilipApps/line.html looks to be little more than my attempts at d-math, though there might be more than one d -- or H, its dual. No doubt one could claim at least three theories: [1] An infinite hierarchy of d < 1/n for all n in N: 0 < ... < d^4 < d^3 < d^2 < d < 1, with a more or less standard algebra (e.g., (1-d)^3 = 1 - 3d + 3d^2 - d^3). [2] d^k = d for some k in N. [3] Some other esoteric condition. I suppose one might even notate this as R[d] -- a standard polynomial group over R, with a slightly weird ordering. And again, I must complain that S. Enterprize is being extremely sloppy here. (Not that I'm all that neat, but hopefully my notation's clear at least.) -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: PROOF that 0.99999... = 1 >> What's a hyper-real number? Do you even know anything about math? >No. But I do know how the hyperrals are You don't even know what a hyper-real number is??? And you are name calling people here like you know everything?????? Why not admit you ARE WRONG! constructed. >Bob Kolker Smart's Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: PROOF that 0.99999... = 1 > What's a hyper-real number? Do you even know anything about math? >>No. But I do know how the hyperrals are > You don't even know what a hyper-real number is??? And you are name calling > people here like you know everything?????? Why not admit you ARE WRONG! Quick. Define an ultra-filter. No, don't look it up. Bob Kolker === Subject: Re: PROOF that 0.99999... = 1 >> What's a hyper-real number? Do you even know anything about math? >No. But I do know how the hyperrals are >> You don't even know what a hyper-real number is??? And you are name >calling >> people here like you know everything?????? Why not admit you ARE WRONG! >Quick. Define an ultra-filter. No, don't look it up. >Bob Kolker Oh this is so hard to understand, I might need to take an asprin for a headache. I'll define it with an example. Suppose you have alot of people here making noise here on this NG and they don't know what they are talking about with .999..., and then comes along an ultrafilter F_Smart1234 with the correct information. What we do is apply ultrafilter F_Smart1234 to the whole set S of noise on the NG, and then just the pure correct answer is shown. The ultrafilter is then said to be a success and has worked very well, and is therefore proven. Smart's Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: PROOF that 0.99999... = 1 >> >> > What's a hyper-real number? Do you even know anything about math? >> >>No. But I do know how the hyperrals are > > You don't even know what a hyper-real number is??? And you are name >>calling > people here like you know everything?????? Why not admit you ARE WRONG! >>Quick. Define an ultra-filter. No, don't look it up. >>Bob Kolker > Oh this is so hard to understand, I might need to take an asprin for a >headache. > I'll define it with an example. Suppose you have alot of people here >making >noise here on this NG and they don't know what they are talking about with >.999..., and then comes along an ultrafilter F_Smart1234 with the correct >information. What we do is apply ultrafilter F_Smart1234 to the whole set S >noise on the NG, and then just the pure correct answer is shown. > The ultrafilter is then said to be a success and has worked very well, and >is therefore proven. Your turn. Perform a ANOVA statistical test between .999... and 1. And of course go into details explaining what the ANOVA test is. hurry hurry don't look... Smart's Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: PROOF that 0.99999... = 1 >> What's a hyper-real number? Do you even know anything about math? >No. But I do know how the hyperrals are >> You don't even know what a hyper-real number is??? And you are name >calling >> people here like you know everything?????? Why not admit you ARE WRONG! >Quick. Define an ultra-filter. No, don't look it up. Oh, but I do have the right to refresh my memory. I even gave you time to do this and you still don't know what a hyper-real number is. >Bob Kolker Smart's Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: PROOF that 0.99999... = 1 >In sci.math, S. Enterprize Company > >In sci.math, S. Enterprize Company > >I made a typing error here, > >10x = 9 + x > >> >> It's still useless. >> >> let x = 2 >> >> 10( 2) = 9 + 2 >> >> 20 =/= 11 >> >Congratulations! You've proven that x=2 is not a root of >the above equation. >Care to try another number, though? :-) >[.sigsnip] >-- >#191, ewill3@earthlink.net >It's still legal to go .sigless. >> Ok, let's look at it in this point of view. >> 10x = 9 + x >> x can equal 1 or .999... . >> There are two roots to the equation. >> x1 = 1 >> x2 = .999... >> but, >> x1 =/= x2 >If x1 != x2, is 10*x1 - 9 - x1 == 10*x2 - 9 - x2? x1 =/= x2 Two different roots. >[rest snipped] >-- >#191, ewill3@earthlink.net >It's still legal to go .sigless. Smart's Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: PROOF that 0.99999... = 1 [megasnip] 1/3 = .33333... = 3x10^-1+3x10^-2+3x10^-3+3x10^-4+3x10^-5+... thus: 3*1/3 = 3*(3x10^-1+3x10^-2+3x10^-3+3x10^-4+3x10^-5+...) which is: 9x10^-1+9x10^-2+9x10^-3+9x10^-4+9x10^-5+... = .99999... therefore .99999... = 3*1/3 = 3/3 = 1 [] === Subject: Re: PROOF that 0.99999... = 1 In sci.math, Dan Weiner : > [megasnip] > 1/3 = .33333... = 3x10^-1+3x10^-2+3x10^-3+3x10^-4+3x10^-5+... > thus: > 3*1/3 = > 3*(3x10^-1+3x10^-2+3x10^-3+3x10^-4+3x10^-5+...) > which is: > 9x10^-1+9x10^-2+9x10^-3+9x10^-4+9x10^-5+... = .99999... > therefore .99999... = 3*1/3 = 3/3 = 1 > [] Not quite rigorous enough, but pretty close. One first has to make sure 3 * (3*10^-1 + 3*10^-2 + ... + 3*10^(-n) + ...) makes sense. To illustrate, I shall prove the nonsense result below: Let x = 1 + 2 + 4 + 8 + ... + 2^n + ... Then 2 * x = 2 * (1 + 2 + 4 + 8 + ... + 2^n + ...) = 2 + 4 + 8 + ... + 2^(n+1) + ... = x - 1 Since 2 * x = x - 1, x = -1. QED. The main flaw of course is the divergent series x. This one's obvious but there are some subtle ones such as the alternate-sign harmonic series that aren't. Fortunately, Cauchy proved that, under certain conditions (which are met by your series, but not by mine :-) ) the distribution of the product does make sense. One can go the long way, if one wishes; define S3_n = .333...3 (n 3's) = (3 * 10^-1 + ... + 3 * 10^-n). Multiplying S3_n by 3 is perfectly valid, and leads to S9_n = .999...9 = (9 * 10^-1 + ... + 9 * 10^-n). As n tends to the limit, one can easily prove that S3_n -> S3 = 1/3, and therefore S9_n -> S9 = 3 * S3 = 3/3 = 1. Not that I expect S. Enterprize to understand much of this, as he seems to think ... = oo and have some very weird notions regarding series -- and no understanding at all regarding limits. -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: PROOF that 0.99999... = 1 > Not quite rigorous enough, but pretty close. One first has > to make sure 3 * (3*10^-1 + 3*10^-2 + ... + 3*10^(-n) + ...) > makes sense. Do you accept 1/3 = (3*10^-1 + 3*10^-2 + ... + 3*10^(-n) + ...) ? It pretty much follows that the series converges.... Anyway, if you REALLY want proof, the series is Cauchy therefore it's OK. QED. Hah! > Not that I expect S. Enterprize to understand much of this, as > he seems to think ... = oo and have some very weird notions > regarding series -- and no understanding at all regarding limits. True... why did we even bother... --D === Subject: Re: PROOF that 0.99999... = 1 In sci.math, S. Enterprize Company >>In sci.math, S. Enterprize Company >> >>In sci.math, S. Enterprize Company >> >>I made a typing error here, >> >>10x = 9 + x >> > > It's still useless. > > let x = 2 > > 10( 2) = 9 + 2 > > 20 =/= 11 > >> >>Congratulations! You've proven that x=2 is not a root of >>the above equation. >> >>Care to try another number, though? :-) >> >>[.sigsnip] >> >>-- >>#191, ewill3@earthlink.net >>It's still legal to go .sigless. >> >> > Ok, let's look at it in this point of view. > 10x = 9 + x > x can equal 1 or .999... . > There are two roots to the equation. > x1 = 1 > x2 = .999... > but, > x1 =/= x2 >>If x1 != x2, is 10*x1 - 9 - x1 == 10*x2 - 9 - x2? > x1 =/= x2 > Two different roots. Oh, so an equation of polynomial degree n (n=1 in this case) can have more than n roots? [rest snipped] -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics > Philosophy argues that A is A, B is B and C is C. They are all part of an > didactic letternomic set with has been assigned arbitrary cultural and > cognitive importance... at least that's what it says here... in this book. Obviously I missed this... Philosophy argues that A is A, B is B and C is C but is 'is' =? === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics <41c10c20.49709196@netnews.att.net> <41c13ddb.56649778@netnews.att.net> said: >Quantum mechanics already is a mechanics. It's not classical >mechanics, but it's quantum mechanics. Classical mechanics can be >shown to be a specialization of quantum mechanics. Not if you take gravity into account. Or, at least, not yet. >And my understanding is that it's impossible to reduce quantum to >classical mechanics because classical mechanics doesn't have the >concept of a superposition of states. Classical EM theory has it, and you can do nonrelativistic QM as a hidden variables theory. Of course, you need spooky action at a distance to do it. >Axioms don't fail-- sets of axioms can fail. They can lead to >internal contradictions, or they can be empirically falsified. What you empirically falsify is not the set of axioms, but rather the claim that it applies in a specific way to a specific physical system. That doesn't keep the axiom system from being Mathematically interesting and relevant, or even keep it from being appropriate for a different physical system. -- Shmuel (Seymour J.) Metz, SysProg and JOAT Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) in [. . .] >>I strongly suspect that I dis quantum theory primarily because it >>morphed physics from no stinkin reasons we can see into just plain >>ol' no stinkin reasons. >So you want to reduce quantum mechanics to classical mechanics, I suppose. >But ignoring whether that's even possible, which unanswered questions >about classical mechanics have you chosen not to ask? >>This is interesting. I think I expect quantum theory can be reduced to >>mechanics even if it can't be exactly reduced to classical mechanics. >Quantum mechanics already is a mechanics. It's not classical mechanics, >but it's quantum mechanics. Classical mechanics can be shown to be a >specialization of quantum mechanics. And my understanding is that it's >impossible to reduce quantum to classical mechanics because classical >mechanics doesn't have the concept of a superposition of states. It has >statistical mixtures of states, but not superpositions. Yeah, here I'd like to say that I consider quantum mechanics not a mechanics at all because it is really just a series of principles not relating things defined by those principles. I consider that there are certainly quantum effects and there is certainly quantum theory. But I don't see quantum theory explaining transitions between and among various quantum effects. And that's what I consider a mechanics does. Newtonian and classical mechanics generally rested their mechanics not on general observations alone but explanations for transitions between and among observations. That's mechanics. QM maintains there is no choose of those open to it as long as conservation principles apply in aggregate whereas classical mechanics considers those conservation laws apply in particular as well as in general. And without particular You just wind up with a non material anthropomorphic probability. As far as superposition of states is concerned, I don't know of any tenet of classical mechanics that precludes it if I understand the point correctly. Certainly there is a superposition of properties. [. . .] >Why is it possible to develop a mechanics based on the concept of the >>I suspect it would be a lot easier to develop a quantum mechanics >>developed such a mechanics it will be a lot easier to develop a >>mechanics consistent with the electrodynamics of moving bodies. >When an object goes from point A to point B, why should we expect it to >occupy in succession every point along a path from A to B rather than >teleporting to its destination? >>We don't necessarily. It's just that the extrapolation of a mechanics >>and B in the same terms we describe every point in between. If you or >>anyone else can explain teleportation in such terms consistent with >>the designation of A and B as points of properties common to those in >>between A and B, we can certainly re examine the issue in mechanical >>terms. >It's a generalization of our experience. We see a baseball occupy a >succession of points from A to B. A Volkswagon Quantum has never been >observed to pass safely through a bridge support or another car. When >I've seen, um, free-thinkers rail against relativity and cry Illogical!, >sometimes the only reasonable interpretation seems to be that they don't >and take a postulate inspired by experience to be a logical necessity >rather than a postulate. But we know that the very fast and the very >small don't act like little baseballs, and I don't know why we should >expect common sense to extend beyond the common phenomena that inspired >it. I agree it's a generalization from experience but that it is also a very poorly analyzed generalization because what is being generalized is an amalgm of geometric and material circumstances having no necessary connection to one another. Our experience superimposes geometric spatiality on material circumstances and then maintains that the cause of material circumstances is only certain aspects of the geometry and not the geometry as a whole. We are the ones who characterize some object as existing at point A and then point B according to experience and then we blithely ellide the points in between as if our geometry of paths weren't based on all points along the path of motion to begin with. The point of lack of experience is often raised against putative debunkers of relativity when they cry foul. But the point is that Einstein's own suppositions debunk SR contraction hypotheses and not any extrapolated material consequences. The logic is simple if not intuitively obvious to the casual observer: A frame of reference in SR is defined in terms of velocity alone (some would argue the term should be relative velocity, but the point is nugatory since velocities are always relative) because all bodies of a common velocity occupy a common frame of reference because they all have zero relative velocity, and bodies with different velocities occupy different frames of reference because they have different relative velocities. Thus the sole mechanical determinant for any frame of reference in SR is velocity and not anything else. However, palpable bodies represent composites of interstitial bodies moving at different velocities relative to one another which means they occupy different frames of reference geometrically overlapping one another. Hence there can be no uniform geometric contraction applying to the body as a whole and thus no geometric explanation for an isotropic frequency dilation as supposed by Einstein in SR. Consequently, as a practical matter there is no geometric contraction at velocity and observed frequency dilations cannot be isotropic and must therefore be anisotropic and be explained by the bidirectional relative velocity of light at right angles to the direction of motion because this factor has exactly the same magnitude as the observed frequency dilation whereas the same factor in the directions of motion does not. Now, the rest of the world can rail against the heavens, but this is what I consider a mechanical explanation for relativistic effects and a mechanical reductio argument against geometric contraction and isotropic frequency dilation in defintive logical form. >Why should time move forward at the same rate at all points in the >universe? Why should time always move forward at all points in the >universe? >>Well, time in what I choose to call material terms is just a temporal >>metric we commonly analyze in terms of EM frequency. In the past it >>has been measured in other terms of pendulum, balance wheel, or solar >>frequency. So, if these metrics change throughout the universe in >>response to velocity or gravitation, I, if not classical mechanics, >>see no reason our measures of time should not change as well. >You're not thinking dramatically enough! This is exactly what makes physicists into drama queens uttering the immortal line: we don't need no stinkin reasons. If you want drama, go on stage. A collateral objective of mine is to get scientists out of drama and back into mundane elementary mechanics instead of acting like a bunch of fairy queens preening and primping on a universal stage. > It's not a priori impossible for >someone to visit The Vortex in Oregon and come back ten years younger than >when he had left. It's not something we observe in practice, but that >sort of thing can't be ruled out a priori. I think you mean ten years younger than we are when he gets back. Nothing will make him any younger than he was when he left. In any event there is nothing apriori preventing this as long as he transits all the space in between at velocities probably not exceeding that of light. >But when you look for stinkin' reasons, you either have to derive them >from a priori necessities, or you've only given a reason in terms of other >postulates that have no stinkin' reason. And like I've suggested above, >a lot of what some people consider a priori necessities are only personal >preferences. No stinkin reason regressions are one of the greatest problems in conventional science. We have to find some logical necessity and way to preclude infinite regression or we have no mechanics or science; we only have more or less self consistent plausibilities. >Consider also the well known point that even if there was a One True >Theory that exists, and even if we find it, we can never really prove that >we've found it. And that's really my basis for a distaste of finding the >real reasons for fundamental physics-- anyone who claims they've found >the real reasons as opposed to the fake reasons that merely make all the >right predictions have duped themselves; they're just guessing, and can >never do more. Quoth the raven, nevermore. And you know this how? All you can really know empirically is what has been found and not what can't be found. What can't be found is a matter of proof and self contradiction and there is none here not born of the frustration of empirical failure to explain.. >The first two questions, at least, can be answered by quantum mechanics. >whose position and momentum uncertainties are much smaller than the >interpretation of the fields is available in the mostly flat and quiescent >spacetime we find ourselves in. >>Well, you don't eliminate issues by failing to address them. You posit >>certain failings of classical mechanics and then assert that these can >>be resolved by quantum theory because quantum theory doesn't rely on >>the issues you posit for classical mechanics. Quantum theory just says >>it has no idea what these things mean in mechanical terms: they're >>just so many empirical ad hoc observations that quantum theory fails >>to justify on any basis of mechanical interconnectedness either. >The philosophical lesson to take away is that a definite trajectory, and >not something the metaphysician can just take for granted. An explanation >shift us to a different set of unanswered questions. according to what mechanical necessity. As long as metaphysicians and this will undoubtedly remain the case. When they get around to analyzing the stinkin reasons for their beliefs and what they imagine to be true, however, all that can change. >All theories have things that are true for no stinkin' reason, including >classical mechanics. They're called postulates. If they could be >derived from more basic notions they wouldn't be postulates. But >sometimes people reach their comfort zone and it doesn't even occur to >them to keep asking what the stinkin' reasons are. >>Yes, well, that's a common failing of axiomatic systems. When axioms >>fail we do indeed tend to move into comfort zones where such questions >>are no longer asked because we no longer feel comfortable asking >>questions our axioms can't answer or even address. That's exactly what >>happened to classical physics. So we replaced these postulates with >>more comfortable ones known as quantum theory and relativity instead >>of resolving the issue of failed axioms. >Axioms don't fail-- sets of axioms can fail. A distinction without a difference. The plural of axiom is axioms meaning sets of axioms. > They can lead to internal >contradictions, or they can be empirically falsified. Falsification >doesn't tell you which axiom was wrong, or even that a single axiom was to >blame. And virtually any axiom can be asserted true if you change the >remainder of the set appropriately. I have yet to see axioms emprically falsified. They're usually only logically contradicted by other axioms when one set fails to explain empirical observations the way another set does. And every set of axioms can be asserted true but that does not make them true. >-- >Experiments are the only means of knowledge at our disposal. The rest is >poetry, imagination. -- Max Planck Yeah, well empirical observation is only one part of tautologies, and the raveled sleeve of reality is knit up by both halves and not by the emprirical pretense of people who substitute axioms for explanations. -- Lester Zick === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) in >[. . .] >I strongly suspect that I dis quantum theory primarily because it >morphed physics from no stinkin reasons we can see into just plain >ol' no stinkin reasons. >> >>So you want to reduce quantum mechanics to classical mechanics, I suppose. >>But ignoring whether that's even possible, which unanswered questions >>about classical mechanics have you chosen not to ask? >This is interesting. I think I expect quantum theory can be reduced to >mechanics even if it can't be exactly reduced to classical mechanics. >>Quantum mechanics already is a mechanics. It's not classical mechanics, >>but it's quantum mechanics. Classical mechanics can be shown to be a >>specialization of quantum mechanics. And my understanding is that it's >>impossible to reduce quantum to classical mechanics because classical >>mechanics doesn't have the concept of a superposition of states. It has >>statistical mixtures of states, but not superpositions. >Yeah, here I'd like to say that I consider quantum mechanics not a >mechanics at all because it is really just a series of principles not >relating things defined by those principles. I consider that there are >certainly quantum effects and there is certainly quantum theory. But I >don't see quantum theory explaining transitions between and among >various quantum effects. And that's what I consider a mechanics does. Apply the time evolution operator. >Newtonian and classical mechanics generally rested their mechanics not >on general observations alone but explanations for transitions between >and among observations. That's mechanics. QM maintains there is no >choose of those open to it as long as conservation principles apply in >aggregate whereas classical mechanics considers those conservation >laws apply in particular as well as in general. And without particular >You just wind up with a non material anthropomorphic probability. Again I warn you against falling prey to your own hidden assumptions. Quantum mechanics maintains that there is no defined path! Not that the even exist in the first place. By insisting that a path be taken, you've already inserted a postulate of your own preference, and I'll immediately ask you the reason that a single path can be taken. >[. . .] >>Why should time move forward at the same rate at all points in the >>universe? Why should time always move forward at all points in the >>universe? >Well, time in what I choose to call material terms is just a temporal >metric we commonly analyze in terms of EM frequency. In the past it >has been measured in other terms of pendulum, balance wheel, or solar >frequency. So, if these metrics change throughout the universe in >response to velocity or gravitation, I, if not classical mechanics, >see no reason our measures of time should not change as well. >>You're not thinking dramatically enough! >This is exactly what makes physicists into drama queens uttering the >immortal line: we don't need no stinkin reasons. If you want drama, >go on stage. A collateral objective of mine is to get scientists out >of drama and back into mundane elementary mechanics instead of acting >like a bunch of fairy queens preening and primping on a universal >stage. Then let your clock come back from The Vortex rewound. Whatever. >> It's >not a priori impossible for >>someone to visit The Vortex in Oregon and come back ten years younger than >>when he had left. It's not something we observe in practice, but that >>sort of thing can't be ruled out a priori. >I think you mean ten years younger than we are when he gets back. >Nothing will make him any younger than he was when he left. In any >event there is nothing apriori preventing this as long as he transits >all the space in between at velocities probably not exceeding that of >light. >>But when you look for stinkin' reasons, you either have to derive them >>from a priori necessities, or you've only given a reason in terms of other >>postulates that have no stinkin' reason. And like I've suggested above, >>a lot of what some people consider a priori necessities are only personal >>preferences. >No stinkin reason regressions are one of the greatest problems in >conventional science. We have to find some logical necessity and way >to preclude infinite regression or we have no mechanics or science; we >only have more or less self consistent plausibilities. What makes you think mechanics or science are more than self-consistent plausibilities? >>Consider also the well known point that even if there was a One True >>Theory that exists, and even if we find it, we can never really prove that >>we've found it. And that's really my basis for a distaste of finding the >>real reasons for fundamental physics-- anyone who claims they've found >>the real reasons as opposed to the fake reasons that merely make all the >>right predictions have duped themselves; they're just guessing, and can >>never do more. >Quoth the raven, nevermore. And you know this how? All you can really >know empirically is what has been found and not what can't be found. >What can't be found is a matter of proof and self contradiction and >there is none here not born of the frustration of empirical failure to >explain.. I know this because you cannot measure every part of every phenomenon with infinite precision from the beginning to the end of time. We've already missed 15 billions years' worth of phenomena! There may be discrepencies hiding a few sig-figs lower than your most precise measurements. There may be conflicts with phenomena that you haven't explored yet. Physical constants might change on time scales that are very long compared with the times over which you've made detailed measurements. May be, might-- you don't know, and you can't know. And measurements aside, different words can be used to describe the same quantitative predictions. E.g. Lorentz's aether theory and special relativity make identical predictions about the observable quantities in electrodynamics. Lorentz's theory has an aether whose properties drop out by the time an observable is calculated. We could call that surplus metaphysical baggage, but we certainly can't say it's been empirically falsified. A pre-The Matrix twist on Descarte's question, from a philosophy class, is how do you know you're not a brain in a jar with memories and sensory data given you by an interactive computer program, and all your experiences lead you to conclude the wrong laws of nature? [...] >>The philosophical lesson to take away is that a definite trajectory, and >>not something the metaphysician can just take for granted. An explanation >>shift us to a different set of unanswered questions. >according to what mechanical necessity. As long as metaphysicians and >this will undoubtedly remain the case. When they get around to >analyzing the stinkin reasons for their beliefs and what they imagine >to be true, however, all that can change. Quantum field theory is a theory of fields, and it is the field, not the DeBroglie's relation and the superposition principle and you can get sudden and finite changes in a momentum or energy or something, that can But that's really just a concrete illustration of the more abstract point -- I'm giving you the chance to look fate in those pretty eyes of hers and say, 'Step off, bitch. This is my party and you're not invited.' -- Chris Shugart, _Testosterone Magazine_ === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) in >>glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) in >>[. . .] >>I strongly suspect that I dis quantum theory primarily because it >>morphed physics from no stinkin reasons we can see into just plain >>ol' no stinkin reasons. > >So you want to reduce quantum mechanics to classical mechanics, I suppose. >But ignoring whether that's even possible, which unanswered questions >about classical mechanics have you chosen not to ask? >> >>This is interesting. I think I expect quantum theory can be reduced to >>mechanics even if it can't be exactly reduced to classical mechanics. >Quantum mechanics already is a mechanics. It's not classical mechanics, >but it's quantum mechanics. Classical mechanics can be shown to be a >specialization of quantum mechanics. And my understanding is that it's >impossible to reduce quantum to classical mechanics because classical >mechanics doesn't have the concept of a superposition of states. It has >statistical mixtures of states, but not superpositions. >>Yeah, here I'd like to say that I consider quantum mechanics not a >>mechanics at all because it is really just a series of principles not >>relating things defined by those principles. I consider that there are >>certainly quantum effects and there is certainly quantum theory. But I >>don't see quantum theory explaining transitions between and among >>various quantum effects. And that's what I consider a mechanics does. >Apply the time evolution operator. Operators are just substitutes for explanations. I understand that Star Trek often employs a Heisenberg compensator. Same principle. >>Newtonian and classical mechanics generally rested their mechanics not >>on general observations alone but explanations for transitions between >>and among observations. That's mechanics. QM maintains there is no >>choose of those open to it as long as conservation principles apply in >>aggregate whereas classical mechanics considers those conservation >>laws apply in particular as well as in general. And without particular >>You just wind up with a non material anthropomorphic probability. >Again I warn you against falling prey to your own hidden assumptions. >Quantum mechanics maintains that there is no defined path! Then what QM maintains is at odds with the geometry underlying your empirical observation of the existence of the same material object at point A then at point B. That empirical observation is what defines the geometry involved. Points A and B don't actually exist any more than the path between them. They're products of geometry in our minds. And if what QM maintains is at variance with the geometry you employ to define points A and B, it is going to need a new geometry with non linear non contiguous points. Then, of course, it'll be difficult to define space. But, what the hey, you can't have everything. > Not that the >even exist in the first place. By insisting that a path be taken, you've >already inserted a postulate of your own preference, and I'll immediately >ask you the reason that a single path can be taken. Because you define it with points A and B. Unless you're inserting a postulate of your own preference to the effect that points can exist without lines, and I'll immediately ask you how that trick is done in mechanically geometric rather than merely arbitrary postulated terms. [. . .] >>No stinkin reason regressions are one of the greatest problems in >>conventional science. We have to find some logical necessity and way >>to preclude infinite regression or we have no mechanics or science; we >>only have more or less self consistent plausibilities. >What makes you think mechanics or science are more than self-consistent >plausibilities? Well, the point of my observation was the standard of plausibility and not self consistency. If all science is based on is plausibility, it, like Euclidean geometry, rests on a rather sandy foundation. >Consider also the well known point that even if there was a One True >Theory that exists, and even if we find it, we can never really prove that >we've found it. And that's really my basis for a distaste of finding the >real reasons for fundamental physics-- anyone who claims they've found >the real reasons as opposed to the fake reasons that merely make all the >right predictions have duped themselves; they're just guessing, and can >never do more. >>Quoth the raven, nevermore. And you know this how? All you can really >>know empirically is what has been found and not what can't be found. >>What can't be found is a matter of proof and self contradiction and >>there is none here not born of the frustration of empirical failure to >>explain.. >I know this because you cannot measure every part of every phenomenon with >infinite precision from the beginning to the end of time. You're talking empiricism not knowledge. Precisely what's wrong with positivism. A thousand years of observations do not an idea make. > We've already >missed 15 billions years' worth of phenomena! There may be discrepencies >hiding a few sig-figs lower than your most precise measurements. There >may be conflicts with phenomena that you haven't explored yet. Physical >constants might change on time scales that are very long compared with >the times over which you've made detailed measurements. May be, might-- >you don't know, and you can't know. And measurements aside, different >words can be used to describe the same quantitative predictions. E.g. >Lorentz's aether theory and special relativity make identical predictions >about the observable quantities in electrodynamics. Lorentz's theory has >an aether whose properties drop out by the time an observable is >calculated. We could call that surplus metaphysical baggage, but we >certainly can't say it's been empirically falsified. Well, since you chose not comment on my discussion of contraction hypotheses, I can't comment on Lorentz except to say that MM can be performed successfully with radiation polarized normal to the plane of rotation and the absolute motion of the earth through space detected. >A pre-The Matrix twist on Descarte's question, from a philosophy class, >is how do you know you're not a brain in a jar with memories and sensory >data given you by an interactive computer program, and all your >experiences lead you to conclude the wrong laws of nature? Probably the same way you can know that you're not standing on your head: logical inference. Science should try it some time. >The philosophical lesson to take away is that a definite trajectory, and >not something the metaphysician can just take for granted. An explanation >shift us to a different set of unanswered questions. >>according to what mechanical necessity. As long as metaphysicians and >>this will undoubtedly remain the case. When they get around to >>analyzing the stinkin reasons for their beliefs and what they imagine >>to be true, however, all that can change. >Quantum field theory is a theory of fields, and it is the field, not the The problem lies in considering anything an irreducible atomic monad. > Throw in >DeBroglie's relation and the superposition principle and you can get >sudden and finite changes in a momentum or energy or something, that can Yes, but can they be interpreted as irreducible atomic monads? >But that's really just a concrete illustration of the more abstract point will find yourself in the enviable position of being able to explain >I'm giving you the chance to look fate in those pretty eyes of hers >and say, 'Step off, bitch. This is my party and you're not invited.' > -- Chris Shugart, _Testosterone Magazine_ I'm not allowed out much anyway because I step on too many toes. -- Lester Zick === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) in >glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) in >[. . .] >I strongly suspect that I dis quantum theory primarily because it >morphed physics from no stinkin reasons we can see into just plain >ol' no stinkin reasons. >> >>So you want to reduce quantum mechanics to classical mechanics, I >suppose. >>But ignoring whether that's even possible, which unanswered questions >>about classical mechanics have you chosen not to ask? > >This is interesting. I think I expect quantum theory can be reduced to >mechanics even if it can't be exactly reduced to classical mechanics. >> >>Quantum mechanics already is a mechanics. It's not classical mechanics, >>but it's quantum mechanics. Classical mechanics can be shown to be a >>specialization of quantum mechanics. And my understanding is that it's >>impossible to reduce quantum to classical mechanics because classical >>mechanics doesn't have the concept of a superposition of states. It has >>statistical mixtures of states, but not superpositions. >Yeah, here I'd like to say that I consider quantum mechanics not a >mechanics at all because it is really just a series of principles not >relating things defined by those principles. I consider that there are >certainly quantum effects and there is certainly quantum theory. But I >don't see quantum theory explaining transitions between and among >various quantum effects. And that's what I consider a mechanics does. >>Apply the time evolution operator. >Operators are just substitutes for explanations. I understand that >Star Trek often employs a Heisenberg compensator. Same principle. Did you think F=ma is any more explanatory? The potential in quantum mechanics has the same role as the index of refraction in optics. And, in a sense, the same role as force in classical mechanics. A wave packet will tend to be drawn toward low spots of the potential and pushed away from high spots. Electrons are bound to atoms because of the attractive force between the two; an atom can transition to a higher energy state when, e.g., another atom bumps into it and pushes the peices around. >Newtonian and classical mechanics generally rested their mechanics not >on general observations alone but explanations for transitions between >and among observations. That's mechanics. QM maintains there is no >choose of those open to it as long as conservation principles apply in >aggregate whereas classical mechanics considers those conservation >laws apply in particular as well as in general. And without particular >You just wind up with a non material anthropomorphic probability. >>Again I warn you against falling prey to your own hidden assumptions. >>Quantum mechanics maintains that there is no defined path! >Then what QM maintains is at odds with the geometry underlying your >empirical observation of the existence of the same material object at >point A then at point B. That empirical observation is what defines >the geometry involved. Points A and B don't actually exist any more >than the path between them. They're products of geometry in our minds. >And if what QM maintains is at variance with the geometry you employ >to define points A and B, it is going to need a new geometry with non >linear non contiguous points. Then, of course, it'll be difficult to >define space. But, what the hey, you can't have everything. >> > Not that the >>even exist in the first place. By insisting that a path be taken, you've >>already inserted a postulate of your own preference, and I'll immediately >>ask you the reason that a single path can be taken. >Because you define it with points A and B. Unless you're inserting a >postulate of your own preference to the effect that points can exist >without lines, and I'll immediately ask you how that trick is done in >mechanically geometric rather than merely arbitrary postulated terms. I didn't realize you were assuming a linear path from A to B even in the classical sense. Paths, linear or otherwise, can be defined in quantum mechanics. But they are, as you said, geometry. It in no way implies wave takes. >[. . .] >No stinkin reason regressions are one of the greatest problems in >conventional science. We have to find some logical necessity and way >to preclude infinite regression or we have no mechanics or science; we >only have more or less self consistent plausibilities. >>What makes you think mechanics or science are more than self-consistent >>plausibilities? >Well, the point of my observation was the standard of plausibility >and not self consistency. If all science is based on is plausibility, >it, like Euclidean geometry, rests on a rather sandy foundation. Get used to it. As Poincare said, science doesn't tell you what things are. It organizes relationships between them, and any theory is a true theory to the extent that if faithfully describes those relationships within the theory's valid regime of application. Read Science and Hypothesis by Poincare, which is usefully close to modern views on the philosophy of science despite being a hundred years old. >>Consider also the well known point that even if there was a One True >>Theory that exists, and even if we find it, we can never really prove that >>we've found it. And that's really my basis for a distaste of finding the >>real reasons for fundamental physics-- anyone who claims they've found >>the real reasons as opposed to the fake reasons that merely make all the >>right predictions have duped themselves; they're just guessing, and can >>never do more. >Quoth the raven, nevermore. And you know this how? All you can really >know empirically is what has been found and not what can't be found. >What can't be found is a matter of proof and self contradiction and >there is none here not born of the frustration of empirical failure to >explain.. >>I know this because you cannot measure every part of every phenomenon with >>infinite precision from the beginning to the end of time. >You're talking empiricism not knowledge. Precisely what's wrong with >positivism. A thousand years of observations do not an idea make. Science is an empirical practice. A theory is good if and only if it stands up to empirical scrutiny. If it can't, the theory is flawed. And if you can't measure every part of every phenomenon with infinite precision from the beginning to the end of time, then you might never know you have a flawed theory. > We've already >>missed 15 billions years' worth of phenomena! There may be discrepencies >>hiding a few sig-figs lower than your most precise measurements. There >>may be conflicts with phenomena that you haven't explored yet. Physical >>constants might change on time scales that are very long compared with >>the times over which you've made detailed measurements. May be, might-- >>you don't know, and you can't know. And measurements aside, different >>words can be used to describe the same quantitative predictions. E.g. >>Lorentz's aether theory and special relativity make identical predictions >>about the observable quantities in electrodynamics. Lorentz's theory has >>an aether whose properties drop out by the time an observable is >>calculated. We could call that surplus metaphysical baggage, but we >>certainly can't say it's been empirically falsified. >Well, since you chose not comment on my discussion of contraction >hypotheses, I can't comment on Lorentz except to say that MM can be >performed successfully with radiation polarized normal to the plane of >rotation and the absolute motion of the earth through space detected. Okay, I'll comment on it. The Lorentz transforms form a group, which means if you think you find a contradiction in them, you owe it to yourself to figure out why you're wrong. In your particular example, the length of an object is defined by the hollow cube, is moving around in there, that doesn't make a bit of difference as long as you know what you want to call the front and what you want to call the back. The gas molecule might itself have a different length contraction than the block as a whole, but if it's between the front and back in any frame, it will be between the front and back in all frames. The object itself isn't even necessary. Pick any two points. >>A pre-The Matrix twist on Descarte's question, from a philosophy class, >>is how do you know you're not a brain in a jar with memories and sensory >>data given you by an interactive computer program, and all your >>experiences lead you to conclude the wrong laws of nature? >Probably the same way you can know that you're not standing on your >head: logical inference. Science should try it some time. How is your logical inference different from the way you want things to be? Logic doesn't tell you what your premises have to be, which is why it's always bugged me when Vulcans in Star Trek go running around saying That's illogical. It's pragmatic to suppose that you're not a brain in a jar being fed false experiences, but you can't prove it from a priori considerations. >>The philosophical lesson to take away is that a definite trajectory, and >>not something the metaphysician can just take for granted. An explanation >>shift us to a different set of unanswered questions. >according to what mechanical necessity. As long as metaphysicians and >this will undoubtedly remain the case. When they get around to >analyzing the stinkin reasons for their beliefs and what they imagine >to be true, however, all that can change. >>Quantum field theory is a theory of fields, and it is the field, not the >The problem lies in considering anything an irreducible atomic monad. Well, I don't ask a model to be a monad. But isn't a monad exactly what you were looking for with a sort-of classical description of quantum mechanics? >> > Throw in >>DeBroglie's relation and the superposition principle and you can get >>sudden and finite changes in a momentum or energy or something, that can >Yes, but can they be interpreted as irreducible atomic monads? www.dictionary.com describes a monad as An indivisible, impenetrable unit of substance viewed as the basic constituent element of physical reality in the metaphysics of Leibnitz. No, a field cannot be interpreted literally as a monad. It's not impenetrable, it probably doesn't make sense to call it divisible or indivisible. And Leibnitz probably had a stronger opinion of the ultimate truthness of it than I would admit in a theory. But more loosely, as the basic quantity in a model from which electrons and atoms and baseballs can be derived, sure. >>But that's really just a concrete illustration of the more abstract point >will find yourself in the enviable position of being able to explain interacting quantum fields. I think you must have meant an explanation with elements that nobody thinks to ask the explanation of. -- Suppose you were an idiot... And suppose you were a member of Congress... But I repeat myself. - Mark Twain === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) in >>glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) in >>glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) in >> >> >>[. . .] >> >>I strongly suspect that I dis quantum theory primarily because it >>morphed physics from no stinkin reasons we can see into just plain >>ol' no stinkin reasons. > >So you want to reduce quantum mechanics to classical mechanics, I >>suppose. >But ignoring whether that's even possible, which unanswered questions >about classical mechanics have you chosen not to ask? >> >>This is interesting. I think I expect quantum theory can be reduced to >>mechanics even if it can't be exactly reduced to classical mechanics. > >Quantum mechanics already is a mechanics. It's not classical mechanics, >but it's quantum mechanics. Classical mechanics can be shown to be a >specialization of quantum mechanics. And my understanding is that it's >impossible to reduce quantum to classical mechanics because classical >mechanics doesn't have the concept of a superposition of states. It has >statistical mixtures of states, but not superpositions. >> >>Yeah, here I'd like to say that I consider quantum mechanics not a >>mechanics at all because it is really just a series of principles not >>relating things defined by those principles. I consider that there are >>certainly quantum effects and there is certainly quantum theory. But I >>don't see quantum theory explaining transitions between and among >>various quantum effects. And that's what I consider a mechanics does. >Apply the time evolution operator. >>Operators are just substitutes for explanations. I understand that >>Star Trek often employs a Heisenberg compensator. Same principle. >Did you think F=ma is any more explanatory? Oh, hell, yes. F, m, and a are explained in terms of one another. QM merely asserts there is some explanation but can't say what it is. >The potential in quantum mechanics has the same role as the index of >refraction in optics. And, in a sense, the same role as force in >classical mechanics. A wave packet will tend to be drawn toward low spots >of the potential and pushed away from high spots. Electrons are bound to >atoms because of the attractive force between the two; an atom can >transition to a higher energy state when, e.g., another atom bumps into it >and pushes the peices around. Yes, yes, this is all very interesting. I always imagined there was some reason electrons didn't go marching down Main Street in unison on New Years Day. QM just doesn't explain what all this anthropomorphism really amount to. >>Newtonian and classical mechanics generally rested their mechanics not >>on general observations alone but explanations for transitions between >>and among observations. That's mechanics. QM maintains there is no >>choose of those open to it as long as conservation principles apply in >>aggregate whereas classical mechanics considers those conservation >>laws apply in particular as well as in general. And without particular >>You just wind up with a non material anthropomorphic probability. >Again I warn you against falling prey to your own hidden assumptions. >Quantum mechanics maintains that there is no defined path! >>Then what QM maintains is at odds with the geometry underlying your >>empirical observation of the existence of the same material object at >>point A then at point B. That empirical observation is what defines >>the geometry involved. Points A and B don't actually exist any more >>than the path between them. They're products of geometry in our minds. >>And if what QM maintains is at variance with the geometry you employ >>to define points A and B, it is going to need a new geometry with non >>linear non contiguous points. Then, of course, it'll be difficult to >>define space. But, what the hey, you can't have everything. > >> Not that the >even exist in the first place. By insisting that a path be taken, you've >already inserted a postulate of your own preference, and I'll immediately >ask you the reason that a single path can be taken. >>Because you define it with points A and B. Unless you're inserting a >>postulate of your own preference to the effect that points can exist >>without lines, and I'll immediately ask you how that trick is done in >>mechanically geometric rather than merely arbitrary postulated terms. >I didn't realize you were assuming a linear path from A to B even in the >classical sense. Paths, linear or otherwise, can be defined in quantum >mechanics. But they are, as you said, geometry. It in no way implies >wave takes. So, your references to points A and B were metaphorical? I don't >>No stinkin reason regressions are one of the greatest problems in >>conventional science. We have to find some logical necessity and way >>to preclude infinite regression or we have no mechanics or science; we >>only have more or less self consistent plausibilities. >What makes you think mechanics or science are more than self-consistent >plausibilities? >>Well, the point of my observation was the standard of plausibility >>and not self consistency. If all science is based on is plausibility, >>it, like Euclidean geometry, rests on a rather sandy foundation. >Get used to it. As Poincare said, science doesn't tell you what things >are. It organizes relationships between them, and any theory is a true >theory to the extent that if faithfully describes those relationships >within the theory's valid regime of application. Read Science and >Hypothesis by Poincare, which is usefully close to modern views on the >philosophy of science despite being a hundred years old. No, no. Science tells us what things are. That's why we keep it around. It's mysticism that doesn't tell us what things are but maintains it needs to be kept around as a substitute for science. >Consider also the well known point that even if there was a One True >Theory that exists, and even if we find it, we can never really prove that >we've found it. And that's really my basis for a distaste of finding the >real reasons for fundamental physics-- anyone who claims they've found >the real reasons as opposed to the fake reasons that merely make all the >right predictions have duped themselves; they're just guessing, and can >never do more. >> >>Quoth the raven, nevermore. And you know this how? All you can really >>know empirically is what has been found and not what can't be found. >>What can't be found is a matter of proof and self contradiction and >>there is none here not born of the frustration of empirical failure to >>explain.. >I know this because you cannot measure every part of every phenomenon with >infinite precision from the beginning to the end of time. >>You're talking empiricism not knowledge. Precisely what's wrong with >>positivism. A thousand years of observations do not an idea make. >Science is an empirical practice. A theory is good if and only if it >stands up to empirical scrutiny. If it can't, the theory is flawed. And >if you can't measure every part of every phenomenon with infinite >precision from the beginning to the end of time, then you might never know >you have a flawed theory. And my point is empiricists never empirically know anything. >> We've already >missed 15 billions years' worth of phenomena! There may be discrepencies >hiding a few sig-figs lower than your most precise measurements. There >may be conflicts with phenomena that you haven't explored yet. Physical >constants might change on time scales that are very long compared with >the times over which you've made detailed measurements. May be, might-- >you don't know, and you can't know. And measurements aside, different >words can be used to describe the same quantitative predictions. E.g. >Lorentz's aether theory and special relativity make identical predictions >about the observable quantities in electrodynamics. Lorentz's theory has >an aether whose properties drop out by the time an observable is >calculated. We could call that surplus metaphysical baggage, but we >certainly can't say it's been empirically falsified. >>Well, since you chose not comment on my discussion of contraction >>hypotheses, I can't comment on Lorentz except to say that MM can be >>performed successfully with radiation polarized normal to the plane of >>rotation and the absolute motion of the earth through space detected. >Okay, I'll comment on it. The Lorentz transforms form a group, which >means if you think you find a contradiction in them, you owe it to >yourself to figure out why you're wrong. >In your particular example, the length of an object is defined by the >hollow cube, is moving around in there, that doesn't make a bit of >difference as long as you know what you want to call the front and what >you want to call the back. The gas molecule might itself have a different >length contraction than the block as a whole, but if it's between the >front and back in any frame, it will be between the front and back in all >frames. The object itself isn't even necessary. Pick any two points. how do we average all these different frames of reference since any average of v's would be linear but the effect is non linear? Einstein's geometric contraction hypothesis applies to a body moving at any particular velocity. Different v's, different contraction factors. So, which geometry applies to the group of interstitial bodies that constitute the body as a whole that Einstein uses to explain frequency dilation at v when there are numerous different v's? Surely, you're not going to say net v for the body as whole because there is no body as a whole; there are only aggregates of interstitial frames of reference with v's of their own. And v's average linearly whereas frequency dilation factors are non linear. >A pre-The Matrix twist on Descarte's question, from a philosophy class, >is how do you know you're not a brain in a jar with memories and sensory >data given you by an interactive computer program, and all your >experiences lead you to conclude the wrong laws of nature? >>Probably the same way you can know that you're not standing on your >>head: logical inference. Science should try it some time. >How is your logical inference different from the way you want things to >be? It doesn't. It's just different from empirical observation. Empirical observation isn't knowledge. Logical inference is. It may be right or wrong knowledge, but it's the only form of knowledge there is. >Logic doesn't tell you what your premises have to be, which is why it's >always bugged me when Vulcans in Star Trek go running around saying >That's illogical. It's pragmatic to suppose that you're not a brain in >a jar being fed false experiences, but you can't prove it from a priori >considerations. Well, you are a brain in a jar, the skull, being fed experiences by the senses, and it's up to the brain in the jar to decipher truth from falsity through logical inference. I don't know what a priori considerations are and neither does anyone else except to say that they're assumptions, neither true nor false. >The philosophical lesson to take away is that a definite trajectory, and >not something the metaphysician can just take for granted. An explanation >shift us to a different set of unanswered questions. >> >>according to what mechanical necessity. As long as metaphysicians and >>this will undoubtedly remain the case. When they get around to >>analyzing the stinkin reasons for their beliefs and what they imagine >>to be true, however, all that can change. >Quantum field theory is a theory of fields, and it is the field, not the >>The problem lies in considering anything an irreducible atomic monad. >Well, I don't ask a model to be a monad. But isn't a monad exactly what >you were looking for with a sort-of classical description of quantum >mechanics? Yeah, therein lies a tale of woe for classical and contemporary physics. What I'm looking for is an Einstein compensator, an effect called anisotropic time, that explains relativistic frequency dilation without the mystic mumbo jumbo of longitudinal contraction and nonsensical geometric distortion, and a Planck compensator that explains the origin of Planck's constant in purely analytical terms of deducible instead of postulated. And both compensators exist. >> Throw in >DeBroglie's relation and the superposition principle and you can get >sudden and finite changes in a momentum or energy or something, that can >>Yes, but can they be interpreted as irreducible atomic monads? >www.dictionary.com describes a monad as An indivisible, impenetrable unit >of substance viewed as the basic constituent element of physical reality >in the metaphysics of Leibnitz. >No, a field cannot be interpreted literally as a monad. It's not >impenetrable, it probably doesn't make sense to call it divisible or >indivisible. And Leibnitz probably had a stronger opinion of the >ultimate truthness of it than I would admit in a theory. But more >loosely, as the basic quantity in a model from which electrons and atoms >and baseballs can be derived, sure. As long as they aren't atomic monads. What's irreducible about >But that's really just a concrete illustration of the more abstract point >>will find yourself in the enviable position of being able to explain >interacting quantum fields. nice that interacting quantum fields can be interpreted one way or >I think you must have meant an explanation with elements that nobody >thinks to ask the explanation of. No, I mean an explanation with elements that nobody needs to ask the explanation of because the explanation is evident in the explanation without interpretation or the divine intercession of an interpreter. >-- >Suppose you were an idiot... And suppose you were a member of >Congress... But I repeat myself. - Mark Twain === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) in >glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) in [...] >>Did you think F=ma is any more explanatory? >Oh, hell, yes. F, m, and a are explained in terms of one another. QM >merely asserts there is some explanation but can't say what it is. >>The potential in quantum mechanics has the same role as the index of >>refraction in optics. And, in a sense, the same role as force in >>classical mechanics. A wave packet will tend to be drawn toward low spots >>of the potential and pushed away from high spots. Electrons are bound to >>atoms because of the attractive force between the two; an atom can >>transition to a higher energy state when, e.g., another atom bumps into it >>and pushes the peices around. >Yes, yes, this is all very interesting. I always imagined there was >some reason electrons didn't go marching down Main Street in unison on >New Years Day. QM just doesn't explain what all this anthropomorphism >really amount to. If you like F=ma, then I don't know why you'd have a problem with a wavefunction attracted to lows and repelled from highs in the potential. Do you know how force and potential energy are related? And did you, by any chance, notice that force in classical mechanics isn't itself explained, but simply defined as that which causes an acceleration? [...] >>What makes you think mechanics or science are more than self-consistent >>plausibilities? >Well, the point of my observation was the standard of plausibility >and not self consistency. If all science is based on is plausibility, >it, like Euclidean geometry, rests on a rather sandy foundation. >>Get used to it. As Poincare said, science doesn't tell you what things >>are. It organizes relationships between them, and any theory is a true >>theory to the extent that if faithfully describes those relationships >>within the theory's valid regime of application. Read Science and >>Hypothesis by Poincare, which is usefully close to modern views on the >>philosophy of science despite being a hundred years old. >No, no. Science tells us what things are. That's why we keep it >around. It's mysticism that doesn't tell us what things are but >maintains it needs to be kept around as a substitute for science. Science has grown out of that viewpoint after a few theories that say what things are have fallen away. But to claim to know what things REALLY are seems far more akin to claiming knowledge about spirits and other unmeasurables to me. [...] >You're talking empiricism not knowledge. Precisely what's wrong with >positivism. A thousand years of observations do not an idea make. >>Science is an empirical practice. A theory is good if and only if it >>stands up to empirical scrutiny. If it can't, the theory is flawed. And >>if you can't measure every part of every phenomenon with infinite >>precision from the beginning to the end of time, then you might never know >>you have a flawed theory. >And my point is empiricists never empirically know anything. You can't have missed the existence of modern technology, so I think you must be using a personal, restrictive definition of what it means to know something. [...] >Well, since you chose not comment on my discussion of contraction >hypotheses, I can't comment on Lorentz except to say that MM can be >performed successfully with radiation polarized normal to the plane of >rotation and the absolute motion of the earth through space detected. >>Okay, I'll comment on it. The Lorentz transforms form a group, which >>means if you think you find a contradiction in them, you owe it to >>yourself to figure out why you're wrong. >>In your particular example, the length of an object is defined by the >>hollow cube, is moving around in there, that doesn't make a bit of >>difference as long as you know what you want to call the front and what >>you want to call the back. The gas molecule might itself have a different >>length contraction than the block as a whole, but if it's between the >>front and back in any frame, it will be between the front and back in all >>frames. The object itself isn't even necessary. Pick any two points. >how do we average all these different frames of reference since any >average of v's would be linear but the effect is non linear? If you want to calculate a length from the frame of each individual observer is not oberving simultaneously from all those different reference frames. The observer observes from his own rest frame. >Einstein's geometric contraction hypothesis applies to a body moving >at any particular velocity. Different v's, different contraction >factors. So, which geometry applies to the group of interstitial >bodies that constitute the body as a whole that Einstein uses to >explain frequency dilation at v when there are numerous different v's? >Surely, you're not going to say net v for the body as whole because >there is no body as a whole; there are only aggregates of interstitial >frames of reference with v's of their own. And v's average linearly >whereas frequency dilation factors are non linear. Einstein's geometric contraction hypothesis relates a length measured in one frame to a length measured in another frame. When you, Lester, measure (taking an example from in front of me) the diameter of a Vanilla Pepsi can, do you need to measure the speeds of each individual atom within the can? Review the derivation of length contraction and the definition of length that is used. >>A pre-The Matrix twist on Descarte's question, from a philosophy class, >>is how do you know you're not a brain in a jar with memories and sensory >>data given you by an interactive computer program, and all your >>experiences lead you to conclude the wrong laws of nature? >Probably the same way you can know that you're not standing on your >head: logical inference. Science should try it some time. >>How is your logical inference different from the way you want things to >>be? >It doesn't. It's just different from empirical observation. Empirical >observation isn't knowledge. Logical inference is. It may be right or >wrong knowledge, but it's the only form of knowledge there is. Logical inference without data is fantasy. If you're not beholden to empirical observation, you can build any kind of abstract world that you like. >>Logic doesn't tell you what your premises have to be, which is why it's >>always bugged me when Vulcans in Star Trek go running around saying >>That's illogical. It's pragmatic to suppose that you're not a brain in >>a jar being fed false experiences, but you can't prove it from a priori >>considerations. >Well, you are a brain in a jar, the skull, being fed experiences by >the senses, and it's up to the brain in the jar to decipher truth from >falsity through logical inference. I don't know what a priori >considerations are and neither does anyone else except to say that >they're assumptions, neither true nor false. A priori means knowable without reference to particular facts or experience. Your logical inference is something that is done with postulates. Where do those postulates come from? As far as I can tell, you want postulates that do not appeal to any particular facts or experiences, and yet say something useful about particular facts or exeriences. Good luck. [...] >>I think you must have meant an explanation with elements that nobody >>thinks to ask the explanation of. >No, I mean an explanation with elements that nobody needs to ask the >explanation of because the explanation is evident in the explanation >without interpretation or the divine intercession of an interpreter. The explanation is evident in the explanation? Go right ahead, but I urge you again to be careful of unstated assumptions without which the explanation would not be so evident. interpretation of quantum fields. There's another view advanced by a a basic result in fluid mechanics, for instance, that two vortices will attract or repel each other, according to the relative directions of their spin axes, analogously to a Coulomb force. He likes to think of a fluid that's ultimately particulate, but fluid mechanics is typically done in the continuum limit and I see no logical problem with assuming the fluid is itself a continuum. and necessary, you're fooling yourself. First, because you know from priori knowledge. Even if there were no concrete examples there will always be possibilities that just haven't occured to you yet, or the possibility that there are possibilities that haven't occured to you yet. -- Then they placed the ark of the Lord on the cart; along with the box containing the golden mice and the images of the hemorrhoids. -- 1 Samuel 6:11 === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) in >>glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) in >>glhansen@steel.ucs.indiana.edu (Gregory L. Hansen) in >[...] >Did you think F=ma is any more explanatory? >>Oh, hell, yes. F, m, and a are explained in terms of one another. QM >>merely asserts there is some explanation but can't say what it is. >The potential in quantum mechanics has the same role as the index of >refraction in optics. And, in a sense, the same role as force in >classical mechanics. A wave packet will tend to be drawn toward low spots >of the potential and pushed away from high spots. Electrons are bound to >atoms because of the attractive force between the two; an atom can >transition to a higher energy state when, e.g., another atom bumps into it >and pushes the peices around. >>Yes, yes, this is all very interesting. I always imagined there was >>some reason electrons didn't go marching down Main Street in unison on >>New Years Day. QM just doesn't explain what all this anthropomorphism >>really amount to. >If you like F=ma, then I don't know why you'd have a problem with a >wavefunction attracted to lows and repelled from highs in the potential. >Do you know how force and potential energy are related? I assume you know how they are related. Do you know why they are related? >And did you, by any chance, notice that force in classical mechanics isn't >itself explained, but simply defined as that which causes an acceleration? Well, force causes acceleration; it doesn't cause an acceleration unless mediated by mass. What kind of explanation do you expect for force? What kind of explanation for force or mass or even acceleration is possible? I don't say no better explanation for force etc. is possible. I do say that Newton explained what he could of force pretty effectively. Mechanics engages in a mechanical reduction of causes by explaining factors in terms of one another. >[...] >What makes you think mechanics or science are more than self-consistent >plausibilities? >> >>Well, the point of my observation was the standard of plausibility >>and not self consistency. If all science is based on is plausibility, >>it, like Euclidean geometry, rests on a rather sandy foundation. >Get used to it. As Poincare said, science doesn't tell you what things >are. It organizes relationships between them, and any theory is a true >theory to the extent that if faithfully describes those relationships >within the theory's valid regime of application. Read Science and >Hypothesis by Poincare, which is usefully close to modern views on the >philosophy of science despite being a hundred years old. >>No, no. Science tells us what things are. That's why we keep it >>around. It's mysticism that doesn't tell us what things are but >>maintains it needs to be kept around as a substitute for science. >Science has grown out of that viewpoint after a few theories that say what >things are have fallen away. None of the macro cause and effect theories I'm familiar with have fallen away. They have merely been supplanted at the micro level by QM postulates which turn out to be true but of ambiguous origin in the sense that they don't explain what they explain or fail to explain. If classical mechanics had taken a similar route, we would still be doing classical mechanics supplemented by arbitrary postulates of dubious orign and ambiguous mechanical significance. >But to claim to know what things REALLY are seems far more akin to >claiming knowledge about spirits and other unmeasurables to me. Not necessarily although I have to admit that such a conclusion certainly mirrors contemporary thinking. Personally I consider that if one can show that alternatives of a specific empirical theory are inherently self contradictory truth of the theory itself is necessary. Conversely, showing self contradiction within an empirical theory shows the theory to be false regardless of what experiments fail to contradict, which is the approach I've taken with special relativity. >[...] >>You're talking empiricism not knowledge. Precisely what's wrong with >>positivism. A thousand years of observations do not an idea make. >Science is an empirical practice. A theory is good if and only if it >stands up to empirical scrutiny. If it can't, the theory is flawed. And >if you can't measure every part of every phenomenon with infinite >precision from the beginning to the end of time, then you might never know >you have a flawed theory. >>And my point is empiricists never empirically know anything. >You can't have missed the existence of modern technology, so I think you >must be using a personal, restrictive definition of what it means to know >something. I think you'll find development of modern technology is largely a reflection of classical mechanics with the enhancement of quantum theories. I don't know many machines that inherently rely on quantum ambiguity. We certainly have things like tunneling diodes that rely on quantum suppositions. But those suppositions themselves are not based on any mechanical understanding of the forces at work. They are just suppositions that turn out to reflect poorly understood mechanisms. >[...] >>Well, since you chose not comment on my discussion of contraction >>hypotheses, I can't comment on Lorentz except to say that MM can be >>performed successfully with radiation polarized normal to the plane of >>rotation and the absolute motion of the earth through space detected. >Okay, I'll comment on it. The Lorentz transforms form a group, which >means if you think you find a contradiction in them, you owe it to >yourself to figure out why you're wrong. >In your particular example, the length of an object is defined by the >hollow cube, is moving around in there, that doesn't make a bit of >difference as long as you know what you want to call the front and what >you want to call the back. The gas molecule might itself have a different >length contraction than the block as a whole, but if it's between the >front and back in any frame, it will be between the front and back in all >frames. The object itself isn't even necessary. Pick any two points. >>how do we average all these different frames of reference since any >>average of v's would be linear but the effect is non linear? >If you want to calculate a length from the frame of each individual >observer is not oberving simultaneously from all those different reference >frames. The observer observes from his own rest frame. OK. Back to the observer paradox. I guess we've regressed to ground zero for relativistic explanations because I don't need an observer to explain and justify my geometric conclusions but relativity does. >>Einstein's geometric contraction hypothesis applies to a body moving >>at any particular velocity. Different v's, different contraction >>factors. So, which geometry applies to the group of interstitial >>bodies that constitute the body as a whole that Einstein uses to >>explain frequency dilation at v when there are numerous different v's? >>Surely, you're not going to say net v for the body as whole because >>there is no body as a whole; there are only aggregates of interstitial >>frames of reference with v's of their own. And v's average linearly >>whereas frequency dilation factors are non linear. >Einstein's geometric contraction hypothesis relates a length measured in >one frame to a length measured in another frame. When you, Lester, >measure (taking an example from in front of me) the diameter of a >Vanilla Pepsi can, do you need to measure the speeds of each individual >atom within the can? Review the derivation of length contraction and the >definition of length that is used. I don't need to measure anything in my spatial geometry because the problem is logic and not geometry. You, on the other hand, need to resort to observers and personalized measurement because the spatial geometry in relativity contradiction hypotheses is self contradictory on its own terms. >A pre-The Matrix twist on Descarte's question, from a philosophy class, >is how do you know you're not a brain in a jar with memories and sensory >data given you by an interactive computer program, and all your >experiences lead you to conclude the wrong laws of nature? >> >>Probably the same way you can know that you're not standing on your >>head: logical inference. Science should try it some time. >How is your logical inference different from the way you want things to >be? >>It doesn't. It's just different from empirical observation. Empirical >>observation isn't knowledge. Logical inference is. It may be right or >>wrong knowledge, but it's the only form of knowledge there is. >Logical inference without data is fantasy. If you're not beholden to >empirical observation, you can build any kind of abstract world that you >like. I never suggested inference without data. Empirical observations are the data. Logical inference is what we make of the data interrelated. >Logic doesn't tell you what your premises have to be, which is why it's >always bugged me when Vulcans in Star Trek go running around saying >That's illogical. It's pragmatic to suppose that you're not a brain in >a jar being fed false experiences, but you can't prove it from a priori >considerations. >>Well, you are a brain in a jar, the skull, being fed experiences by >>the senses, and it's up to the brain in the jar to decipher truth from >>falsity through logical inference. I don't know what a priori >>considerations are and neither does anyone else except to say that >>they're assumptions, neither true nor false. >A priori means knowable without reference to particular facts or >experience. Your logical inference is something that is done with >postulates. Where do those postulates come from? As far as I can tell, >you want postulates that do not appeal to any particular facts or >experiences, and yet say something useful about particular facts or >exeriences. Good luck. Goodness has nothing to do with it. Nor do postulates. What you do need is mechanically irreducible and unregressable truth, by which I mean some empirical observation which is irreducible because alternatives are self contradictory. Logical inference is then done with that mechanism on other reducible empirical observations to determine their relations to one another. >[...] >I think you must have meant an explanation with elements that nobody >thinks to ask the explanation of. >>No, I mean an explanation with elements that nobody needs to ask the >>explanation of because the explanation is evident in the explanation >>without interpretation or the divine intercession of an interpreter. >The explanation is evident in the explanation? Go right ahead, but I >urge you again to be careful of unstated assumptions without which the >explanation would not be so evident. Well, you just start with the empirical observation [not] and work up from there. T:[not][not not] is a tautology and tautologies are always true because they include all possibilities and exclude no possibilities. Then we find that the empirical observation [not] is itself always true because its logical regression [not not] is self contradictory. Which implies that [not] is the logic mechanism we're looking for because it is irreducible and unregressable. >interpretation of quantum fields. There's another view advanced by a >a basic result in fluid mechanics, for instance, that two vortices will >attract or repel each other, according to the relative directions of their >spin axes, analogously to a Coulomb force. He likes to think of a fluid >that's ultimately particulate, but fluid mechanics is typically done in >the continuum limit and I see no logical problem with assuming the fluid >is itself a continuum. I agree completely and I'll even give it a name: the ether. >and necessary, you're fooling yourself. First, because you know from >priori knowledge. Even if there were no concrete examples there will >always be possibilities that just haven't occured to you yet, or the >possibility that there are possibilities that haven't occured to you yet. There are always possibilities that just haven't occurred to me yet or to you or to others. Quantum theory is filled with them and their symmetries and with quantum theorists hunting for them. I take nothing except basic truth T:[not][not not] as self evident and that only because the term self evident means evident of itself in that alternatives are inherently self contradictory. Maybe there are other definitions for self evidence, but that is the one I use. You seem to think I'm whistling in the dark regarding the existence of of the mechanical origin of Planck's constant, I'm enclosing a copy: > Planck's Constant >Previously in the thread Angular Momentum in Rotating Bodies, I >presented an analytical framework for the interpretation of dr/dt in >circular rotation of a point mass m at velocity v and radius r. No one >I know of agrees with my interpretation of dr/dt. However, in the >interests of further establishing this general framework, I would like >to pursue general developement of the idea which culminates in the >analytical definition of Planck's constant. >We begin by noting that in cases of circular rotation at constant >angular velocity we have a centripetally directed dr/dt acting on >point mass m of a magnitude equal to tangential velocity v. This is >what causes the rotation of v and produces r as a consequence of >rotation. >We then integrate dr/dt along r which produces 1/2 mvr/2pi with units >of measure equal to rr/t. Now, I have been cautioned on several >occasions not to suggest that this quantity represents angular >momentum in conventional terms and I agree. Perhaps we should simply >call it rotational momentum to prevent confusion. >What we notice immediately however is that it bears the same form as >the quantity mvr corresponding to Planck's constant. However, we have >to straighten certain things out in this connection. >In conventional macro angular rotation such as flywheels we have a >centripetal dr/dt and tangential v which are equal to each other. They >are effectively bound up through tensile forces internal to the body >undergoing rotation. In celestial angular mechanics on the other hand >we have a wide variety of potential dr/dt's and tangential orbital >velocities operating in various combinations. >different situation. The tangential velocity of rotation v is constant >under all circumstances. In other words, v = c. Thus dr/dt operates >mass. >second) times an analytical masslet, m0 (kg-sec) and interpret the >quantity mvr as a multiple of nm0vr. Further we can interpret r as a >function of c/n such that Planck's constant = m0cc. In other words, m0 >is roughly on the order of 10^-50 kg-sec in magnitude and Planck's >constant corresponds to the multiple of m0 and the square of the >velocity of light. >We notice several things about rotational momentum. In linear motion >at constant velocity rotational momentum is zero because dr/dt and mvr >are both zero. And in circular rotation at a constant angular velocity >rotational momentum is constant because mvr is constant. This >represents the analytical distinction between circular and linear >motion. >Further we notice that dr/dt can be of any magnitude. It is not bound >by the constancy of the velocity of light as an upper limit because it >doesn't go anywhere. It only produces rotation in relation to actual >tangential motion v = c. >mass and radius of rotation are inversely proportional, that is that >remove DEL in address for email Linear versus Analytical Mechanics One of the really unfortunate aspects of Newton's choice of a linear frame of reference for the analysis of mechanics is that r is poorly defined and t is not defined at all. In other words, r is only defined in direction and t is not defined by any consideration pertinent to the analytical frame of reference. And this had a pernicious impact on the subsequent development of angular mechanics as well as relativistic considerations and quantum mechanics in the twentieth century. The problem is that r and t and their combinations are all we have to work with. Taken to the second level of compounding we have six combinations: r, 1/t, r/t, r/tt, rr/t, and rr/tt. However, in the linear analytical frame of reference the next to last combination rr/t was overlooked because there is no apparent application for it in linear mechanical contexts. On the other hand, in angular frames of reference we have applications for all combinations and all the elements are well defined. The radius of rotation is well defined in terms of direction and magnitude and time is well defined in analytical terms as whatever time is needed for 2pi radians of rotation. The rr/t combination is also well defined in angular terms. However, in extrapolating the idea of rr/t from linear to angular contexts in classical mechanics, whoever devised the analytical approach made the mistake of trying to emulate linear mechanics in the sense of explaining rotation as a linear progression of r instead of a simple radial v in combination with tangential v. This is more akin to an anachronistic pre Newtonian view of mechanics. Kepler thought that some force of angels was needed to keep planets in orbit around the sun and regarded that force as tangential in direction. Newton on the other hand recognized that the only force needed was centripetal in nature and not tangential. But whoever devised the analytical considerations underlying angular mechanics apparently never considered the Newtonian perspective and presumably relied on the pre Newtonian rationale. Thus we wind up with a conceptual schism among the various realms of angular mechanics. On the one hand we have orbital angular mechanics, the macro realm of ordinary angular mechanics, and the micro realm of quantum effects. And unfortunately there is no conceptual integration among them. We are convinced that all represent mechanical realms but we have no basis for comprehending each in terms of the others. Orbital angular mechanics represents the realm of remote interactions dealt with in terms of inverse square centripetal forces and tangential orbital velocities. Whereas the macro realm of ordinary angular mechanics deals with linear analogs such as moments of inertia instead of mass, torque instead of force, and angular acceleration and velocity instead of their linear analogs. The micro realm of angular mechanics on the other hand is dealt with on the merely descriptive basis of formalisms. This is the realm of quantum mechanics - QM - or as I prefer to call it quantum magic where things don't seem to happen for any definite mechanical reason at all. However with the redefinition of macro angular momentum and Planck's constant in circular rotation we are at last in a position to understand the mechanical differences among the realms in conceptual terms. The micro realm of quantum effects is one of constant tangential velocity of rotation v = c and a variable radial dr/dt. The macro realm of ordinary angular mechanics on the other hand is one in which the tangential velocity of rotation is variable but tangential v = radial dr/dt and both are kept in strict synchronization by internal tensile forces. And finally orbital angular mechanics is defined by various combinations of tangential v and radial dr/dt. This is normally thought of in celestial terms but in point of fact applies equally to the atomic realm as well. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics [...] > Because you define it with points A and B. Unless you're inserting a > postulate of your own preference to the effect that points can exist > without lines, and I'll immediately ask you how that trick is done in > mechanically geometric rather than merely arbitrary postulated terms. [...] Lester once again displaying his unwillingness to believe that concepts he can't understand nevertheless makes sense. Lester, Lester, you're turning into a first class crank. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics > Lester once again displaying his unwillingness to believe that concepts > he can't understand nevertheless makes sense. > Lester, Lester, you're turning into a first class crank. Actually, the more of Lester's posts I read, the more his arguments make sense. I guess I'm a crank also. I don't like my science seasoned with magic. -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics > >> Lester once again displaying his unwillingness to believe that concepts >> he can't understand nevertheless makes sense. >> Lester, Lester, you're turning into a first class crank. >Actually, the more of Lester's posts I read, the more his >arguments make sense. I guess I'm a crank also. I don't like my >science seasoned with magic. Very, very good, Albert. lol (lots of hilarity). I begin to think very well of you. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics [...] > Actually, the more of Lester's posts I read, the more his arguments make > sense. I guess I'm a crank also. I don't like my science seasoned with > magic. Where's the magic? === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics > [...] >> Actually, the more of Lester's posts I read, the more his arguments >> make sense. I guess I'm a crank also. I don't like my science >> seasoned with magic. > Where's the magic? Well, one good example is available in my reply to your reply to Stlbl above: your assertion that any unfalsified theory was evidence. A second example might be your statement to Lester that: The 'calculations', as you call them, _are_ the explanations. This assumed link between arbitrary calculations and reality is pretty magical sounding to me. But I'm sure that Lester will explain that to you. -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >> [...] > Actually, the more of Lester's posts I read, the more his arguments > make sense. I guess I'm a crank also. I don't like my science > seasoned with magic. >> Where's the magic? >Well, one good example is available in my reply to your reply to >Stlbl above: your assertion that any unfalsified theory was >evidence. >A second example might be your statement to Lester that: The >'calculations', as you call them, _are_ the explanations. This >assumed link between arbitrary calculations and reality is pretty >magical sounding to me. But I'm sure that Lester will explain >that to you. Wolf is rather difficult to reach on issues in mechanics. He prefers philosophy to science both in behavior analysis and mathematics and constantly regales us with his beliefs instead of his knowledge and berates me, at least, for failing to believe in his beliefs as opposed to his science. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics > A second example might be your statement to Lester that: The > 'calculations', as you call them, _are_ the explanations. This assumed > link between arbitrary calculations and reality is pretty magical > sounding to me. But I'm sure that Lester will explain that to you. It isn't magic when it leads to good technology. Technology is solid and sound reason for using a theory. That is why quantum field theory is a winner. It is behind most of our best technology. And theories do not -explain- anything. They predict. They also function as good intuition pumps for making more good theories and promoting engineering and applications. We cannot know the Real Reality (i.e. the reality beyond the appearences). The only things we really -know- (as opposed to guess at or infer or hypothesize) is direct experience. Bob Kolker === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >> A second example might be your statement to Lester that: The >> 'calculations', as you call them, _are_ the explanations. This >> assumed link between arbitrary calculations and reality is pretty >> magical sounding to me. But I'm sure that Lester will explain that >> to you. > It isn't magic when it leads to good technology. Technology is solid > and sound reason for using a theory. That is why quantum field theory > is a winner. It is behind most of our best technology. > And theories do not -explain- anything. Which begs the question - is there anything that explains anything? Whats *is* an explanation? They predict. They also function > as good intuition pumps for making more good theories and promoting > engineering and applications. We cannot know the Real Reality (i.e. > the reality beyond the appearences). Appearances are real realities. To call them appearances instead of real realities might be caused by..[explanation] > The only things we really -know- (as opposed to guess at or infer or > hypothesize) is direct experience. But guessing is very much part of direct experience since [expectancy]->[confirmation yes/no] cycling is what direct experience is all about. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >> A second example might be your statement to Lester that: The >> 'calculations', as you call them, _are_ the explanations. This assumed >> link between arbitrary calculations and reality is pretty magical >> sounding to me. But I'm sure that Lester will explain that to you. >It isn't magic when it leads to good technology. Technology is solid and >sound reason for using a theory. That is why quantum field theory is a >winner. It is behind most of our best technology. Sure. Behind most of our best and worst technology lie intuition pumps. >And theories do not -explain- anything. They predict. They also function >as good intuition pumps for making more good theories and promoting >engineering and applications. We cannot know the Real Reality (i.e. the >reality beyond the appearences). The only things we really -know- (as >opposed to guess at or infer or hypothesize) is direct experience. Is all this theory or speculation? The difference between speculation and theory is that theory tells us how speculations actually work. All speculation is is educated guesses based on collateral factors. Theory makes speculation explicit by making various middle terms objective. So far as I know, QM is still pretty much stuck in the speculation phase unable to say how or why various symmetries work or don't work. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >> A second example might be your statement to Lester that: The >> 'calculations', as you call them, _are_ the explanations. This >> assumed link between arbitrary calculations and reality is pretty >> magical sounding to me. But I'm sure that Lester will explain that to >> you. > It isn't magic when it leads to good technology. Damn, Bob. Please read for comprehension and note the context. I never said nor implied otherwise. > Technology is solid and > sound reason for using a theory. That is why quantum field theory is a > winner. It is behind most of our best technology. You talk about QM as if it were explained by a monolithic single theory and all aspects of that single theory have been empirically proven. > And theories do not -explain- anything. They predict. And are useful *only* to the extent they predict. Please explain this to Wolf. > They also function > as good intuition pumps for making more good theories and promoting > engineering and applications. We cannot know the Real Reality (i.e. the > reality beyond the appearences). The only things we really -know- (as > opposed to guess at or infer or hypothesize) is direct experience. Please explain this to Wolf et al. -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics <41c13ddb.56649778@netnews.att.net> <41c258f7.67644674@netnews.att.net> <41c287c2.75781421@netnews.att.net> <09qwd.117$ql2.71@okepread04> A second example might be your statement to Lester that: The 'calculations', as you call them, _are_ the explanations. This assumed link between arbitrary calculations and reality is pretty magical sounding to me. But I'm sure that Lester will explain that to you. > It isn't magic when it leads to good technology. Technology is solid and > sound reason for using a theory. That is why quantum field theory is a > winner. It is behind most of our best technology. > And theories do not -explain- anything. They predict. They also function > as good intuition pumps for making more good theories and promoting > engineering and applications. We cannot know the Real Reality (i.e. the > reality beyond the appearences). The only things we really -know- (as > opposed to guess at or infer or hypothesize) is direct experience. > Bob Kolker What you write is exactly what I understand science to be. You end with The only things we really -know- ... is direct experience. I would say all we really -know- is our conscious experience Now, (qualia). We don't really -know- anything beyond that. Without that there is, for the individual, nothing. John Casey === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics > Well, one good example is available in my reply to your reply to Stlbl > above: your assertion that any unfalsified theory was evidence. I never said or implied that. We can never know when a theory is true. We can only know when it is falsified. A non-falsified theory with a good record of predictions (many kinds of things predicted or related) is sufficient reason for staying with the theory. If the theory leads to good technology that is even more reason for staying with the theory. We stay with a theory because it is productive and useful, not because it is true. Bob Kolker === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >> Well, one good example is available in my reply to your reply to Stlbl >> above: your assertion that any unfalsified theory was evidence. > I never said or implied that. I never said you did, Bob. There is something wrong with your reader. I was replying to Wolf. > We can never know when a theory is true. > We can only know when it is falsified. A non-falsified theory with a > good record of predictions (many kinds of things predicted or related) > is sufficient reason for staying with the theory. If the theory leads to > good technology that is even more reason for staying with the theory. We > stay with a theory because it is productive and useful, not because it > is true. Agreed. -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics : :> Lester once again displaying his unwillingness to believe that concepts :> he can't understand nevertheless makes sense. :> Lester, Lester, you're turning into a first class crank. : Actually, the more of Lester's posts I read, the more his : arguments make sense. I guess I'm a crank also. I don't like my : science seasoned with magic. Do you think that the set {1,2,3} contains the element 0? :) Stephen === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >: >:> Lester once again displaying his unwillingness to believe that concepts >:> he can't understand nevertheless makes sense. >:> >:> Lester, Lester, you're turning into a first class crank. >: Actually, the more of Lester's posts I read, the more his >: arguments make sense. I guess I'm a crank also. I don't like my >: science seasoned with magic. >Do you think that the set {1,2,3} contains the element 0? :) When you suggested a photograph of a class as a set, I asked what the cardinality of a photograph of 42 students was and suggested that 0 just photograph very well. With a cardinality of equal differences, we find that 1-1=0 and that 1-(1-1) is equal to 2-1 and 3-2. Of course, if your particular brand of cardinality contains unequal differences or no differences, I can readily appreciate that you have a problem. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics > : > :> Lester once again displaying his unwillingness to believe that concepts > :> he can't understand nevertheless makes sense. > :> > :> Lester, Lester, you're turning into a first class crank. > : Actually, the more of Lester's posts I read, the more his > : arguments make sense. I guess I'm a crank also. I don't like my > : science seasoned with magic. > Do you think that the set {1,2,3} contains the element 0? :) Do you think that is what I was talking about? -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics :> : :> :> Lester once again displaying his unwillingness to believe that concepts :> :> he can't understand nevertheless makes sense. :> :> :> :> Lester, Lester, you're turning into a first class crank. :> : Actually, the more of Lester's posts I read, the more his :> : arguments make sense. I guess I'm a crank also. I don't like my :> : science seasoned with magic. :> Do you think that the set {1,2,3} contains the element 0? :) : Do you think that is what I was talking about? You mentioned Lester's arguments and that is one of Lester's arguments. Stephen === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >:> : >:> :> Lester once again displaying his unwillingness to believe that concepts >:> :> he can't understand nevertheless makes sense. >:> :> >:> :> Lester, Lester, you're turning into a first class crank. >:> >:> : Actually, the more of Lester's posts I read, the more his >:> : arguments make sense. I guess I'm a crank also. I don't like my >:> : science seasoned with magic. >:> >:> Do you think that the set {1,2,3} contains the element 0? :) >: Do you think that is what I was talking about? >You mentioned Lester's arguments and that is one of Lester's arguments. One of many of Lester's arguments relating to mathematics and differences and differences between differences, none of which you seem to prefer to conventional mathematical mysticism, sometimes known as pythagoreanism or platonism. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics > Do you think that the set {1,2,3} contains the element 0? :) Trick question :) The set {1,2,3} contains { }. Some people confuse Null with Empty with Zero; the concepts are in the same general vicinity. Like accidentally putting salt in your coffee, or thinking you woke up late for work when it's your day off. -:|:- AngleWyrm === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >> Do you think that the set {1,2,3} contains the element 0? :) >Trick question :) >The set {1,2,3} contains { }. Some people confuse Null with Empty with Zero; the >concepts are in the same general vicinity. >Like accidentally putting salt in your coffee, or thinking you woke up late for >work when it's your day off. Decent observation, but I prefer my collateral reply to Stephen regarding differences in sets and the determination of cardinality. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics :> Do you think that the set {1,2,3} contains the element 0? :) : Trick question :) No. Just part of the definition of Lesternality. : The set {1,2,3} contains { }. Some people confuse Null with Empty with Zero; the : concepts are in the same general vicinity. { } is a subset of { 1, 2, 3}. {} is not an element of { 1, 2, 3 }. Stephen === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >[...] >> Because you define it with points A and B. Unless you're inserting a >> postulate of your own preference to the effect that points can exist >> without lines, and I'll immediately ask you how that trick is done in >> mechanically geometric rather than merely arbitrary postulated terms. >[...] >Lester once again displaying his unwillingness to believe that concepts >he can't understand nevertheless makes sense. >Lester, Lester, you're turning into a first class crank. Yes, yes, Wolf, but at least I don't have the pompous effrontery to reply to others with exactly the point they're making then instruct them to do some reading. What I display is not my unwillingness to believe in concepts that I don't understand but my unwillingness to believe in concepts that you don't understand. I just have a tough time believing in your beliefs. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics > Operators are just substitutes for explanations. I understand that > Star Trek often employs a Heisenberg compensator. Same principle. How about predictions. Quantum Field Theory predicts everything that happens outside the atomic nucleus except gravity to accuracies of 12 to 15 decimal places. And it doesn't explain a thing in the sense that causes are not postulated. I am of the opinion that explanations (so called) are no more than hypotheses and the only thing that matters is that all quantitative predictions flowing therefrom are experimentally correct. Bob Kolker === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >> Operators are just substitutes for explanations. I understand that >> Star Trek often employs a Heisenberg compensator. Same principle. >How about predictions. Quantum Field Theory predicts everything that >happens outside the atomic nucleus except gravity to accuracies of 12 to >15 decimal places. And it doesn't explain a thing in the sense that >causes are not postulated. Well, that's exactly the complaint. Science is about explanations and not just calculations in which we can't tell what is being calculated. >I am of the opinion that explanations (so called) are no more than >hypotheses and the only thing that matters is that all quantitative >predictions flowing therefrom are experimentally correct. Well, it would help to know what the objects of empirically correct calculations are so we know what empirically correct calculations apply to. Of course, we can always just say that empirically correct calculations apply to whatever they apply to and let it go at that. Just not very satisfying scientifically to say we have a 3.14159 . . . here and a 6.27 there. Oh well, I suppose we all do the best we can. Close enough for government work I expect. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics [...] > Well, that's exactly the complaint. Science is about explanations and > not just calculations in which we can't tell what is being calculated. The calculations, as you call them, _are_ the explanations. What is explained by saying F = m * a? Nothing. The statement, whether rendered in alegbra or in English merely asserts a relationship between measurable quantities. And what's a measurable quantity, anyhow? The end result of a peculiar behaviour we call measuring. EG, when I measure the length of a piece of wood, I put a tape measure up alongside it, and look at the numbers at both ends of the tape measure, and perform a calculation. To make it easier, tape measures have a zero at one end, so the calculation amounts to reading the second number. If I use a metric tape, I might get 904mm. If I use an imperial tape, I might get 2ft 11-1/2 plus a smidgen. Does this procedure explain what length is? Nope. Does it explain why some objects, such as sticks of wood, have length and others, such as electric curtrents, don't? Nope. Length is what we measure with meter sticks my physics teacher told us. That's all we know about it. I think he was right. The scientific attitude is one of profound humility in the face of the unknowable. It requires accepting the limits of our knowledge. It refuses to deal with things we cannot know, or to answer questions that can have no answers. It's scientific to say Length is what we measure with meter sticks. It's unscientifc to say Length is really..... (finish that sentence any way you like.) St Augustine commented that How long is a piece of string? has no answer, while How long is this piece of string? does. Good comment IMO. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics > And what's a measurable quantity, anyhow? The end result of a peculiar > behaviour we call measuring. EG, when I measure the length of a piece > of wood, I put a tape measure up alongside it, and look at the numbers > at both ends of the tape measure, and perform a calculation. To make it > easier, tape measures have a zero at one end, so the calculation amounts > to reading the second number. If I use a metric tape, I might get 904mm. > If I use an imperial tape, I might get 2ft 11-1/2 plus a smidgen. Does > this procedure explain what length is? Nope. Does it explain why some > objects, such as sticks of wood, have length and others, such as > electric curtrents, don't? Nope. Length is what we measure with meter > sticks my physics teacher told us. That's all we know about it. I > think he was right. My definition of measuring: A communication, where the sender relates a description of a subject to the receiver in terms that the receiver can recognize, but with an additional convenient property: The comparison between subject and familiar ground is indirectly related to the receiver's knowledge. Outside my window right now, the sky is 100% overcast. The thermometer on the wall reads 74 degrees. Something I don't seem to have a good measurement for is the amount of disorganization I see in the things laying about on my desk. My desk is 10% organized? 90% entropy? 2 standard deviations from organized? -:|:- AngleWyrm === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >[...] >> Well, that's exactly the complaint. Science is about explanations and >> not just calculations in which we can't tell what is being calculated. >The calculations, as you call them, _are_ the explanations. Sure, just like behaviorism's dictum to wit the history of behavior is the explanation of behavior. I should have thought you would have jumped for joy at my discovery of an extensional rationale for cardinality and some theoretical confirmation for naturalized epistemology. Of course, that would take us back to counting on our fingers and toes. But positivism, materialism, and behaviorism have pretty much done that already. >What is explained by saying F = m * a? Nothing. The statement, whether >rendered in alegbra or in English merely asserts a relationship between >measurable quantities. Sunofagun! I'll be sure to pass the word that f=ma explains nothing about f, m, and a in relation to one another. Maybe that relationship is an explanation? But if you insist on calculation, how about f=2 dynes? That's an empirical calculation that I defy you to falsify. >And what's a measurable quantity, anyhow? The end result of a peculiar >behaviour we call measuring. EG, when I measure the length of a piece >of wood, I put a tape measure up alongside it, and look at the numbers >at both ends of the tape measure, and perform a calculation. To make it >easier, tape measures have a zero at one end, so the calculation amounts >to reading the second number. If I use a metric tape, I might get 904mm. >If I use an imperial tape, I might get 2ft 11-1/2 plus a smidgen. Does >this procedure explain what length is? Nope. Does it explain why some >objects, such as sticks of wood, have length and others, such as >electric curtrents, don't? Nope. Length is what we measure with meter >sticks my physics teacher told us. That's all we know about it. I >think he was right. No doubt you both do. Not too keen on geometric inference, are you? >The scientific attitude is one of profound humility in the face of the >unknowable. Yes, I've routinely noticed your profound humility in the face of all my suggestions. I would sure hate to see your pride. > It requires accepting the limits of our knowledge. It >refuses to deal with things we cannot know, or to answer questions that >can have no answers. Well, it's clear at least that there are questions than can have no answers for you, among others the length of a piece of string, and the mechanics of intelligence. However, I'm not so sure such questions have no answers for others of greater acumen. > It's scientific to say Length is what we measure >with meter sticks. It's unscientifc to say Length is really..... >(finish that sentence any way you like.) >St Augustine commented that How long is a piece of string? has no >answer, while How long is this piece of string? does. Good comment IMO. And I'm sure this parable is noted in canonic apologetics everywhere. The length of a piece of string is a piece. You just said that was the metric just like the length of a foot of string is a foot. But please don't tell canon authorities or I might be called before some unholy office of the inquisition. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >The calculations, as you call them, _are_ the explanations. I once had a Physics Prof announce that In Physics meaning has no meaning, the only thing that matters is whether we can manipulate the formalisms. I'm not sure what this contributes to the thread, but it's an interesting quote. Dark skies, tom -- We have discovered a therapy ( NOT a cure ) for the common cold. Play tuba for an hour. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >>The calculations, as you call them, _are_ the explanations. >I once had a Physics Prof announce that In Physics meaning >has no meaning, the only thing that matters is whether we can >manipulate the formalisms. The we don't need no stinkin reasons school of physics. >I'm not sure what this contributes to the thread, but it's >an interesting quote. >Dark skies, >tom >-- >We have discovered a therapy ( NOT a cure ) >for the common cold. Play tuba for an hour. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >>The calculations, as you call them, _are_ the explanations. > I once had a Physics Prof announce that In Physics meaning > has no meaning, the only thing that matters is whether we can > manipulate the formalisms. > I'm not sure what this contributes to the thread, but it's > an interesting quote. > Dark skies, > tom He was right. But of course the really odd thing is that those manipulated formalisms somehow refer to the real world.... === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >The calculations, as you call them, _are_ the explanations. >> I once had a Physics Prof announce that In Physics meaning >> has no meaning, the only thing that matters is whether we can >> manipulate the formalisms. >> I'm not sure what this contributes to the thread, but it's >> an interesting quote. >> Dark skies, >> tom >He was right. >But of course the really odd thing is that those manipulated formalisms >somehow refer to the real world.... Unfortunately, QM doesn't explain the somehow, which is what a mechanics is supposed to do. And I daresay there are a multitude of formalisms that somehow don't apply for reasons QM is too busy providing intuition pumps to supply. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics > Unfortunately, QM doesn't explain the somehow, which is what a > mechanics is supposed to do. Tell me. How does a Lagrangian formulation of mechanics explain anything? What it does is imply some laws which when applied produce good predictions. What makes you think there are entities behind the appearences that behave mechanically? What evidence do you have. Classical physics which was solidly mechanics is an empirical failure which is why we have quantum theory and relativity. The truly mechanical theories such as statistical mechanics are heuristics, not fundemental models of reality. QM works. Relativity works. What more do you want? Bob Kolker === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >> Unfortunately, QM doesn't explain the somehow, which is what a >> mechanics is supposed to do. >Tell me. How does a Lagrangian formulation of mechanics explain >anything? What it does is imply some laws which when applied produce >good predictions. What makes you think there are entities behind the >appearences that behave mechanically? What makes you think there aren't. By, George, I do believe you've just confessed that QM isn't mechanics. > What evidence do you have. What evidence do you have that there is no mechanics? All QM has at presence are educated speculations, some of which turn out to be correct and many of which turn out to be incorrect, and no mechanics to explain why any of their guesses are correct. >Classical physics which was solidly mechanics is an empirical failure Well, not exactly. Quantum theorists just took advantage of emprical failures of classical mechanics to suggest there is no mechanics and informed intuition or educated guesses were superior. >which is why we have quantum theory and relativity. The truly mechanical >theories such as statistical mechanics are heuristics, not fundemental >models of reality. QM works. Relativity works. What more do you want? Mechanics. Something that shows how they work. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >> Unfortunately, QM doesn't explain the somehow, which is what a >> mechanics is supposed to do. > Tell me. How does a Lagrangian formulation of mechanics explain > anything? What it does is imply some laws which when applied produce > good predictions. What makes you think there are entities behind the > appearences that behave mechanically? What evidence do you have. > Classical physics which was solidly mechanics is an empirical failure > which is why we have quantum theory and relativity. The truly mechanical > theories such as statistical mechanics are heuristics, not fundemental > models of reality. QM works. Relativity works. What more do you want? > Bob Kolker Lester want to be recognised as the one who explains the _reasons why_ everything is the way it is. But just why mecvhanism should be the reasons why, he can't say; nor why the mechnaism of differences between doifferences is the fundamental mechanism. IOW, Lester can't accept that things are simply what they are, and the best we can do is describe how things happen. IOW, he can't accept what I call the Fundamnetal Ignorance Theorem (first expressed by Bertrand Russell): When we know that we say is true, we don't know what it means. When we know what mean, we don't know if it's true. Now, I wonder how Lester will blast me for that piece of heresy. :-) === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics : Lester want to be recognised as the one who explains the _reasons why_ : everything is the way it is. But just why mecvhanism should be the : reasons why, he can't say; nor why the mechnaism of differences between : doifferences is the fundamental mechanism. The following is from the preface for the second edition of Newton's Principia: I can hear some people disagreeing with this conclusion and muttering something or other about occult qualities. They are always prattling on and on to the effect that gravity is something occult, and that occult causes are to be banished completely from philosophy. But it is easy to answer them: occult causes are not those causes whose existence is very clearly demonstrated by observations, but only those whose existence is occult, imagined, and not yet proved. Therefore gravity is not an occult cause of celestial motions, since it has been shown from phenomena that this force really exists. Rather, occult causes are the refuge of those who assign the governing of these motions to some sort of vortices of a certain matter utterly fictitious and completely imperceptible to the senses. But will gravity be called an occult cause and be cast out of natural philosophy on the grounds that the cause of gravity itself is occult and not yet found? Let those who so believe take care lest they believe in an absurdity that, in the end may overthrow the foundations of all philosophy. For causes generally proceed in a continuous chain from compound to more simple; when you reach the simplest cause, you will not be able to proceed any further. Therefore no mechanical explanation can be given for the simplest cause; for if it could, the cause would not yet be the simplest. Will you accordingly call these simplest causes occult, and banish them? It was written by Roger Cotes in 1713. Stephen === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >: Lester want to be recognised as the one who explains the _reasons why_ >: everything is the way it is. But just why mecvhanism should be the >: reasons why, he can't say; nor why the mechnaism of differences between >: doifferences is the fundamental mechanism. >The following is from the preface for the second edition of Newton's >Principia: > I can hear some people disagreeing with this conclusion and > muttering something or other about occult qualities. They are always > prattling on and on to the effect that gravity is something occult, and > that occult causes are to be banished completely from philosophy. But > it is easy to answer them: occult causes are not those causes whose > existence is very clearly demonstrated by observations, but only those > whose existence is occult, imagined, and not yet proved. Therefore > gravity is not an occult cause of celestial motions, since it has been > shown from phenomena that this force really exists. Rather, occult > causes are the refuge of those who assign the governing of these motions > to some sort of vortices of a certain matter utterly fictitious and > completely imperceptible to the senses. > But will gravity be called an occult cause and be cast out of natural > philosophy on the grounds that the cause of gravity itself is occult > and not yet found? Let those who so believe take care lest they believe > in an absurdity that, in the end may overthrow the foundations of > all philosophy. For causes generally proceed in a continuous chain > from compound to more simple; when you reach the simplest cause, you will > not be able to proceed any further. Therefore no mechanical explanation > can be given for the simplest cause; for if it could, the cause would > not yet be the simplest. Will you accordingly call these simplest > causes occult, and banish them? >It was written by Roger Cotes in 1713. With which I agree and maintain that if quantum observations are in fact the simplest cause, the reason why has yet to be shown, nor has the reason they are not the simplest cause because quantum theory doesn't deal in causes for quantum observations. Quantum theory as I understand it maintains there are no causes. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics > Unfortunately, QM doesn't explain the somehow, which is what a > mechanics is supposed to do. >> Tell me. How does a Lagrangian formulation of mechanics explain >> anything? What it does is imply some laws which when applied produce >> good predictions. What makes you think there are entities behind the >> appearences that behave mechanically? What evidence do you have. >> Classical physics which was solidly mechanics is an empirical failure >> which is why we have quantum theory and relativity. The truly >> mechanical theories such as statistical mechanics are heuristics, not >> fundemental models of reality. QM works. Relativity works. What more >> do you want? >> Bob Kolker > Lester want to be recognised as the one who explains the _reasons why_ > everything is the way it is. But just why mecvhanism should be the > reasons why, he can't say; nor why the mechnaism of differences between > doifferences is the fundamental mechanism. > IOW, Lester can't accept that things are simply what they are, and the > best we can do is describe how things happen. IOW, he can't accept what > I call the Fundamnetal Ignorance Theorem (first expressed by Bertrand > Russell): When we know that we say is true, we don't know what it means. > When we know what mean, we don't know if it's true. To me, Fundamental Ignorance is that willed ignorance that you display when you argue *against* explaining the _reasons why_ things are the way they are. An excellent reply for a novice hoping for entry into the priesthood of Scientific Faith. BTW, I prefer to judge Lester by his own words and not by your revisionist interpretations of them. -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics [...] > To me, Fundamental Ignorance is that willed ignorance that you display > when you argue *against* explaining the _reasons why_ things are the way > they are. An excellent reply for a novice hoping for entry into the > priesthood of Scientific Faith. I don't understand the point of your comment. Are you claiming that I am trying to enter the Pristehood? Whatever gave you that idea? Are you implying that I am a novice? Since I don't know what you mean by novice, I have no idea what imnplication might mean, but I'm pretty sure it's not intended to be complimentary. Are you claiming that I refuse to explain the reasons-why? That depends on what reasons-why are being offered, or more precisely, on what the speaker intends with why. I do detect something rather more than annoyance in your remarks. What's your game? Why do my remarks offend you? Are you suggesting that I refuse to accept that we can ever know why there is something rather than nothing? In the narrow sense of How did the Something within we exist come about? a partial answer can be given. Currently, it starts with a big bang, but since the maths around that singularity are rather, er singular, even the physicists are careful to hedge their bets by pointing out that this the best they can sofar. The story of what happened after the bang is somewhat better grounded. In the wider sense of Howcome did it get started at all? there is no scientific answer (though there are plenty of more or less plausible and entertaining sci-fi speculations.) Profound ignorance is what we must admit at this point. In the rather different sense of Is there some reason or purpose for it all? there is no answer, there are only statements of belief. You're welcome to guess what my statement of belief might be. :-) > BTW, I prefer to judge Lester by his own words and not by your > revisionist interpretations of them. > Well, any judgement you make will be your revisionist interpretation, is all. I don't see that it makes any difference. Oh, that phrase willed ignorance belongs to theology, at least as far as my admittedly limited reading in that field indicates. Are you trying to bring in some theological version of the reasons-why? I can see that if you commit to theological reasons, you would be annoyed at any remark that implies that such reasons-why aren't reasons at all. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics > [...] >> To me, Fundamental Ignorance is that willed ignorance that you display >> when you argue *against* explaining the _reasons why_ things are the >> way they are. An excellent reply for a novice hoping for entry into >> the priesthood of Scientific Faith. > I don't understand the point of your comment. Are you claiming that I am > trying to enter the Pristehood? Whatever gave you that idea? Are you > implying that I am a novice? Since I don't know what you mean by > novice, I have no idea what imnplication might mean, but I'm pretty > sure it's not intended to be complimentary. Are you claiming that I > refuse to explain the reasons-why? That depends on what reasons-why are > being offered, or more precisely, on what the speaker intends with > why. I do detect something rather more than annoyance in your remarks. > What's your game? You must be severely challenged in the interpretation of metaphor. I have no 'game'. >Why do my remarks offend you? Well, now. Aren't you the paragon of innocence. > Are you suggesting that I refuse to accept that we can ever know why > there is something rather than nothing? In the narrow sense of How did > the Something within we exist come about? a partial answer can be > given. Currently, it starts with a big bang, but since the maths around > that singularity are rather, er singular, even the physicists are > careful to hedge their bets by pointing out that this the best they can > sofar. The story of what happened after the bang is somewhat better > grounded. > In the wider sense of Howcome did it get started at all? there is no > scientific answer (though there are plenty of more or less plausible and > entertaining sci-fi speculations.) Profound ignorance is what we must > admit at this point. If you did in fact admit profound ignorance, rather than defend the more or less plausible and entertaining sci-fi speculations as truth, then we would have nothing to discuss. > In the rather different sense of Is there some reason or purpose for it > all? there is no answer, there are only statements of belief. You're > welcome to guess what my statement of belief might be. :-) I have no choice but to infer your beliefs from your posts. >> BTW, I prefer to judge Lester by his own words and not by your >> revisionist interpretations of them. >> > Well, any judgement you make will be your revisionist interpretation, is > all. I don't see that it makes any difference. > Oh, that phrase willed ignorance belongs to theology, at least as far > as my admittedly limited reading in that field indicates. Your admittedly limited reading in that field leads you to serious mistakes. The statement has nothing to do with theology. > Are you trying > to bring in some theological version of the reasons-why? I can see that > if you commit to theological reasons, you would be annoyed at any remark > that implies that such reasons-why aren't reasons at all. LOL. Sorry, Wolf, but you simply lack the rhetorical skill to tar me with that brush. I have not offered any theological arguments in my observations about your beliefs concerning science. I have argued only for the Scientific Method in the search for scientific evidence. -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics [...] >> Oh, that phrase willed ignorance belongs to theology, at least as >> far as my admittedly limited reading in that field indicates. > Your admittedly limited reading in that field leads you to serious > mistakes. The statement has nothing to do with theology. Sorry, but I've never before encountered that phrase in discussions about scientific evidence for or against some claim, only in theological contexts, where it is considered a sin. Since you imply that my comments constitute some sort of heresy against the Scientific Faith, my inference isn't as far wrong as you may wish to claim. >> Are you trying to bring in some theological version of the >> reasons-why? I can see that if you commit to theological reasons, you >> would be annoyed at any remark that implies that such reasons-why >> aren't reasons at all. > LOL. Sorry, Wolf, but you simply lack the rhetorical skill to tar me > with that brush. I have not offered any theological arguments in my > observations about your beliefs concerning science. I have argued only > for the Scientific Method in the search for scientific evidence. Sofar, you haven't once said anything that touches on my beliefs, only on my comments, from which you claim to be able to infer my beliefs. Since you capitalise Scientific Method, am I correct in inferring that you believe it is a path to Truth? If not, what do you intend by capitalising the phrase? In any case, I'm not at all clear just what you object to and why. It seems to have something to do with the origin of life, but you haven't as far as I can make out made your objections clear. On the one hand you claim that the Scietific Method leads to truth (or Truth, I dunno), OTOH, you object to maybe-explanations of the o-o-l on the grounds that they're not proven, or not evidence, or something. But these maybe-explanations are attempts to account for the facts as known. That they aren't any better than maybe-explanations is just the way it is. For now, anyway. Maybe for the foreseeable future. I dunno. But I'm pretty sure that evcentually some scientific account will be available. (There, I've actually _stated_ a belief - you needn't infer it. Saved you a lot of work, no doubt.) Oh, re your LOLs -- great rhetorical skills there. You must have laboured long and hard to come up with such witty ripostes. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics > [...] > Oh, that phrase willed ignorance belongs to theology, at least as > far as my admittedly limited reading in that field indicates. >> Your admittedly limited reading in that field leads you to serious >> mistakes. The statement has nothing to do with theology. > Sorry, but I've never before encountered that phrase in discussions > about scientific evidence for or against some claim, only in theological > contexts, where it is considered a sin. I've never seen it in either context. I simply stuck two words together to mean a type of ignorance that is not accidental, but rather willed, usually done to avoid looking at facts. If you examine previous posts you will see that I invented the phrase as a mockery of your intended insult of Lester where you said: [Lester] can't accept what I call the Fundamnetal Ignorance Theorem (first expressed by Bertrand Russell): When we know that [what] we say is true, we don't know what it means. When we know what [it] mean[s], we don't know if it's true. I think my phrase is as valid as yours, and much more descriptive of your replies. You seem determined to lead this discussion away from science and into religion. However, if you accept the definition of sin as 'missing the mark', then I suppose you could say that 'willed ignorance' is shooting your arrow in a direction guaranteed to miss the mark. > Since you imply that my comments > constitute some sort of heresy against the Scientific Faith, my > inference isn't as far wrong as you may wish to claim. No, on the contrary. Your comments are not 'heresy' against the Scientific Faith, but rather the ingratiating repetitions of a toady hoping to become a high priest of that faith. I personally believe that properly done science should require no faith. > Are you trying to bring in some theological version of the > reasons-why? I can see that if you commit to theological reasons, you > would be annoyed at any remark that implies that such reasons-why > aren't reasons at all. >> LOL. Sorry, Wolf, but you simply lack the rhetorical skill to tar me >> with that brush. I have not offered any theological arguments in my >> observations about your beliefs concerning science. I have argued >> only for the Scientific Method in the search for scientific evidence. > Sofar, you haven't once said anything that touches on my beliefs, only > on my comments, Which leads unavoidably to the conclusion that your comments have all been lies. > from which you claim to be able to infer my beliefs. Yes, I made the apparently erroneous assumption that your comments arose from your beliefs. Would it be safe to continue in the now corrected assumption that everything you post henceforth is also a lie. > Since you capitalise Scientific Method, am I correct in inferring that > you believe it is a path to Truth? Not Universal Truth. But certainly the accepted method for discovering scientific truth. > If not, what do you intend by capitalising the phrase? What did you intend by capitalizing the phrase Fundamnetal Ignorance Theorem? > In any case, I'm not at all clear just what you object to and why. I am aware of your disability. Remember? I called it 'willed ignorance.' It does indeed have serious consequences. > It > seems to have something to do with the origin of life, but you haven't > as far as I can make out made your objections clear. Not even warm. > On the one hand you > claim that the Scietific Method leads to truth (or Truth, I dunno), > OTOH, you object to maybe-explanations of the o-o-l on the grounds that > they're not proven, or not evidence, or something. But these > maybe-explanations are attempts to account for the facts as known. 'Maybe-explanations' are speculation, which is a valid word and quite proper in the context. And speculation does not in any way account for 'the facts as known.' Which is what this whole stupid thread is about and why I accuse you of 'willed ignorance.' > That > they aren't any better than maybe-explanations is just the way it is. > For now, anyway. Maybe for the foreseeable future. I dunno. But I'm > pretty sure that evcentually some scientific account will be available. > (There, I've actually _stated_ a belief - you needn't infer it. Saved > you a lot of work, no doubt.) account will be available. Now if you would just quit referring to such speculation as fact in the future, then possibly we can drop this stupid thread. > Oh, re your LOLs -- great rhetorical skills there. You must have > laboured long and hard to come up with such witty ripostes. Not at all. Laughing out loud was indeed my response to many of your statements. It was not a rhetorical device. -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >> [...] [re: wiklled ignorance:] > I've never seen it in either context. I simply stuck two words together > to mean a type of ignorance that is not accidental, but rather willed, > usually done to avoid looking at facts. OK. > If you examine previous posts > you will see that I invented the phrase as a mockery of your intended > insult of Lester where you said: > [Lester] can't accept what I call the Fundamnetal Ignorance Theorem > (first expressed by Bertrand Russell): > When we know that [what] we say is true, we don't know what it means. > When we know what [it] mean[s], we don't know if it's true. > I think my phrase is as valid as yours, and much more descriptive of > your replies. Only becaues you operate with what IMO are incorrect notions of beliefs and lies. (I've snipped that part of your post.) As explicitly as I can say it: The contrary of a belief statement is not a lie. [...] > No, on the contrary. Your comments are not 'heresy' against the > Scientific Faith, but rather the ingratiating repetitions of a toady > hoping to become a high priest of that faith. I personally believe that > properly done science should require no faith. Me, a toady? Oh, Albert, Albert, how low you have fallen. If anything, you're the toady, with your reiterated references to a Scientific Method, in capitals, yet! [...] >> Sofar, you haven't once said anything that touches on my beliefs, only >> on my comments, > Which leads unavoidably to the conclusion that your comments have all > been lies. Only by your defintion, which I find odd, to say the least. The usual claim is that belief statements are not truthfunctions. Hnece their relationship to lies is, um unclear, to put it mildly. I try to avoid belief statements, though I know that's difficult. If you are claiming that I assert as truths statements I don't beleive in, I have two comments on that. A) What does belief have to do with truth? B) I offer commentys as more or less probbaly true - and I assume that the careful reader will be able to asses probbale truths. Your unpleasant attacks started when I offreed _possible_ scenarios for the origin of loifge. I didn't claim they were true. I didn't claim they were proven. I didn't claim I believed them. All these things you read into my comments because of your quaint notion that in discussions one only utters statments that one believes (whatver you mean by that - I've found it difficult to figure what people mean when they use that word.) >> from which you claim to be able to infer my beliefs. > Yes, I made the apparently erroneous assumption that your comments arose > from your beliefs. Would it be safe to continue in the now corrected > assumption that everything you post henceforth is also a lie. No, because I don't claim knowlege of truth or Truth, just tentative attempts. >> Since you capitalise Scientific Method, am I correct in inferring that >> you believe it is a path to Truth? > Not Universal Truth. But certainly the accepted method for discovering > scientific truth. Which is not a very great truth, but is the best we can do (I don't believe in Universal Truth, just in local and limited truths, of varying imnportance. The truths in personal relationships are very important. That's naother beleif statement, in case you missed it.) >> If not, what do you intend by capitalising the phrase? > What did you intend by capitalizing the phrase Fundamnetal Ignorance > Theorem? Irony. Some ironies point the way to a truth (small t.) >> In any case, I'm not at all clear just what you object to and why. > I am aware of your disability. Remember? I called it 'willed > ignorance.' It does indeed have serious consequences. >> It seems to have something to do with the origin of life, but you >> haven't as far as I can make out made your objections clear. > Not even warm. >> On the one hand you claim that the Scietific Method leads to truth (or >> Truth, I dunno), OTOH, you object to maybe-explanations of the o-o-l >> on the grounds that they're not proven, or not evidence, or something. >> But these maybe-explanations are attempts to account for the facts as >> known. > 'Maybe-explanations' are speculation, which is a valid word and quite > proper in the context. And speculation does not in any way account for > 'the facts as known.' Which is what this whole stupid thread is about > and why I accuse you of 'willed ignorance.' Ok, now that you;ve made yourself clear, kindly point me to what I am deliberately ignorant of. >> That they aren't any better than maybe-explanations is just the way it >> is. For now, anyway. Maybe for the foreseeable future. I dunno. But >> I'm pretty sure that evcentually some scientific account will be >> available. (There, I've actually _stated_ a belief - you needn't infer >> it. Saved you a lot of work, no doubt.) > will be available. Now if you would just quit referring to such > speculation as fact in the future, then possibly we can drop this stupid > thread. I never did - that's what you read into my comments. >> Oh, re your LOLs -- great rhetorical skills there. You must have >> laboured long and hard to come up with such witty ripostes. > Not at all. Laughing out loud was indeed my response to many of your > statements. It was not a rhetorical device. Chuckle. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics > Unfortunately, QM doesn't explain the somehow, which is what a > mechanics is supposed to do. >> Tell me. How does a Lagrangian formulation of mechanics explain >> anything? What it does is imply some laws which when applied produce >> good predictions. What makes you think there are entities behind the >> appearences that behave mechanically? What evidence do you have. >> Classical physics which was solidly mechanics is an empirical failure >> which is why we have quantum theory and relativity. The truly mechanical >> theories such as statistical mechanics are heuristics, not fundemental >> models of reality. QM works. Relativity works. What more do you want? >> Bob Kolker >Lester want to be recognised as the one who explains the _reasons why_ >everything is the way it is. But just why mecvhanism should be the >reasons why, he can't say; nor why the mechnaism of differences between >doifferences is the fundamental mechanism. Wolf's reading comprehension is rather poor not to mention his rather comical editing skills. Probably stream of consciousness appraisals of what he wants to imagine he's read. >IOW, Lester can't accept that things are simply what they are, Once more Wolf universalizes a personal observation. What bothers Wolf is that I can't accept that things simply are what he believes they are because I'm not particularly partial to beliefs in any form as a substitute for science. > and the >best we can do is describe how things happen. IOW, he can't accept what >I call the Fundamnetal Ignorance Theorem (first expressed by Bertrand >Russell): When we know that we say is true, we don't know what it means. >When we know what mean, we don't know if it's true. Yes, well, I daresay when we don't understand what you mean, all we have to do is reread this particular comment. >Now, I wonder how Lester will blast me for that piece of heresy. :-) Heresy, what heresy? Just a rather confused description of your beliefs as far as I can tell. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >>IOW, Lester can't accept that things are simply what they are, > Once more Wolf universalizes a personal observation. What bothers > Wolf is that I can't accept that things simply are what he believes > they are because I'm not particularly partial to beliefs in any form > as a substitute for science. It's nice to know that you know what I believe what things are, because I don't. Enlighten me. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >IOW, Lester can't accept that things are simply what they are, >> Once more Wolf universalizes a personal observation. What bothers >> Wolf is that I can't accept that things simply are what he believes >> they are because I'm not particularly partial to beliefs in any form >> as a substitute for science. >It's nice to know that you know what I believe what things are, because >I don't. Enlighten me. All you have to do is read your own posts, Wolf. You're pretty long on observation and pretty short on explanations. === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics > The calculations, as you call them, _are_ the explanations. >> I once had a Physics Prof announce that In Physics meaning has no >> meaning, the only thing that matters is whether we can manipulate the >> formalisms. >> I'm not sure what this contributes to the thread, but it's >> an interesting quote. >> Dark skies, >> tom > He was right. > But of course the really odd thing is that those manipulated formalisms > somehow refer to the real world.... Somehow? LOL. Well, sometimes they do and sometimes they don't. That area between mathematics and the real world that you refer to as 'somehow' I assume defines where your faith comes into play. -- Don't you see that the whole aim of Newspeak is to narrow the range of thought? In the end we shall make thoughtcrime literally impossible, because there will be no words in which to express it. -- George Orwell as Syme in 1984t === Subject: Re: Science, Philosophy, Mysticism, Art, Mathematics, and Physics >> The calculations, as you call them, _are_ the explanations. > I once had a Physics Prof announce that In Physics meaning has no > meaning, the only thing that matters is whether we can manipulate the > formalisms. > I'm not sure what this contributes to the thread, but it's > an interesting quote. > Dark skies, > tom >> He was right. >> But of course the really odd thing is that those manipulated >> formalisms somehow refer to the real world.... > Somehow? LOL. Well, sometimes they do and sometimes they don't. That > area between mathematics and the real world that you refer to as > 'somehow' I assume defines where your faith comes into play. To quote you: Damn, Albert, read for comprehension. Somehow does not deny sometimes they do and sometimes they don't. The issue is, why do they refer to the real world at all, not what proportion does so. Though it is odd, isn't it, that so much math that began as pure manipulation of the formalisms turned out and still turns out to be applicable to the real world. I guess I must be on your bad-guy list, since you react with foaming mouth and irrelevant comments to just about everything I say - even when I make a point that doesn't disagree with you. === Subject: Re: Existence of countable Hamel basis by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBHDJ5P23976; >>All normed spaces as vector spaces have basis,so called >>Hamel basis. >>Hamel bases in infinite dimensional Banach spaces >>are uncountable.This follows from Baire category theorem. >>What if the infinite dimensional space isn't complete? >>Does it always contain countable Hamel basis? >Of course not; it might, for instance, contain a complete >infinite dimensional subspace even though it itself isn't >complete. === Subject: Re: Existence of countable Hamel basis >All normed spaces as vector spaces have basis,so called >Hamel basis. >Hamel bases in infinite dimensional Banach spaces >are uncountable.This follows from Baire category theorem. >What if the infinite dimensional space isn't complete? >Does it always contain countable Hamel basis? >>Of course not; it might, for instance, contain a complete >>infinite dimensional subspace even though it itself isn't >>complete. Yes. Don't waste your time thinking about subtleties; consider appropriate direct sums of two suitable spaces. Lee Rudolph === Subject: Re: Counting set covers by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBHDOgn24792; >Hello all. I was wondering if there is some relatively simple way >to count the number of set covers of a set of size n that satisfy >the following properties: >1. No set in the cover is a subset of another set in the cover >2. For all sets A(i) in the cover: > A(i) (union of all A(k) (k != i)) = {} >> S = /_j A_j >> nulset = A_k - (/_j A_j - A_k) = A_k - SA_k = A_k >Unfortunately (/_j A_j - A_k) != SA_k. Actually, the problem is that /_{j != k} A_j != /_j A_j - A_k. Consider the cover {{1, 2}, {2, >3}, {1, 3}} of {1, 2, 3}. It satisfies the above two properties, yet >none of the sets are empty. It is incidentally the only such cover of >{1, 2, 3} besides {{1, 2, 3}}. It looks like {{1, 2, 3}} doesn't satisfy the second condition, since the union of an empty collection is empty and this would force {1, 2, 3} to be included in the empty set. Todd Trimble === Subject: Re: Counting set covers >>Hello all. I was wondering if there is some relatively simple way >>to count the number of set covers of a set of size n that satisfy >>the following properties: >>1. No set in the cover is a subset of another set in the cover >>2. For all sets A(i) in the cover: >> A(i) (union of all A(k) (k != i)) = {} >S = /_j A_j >nulset = A_k - (/_j A_j - A_k) = A_k - SA_k = A_k >>Unfortunately (/_j A_j - A_k) != SA_k. > Actually, the problem is that > /_{j != k} A_j != /_j A_j - A_k. > Consider the cover {{1, 2}, {2, >>3}, {1, 3}} of {1, 2, 3}. It satisfies the above two properties, yet >>none of the sets are empty. It is incidentally the only such cover of >>{1, 2, 3} besides {{1, 2, 3}}. > It looks like {{1, 2, 3}} doesn't satisfy the second condition, > since the union of an empty collection is empty and this would > force {1, 2, 3} to be included in the empty set. Ah, yes. I meant to include 'the set of the set' into the count as well, so I should have mentioned that. But, the problem is largely the same anyway. -- Daniel Sj.9ablom === Subject: Re: Counting set covers by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBHDOgC24796; === >Subject: Counting set covers >> Hello all. I was wondering if there is some relatively simple way >> to count the number of set covers of a set of size n that satisfy >> the following properties: >> 1. No set in the cover is a subset of another set in the cover >> 2. For all sets A(i) in the cover: >> A(i) (union of all A(k) (k != i)) = {} >S = /_j A_j >nulset = A_k - (/_j A_j - A_k) = A_k - SA_k = A_k >> The second property could also be formulated as: There is no set in >> the cover that contains an element not contained in another set in >> the cover. >A_k subset /_j A_j - A_k = SA_k. Thus again A_k = nulset. >The second property requires all A_k to be empty. >> There are still a huge amount of such covers, but I would be >> interested in comparing the amount of such covers to the amount of >> all possible set covers. >There are none unless the set S being covered is empty. Nonsense. For S = {1, 2, 3}, the cover { {1, 2}, {1, 3}, {2, 3} } meets the OP's conditions. Todd Trimble === Subject: Geometry software Hi! I am looking for a software that letÇs you draw geometric figures and that the software calculates lengths and ratios. Do you know of such a software? Must be many but you probably have more experience than me. === Subject: Re: Geometry software > Hi! I am looking for a software that letÇs you draw geometric figures and > that the software calculates lengths and ratios. Do you know of such a > software? Must be many but you probably have more experience than me. A commercial software popular with educators is Geometer's Sketchpad. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Geometry software Hi! I am looking for a software that letÇs you draw geometric figures and that the software calculates lengths and ratios. Do you know of such a software? Must be many but you probably have more experience than me. === Subject: Re: Geometry software > Hi! I am looking for a software that let.8bs you draw geometric figures and > that the software calculates lengths and ratios. Do you know of such a > software? Must be many but you probably have more experience than me. http://www.euclidraw.com/ -- I. N. G. --- http://users.forthnet.gr/ath/jgal/ === Subject: Lebesgue measurable functions I need to see an example of a Lebesgue measurable function f: R -->R such that inverse image of a Lebesgue measurable set is not Lebesgue measurable. Examples through cardinality argument will be accepted. === Subject: Re: Lebesgue measurable functions > I need to see an example of a Lebesgue measurable function f: R -->R such > that inverse image of a Lebesgue measurable set is not Lebesgue measurable. > Examples through cardinality argument will be accepted. Can you find a Lebesgue measurable f that maps [0,1] bijectively to the Cantor set? Every subset of the Cantor set is Lebesgue measurable, but not every subset of [0,1] is. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: Lebesgue measurable functions I need to see an example of a Lebesgue measurable function f: R -->R such that inverse image of a Lebesgue measurable set is not Lebesgue measurable. Examples through cardinality argument will be accepted. > Can you find a Lebesgue measurable f that maps [0,1] bijectively to the > Cantor set? Every subset of the Cantor set is Lebesgue measurable, but > not every subset of [0,1] is. > -- > G. A. Edgar http://www.math.ohio-state.edu/~edgar/ There is also a continuous (strictly increasing) bijection of [0,1] onto [0,2] which maps the Cator set (of Lebesgue measure 0) onto a Cantor-like set of Lebesgue measure 1. It appeared recently: x+C(x) where C is the Cantor function. === Subject: Re: Lebesgue measurable functions > I need to see an example of a Lebesgue measurable function f: R -->R such > that inverse image of a Lebesgue measurable set is not Lebesgue measurable. > Examples through cardinality argument will be accepted. > Can you find a Lebesgue measurable f that maps [0,1] bijectively to the Cantor set? Every subset of the Cantor set is Lebesgue measurable, but not every subset of [0,1] is. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ > There is also a continuous (strictly increasing) bijection of [0,1] onto > [0,2] which maps the Cator set (of Lebesgue measure 0) onto a Cantor-like > set of Lebesgue measure 1. It appeared recently: x+C(x) where C is the > Cantor function. Cator set is of course my private nickname for the Cantor set. (Sitting in my office on Saturday night, ready to answer questions from students writing a final exam...) === Subject: Convexity (x^2+y^2)^a + (x*y)^2 <= 1 What is the minimal a for which the set consisted of the solutions of the above inequality is convex? How to solve this and similar problems? Niles W. === Subject: Re: Convexity > (x^2+y^2)^a + (x*y)^2 <= 1 > What is the minimal a for which the set consisted of the solutions of > the above inequality is convex? Is the question unclear/incorrect or the answer trivial? Niles W. === Subject: Re: Convexity (x^2+y^2)^a + (x*y)^2 <= 1 What is the minimal a for which the set consisted of the solutions of the above inequality is convex? > Is the question unclear/incorrect or the answer trivial? None of the above, I think. It's just hard. I'll leave consideration of a <= 0 to you, and suppose a > 0. By symmetry it's enough to consider the first quadrant, where we can take y as a function of x for 0 <= x <= 1, y(0) = 1 and y(1) = 0. It looks to me like when we decrease a, the first place y'' becomes positive would be where x=y, where (2 x^2)^a + x^4 = 1. I think the answer is the value of a in the solution of the system of equations (2 x^2)^a + x^4 = 1 -a + (a + 2) x^4 = 0 which is approximately 0.21406286037879413377. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: .99999... still=/= 1 by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBHFa5605219; >> So from another point of view, >> .999... =/= 1 >The locution .999.... is a meaningless string of symbols. It is not a >properly expressed mathematical entity. Whereas: >SUM [n >=1] (9/10^n) >is a proper mathematical expression (it is equal to 1) >So the sentence .999... =/= 1 is meaningless. >You are a troll and a moron. >Bob Kolker Bob, Please avoid the name calling. It gives sci.math a bad reputation. You may not agree with SE's logic (I do not agree with it either), but it is not OK to call him names because of your disagreement with him on things mathmatical. Let's keep it friendly, and on topic. Please, no more name calling. - MO === Subject: Re: .99999... still=/= 1 > So from another point of view, > > .999... =/= 1 >>The locution .999.... is a meaningless string of symbols. It is not a >>properly expressed mathematical entity. Whereas: >>SUM [n >=1] (9/10^n) >>is a proper mathematical expression (it is equal to 1) >>So the sentence .999... =/= 1 is meaningless. >>You are a troll and a moron. >>Bob Kolker >Bob, >Please avoid the name calling. It gives sci.math a bad reputation. >You may not agree with SE's logic (I do not agree with it either), >but it is not OK to call him names because of your disagreement >with him on things mathmatical. >Let's keep it friendly, and on topic. Please, no more name calling. about the math part. >- MO Smart's Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 > Bob, > Please avoid the name calling. It gives sci.math a bad reputation. > You may not agree with SE's logic (I do not agree with it either), > but it is not OK to call him names because of your disagreement > with him on things mathmatical. > Let's keep it friendly, and on topic. Please, no more name calling. Enterprise has bone told again and again what a series is, what convergence is and what a limit is. He does not seem to learn. That justifies the term moron, a low level intellect. He is also a troll because he repeats his errors again and again. So the appleation is well justified. Bob Kolker === Subject: Re: .99999... still=/= 1 >> Bob, >> Please avoid the name calling. It gives sci.math a bad reputation. >> You may not agree with SE's logic (I do not agree with it either), >> but it is not OK to call him names because of your disagreement >> with him on things mathmatical. >> Let's keep it friendly, and on topic. Please, no more name calling. >Enterprise has bone told again and again what a series is, what >convergence is and what a limit is. He does not seem to learn. That >justifies the term moron, a low level intellect. >He is also a troll because he repeats his errors again and again. >So the appleation is well justified. >Bob Kolker I have been using a non-standard approach to this math problem. And usually the way to expansion to new frontiers in math. is to challenge what is usually accepted as correct but may not necessarily be correct. If it wasn't for Columbus, you people might have fallen off the edge of the earth. Smart's Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 > I have been using a non-standard approach to this math problem. You are full of . You don't know what non-standard arithmetic is andyou would not know a hyperreal number if it bit your nose. Bob Kolker === Subject: Re: .99999... still=/= 1 Originator: richard@cogsci.ed.ac.uk (Richard Tobin) >You may not agree with SE's logic (I do not agree with it either), >but it is not OK to call him names because of your disagreement >with him on things mathmatical. They don't have a disagreement about anything mathematical. It should be clear by now that SE is not arguing in good faith, but is just trolling. -- Richard === Subject: Re: .99999... still=/= 1 >>You may not agree with SE's logic (I do not agree with it either), >>but it is not OK to call him names because of your disagreement >>with him on things mathmatical. >They don't have a disagreement about anything mathematical. It should >be clear by now that SE is not arguing in good faith, but is just >trolling. >-- Richard I sincerely have been debating this topic the best way I perceive it. For crying out loud, do you actually believe a real number equals a number that isn't real? Smart's Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 In sci.math, S. Enterprize Company >You may not agree with SE's logic (I do not agree with it either), >but it is not OK to call him names because of your disagreement >with him on things mathmatical. >>They don't have a disagreement about anything mathematical. It should >>be clear by now that SE is not arguing in good faith, but is just >>trolling. >>-- Richard > I sincerely have been debating this topic the best way I > perceive it. For crying out loud, do you actually believe a > real number equals a number that isn't real? Well, here's a fine mess you've gotten us into... :-) First, what is a real number anyway? Despite our many valiant attempts to beat you over the head with the club of reason, it's clear you're resisting -- and it may very well because the club of reason is actually a thin, wispy, non-existent fog of a metaphor. In short, real numbers are about as real as pink elephants. They do not exist. Oh, sure, one can blather on about measuring 1.25 inches or 3 1/2 cups of coffee or an air pressure of 101 kPa during a nice sunny day -- but those are physical measurements, not numbers per se. Try catching a number in a butterfly net -- or any other kind of net, for that matter. One can't do it; the best one might get is a pair of swallowtailed yellowbeaks. Or something. (Or was that yellowtailed swallowbeaks? Does one count an egg as half of a bird, or a third? Well, never mind that; ornithology was never my strongest subject.) The best we can do is lay a groundwork of phantom assumptions, such as Peano's Axioms and Dedekind cuts, but here is where the problems start. First, what does '=' actually mean? For most of us, we simply accept that limits make sense -- but there are far deeper issues here; the partial sums, for example, of the series 1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + ... are all rational, but the result is pi^2/6. Hey, waitaminnit, that ain't in Q! What happened? Good question. Or one can try 1 + 1/2 + 1/3 + ... All your partial sums are belong to Q, but the series diverges; this one has no limit, although one might make a case that 1 + 1/2 + 1/3 + ... = +oo. but remember that +oo is not a real number. Of course, the usage of the ellipsis ('...') is a bit tricky. Take, for example, this definition: e = 2.718281828... Now before one asks uh, is that rational? -- turns out this is Euler's Number, which equates to the series e = 1/0! + 1/1! + 1/2! + 1/3! + ... and no, it is not rational (although all of the partial sums are!); a more honest ellipsis might be e = 2.718281828459... So we fall back on convention. You may have heard the expression what I say three times is true. (It's not quite true, admittedly, but never mind that.) So one might write 1/7 = 0.142857142857142857... with the understanding that the last 6 digits repeat endlessly. (It turns out they do; a long-division proof is left to the interested reader, or one can simply multiply 142857*7 = 999999 = 10^6 - 1 and/or note that 1/(1-x) = 1+x+x^2+x^3+... for any x between -1 and +1.) Or 1/11 = 0.090909... or even 1/9 = 0.111... So what does 0.999... really mean? Endless 9's, in this case. A proper treatment of such a number might involve summation of the infinite series sum(i=1,+oo) (9 * 10^(-i)). If numbers are an abstract_1 concept, this sum is an abstract_3 (simple expressions are abstract_2, in some sense). The partial sums of this series are of course sum(i=1,n) (9 * 10^(-i)); a little work with a competent algebra book or symbolic calculator show that a partial sum sum(i=1,n) (9 * 10^(-i)) = 1 - 10^n = 0.999...9. There's that ellipsis again, this time in a different context; the 9's do *not* repeat endlessly in this case (unless one is Gary Denke, but one doesn't really want to know the details there). In this case, there are n 9's total. Of course, I've pointed this out before to you, and so far it's yet to sink in, unlike the iron spike in the case of Phinehas Gage. (ObOuch: Ouch.) Unfortunately, to most calculators, all numbers are rationals -- furthermore, to modern computers, all numbers are multiples of a power of 2, with some fudging on the arithmetic side to make it look otherwise. (The standard representation is r = 2^e * M, where M is up to about 53 bits and e can range from -2048 to +2047 or thereabouts.) Therefore, a partial sum may yield anomalies if not done carefully. For example, to a computer, 1/3 = 0x3fd5555555555555 . This is readily verified using C and a hack such as the following: #include int main() { union { unsigned char c[sizeof(double)]; double d; } u; int i; u.d = 1.0/3.0; printf(1/3 = 0x); for(i=0;i and #if BYTE_ORDER == BIG_ENDIAN in appropriate spots.) The first bit is the sign bit; the next 11 is a modified exponent, and the rest are the mantissa, with a hidden-1. For example, 1.0 = 0x3ff0000000000000 2.0 = 0x4000000000000000 3.0 = 0x4008000000000000 4.0 = 0x4010000000000000 5.0 = 0x4014000000000000 In a very real sense, to a computer 1/3 = 6004799503160661/18014398509481984 = .333333333333333314829616256247390992939472198486328125 . (Note that 15555555555555(16) = 6004799503160661(10).) One can use an infinite-precision calculator such as bc to derive these results: $ bc <In sci.math, S. Enterprize Company > >>You may not agree with SE's logic (I do not agree with it either), >>but it is not OK to call him names because of your disagreement >>with him on things mathmatical. >They don't have a disagreement about anything mathematical. It should >be clear by now that SE is not arguing in good faith, but is just >trolling. >-- Richard >> I sincerely have been debating this topic the best way I >> perceive it. For crying out loud, do you actually believe a >> real number equals a number that isn't real? >Well, here's a fine mess you've gotten us into... :-) >First, what is a real number anyway? Despite our many valiant >attempts to beat you over the head with the club of reason, >it's clear you're resisting -- and it may very well because >the club of reason is actually a thin, wispy, non-existent fog >of a metaphor. >In short, real numbers are about as real as pink elephants. >They do not exist. Oh, sure, one can blather on about >measuring 1.25 inches or 3 1/2 cups of coffee or an >air pressure of 101 kPa during a nice sunny day -- but >those are physical measurements, not numbers per se. >Try catching a number in a butterfly net -- or any other kind >of net, for that matter. One can't do it; the best one might >get is a pair of swallowtailed yellowbeaks. Or something. >(Or was that yellowtailed swallowbeaks? Does one count an >egg as half of a bird, or a third? Well, never mind that; >ornithology was never my strongest subject.) >The best we can do is lay a groundwork of phantom assumptions, >such as Peano's Axioms and Dedekind cuts, but here is where >the problems start. >First, what does '=' actually mean? For most of us, we simply >accept that limits make sense -- but there are far deeper issues >here; the partial sums, for example, of the series >1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + ... >are all rational, but the result is pi^2/6. Hey, waitaminnit, that >ain't in Q! What happened? >Good question. >Or one can try >1 + 1/2 + 1/3 + ... >All your partial sums are belong to Q, but the series diverges; >this one has no limit, although one might make a case that >1 + 1/2 + 1/3 + ... = +oo. >but remember that +oo is not a real number. >Of course, the usage of the ellipsis ('...') is a bit tricky. Take, >for example, this definition: >e = 2.718281828... >Now before one asks uh, is that rational? -- turns out this is >Euler's Number, which equates to the series >e = 1/0! + 1/1! + 1/2! + 1/3! + ... >and no, it is not rational (although all of the partial sums are!); >a more honest ellipsis might be >e = 2.718281828459... >So we fall back on convention. You may have heard the expression >what I say three times is true. (It's not quite true, admittedly, >but never mind that.) So one might write >1/7 = 0.142857142857142857... >with the understanding that the last 6 digits repeat endlessly. >(It turns out they do; a long-division proof is left to the >interested reader, or one can simply multiply 142857*7 = 999999 = 10^6 - 1 >and/or note that 1/(1-x) = 1+x+x^2+x^3+... for any x between -1 and +1.) >1/11 = 0.090909... >or even >1/9 = 0.111... >So what does 0.999... really mean? Endless 9's, in this case. >A proper treatment of such a number might involve summation of the >infinite series sum(i=1,+oo) (9 * 10^(-i)). If numbers are >an abstract_1 concept, this sum is an abstract_3 (simple expressions >are abstract_2, in some sense). The partial sums of this series >are of course sum(i=1,n) (9 * 10^(-i)); a little work with a competent >algebra book or symbolic calculator show that a partial sum >sum(i=1,n) (9 * 10^(-i)) = 1 - 10^n = 0.999...9. There's that >ellipsis again, this time in a different context; the 9's do *not* >repeat endlessly in this case (unless one is Gary Denke, but one >doesn't really want to know the details there). In this case, >there are n 9's total. >Of course, I've pointed this out before to you, and so far it's >yet to sink in, unlike the iron spike in the case of Phinehas Gage. >(ObOuch: Ouch.) >Unfortunately, to most calculators, all numbers are rationals -- >furthermore, to modern computers, all numbers are multiples of a >power of 2, with some fudging on the arithmetic side to make it >look otherwise. (The standard representation is r = 2^e * M, where >M is up to about 53 bits and e can range from -2048 to +2047 >or thereabouts.) Therefore, a partial sum may yield anomalies >if not done carefully. >For example, to a computer, >1/3 = 0x3fd5555555555555 . >This is readily verified using C and a hack such as the following: >#include >int main() > union { unsigned char c[sizeof(double)]; double d; } u; > int i; > u.d = 1.0/3.0; > printf(1/3 = 0x); > for(i=0;i printf(n); > return 0; >(Depending on machine the hex values will print either >forwards or backwards. If it's a worry, use something like >#include >and >#if BYTE_ORDER == BIG_ENDIAN >in appropriate spots.) >The first bit is the sign bit; the next 11 is a modified exponent, >and the rest are the mantissa, with a hidden-1. For example, >1.0 = 0x3ff0000000000000 >2.0 = 0x4000000000000000 >3.0 = 0x4008000000000000 >4.0 = 0x4010000000000000 >5.0 = 0x4014000000000000 >In a very real sense, to a computer >1/3 = 6004799503160661/18014398509481984 >= .333333333333333314829616256247390992939472198486328125 . >(Note that 15555555555555(16) = 6004799503160661(10).) >One can use an infinite-precision calculator such as bc to >derive these results: >$ bc <ibase=16 >15555555555555 >6004799503160661 >Now multiply by 3, and one gets >18014398509481983/18014398509481984 >= .999999999999999944488848768742172978818416595458984375 . >Is 1 = .999999999999999944488848768742172978818416595458984375 ? >Certainly not. But computers aren't as bright as one might think. :-) >I'll also mention a little bug in Microsoft's calculator. Microsoft >(or an engineer therein) was apparently somewhat naive, and one got >3.11 - 3.10 = 0.00 >in their calculator in Win95. >It turns out that >3.11 = 0x4008e147ae147ae1 >3.10 = 0x4008cccccccccccd >diff = 0x3f847ae147ae1400 >0.01 = 0x3f847ae147ae147b >so, to a computer, (3.11 - 3.10) is just a smidge less than 0.01. >With rounding this ordinarily isn't a problem, but if one forgets >to round -- well, one very well might get 0.00, with an invisible >9 for one's trouble. >The bug finally got fixed some time ago, but it took awhile. >Another illustration is 9007199254740992+1 = 9007199254740992, >if one uses double. Not a lot one can do about it without >switching away from double (e.g., one might use long long >instead), as the bit gets lost; 2^53 = 9007199254740992. >It gets worse. >Remember that the exact representation of 1/3 using double >precision is >1/3 = .333333333333333314829616256247390992939472198486328125 >A naive C program using %.60f (which is waaay too much precision >and not enough accuracy!) gives >1/3 = .333333333333333314829616256247390992939472198486328125 >which actually turns out to be the right answer -- if one can >call this a right answer. >However, a naive progressive digitation algorithm prints out >1/3 = .333333333333333303727386009995825588703155517578125 >Yipes. >The conclusion: don't depend on partial sums generated by computer, >unless you know exactly what it's doing -- and what you're doing. >[.sigsnip] >-- >#191, ewill3@earthlink.net >It's still legal to go .sigl LEARN MATH. http://mathworld.wolfram.com/HyperrealNumber.html .999... < 1 .999... =/= 1 Smart's Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 >I won this debate years ago. So what is the reason to start it again? === Subject: Re: .99999... still=/= 1 > At one digit less than oo, ( assuming you really reach infinity) the nth > term reaches 0 so, .999... reaches 0 not 1. And what is the decimal digit of pi at the infinity? > Even if you go past oo by one digit, it still doesn't reach 1. In which direction the wind blows at that point? === Subject: Re: .99999... still=/= 1 >> At one digit less than oo, ( assuming you really reach infinity) >the nth >> term reaches 0 so, .999... reaches 0 not 1. >And what is the decimal digit of pi at the infinity? I already answered that. First of all pi ISSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS an irrational number, the decimal values vary. It can not be determined. But with a repeating decimal, you can use a non-standard approach and use one digit less than oo then, n-->oo -1 lim 9/10^n ---> 90/10^oo ^ | last digit seen is zero right before infinity. It never reaches 1. >> Even if you go past oo by one digit, it still doesn't reach 1. >In which direction the wind blows at that point? Smart's Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 > At one digit less than oo, ( assuming you really reach infinity) the nth > term reaches 0 so, .999... reaches 0 not 1. And what is the decimal digit of pi at the infinity? > I already answered that. First of all pi > ISSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS an > irrational number, the decimal values vary. It can not be determined. But with > a repeating decimal, you can use a non-standard approach and use one digit less > than oo then, > n-->oo -1 > lim 9/10^n ---> 90/10^oo > ^ > | > last digit seen is zero right before infinity. It never reaches 1. There is no right before infinity. === Subject: Re: .99999... still=/= 1 >> At one digit less than oo, ( assuming you really reach infinity) >the nth >> term reaches 0 so, .999... reaches 0 not 1. > >And what is the decimal digit of pi at the infinity? >> I already answered that. First of all pi >> ISSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS an >> irrational number, the decimal values vary. It can not be determined. But >with >> a repeating decimal, you can use a non-standard approach and use one digit >less >> than oo then, >> n-->oo -1 >> lim 9/10^n ---> 90/10^oo >> ^ >> | >> last digit seen is zero right before infinity. It never reaches 1. >There is no right before infinity. I said the digit right before oo. REFERENCE MATHCAD PROFESSIONAL Smart's Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 > | > last digit seen is zero right before infinity. It never reaches 1. There is no right before infinity, numbskull. Your comprehension of mathematics is infitesimally small. A hyperreal equivalent of 0. Bob Kolker === Subject: Re: .99999... still=/= 1 >> | >> last digit seen is zero right before infinity. It never reaches 1. >There is no right before infinity, numbskull. Hey, there are probably about 1 million supporters using MathCAD Professional. This is an industry standard for math for scientists and engineers. A non-standard approach using MathCAD clearly shows that, n-->oo -1 lim 9/10^n ---> 90/10^n The digit right before oo for the hyperreal number or series .999... is 0. Therefore, there is a space existing between, .999... and 1 so, as the definition of a hyper-real number implies, .999... < 1 A hyper-real number causes a space to exist between it and a real number. 1 is the real number .999... is the hyper-real number that forms the space between the two. So, .999... =/= 1 .999... < 1 >Your comprehension of mathematics is infitesimally small. A hyperreal >equivalent of 0. >Bob Kolker Smart's Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 In sci.math, S. Enterprize Company > | > > last digit seen is zero right before infinity. It never reaches 1. >>There is no right before infinity, numbskull. > Hey, there are probably about 1 million supporters using MathCAD > Professional. This is an industry standard for math for scientists and > engineers. A non-standard approach using MathCAD clearly shows that, > n-->oo -1 > lim 9/10^n ---> 90/10^n > The digit right before oo for the hyperreal number or > series .999... is 0. Therefore, there is a space existing between, > .999... and 1 so, > as the definition of a hyper-real number implies, > .999... < 1 An interesting notion, that. So D[.999..., w-1] = 0, eh? [*] What is D[.999..., w-2]? How about D[.999..., w/2]? It's easily proven that, if D[.999..., n] = 9, then D[.999...., n+1] = 9 as well (the simplest method arguably is to evaluate D[x*10, n]), for any finite n. Not sure if w-1 is finite or not -- or even meaningful. As for MathCAD: that's a program, an approximation of reality. Not that real numbers are all that real, anyway -- they're mathematical/symbolic abstractions, there because Dedekind, Cauchy, and Cantor and others needed more numbers for set theory. In light of what I've written before regarding 1/3, one might have to verify the results carefully. > A hyper-real number causes a space to exist between it and a real number. > 1 is the real number > .999... is the hyper-real number that forms the space between the two. > So, > .999... =/= 1 > .999... < 1 Your logic is extremely sloppy, though your conclusion is interesting. I'm just not sure which realm it exists in, although the standard real realm does not contain it (the standard realm doesn't contain any numbers between 0 and all 1/n, n > 0, n in J: the hyperreal realm, however, does). [.sigsnip] [*] I don't have an omega, so I'm using 'w' here to indicate the first transfinite ordinal. Is there a w_0, analogous to the cardinal aleph_0? This gets a bit messy. D[r,n] = the digit associated with the n'th decimal place after the decimal point (e.g., D[.98765, 4] = 6). -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: .99999... still=/= 1 >In sci.math, S. Enterprize Company > >> | >> >> last digit seen is zero right before infinity. It never reaches 1. >There is no right before infinity, numbskull. >> Hey, there are probably about 1 million supporters using MathCAD >> Professional. This is an industry standard for math for scientists and >> engineers. A non-standard approach using MathCAD clearly shows that, >> n-->oo -1 >> lim 9/10^n ---> 90/10^n >> The digit right before oo for the hyperreal number or >> series .999... is 0. Therefore, there is a space existing between, >> .999... and 1 so, >> as the definition of a hyper-real number implies, >> .999... < 1 >An interesting notion, that. So D[.999..., w-1] = 0, eh? [*] >What is D[.999..., w-2]? How about D[.999..., w/2]? Using MathCAD I get, n-->oo - 2 lim 9/10^n ---> 900/10^oo How can this be interpreted? In a non-standard analysis, in my opinion, it could be mean we are moving further away from 1, but we still maintain the surreal number or hyperreal number. For n digits less than infinity .999..n | 1 you still get, .999... < 1 For the case 1/2 a digit less than infinity, MathCAD calls this a bidirectional limit. n-->oo/2 lim 9/10^n --> 9/((10^oo)^1/2) But I don't think this applies to a surreal or a hyperreal number. .999... doesn't really exist anymore. So I don't think you can say this. >It's easily proven that, if D[.999..., n] = 9, then D[.999...., n+1] = 9 Ok, because, If the mth digit less than oo = oo n--> oo - m lim 9/10^oo-oo ---> 9 What can this be interpreted as in regard to a surreal number? It looks like to me, in this case we have totally removed the original hyperreal or surreal number. I think there is a limit to how far you can go minus digits before you change, .999... totally from it original form. In this case, you no longer have, .999..., you have something else. You have just 9. This isn't surreal or hyperreal. >as well (the simplest method arguably is to evaluate D[x*10, n]), >for any finite n. Not sure if w-1 is finite or not -- or even meaningful. >As for MathCAD: that's a program, an approximation of reality. Not that >real numbers are all that real, anyway -- they're mathematical/symbolic >abstractions, there because Dedekind, Cauchy, and Cantor and others needed >more numbers for set theory. In light of what I've written before >regarding 1/3, one might have to verify the results carefully. >> A hyper-real number causes a space to exist between it and a real number. >> 1 is the real number >> .999... is the hyper-real number that forms the space between the two. >> So, >> .999... =/= 1 >> .999... < 1 >Your logic is extremely sloppy, though your conclusion is interesting. Why would it be sloppy to you? Do you mean as far as accuracy is concerned? >I'm just not sure which realm it exists in, although the standard >real realm does not contain it (the standard realm doesn't >contain any numbers between 0 and all 1/n, n > 0, n in J: the hyperreal >realm, however, does). >[.sigsnip] >[*] I don't have an omega, so I'm using 'w' here to indicate the > first transfinite ordinal. Is there a w_0, analogous to > the cardinal aleph_0? This gets a bit messy. > D[r,n] = the digit associated with the n'th decimal place > after the decimal point (e.g., D[.98765, 4] = 6). >-- >#191, ewill3@earthlink.net >It's still legal to go .sigless. Smart's Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 >> Yes the limit is exactly 1 in reals. > And also in the hyperreals. >> Yes, if you count the limit over reals plus over hyperreals. Sorry - I >> cannot find better word than over. Find a better word as your mother >> language is English. >> I try to specify: >> As you count the limit in reals, then N --> oo, where every N is finite >> integer. As you count the limit over hyperreals, then N_inf -->oo_inf, >> where >> oo_inf has higher cardinality as the cardinality of N_inf >N. > No, this is not about cardinality at all. The infinitely large integers > of NSA are not the same as transfinite cardinals. It's necessary to observe: As the cardinality of reals R > cardinality of integers N, so the cardinality of N_inf > N. As you count the limit N --->oo instead of N-inf, then you certainly omit something - namely hyperreal part. >> N (finite integers) does not cover hyperreal area, i.e. the numbers >> smaller >> than reals, because N hardly covers real area as discussed under the >> thread >> Are reals well-ordered. >> As we count the real limit in NSA, then the hyperreals exist, but they >> are >> not counted in limit as N -->oo, but not as N_inf -->oo. Hyperreals are >> omitted and therefore 0.999...<1 in hyperreals, though the real part of >> the >> limit equals to 1. > I take it you mean the standard part of the limit is 1. Well, we can talk about standard part and non-standard part, if you prefer this. > That happens to > be true, but for a trivial reason. The limit itself is exactly 1, and > the standard part of 1 is simply 1. Cases: 1)Yes, the limit is exactly 1only if you count the limit including the non standard part as N-inf --->oo. Then your reference set is N_inf. 2)Yes, the limit is exactly 1only if you count the limit including the standard part as N --->oo. Then your reference set is N. 3) But..., if you count the limit including the standard part as N --->oo and your reference set is N_inf, then you omit non-standard part. As a consequence in this last case 0.999...<1 in N_inf. >> NSA expands the concept of numbers to >> the numbers that are smaller than any real, i.e epsilon environment. >> This >> is >> equivalent with the concept of epsilon delta theorem. Read literally >> what >> epsilon delta theorem says. This was learnt us already in 70Çs in >> university. Are you back in 50's? > Which part of my statement do you not accept? Do you disagree with the > definition I gave? >> Explained above. > Try again. You didn't mention any part of the definition, let alone say > which part you disagreed with. > Definition. Let { a_k } be a sequence and let L be a real number. > We say lim_{k->oo} a_k = L if, for every epsilon > 0, there exists > N > 0 such that | a_k - L | < epsilon for every k > N. > Remark 1. Exactly the same definition applies to standard analysis and > to nonstandard analysis, with the proviso that in NSA the epsilon > 0 > is allowed to be an infinitesimal and the N > 0 is allowed to be > infinitely large. Yes, agreed. > Remark 2. The definition does not say what it means for the limit to be > close to L. The definition only says what it means for the limit to be > equal to L. Either the definition is satisfied, or it isn't. OK. > Now, the questions: (1) Do you agree with the definition? (2) Do you > agree that according to this definition the limit is exactly 1, even in > NSA? If you don't agree, explain why not. Yes, I do agree as explained above in the cases 1 and 2. You did not consider at all the case 3 above. How does your definitions should be applied on the case 3? You should also note that the limit is the upper boundary value. As you will see below, it's not the question about the limit but about AC and the point of reference as we construct the numbers integers, infinite integers, reals, hyperreals etc. > For example, try to give a > particular value of epsilon > 0 such that the definition is not > satisfied. Hint: choosing an infinitesimal epsilon is allowed, but it > won't help your case. The definition still works. What about case 3 above as you stop epsilons in standard part omitting non-standard part. >> I would like to ask the same from You. :-). Is it the point >> of reference that is strange concept for You? > Point of reference is an undefined concept and does not appear in the > definition of limit, quoted above. I won't comment on whether it is > strange, since things have to be defined first before they can possibly > qualify as strange. The point of reference is counting point reference, which also separates infinities. The standard point of reference is the normal decimal dot that separates the integer part and the decimal part, which can be infinite long string. Without the point of reference you do not know which part is integer part and which one is the decimal part. What ever you calculate you always refer your calculations to some point of reference. >> In fact, there seems to be a slight conceptual difference in our >> argumentation. The difference is analocigally the same as we talk about >> finite decimal numbers. You accept that finite 0.99999 <1, as there are >> no >> infinite 9's. As hyperreals are omitted in the real limit counting (not >> counted over hyperreals), then in reals the limit equals to 1, i.e. >> 0.999...=1. But as hypereals exist, but omitted in the limit calculation, >> then 0,999...<1 in NSA. Simple as possible. Reals are finite from the >> hypereal point of view. Therefore 0.999...<1 - in hyperreals as only real >> area is considered. > But in NSA there is no such thing as a .999... that extends only through > finite digit positions. If the string is infinitely long, then it > necessarily extends to infinite digit positions in NSA. That may sound > vaguely similar to an argument commonly made by cranks, but the > difference is that the cranks are not talking about NSA. used the case 3 as an example to point out how the people do not think as their argumentation contains hidden asumptions. So - if there exist non-standard part, which is purposely omitted, then you do not calculate the total or over-all limit. In this case you have to observe that there are numbers smaller than any real number and as a consequence 0.999... cannot be equal to 1, though the limit equals to 1. What is this paradox, which has been the reason to this thread, and how do we fix it? The solution is to point out that the limit is a mathematical upper boundary value, but does not tell the real value of the string but the next, i.e. successor, value of the string, because the limit calculation is based on the epsilon delta theorem. > The definition of a limit in NSA may be stated in various ways, but all > of them are equivalent to the usual epsilon-delta definition, with the > proviso that epsilons and deltas are allowed to be infinitesimal, and N > is allowed to be infinitely large. >> Yes, that's what I mean - too, assuming we mean the same thing. If you >> limit >> your calculation into the real area and omit the hyperreal area, > You can't do that. That's the whole point. LetÇs demontrate it shortly now so simply as possible (like Donald Knuth 1) It is assumed that the digits [0,1...9] in 10-base system are constructed. I leave it for the home work. 2) Then we apply AC so that we have infinite many placeholders (or hooks) with the first one and the last one (the start and the end), just like in [0,1]. (by the way this stops the discussion about the last digit in the infinite decimal string. :-)) For us it's enough that is just possible. 3) Pick-up with the aid of AC some digits in every placeholder. What number do you have? Actually you have just digits, but not a number - yet. Why? Because you have not defined the point of reference! You have only digits in every place holder, i.e. in every hook. 4) Adjust the digits, you have picked up by hooks, into a linear string. It's infinite, but it has the start and the end. Additionally we define (in this particular 10 base case) what is the relation of the adjacent placeholders. What number do you have now? No number, because you just have the linear infinite string without the point of reference. The point of reference defines the infinite string area, but we do not have it yet, thus as a consequence - no number area. 5) Apply the the point of reference. How? You can insert the point of reference anywhere in the string, but let's concentrate our attention into two special case because of simplicity: to the left hand side (lhs) of the infinite string or to the right hand side (rhs). What kind of point of reference? Any defined kind, but let's apply because of the simplicity the standard point of reference that is called the decimal dot (or mark). Let's insert the standard point of reference into lhs of the string. What number do you have? It is familiar decimal number depending on the values of digits in the string. Is it rational, irrational or transcendental depending on the digits. If there are only zeros on rhs starting from some placeholders, then the decimal part is called finite. Let's insert the standard point of reference into rhs of the string. What number do you have? It looks like now an integer, but isn't it integer?. It's infinite long with start and end. It's somehow finite, but anyway infinite. Integers cannot be infinite? Yes, they can. Actually every classic finite integer (N) can be written as infinite, if there are only zeros on lhs starting from some placeholders. In other case we call them infinite integers (N_inf). As an observation we recognize immediately that there is one-to-one bijection between N_inf and decimal part of R. The only difference is that N_inf has the standard point of reference on rhs and R (decimal part) has the standard point of reference on lhs. The standard point of reference, i.e. the decimal dot separates to two infinite long strings. Both parts are constructed by AC. The only difference is the point of reference on lhs or on rhs. As finite integers (N) are the subset of infinite integers (N_inf). AC, Zorn's lemma and well-ordering are equivalent. Integers and infinite integers are well-ordered. By moving the point of reference from rhs (in N_inf) to lhs (in R, decimal part) it's trivial to recognize that the decimal part is also well-ordered. Considering the limit, it's easy to observe that if the point of reference is on lhs of the string of infinite 9's, i.e. 0.999..., the limit equals to 1. But if the point of the reference is on rhs of the same string, then there is no limit, because we cannot calculate it for N_inf like ...999. We can add 1 to ...999, but then - as all the placeholders were occupied with 9's in the infinite long string - the successor equals to omega 1. Thus the limit, i.e. the upper boundary value, describes the successor, not the sum of the string itself as it should count. Successor equals never with the precessor and therefore omega 1> ...999 and also 1>0.999... The reason is AC and well-ordering. Thus for example a general number can be described omega-area (separator) infinite integer area (separator, usually dot), decimal area (separator) non-standard area. Each area is infinite and well-ordered constructed with the aid of AC. We don't have to use the standard approach one first defines Z, then defines Q, then defines R, each in different ways. Dedekind cut is not necessary but useful, which is another thing. All we need is AC and the point of reference. Tapio > -- > Dave Seaman > Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. > === Subject: Re: .99999... still=/= 1 > Yes, if you count the limit over reals plus over hyperreals. Sorry - I > cannot find better word than over. Find a better word as your mother > language is English. > I try to specify: > As you count the limit in reals, then N --> oo, where every N is finite > integer. As you count the limit over hyperreals, then N_inf -->oo_inf, > where > oo_inf has higher cardinality as the cardinality of N_inf >N. >> No, this is not about cardinality at all. The infinitely large integers >> of NSA are not the same as transfinite cardinals. > It's necessary to observe: As the cardinality of reals R > cardinality of > integers N, so the cardinality of N_inf > N. As you count the limit > N --->oo instead of N-inf, then you certainly omit something - namely > hyperreal part. All very confused. Assuming you mean N_inf = *N, the set of hyperintegers, then it's true that *N as an external set has the cardinality of the reals, but the internal set *N has a *cardinality equal to *aleph_0. This certainly doesn't mean that the hypernaturals are the same as transfinite cardinals. For one thing, there is a smallest transfinite cardinal (aleph_0), but there is no such thing as a smallest infinite hypernatural. If n is an infinitely large integer, then so is n-1. And none of this has anything to do with limits. You seem to think that infinity has only one meaning in all of mathematics. Wrong. > N (finite integers) does not cover hyperreal area, i.e. the numbers > smaller > than reals, because N hardly covers real area as discussed under the > thread > Are reals well-ordered. > As we count the real limit in NSA, then the hyperreals exist, but they > are > not counted in limit as N -->oo, but not as N_inf -->oo. Hyperreals are > omitted and therefore 0.999...<1 in hyperreals, though the real part of > the > limit equals to 1. >> I take it you mean the standard part of the limit is 1. > Well, we can talk about standard part and non-standard part, if you prefer > this. I'm trying to guess what you might mean by the real part, since you have not defined your meaning. We are not talking about complex numbers here. >> That happens to >> be true, but for a trivial reason. The limit itself is exactly 1, and >> the standard part of 1 is simply 1. > Cases: > 1)Yes, the limit is exactly 1only if you count the limit including the non > standard part as N-inf --->oo. Then your reference set is N_inf. I don't know what you mean by the reference set is N_inf. The sum is over *N. > 2)Yes, the limit is exactly 1only if you count the limit including the > standard part as N --->oo. Then your reference set is N. Are you talking about standard analysis here, or nonstandard? The set N (consisting of the finite naturals) is not a set in the internal set theory of NSA. Its counterpart is *N, which includes the infinitely large naturals. If you are forming a sum in NSA, then you can sum over *N, but not over N. > 3) But..., if you count the limit including the standard part as N --->oo > and your reference set is N_inf, then you omit non-standard part. As a > consequence in this last case 0.999...<1 in N_inf. That's all very confused, but I think you are trying to say that the limit of the sequence { 1 - 1/10^n } in NSA has a nonzero nonstandard part. False. The limit is exactly 1 according to the definition I gave previously. Perhaps what is confusing you is the following fact from NSA: Theorem. Let { a_k } be a sequence and L a real number. Then the following statements are equivalent: (1) lim_{k->oo} a_k = L. (2) The difference | a_k - L | is an infinitesimal whenever k is infinitely large. Notice, however, that neither (1) nor (2) says anything about the limit differing from L by an infinitesimal. Statement (1) mentions the limit, but it's a statement of exact equality. Statement (2) mentions something that differs from L by an infinitesimal, but it makes no mention of the limit whatsoever. Either way you look at it, neither of these statements supports your conclusion for the case a_k = 1 - 1/10^k and L = 1. > NSA expands the concept of numbers to > the numbers that are smaller than any real, i.e epsilon environment. > This > is > equivalent with the concept of epsilon delta theorem. Read literally > what > epsilon delta theorem says. This was learnt us already in 70Çs in > university. Are you back in 50's? >> Which part of my statement do you not accept? Do you disagree with the >> definition I gave? > Explained above. >> Try again. You didn't mention any part of the definition, let alone say >> which part you disagreed with. >> Definition. Let { a_k } be a sequence and let L be a real number. >> We say lim_{k->oo} a_k = L if, for every epsilon > 0, there exists >> N > 0 such that | a_k - L | < epsilon for every k > N. >> Remark 1. Exactly the same definition applies to standard analysis and >> to nonstandard analysis, with the proviso that in NSA the epsilon > 0 >> is allowed to be an infinitesimal and the N > 0 is allowed to be >> infinitely large. > Yes, agreed. >> Remark 2. The definition does not say what it means for the limit to be >> close to L. The definition only says what it means for the limit to be >> equal to L. Either the definition is satisfied, or it isn't. > OK. >> Now, the questions: (1) Do you agree with the definition? (2) Do you >> agree that according to this definition the limit is exactly 1, even in >> NSA? If you don't agree, explain why not. > Yes, I do agree as explained above in the cases 1 and 2. You did not > consider at all the case 3 above. How does your definitions should be > applied on the case 3? There is no case 3 above. I don't know what you are talking about. > You should also note that the limit is the upper boundary value. As you will > see below, it's not the question about the limit but about AC and the point > of reference as we construct the numbers integers, infinite integers, reals, > hyperreals etc. >> For example, try to give a >> particular value of epsilon > 0 such that the definition is not >> satisfied. Hint: choosing an infinitesimal epsilon is allowed, but it >> won't help your case. The definition still works. > What about case 3 above as you stop epsilons in standard part omitting > non-standard part. There is no case 3 above. I have explained that the sum over N is not a sum in NSA, since N is not an internal set in NSA. That's why we sum over *N instead. > I would like to ask the same from You. :-). Is it the point > of reference that is strange concept for You? >> Point of reference is an undefined concept and does not appear in the >> definition of limit, quoted above. I won't comment on whether it is >> strange, since things have to be defined first before they can possibly >> qualify as strange. > The point of reference is counting point reference, which also separates > infinities. That is not a definition. For an example of what I mean by a definition, look at my definition of what it means to say that lim_{k->oo} a_k = L. Mathematical definitions leave no room for fuzzy language or vague concepts. Try again. > The standard point of reference is the normal decimal dot that separates the > integer part and the decimal part, which can be infinite long string. > Without the point of reference you do not know which part is integer part > and which one is the decimal part. What ever you calculate you always refer > your calculations to some point of reference. The common definitions of the real numbers (via Dedekind cuts or Cauchy sequences) do not mention decimal points at all and do not depend on any such concepts as integer part and decimal part. For any real number x, the integer part of x may be defined as floor(x) = max { n in N : n <= x }. I had no need for any vague concepts such as point of reference in writing that definition. > In fact, there seems to be a slight conceptual difference in our > argumentation. The difference is analocigally the same as we talk about > finite decimal numbers. You accept that finite 0.99999 <1, as there are > no > infinite 9's. As hyperreals are omitted in the real limit counting (not > counted over hyperreals), then in reals the limit equals to 1, i.e. > 0.999...=1. But as hypereals exist, but omitted in the limit calculation, > then 0,999...<1 in NSA. Simple as possible. Reals are finite from the > hypereal point of view. Therefore 0.999...<1 - in hyperreals as only real > area is considered. You need to be precise about what you mean by 0.999... in NSA. I have been taking it to mean the sum of 9/10^n for all n in *N, n > 0. You evidently mean something different. In particular, it can't possibly mean the sum over all finite values of n > 0, because that is not a set in NSA. >> But in NSA there is no such thing as a .999... that extends only through >> finite digit positions. If the string is infinitely long, then it >> necessarily extends to infinite digit positions in NSA. That may sound >> vaguely similar to an argument commonly made by cranks, but the >> difference is that the cranks are not talking about NSA. > used the case 3 as an example to point out how the people do not think as > their argumentation contains hidden asumptions. So - if there exist > non-standard part, which is purposely omitted, then you do not calculate the > total or over-all limit. In this case you have to observe that there are > numbers smaller than any real number and as a consequence 0.999... cannot be > equal to 1, though the limit equals to 1. > What is this paradox, which has been the reason to this thread, and how do > we fix it? If you do not sum 9/10^n for all n > 0, then you are not computing 0.999.... The indicated sum is exactly 1. > The solution is to point out that the limit is a mathematical upper boundary > value, but does not tell the real value of the string but the next, i.e. > successor, value of the string, because the limit calculation is based on > the epsilon delta theorem. The string I am talking about has no successor values. It's already defined for all positions n, including the ones that are infinitely large. >> The definition of a limit in NSA may be stated in various ways, but all >> of them are equivalent to the usual epsilon-delta definition, with the >> proviso that epsilons and deltas are allowed to be infinitesimal, and N >> is allowed to be infinitely large. > Yes, that's what I mean - too, assuming we mean the same thing. If you > limit > your calculation into the real area and omit the hyperreal area, >> You can't do that. That's the whole point. > LetÇs demontrate it shortly now so simply as possible (like Donald Knuth > 1) It is assumed that the digits [0,1...9] in 10-base system are > constructed. I leave it for the home work. > 2) Then we apply AC so that we have infinite many placeholders (or hooks) > with the first one and the last one (the start and the end), just like in > [0,1]. We don't need AC here. We are defining d_0 = 0 and d_k = 9 for all k > 0 in *N. > (by the way this stops the discussion about the last digit in the infinite > decimal string. :-)) For us it's enough that is just possible. I was not aware that there was any such discussion. > 3) Pick-up with the aid of AC some digits in every placeholder. Again, AC is irrelevant. We already have our hyperinfinite string. > What number do you have? Actually you have just digits, but not a number - > yet. Why? Because decimal digit strings are not numbers. They merely represent numbers. > Because you have not defined the point of reference! You have only digits in > every place holder, i.e. in every hook. Nonsense. If { d_k } is a decimal digit string, then the number represented by the string is sum_{k in *N} d_k * 10^(-k). No point of reference is needed. [ snip nonsense about point of reference ] -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: .99999... still=/= 1 >> Yes, if you count the limit over reals plus over hyperreals. Sorry - I >> cannot find better word than over. Find a better word as your mother >> language is English. >> I try to specify: >> As you count the limit in reals, then N --> oo, where every N is finite >> integer. As you count the limit over hyperreals, then N_inf -->oo_inf, >> where >> oo_inf has higher cardinality as the cardinality of N_inf >N. > No, this is not about cardinality at all. The infinitely large integers > of NSA are not the same as transfinite cardinals. We talk about ordinals. That above was just a hint to compare the cardinality of N_inf and N. >> It's necessary to observe: As the cardinality of reals R > cardinality of >> integers N, so the cardinality of N_inf > N. As you count the limit >> N --->oo instead of N-inf, then you certainly omit something - namely >> hyperreal part. > All very confused. Assuming you mean N_inf = *N, the set of > hyperintegers, then it's true that *N as an external set has the > cardinality of the reals, but the internal set *N has a *cardinality > equal to *aleph_0. This is of-course OK! > This certainly doesn't mean that the hypernaturals > are the same as transfinite cardinals. For one thing, there is a > smallest transfinite cardinal (aleph_0), Ok! > but there is no such thing as a > smallest infinite hypernatural. If n is an infinitely large integer, > then so is n-1. Omega was defined in this discussion earlier as follows: The number that is greater than any infinite integer. That is the smallest transinfinite number. Omega is the successor of ...999, i.e. n+1. The precessor (n-1) of ...999 is ...998. > And none of this has anything to do with limits. You seem to think that > infinity has only one meaning in all of mathematics. Wrong. Your opinion. :-) >> N (finite integers) does not cover hyperreal area, i.e. the numbers >> smaller >> than reals, because N hardly covers real area as discussed under the >> thread >> Are reals well-ordered. >> As we count the real limit in NSA, then the hyperreals exist, but they >> are >> not counted in limit as N -->oo, but not as N_inf -->oo. Hyperreals are >> omitted and therefore 0.999...<1 in hyperreals, though the real part of >> the >> limit equals to 1. > I take it you mean the standard part of the limit is 1. >> Well, we can talk about standard part and non-standard part, if you >> prefer >> this. > I'm trying to guess what you might mean by the real part, since you > have not defined your meaning. the real part was the decimal part - as you certainly knew. > We are not talking about complex numbers > here. Exactly! > That happens to > be true, but for a trivial reason. The limit itself is exactly 1, and > the standard part of 1 is simply 1. >> Cases: >> 1)Yes, the limit is exactly 1only if you count the limit including the >> non >> standard part as N-inf --->oo. Then your reference set is N_inf. > I don't know what you mean by the reference set is N_inf. The sum is > over *N. That is exactly what I meant above if N_inf=*N. >> 2)Yes, the limit is exactly 1only if you count the limit including the >> standard part as N --->oo. Then your reference set is N. > Are you talking about standard analysis here, or nonstandard? Standard analysis as You correctly observed. > The set N > (consisting of the finite naturals) is not a set in the internal set > theory of NSA. Coorect, but it should be as *N is the extension of the set N. > Its counterpart is *N, which includes the infinitely > large naturals. If you are forming a sum in NSA, then you can sum over > *N, but not over N. We can sum over subset too. >> 3) But..., if you count the limit including the standard part as >> N --->oo >> and your reference set is N_inf, then you omit non-standard part. As a >> consequence in this last case 0.999...<1 in N_inf. > That's all very confused, but I think you are trying to say that the > limit of the sequence { 1 - 1/10^n } in NSA has a nonzero nonstandard > part. False. The limit is exactly 1 according to the definition I gave > previously. N is the subset of *N. You can count the limit over N or alternatively over *N as You can count N --> 0 to 1000 as a subset of N or alternatively N --->0 to oo. > Perhaps what is confusing you is the following fact from NSA: > Theorem. Let { a_k } be a sequence and L a real number. Then the > following statements are equivalent: > (1) lim_{k->oo} a_k = L. > (2) The difference | a_k - L | is an infinitesimal > whenever k is infinitely large. > Notice, however, that neither (1) nor (2) says anything about the limit > differing from L by an infinitesimal. Yes, as long as you count over *N > Statement (1) mentions the limit, > but it's a statement of exact equality. Statement (2) mentions something > that differs from L by an infinitesimal, but it makes no mention of the > limit whatsoever. Except Yount count over N instead of *N. > Either way you look at it, neither of these statements > supports your conclusion for the case a_k = 1 - 1/10^k and L = 1. >> NSA expands the concept of numbers to >> the numbers that are smaller than any real, i.e epsilon environment. >> This >> is >> equivalent with the concept of epsilon delta theorem. Read literally >> what >> epsilon delta theorem says. This was learnt us already in 70Çs in >> university. Are you back in 50's? > Which part of my statement do you not accept? Do you disagree with > the > definition I gave? >> Explained above. > Try again. You didn't mention any part of the definition, let alone say > which part you disagreed with. > Definition. Let { a_k } be a sequence and let L be a real number. > We say lim_{k->oo} a_k = L if, for every epsilon > 0, there exists > N > 0 such that | a_k - L | < epsilon for every k > N. > Remark 1. Exactly the same definition applies to standard analysis and > to nonstandard analysis, with the proviso that in NSA the epsilon > 0 > is allowed to be an infinitesimal and the N > 0 is allowed to be > infinitely large. >> Yes, agreed. > Remark 2. The definition does not say what it means for the limit to be > close to L. The definition only says what it means for the limit to be > equal to L. Either the definition is satisfied, or it isn't. >> OK. > Now, the questions: (1) Do you agree with the definition? (2) Do you > agree that according to this definition the limit is exactly 1, even in > NSA? If you don't agree, explain why not. >> Yes, I do agree as explained above in the cases 1 and 2. You did not >> consider at all the case 3 above. How does your definitions should be >> applied on the case 3? > There is no case 3 above. I don't know what you are talking about. >> You should also note that the limit is the upper boundary value. As you >> will >> see below, it's not the question about the limit but about AC and the >> point >> of reference as we construct the numbers integers, infinite integers, >> reals, >> hyperreals etc. > For example, try to give a > particular value of epsilon > 0 such that the definition is not > satisfied. Hint: choosing an infinitesimal epsilon is allowed, but it > won't help your case. The definition still works. >> What about case 3 above as you stop epsilons in standard part omitting >> non-standard part. > There is no case 3 above. I have explained that the sum over N is not > a sum in NSA, since N is not an internal set in NSA. That's why we sum > over *N instead. Because You don't see N as subset of *N. :-) >> I would like to ask the same from You. :-). Is it the point >> of reference that is strange concept for You? > Point of reference is an undefined concept and does not appear in the > definition of limit, quoted above. I won't comment on whether it is > strange, since things have to be defined first before they can > possibly > qualify as strange. >> The point of reference is counting point reference, which also separates >> infinities. > That is not a definition. For an example of what I mean by a definition, > look at my definition of what it means to say that lim_{k->oo} a_k = L. > Mathematical definitions leave no room for fuzzy language or vague > concepts. Try again. >> The standard point of reference is the normal decimal dot that separates >> the >> integer part and the decimal part, which can be infinite long string. >> Without the point of reference you do not know which part is integer part >> and which one is the decimal part. What ever you calculate you always >> refer >> your calculations to some point of reference. > The common definitions of the real numbers (via Dedekind cuts or Cauchy > sequences) do not mention decimal points at all and do not depend on > any such concepts as integer part and decimal part. For any real > number x, the integer part of x may be defined as floor(x) = max { n in > N : n <= x }. I had no need for any vague concepts such as point of > reference in writing that definition. Certainly not. It's alternative way to construct numbers from AC. The point of reference is in-constructed assumption in Dedekinds cut or Cauchy sequences, because they assumed so or they did not recognize the point of reference. >> In fact, there seems to be a slight conceptual difference in our >> argumentation. The difference is analocigally the same as we talk about >> finite decimal numbers. You accept that finite 0.99999 <1, as there are >> no >> infinite 9's. As hyperreals are omitted in the real limit counting (not >> counted over hyperreals), then in reals the limit equals to 1, i.e. >> 0.999...=1. But as hypereals exist, but omitted in the limit >> calculation, >> then 0,999...<1 in NSA. Simple as possible. Reals are finite from the >> hypereal point of view. Therefore 0.999...<1 - in hyperreals as only >> real >> area is considered. > You need to be precise about what you mean by 0.999... in NSA. I have > been taking it to mean the sum of 9/10^n for all n in *N, n > 0. You > evidently mean something different. In particular, it can't possibly > mean the sum over all finite values of n > 0, because that is not a set > in NSA. N is the subset of *N. I and You can count over subset of *N just like in the case of any subset of N. > But in NSA there is no such thing as a .999... that extends only through > finite digit positions. If the string is infinitely long, then it > necessarily extends to infinite digit positions in NSA. That may sound > vaguely similar to an argument commonly made by cranks, but the > difference is that the cranks are not talking about NSA. >> evidence. I >> used the case 3 as an example to point out how the people do not think as >> their argumentation contains hidden asumptions. So - if there exist >> non-standard part, which is purposely omitted, then you do not calculate >> the >> total or over-all limit. In this case you have to observe that there are >> numbers smaller than any real number and as a consequence 0.999... cannot >> be >> equal to 1, though the limit equals to 1. >> What is this paradox, which has been the reason to this thread, and how >> do >> we fix it? > If you do not sum 9/10^n for all n > 0, then you are not computing > 0.999.... The indicated sum is exactly 1. Shortly: That's what we have already agreed. I never denied that. What can be done: over N, over *N or over N as a subset of *N, and only in the last case the limit over N is smaller than 1 as You should count over *N in hyperreals. Therefore in principle 0.999... <1 in the last mentioned case. >> The solution is to point out that the limit is a mathematical upper >> boundary >> value, but does not tell the real value of the string but the next, i.e. >> successor, value of the string, because the limit calculation is based on >> the epsilon delta theorem. > The string I am talking about has no successor values. It's already > defined for all positions n, including the ones that are infinitely > large. > The definition of a limit in NSA may be stated in various ways, but > all > of them are equivalent to the usual epsilon-delta definition, with the > proviso that epsilons and deltas are allowed to be infinitesimal, and > N > is allowed to be infinitely large. >> Yes, that's what I mean - too, assuming we mean the same thing. If you >> limit >> your calculation into the real area and omit the hyperreal area, > You can't do that. That's the whole point. >> LetÇs demontrate it shortly now so simply as possible (like Donald Knuth >> 1) It is assumed that the digits [0,1...9] in 10-base system are >> constructed. I leave it for the home work. >> 2) Then we apply AC so that we have infinite many placeholders (or hooks) >> with the first one and the last one (the start and the end), just like in >> [0,1]. > We don't need AC here. We are defining d_0 = 0 and d_k = 9 for all k > 0 > in *N. >> (by the way this stops the discussion about the last digit in the >> infinite >> decimal string. :-)) For us it's enough that is just possible. > I was not aware that there was any such discussion. It was earlier and it arises up from time to time, but not in our mutual discussion. >> 3) Pick-up with the aid of AC some digits in every placeholder. > Again, AC is irrelevant. We already have our hyperinfinite string. >> What number do you have? Actually you have just digits, but not a >> number - >> yet. Why? > Because decimal digit strings are not numbers. They merely represent > numbers. >> Because you have not defined the point of reference! You have only digits >> in >> every place holder, i.e. in every hook. > Nonsense. If { d_k } is a decimal digit string, then the number > represented by the string is sum_{k in *N} d_k * 10^(-k). No point of > reference is needed. You cut (=snip) out the climax of my explanation. What does ^(-k) refer? Please answer to that simple question! Why minus - what does it refer? > [ snip nonsense about point of reference ] Your opinion, because You could not consider the successor of ...999. What is the successor of ...999 as all the placeholders are infinitely occupied with the maximal digit 9? Tapio > -- > Dave Seaman > Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. > === Subject: Re: .99999... still=/= 1 > It's necessary to observe: As the cardinality of reals R > cardinality of > integers N, so the cardinality of N_inf > N. As you count the limit > N --->oo instead of N-inf, then you certainly omit something - namely > hyperreal part. >> All very confused. Assuming you mean N_inf = *N, the set of >> hyperintegers, then it's true that *N as an external set has the >> cardinality of the reals, but the internal set *N has a *cardinality >> equal to *aleph_0. > This is of-course OK! >> This certainly doesn't mean that the hypernaturals >> are the same as transfinite cardinals. For one thing, there is a >> smallest transfinite cardinal (aleph_0), > Ok! >> but there is no such thing as a >> smallest infinite hypernatural. If n is an infinitely large integer, >> then so is n-1. > Omega was defined in this discussion earlier as follows: > The number that is greater than any infinite integer. That is the smallest > transinfinite number. Omega is the successor of ...999, i.e. n+1. The > precessor (n-1) of ...999 is ...998. It's possible to define the hyperordinals and to talk about *omega in NSA, but this *omega has no relation to anything you said in that paragraph. For one thing, *omega is not a member of *N. (In fact, *omega is identical to the set *N itself). For another, there is no connection between the hyperordinals and strings (even hyperstrings!) of decimal digits. I don't accept your definition. Are you under the impression that ...999 is a hyperinteger? It isn't, unless you explain which equivalence class of sequences of integers you are talking about. I can think of ways to make such a correspondence, but you haven't said what you mean. >> And none of this has anything to do with limits. You seem to think that >> infinity has only one meaning in all of mathematics. Wrong. > Your opinion. :-) Infinity has many meanings. It can refer to a compactification of the real line or of the complex plane, or to transfinite ordinals or cardinals, or to hyperreals, or to surreals. All of those are different. And this is not intended to be a complete list. > N (finite integers) does not cover hyperreal area, i.e. the numbers > smaller > than reals, because N hardly covers real area as discussed under the > thread > Are reals well-ordered. > As we count the real limit in NSA, then the hyperreals exist, but they > are > not counted in limit as N -->oo, but not as N_inf -->oo. Hyperreals are > omitted and therefore 0.999...<1 in hyperreals, though the real part of > the > limit equals to 1. >> I take it you mean the standard part of the limit is 1. > Well, we can talk about standard part and non-standard part, if you > prefer > this. >> I'm trying to guess what you might mean by the real part, since you >> have not defined your meaning. > the real part was the decimal part - as you certainly knew. No, I didn't know what you meant, and I still am not sure. What is the real part of sqrt(2), for example. It sounds like you trying to say the real part is sqrt(2) - 1, or approximately 0.41421. That was not one of my first two guesses. I would call that the fractional part. >> We are not talking about complex numbers >> here. > Exactly! That was my first guess, but my second guess was standard part. That is evidently not what you meant, either. In a similar fashion, you keep assuming that I must know what you mean by ...999. However, I assure you that I don't. > 2)Yes, the limit is exactly 1only if you count the limit including the > standard part as N --->oo. Then your reference set is N. >> Are you talking about standard analysis here, or nonstandard? > Standard analysis as You correctly observed. Then you are not discussing the value of 0.999... in NSA at all, as I previously thought. Looks like yet another example of miscommunication. Let's summarize: (1) We have 0.999... = 1 in standard analysis, because the sum is over N. (2) We have 0.999... = 1 in NSA, because the sum is over *N. (3) It is not possible to sum over *N in standard analysis, because *N is not a set in standard analysis. (4) It is not possible to sum over N in NSA, because N is not a set in NSA. Do you agree? >> The set N >> (consisting of the finite naturals) is not a set in the internal set >> theory of NSA. > Coorect, but it should be as *N is the extension of the set N. No, it should not be. There is an important principle involved, known as the transfer principle. This says that every theorem of standard analysis is also a theorem of NSA, provided we make the appropriate substitutions (such as substituting *N for each occurence of N). The transfer principle is extremely important. Without it, NSA loses most of its value as a tool of analysis. It turns out that if all sets in standard analysis are allowed to be internal sets in NSA, then we lose the transfer principle. That's why things are defined the way they are. >> Its counterpart is *N, which includes the infinitely >> large naturals. If you are forming a sum in NSA, then you can sum over >> *N, but not over N. > We can sum over subset too. Yes, but we can't sum over things that fail to be sets at all. That's the point. >> There is no case 3 above. I have explained that the sum over N is not >> a sum in NSA, since N is not an internal set in NSA. That's why we sum >> over *N instead. > Because You don't see N as subset of *N. :-) No, that is not the reason. The reason is that Abraham Robinson, the founder of NSA, did not see N as an internal set. And he had very good reasons. >> Nonsense. If { d_k } is a decimal digit string, then the number >> represented by the string is sum_{k in *N} d_k * 10^(-k). No point of >> reference is needed. > You cut (=snip) out the climax of my explanation. What does ^(-k) refer? > Please answer to that simple question! Why minus - what does it refer? The symbol ^ means that what follows is an exponent. Thus, 10^(-k) means ten to the power of (-k), which is the same as 1/10^k. We can write 0.999... as sum_{k=1}^oo 9*10^(-k) = sum_{k=1}^oo 9/10^k. >> [ snip nonsense about point of reference ] > Your opinion, because You could not consider the successor of ...999. What > is the successor of ...999 as all the placeholders are infinitely occupied > with the maximal digit 9? You haven't said what ...999 is. How am I supposed to answer questions about it if you don't define it? -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: .99999... still=/= 1 > LEARN MATH. His math is sound. Yours is non-existent. Bob Kolker === Subject: Re: .99999... still=/= 1 In sci.math, robert j. kolker : >> LEARN MATH. > His math is sound. Yours is non-existent. the non-standard analysis arena, an area which is not all that familiar to me except as a crudely expressed d-math, which I can occasionally use (how correctly, I've no idea!) to expound on various concepts. It's clear that his definition of '=' is a bit different from the rest of ours, and limit theory has some problems. For example, take .999..., the much-balleyhooed expression. Express it as the series: S_1 = .9 S_2 = .9 + .09 S_3 = .9 + .09 + .009 ... S_n = sum(i=1,n) (9 * 10^(-i)) = 1 - 10^(-n) (easily proved by induction, if one cares to bother). More or less standard stuff, up to this point. Under normal circumstances one can play the N-epsilon [*] game and get the following proof (or outline thereof). Oh, you have an epsilon > 0 for me? Fine, I'll take N = ceil(log10(1/epsilon)). I can now prove that, for any n > N, S_n > 1 - epsilon, but less than 1. Since S_n = 1 - 10^(-n), if n > N, then 10^(-n) < 10^(-ceil(log10(1/epsilon))) <= 10^(-log10(1/epsilon)) <= epsilon. and then the jump: Hence, lim(n->+oo) S_n = 1. QED. Now enter hypperreals. Set epsilon = d, where d is a number greater than 0 but less than all 1/n, n > 0, n in J. This proof goes out the window, as 1/d is a meaningless expression (though one could generate another class of numbers, maybe called quasi-infinities, which would be greater than any integer N but less than aleph_0, or something equally strange; the main intent is to be dual to the hyperreals). One could claim 'd' is a ridiculous concept (and it is to some extent as lim(n->+oo) (1/n) = 0 anyway, in standard analysis), but it does lead to some interesting questions as to how to get around this obstacle without simply claiming well, it's obviously nonsense or well, we've always done it that way. It's a bit like Lobachevskian geometry in that respect -- not a contradiction, but a new realm of number. Not a horribly useful one, to be sure -- 21 or so digits of pi or e are enough to define the Earth's orbit (1.5 * 10^11 m) to the width of an atom (2 * 10^-10 m); the rest is gravy -- but interesting to some, and useful for testing microprocessors. This is not to say S. Enterprize's arguments are any good; they're extremely sloppy, in fact. > Bob Kolker [*] there are four variants of this game (each with three or nine subvariants), depending on what one is proving. delta-epsilon: lim(x->a) f(x) = b lim(x->a+) f(x) = b lim(x->a-) f(x) = b N-epsilon: lim(x->oo) f(x) = b lim(x->+oo) f(x) = b lim(x->-oo) f(x) = b delta-M: lim(x->a) f(x) = oo (or +oo or -oo) lim(x->a+) f(x) = oo (or +oo or -oo) lim(x->a-) f(x) = oo (or +oo or -oo) N-M: lim(x->oo) f(x) = oo (or +oo or -oo) lim(x->+oo) f(x) = oo (or +oo or -oo) lim(x->-oo) f(x) = oo (or +oo or -oo) -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: .99999... still=/= 1 > Under normal circumstances one can play the N-epsilon [*] game > and get the following proof (or outline thereof). > Oh, you have an epsilon > 0 for me? Fine, I'll take > N = ceil(log10(1/epsilon)). I can now prove that, > for any n > N, S_n > 1 - epsilon, but less than 1. > Since S_n = 1 - 10^(-n), if n > N, then > 10^(-n) < 10^(-ceil(log10(1/epsilon))) > <= 10^(-log10(1/epsilon)) <= epsilon. > and then the jump: > Hence, lim(n->+oo) S_n = 1. QED. > Now enter hypperreals. Set epsilon = d, where d is > a number greater than 0 but less than all 1/n, n > 0, n in J. > This proof goes out the window, as 1/d is a meaningless > expression (though one could generate another class of > numbers, maybe called quasi-infinities, which would be > greater than any integer N but less than aleph_0, or something > equally strange; the main intent is to be dual to the hyperreals). Actually, the hyperreals *R form a field. Yes, 1/d is infinitely large if d is an infinitesimal. Also, it should be noted that you can use the standard definition for the limit of a sequence: Definition. Let { a_k } be a sequence and L in R. Then we say that lim_{k->oo} a_k = L if, for every epsilon > 0, there exists N > 0 such that | a_k - L | < epsilon for every k > N. The only difference is in the interpretation of that the terms mean. In nonstandard analysis (NSA), when we say for every epsilon > 0, we mean for every positive epsilon in *R, which means epsilon is allowed to be an infinitesimal, for example. And when we say there exists N > 0 we mean that N is allowed to be infinitely large. It's a theorem of NSA that the following two statements are equivalent: (1) lim_{k->oo} a_k = L (as defined above), and (2) The difference | a_k - L | is an infinitesimal whenever k is infinitely large. Oh, and by the way, it's also true in NSA that sum_{n in *N, n >0} 9/10^n = 1. It's a geometric series. Thus we can say .999... = 1, even in the hyperreals. And no, it's not possible to confine the sum to just the finite digit positions, because the summation can only be carried out over a set, and the finite naturals are not a set according to the internal set theory of NSA. When we say something is infinitely large in NSA, that's only the external view. Within the model, all the members of *R are finite. -- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. === Subject: Re: .99999... still=/= 1 In sci.math, Dave Seaman : >> Under normal circumstances one can play the N-epsilon [*] game >> and get the following proof (or outline thereof). >> Oh, you have an epsilon > 0 for me? Fine, I'll take >> N = ceil(log10(1/epsilon)). I can now prove that, >> for any n > N, S_n > 1 - epsilon, but less than 1. >> Since S_n = 1 - 10^(-n), if n > N, then >> 10^(-n) < 10^(-ceil(log10(1/epsilon))) >> <= 10^(-log10(1/epsilon)) <= epsilon. >> and then the jump: >> Hence, lim(n->+oo) S_n = 1. QED. >> Now enter hypperreals. Set epsilon = d, where d is >> a number greater than 0 but less than all 1/n, n > 0, n in J. >> This proof goes out the window, as 1/d is a meaningless >> expression (though one could generate another class of >> numbers, maybe called quasi-infinities, which would be >> greater than any integer N but less than aleph_0, or something >> equally strange; the main intent is to be dual to the hyperreals). > Actually, the hyperreals *R form a field. Yes, 1/d is infinitely large > if d is an infinitesimal. Also, it should be noted that you can use the > standard definition for the limit of a sequence: > Definition. Let { a_k } be a sequence and L in R. > Then we say that lim_{k->oo} a_k = L if, for every epsilon > 0, there > exists N > 0 such that | a_k - L | < epsilon for every k > N. > The only difference is in the interpretation of that the terms mean. In > nonstandard analysis (NSA), when we say for every epsilon > 0, we mean > for every positive epsilon in *R, which means epsilon is allowed to be > an infinitesimal, for example. And when we say there exists N > 0 we > mean that N is allowed to be infinitely large. > It's a theorem of NSA that the following two statements are equivalent: > (1) lim_{k->oo} a_k = L (as defined above), and > (2) The difference | a_k - L | is an infinitesimal whenever k is > infinitely large. > Oh, and by the way, it's also true in NSA that sum_{n in *N, n >0} 9/10^n >= 1. It's a geometric series. Thus we can say .999... = 1, even in the > hyperreals. > And no, it's not possible to confine the sum to just the finite digit > positions, because the summation can only be carried out over a set, and > the finite naturals are not a set according to the internal set theory of > NSA. When we say something is infinitely large in NSA, that's only the > external view. Within the model, all the members of *R are finite. Interesting. So even in the hyperreals, S. Enterprize is simply wrong. :-) -- #191, ewill3@earthlink.net It's still legal to go .sigless. === Subject: Re: .99999... still=/= 1 > Interesting. So even in the hyperreals, S. Enterprize is simply wrong. :-) I told you. Bob Kolker === Subject: Re: .99999... still=/= 1 > This is not to say S. Enterprize's arguments are any good; > they're extremely sloppy, in fact. His arguments are not even wrong. They are non-existent. Stringing words together doth not an argument make. Bob Kolker === Subject: Re: .99999... still=/= 1 > They don't have a disagreement about anything mathematical. It should > be clear by now that SE is not arguing in good faith, but is just > trolling. Bob Kolker === Subject: Re: .99999... still=/= 1 >> They don't have a disagreement about anything mathematical. It should >> be clear by now that SE is not arguing in good faith, but is just >> trolling. >Bob Kolker If you notice they still haven't proven, .999...(an unreal number) = 1 ( a real number). Smart's Alt. Physics News Group http://pub39.bravenet.com/forum/show.php?usernum=3320272813&cpv=1 S. Enterprize (Science Journal) http://smart1234.s-enterprize.com/ === Subject: Re: .99999... still=/= 1 > If you notice they still haven't proven, > .999...(an unreal number) = 1 ( a real number). Meaningless. This is not a mathematical experession. Bob Kolker === Subject: Re: Proof of Sum_{i=1...n} i^k is a polynomial expression over n by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBHFa5n05215; >Say S_k(n) = Sum_{i=1...n} i^k = 1^k + 2^k + .... + n^k. >I want to prove that S_k(n) is a polynomial expression over n, that >is, that there exists a polynomial p_k(x) in R[x] such that S_k(n) = >p_k(n) for all n (and p_k only depends of k). >I prefer proofs by induction and elementals. I know that it could be >proved using the Bernoulli polynomials, but I want a proof without >that (I want more elemental proof). Could it be?. I tried but I did >not get it. >Xan. N.94 H.89o Xan, 'non-polynomial differential ' could help you. In fact sum(n=1 to n=x n^k ) verifies the following simple equation: p(x)-p(x-1)=x^k or (I-exp(-D))Áp(x)=x^k or p(x)=I/(I-exp(-D))Áx^k . For program reason we must enter:instead of x^k x^(k+1)/(k+1) The given formula corresponds to a polyniomial. === Subject: Re: .99999... still=/= 1 by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBHFa5005223; >Dedekind is credited with giving the first mathematical definition >of the real numbers. >> Is this so? >Yes and no. The set of real numbers was implicitly defined by >algebraic laws on the two basic operations + and *. A rigorous >definition (or reduction) of real numbers to the rationals was done in >the latter half of the 19-th century along with definitions of >continuous functions, differentiable functions and limits. The key to >the whole business is the rigorous definition of limit. >The reals are the topological closure of the rationals. The key to this >is a definition of convergence in which a limit is not explicitly >required. This is attributed to Cauchy. >Bob Kolker I believe you mean topological _completion_, rather than closuer. - MO === Subject: Re: .99999... still=/= 1 > I believe you mean topological _completion_, rather than closuer. Quite so. Dediking and Cauchy found a way of adding the limit points to the set of rationals. Bob Kolker === Subject: Parameter Estimamtion of a Exponential Dist. P(x)=1/A exp(-x/A) is known as an exponential distribution, where Mean=A and Variance=A*A. Using Maximum Likelihood Method, A'=(Sample Mean) can be found. However, I plan to use A'=sqrt(Sample Variance). Does any one have an idea about the robustness of the estimation ? === Subject: Re: Parameter Estimamtion of a Exponential Dist. > However, I plan to use A'=sqrt(Sample Variance). Does any one have > an idea about the robustness of the estimation ? Sample variance is a biased estimator of population variance: = A^2 (n-1) / n. I can prove that A' is a biased estimator of A, but haven't been able to derive a closed expression for /A in terms of n. Unfortunately, setting A' = sqrt(Sample Variance * n / (n-1)) does not remove the bias. - Tim === Subject: Re: weighted arithmetic and geometric means by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id iBHFxcn07339; >Hello >Suppose x_1,....x_n and w_1,...w_n are postive numbers and define >a = (Sum(i=1,n)(w_i*x_i))/(Sum(i=1,n)(w_i)) and >g = (Product(i=1,n)(x_i)^(w_i))^(1/Sum(i=1,n)(w_i))) >I want to prove that there holds an inequalty similar to that related >to the arithmetic and geometric means, that is, a>= g, with equality >if, and only if, x_1 = ....x_n. >Since I already now that the arithmetic/geometric means inequalty is >true, I tried to do as follows. >First, if all the w_i's are positive integers, then we see readily see >that a is the arithmetic mean of numbers x_1,...x_n if if we we take >each x_i w_i times. Since a similar conclusion is true of g, we apply >the a/g means inequality to conclude that, if all the w_i's are >integer then the propostion is true for a and g. >If all the w_i's are rational, then, representing each w_i as the >ratio between 2 positive integers and doing some elementary algebraic >transformations, we see a =a' and g =g', where a' and g' are weighted >means similar to a and g corresponding, now, to integer weights. >Therefore, we are sent back to the integer case, which shows the >proposition still holds if all the w_i's are positive rationals. >If the w_i's are real positive integers, then, for a fixed but >arbitray (x_1,...x_n) , the functions (w_1,...w_n) -> a(w_1,...w_n) >and (w_1,...w_n) -> g(w_1,...w_n,)defined on the subset of R^n >composed of their points with positive coordinates, are continuous. >Since a>=g in the subset of R^n composed of their points with >positive and rational coordinates and since this latter subset is >dense in the former, it follows that a(w_1,...w_n) >= g(w_1,...w_n) >for every positive w_1,...w_n. Since this holds for arbitray positive >x_1,...x_n, we have proved that, in fact, a>=g. But we are not done, >because these arguments do not imply that equality occurs if and only >if x_1...= x_n. >Following my reasoning, can anyone suggest how I can complete the >proof? Or, maybe, it's better to start at the very beginning, without >supposing the a/g means inequalty is known. >Amanda The classical reference for these sorts of inequalities is (amazingly entitled) _Inequalities_, by Hardy, Littlewood, & Polya. Chapters 1 and 2 are all(!) you need. Nick === Subject: Question about President's Social Security plan by the amount of $2.7 trillion in 75 years. Bush administration has a plan. The plan is to privatize some parts of social security. This will cost $2 trillion to set up. It is not guaranteed to fix things, but is only one part of an integrated plan. So my question is, wouldn't it make more sense to just GIVE that $2 trillion to social security, which is guaranteed to fix things by exactly $2 trillion, just leaving a small $0.7 trillion shortfall after 75 years? === Subject: Re: Question about President's Social Security plan >by the amount of $2.7 trillion in 75 years. >Bush administration has a plan. The plan is >to privatize some parts of social security. >This will cost $2 trillion to set up. >It is not guaranteed to fix things, but >is only one part of an integrated plan. >So my question is, wouldn't it make more >sense to just GIVE that $2 trillion to >social security, which is guaranteed >to fix things by exactly $2 trillion, just >leaving a small $0.7 trillion shortfall after >75 years? You mean other then this train wreck if the media ever made this information Public?! I mean name me just ONE Democrat that would still be left in office if the public found out that they had been lied to and that the Social Security funds were being stolen by the Democrats for the last 60+ years? Here are the REAL facts Social Security.89s vaunted trust fund doesn.89t exist; taxes paid into Social Security are merely being handed over as benefits to other people. On May 2nd, President Bush announced the formation of a presidential commission to deal with the Social Security crisis. The last such commission, in 1983, had as its chairman Alan Greenspan. It recommended enormous increases in taxes, and Congress dutifully complied.Government spokesmen immediately patted themselves on their backs for having saved Social security. We are likely in for more of the same. The new commission.89s name is the Commission to Strengthen Social Security. That can only mean more taxes. The Bush administration, though, should not be trying to save Social Security. For decades americans have been deceived by this program, believing that their Social Security taxes are pouring into a fund set aside expressly for the purpose of old age insurance for the support of the elderly. This is a colossal lie and it is nothing short of scandalous. The truth is, the vaunted Social Security trust fund does not exist. Never has, never will. Haven.89t we been told for decades that the government scrupulously, almost religiously, maintains the Social Security trust fund? For the truth, we have to go back to 1935, shortly after the creation of Social Security. A man named George P. Davis contested the program in court, claiming that taxation to support it was unconstitutional, trampled on states.89 rights, and imposed an unjust monetary burden on firms in which he had invested. A federal court rejected his claims, but he won a favorable ruling from the First Circuit Court of Appeals in Boston. Had that Appeals Court decision remained in force, the entire Social Security Act would have been voided at the outset. As expected, however, the federal government.89s Commissioner of Internal Revenue, Guy T. Helvering, immediately appealed the matter to the Supreme Court. Lawyers for Mr. Davis contended that Social Security taxes were collected for a particular purpose (for unemployed and older Americans) and not for the constitutionally acceptable purpose of acquiring revenue for the general welfare of the nation as a whole. They also contended that the tax was not a constitutionally allowable excise tax, a type of taxation defined of use or consumption, not a tax on wages. In Helvering v. Davis, the Supreme Court.89s decision made reference to a related case wherein the court had tortuously maintained that the tax was indeed an excise tax and therefore legitimate. But the Court also agreed with the government.89s lawyers who had openly stated that social security taxes were not collected for a particular purpose but are paid into the Treasury as internal revenue collections, available for the general support of the government. That contention, forming the government.89s major argument against the claims by Mr. Davis, received official endorsement when the Supreme Court ruled in Helvering that the proceeds of the unemployment and old-age taxes are to be paid into the Treasury like internal- revenue taxes generally, and are not earmarked in any way. (Emphasis added.) This decision has never been overturned. All talk about the social security trust fund containing social security tax revenue is pure, unadulterated blather. In 1975, former Secretary of Commerce and former Director of the Budget Maurice Stans hit the nail on the head when he stated: Social Security payments rest upon the general credit of the Government of the United States, upon its taxing power, and not upon any accumulations in a trust fund. In 1976, then-Secretary of the contributors to the system have not been building a fund at all. The taxes they are paying into Social Security are merely being handed over as benefits to other people. In other words, Social Security is a huge Ponzi scheme. So, what the Bush administration is trying to do through the new Commission to Strengthen Social Security is to buttress a lie. The commission will surely attract more attention as it continues recommending placebos to treat a disease that will eventually prove fatal if not properly addressed. What should be done is really quite simple: Phase Social Security out. Allow freedom of choice and watch how few young Americans will stick with Social security. Programs doing precisely this have already been implemented in Chile and elsewhere with stunningly beneficial results. Once the people of Chile were given the choice in 1981, they opted out, put their money into private programs, saw those funds spark an economic surge that became the envy of all of Latin America, and destroyed much of their nation.89s harmful government paternalism. But America.89s leaders continue to insist that Social Security is a success and that only careful management of its trust funds is needed to insure its viability. In his 1975 book Social Security: The Fraud in Your Future, author Warren Shore concluded that claims about the existence of trust funds are made because they help foster the public notion that Social Security is like insurance with its premium pools [available] to support promises made. But the public has been misled. Sad to say, the current activity in Washington shows no signs of addressing this fraud. Go to http://www.ssa.gov and then do a search on Helvering v Davis and be amazed at the over 1000 different links that this case is brought up. NOW do you see why the Democrats are so terrified about this SS reform? They would be hanged and drawn'n'quartered if the seniors ever found out that they were stealing from them. The Democrats would never be elected to dog warden ever again if this was made public, and part of the reform is to MAKE THIS PUBLIC! You Democrats are staring down the barrel of a loaded gun in your own hands, and for some reason your pulling the trigger..... === Subject: Re: Question about President's Social Security plan >>by the amount of $2.7 trillion in 75 years. >>Bush administration has a plan. The plan is >>to privatize some parts of social security. >>This will cost $2 trillion to set up. >>It is not guaranteed to fix things, but >>is only one part of an integrated plan. >>So my question is, wouldn't it make more >>sense to just GIVE that $2 trillion to >>social security, which is guaranteed >>to fix things by exactly $2 trillion, just >>leaving a small $0.7 trillion shortfall after >>75 years? Of course that would make sense. But you have to understand that this will not help Bush and his rich friends and so it will not be supported by the Republicans. > You mean other then this train wreck if the media ever made this > information Public?! I think that the actuarial problems of the current system have been well publicized. But there will always be a few stupeeedos who actually think they have knowledge that is special. > I mean name me just ONE Democrat that would still > be left in office if the public found out that they had been lied to > and that the Social Security funds were being stolen by the Democrats > for the last 60+ years? (snicker) Your indictment of Johnson is correct, but most of this thievery has been by Repugnicans for the last 25 years. They are the big deficit creators, not the Dems. > Here are the REAL facts > Social Security.92s vaunted trust fund doesn.92t exist; taxes paid into > Social Security are merely being handed over as benefits to other > people. There are probably 10 people in this country that do not already KNOW this. You seem to be one of those who have only recently become aware of this fact. > On May 2nd, President Bush announced the formation of a presidential > commission to deal with the Social Security crisis. The last such > commission, in 1983, had as its chairman Alan Greenspan. It > recommended enormous increases in taxes, and Congress dutifully > complied. Yes. That would be the Republican Greenspan sucking up to the Republican Reagan. http://GreaterVoice.org/econ/glossary/The_Great_Ray_Gun_Rip_Off.php ACCELERATES THE SCHEDULE OF TAX HIKES IN SOCIAL SECURITY ORIGINALLY PASSED IN 1977. THE SCHEDULE IS TO BE COMPLETED BY 1990 INSTEAD OF THE YEAR 2030.** . >Government spokesmen immediately patted themselves on their > backs for having saved Social security. We are likely in for more of > the same. The new commission.92s name is the Commission to Strengthen > Social Security. That can only mean more taxes. Sure! We have a lying Republican in the White House and a compliant Greenspan at the Fed. What else would you expect accept some piece of crap that will send billions of dollars to the people who financed Georgie's campaign. > The Bush administration, though, should not be trying to save Social > Security. For decades americans have been deceived by this program, > believing that their Social Security taxes are pouring into a fund set > aside expressly for the purpose of old age insurance for the > support of the elderly. This is a colossal lie and it is nothing short > of scandalous. The truth is, the vaunted Social Security trust fund > does not exist. Never has, never will. And most people of even minimal intelligence realize this. > Haven.92t we been told for decades that the government scrupulously, > almost religiously, maintains the Social Security trust fund? For the > truth, we have to go back to 1935, shortly after the creation of > Social Security. A man named George P. Davis contested the program in > court, claiming that taxation to support it was unconstitutional, > trampled on states.92 rights, and imposed an unjust monetary burden on > firms in which he had invested. A federal court rejected his claims, > but he won a favorable ruling from the First Circuit Court of > Appeals in Boston. Had that Appeals Court decision remained in force, > the entire Social Security Act would have been voided at the outset. > As expected, however, the federal government.92s Commissioner of > Internal Revenue, Guy T. Helvering, immediately appealed the matter to > the Supreme Court. Lawyers for Mr. Davis contended that Social > Security taxes were collected for a particular purpose (for > unemployed and older Americans) and not for the constitutionally > acceptable purpose of acquiring revenue for the general welfare of the > nation as a whole. They also contended that the tax was not a > constitutionally allowable excise tax, a type of taxation defined > of use or consumption, not a tax on wages. > In Helvering v. Davis, the Supreme Court.92s decision made reference to > a related case wherein the court had tortuously maintained that the > tax was indeed an excise tax and therefore legitimate. But the Court > also agreed with the government.92s lawyers who had openly stated > that social security taxes were not collected for a particular purpose > but are paid into the Treasury as internal revenue collections, > available for the general support of the government. > That contention, forming the government.92s major argument against the > claims by Mr. Davis, received official endorsement when the Supreme > Court ruled in Helvering that the proceeds of the unemployment and > old-age taxes are to be paid into the Treasury like internal- > revenue taxes generally, and are not earmarked in any way. (Emphasis > added.) This decision has never been overturned. All talk about the > social security trust fund containing social security tax revenue is > pure, unadulterated blather. Yep. <<<<> > NOW do you see why the Democrats are so terrified about this SS > reform? They would be hanged and drawn'n'quartered if the seniors ever > found out that they were stealing from them. The government has been stealing from the working people of this country ever since the 1980 hike in FICA tax by the Reagan thieves. > The Democrats would never > be elected to dog warden ever again if this was made public, and part > of the reform is to MAKE THIS PUBLIC! You Democrats are staring down > the barrel of a loaded gun in your own hands, and for some reason your > pulling the trigger..... There is little most of us want more than to change the funding side of SS without touching the benefits side. -- I know no safe depository of the ultimate powers of society but the people themselves; and if we think them not enlightened enough to exercise their control with a wholesome discretion, the remedy is not to take it from them, but to inform their discretion by education. - Thomas Jefferson. http://GreaterVoice.org === Subject: Re: Question about President's Social Security plan >So my question is, wouldn't it make more >sense to just GIVE that $2 trillion to >social security, which is guaranteed >to fix things by exactly $2 trillion, just >leaving a small $0.7 trillion shortfall after >75 years? If we had dealt with the problem 20 years ago it would have cost a lot less to fix. Reagan tried but Congress refused to rein in entitlements. They simply raised the FICA and threw more money at it hoping that by the time anyone got wise they'd all be retired (on a FEDERAL pension) themselves. Clinton hemmed and hawed for 8 years and held blue ribbon commissions whose recommendations he quietly filed away for the next Administration to act on. Meanwhile the problem got bigger, not better. Setting aside a small fraction of the FICA taxes to allow workers to invest in their own pension plans doesn't actually cost Social Security anything. There is no lockbox full of money being set aside to pay retirees; it all comes out of the General Revenue fund anyway. These retirement accounts will generate enough taxable revenue to offset the loss in FICA taxes going into the system; they'll create new revenue streams into the Treasury. Put it like this. Right now you get ~ 1.75% annual return on your FICA investment, paid out of the Treasury when you retire. If you invested that money privately you'd get at least that and probably much more, paid out of the growth in the GDP between now and retirement. In effect it's a redistribution of wealth (which should make liberals happy but go figure) because what you don't tap from that revenue stream goes into the pockets of wealthy investors and multinational corporations. -- Iraq was a brilliant campaign fought with minimal casualties, 11 September was a humiliating failure by government to fulfill its primary role of national defence. But Democrats who complained that Bush was too slow to act on doubtful intelligence re 9/11 now profess to be horrified that he was too quick to act on doubtful intelligence re Iraq. This is not a serious party. === Subject: Re: Question about President's Social Security plan <57u6s0tmu6hrfdaqsjirlk290r7u5p3v4l@4ax.com> > Put it like this. Right now you get ~ 1.75% annual return on your > FICA investment, paid out of the Treasury when you retire. If you > invested that money privately you'd get at least that and probably > much more, paid out of the growth in the GDP between now and I figure it differently. If this applied to one or two individuals, it would indeed work the way you say. But when applied to entire populations, I think it would work another way. All this money is going to be put in one particular area of the economy - the stock market. The stock market operates on demand and supply. While the money is going in, the stock market goes up and absorbs all the money. Now recall that the whole problem is that at some point, there are going to be many more retirees than contributors to the system. Shifting it to the stock market is not going to change that fundamental characteristic. At some point, there simply will be many more sellers than buyers. When that happens, the stock market adjusts by going down. Given the volume of the movements, and the fact that the smart money will have flown out of the stock market a little earlier, it will crash. So the situation is, instead of the ~1.75% return there will be tremendous loss of capital. > retirement. In effect it's a redistribution of wealth (which should Yes, in effect it is indeed nothing but a redistribution of wealth. But I don't agree with the direction you seem to think it works in. === Subject: Re: Question about President's Social Security plan >> Put it like this. Right now you get ~ 1.75% annual return on your >> FICA investment, paid out of the Treasury when you retire. If you >> invested that money privately you'd get at least that and probably >> much more, paid out of the growth in the GDP between now and >I figure it differently. If this applied to one or two individuals, >it would indeed work the way you say. But when applied to entire >populations, I think it would work another way. >All this money is going to be put in one particular area of >the economy - the stock market. or the real estate market, or the municipal bond market, or any of a hundred other investments. Historically all of these have outperformed Social Security over time. Most pension plans don't put all their eggs in one basket, like Social Security does. SS is REQUIRED BY LAW to invest in the infusion of new workers over time and we know that ain't gonna happen; just the opposite is happening. -- Iraq was a brilliant campaign fought with minimal casualties, 11 September was a humiliating failure by government to fulfill its primary role of national defence. But Democrats who complained that Bush was too slow to act on doubtful intelligence re 9/11 now profess to be horrified that he was too quick to act on doubtful intelligence re Iraq. This is not a serious party. === Subject: Re: Question about President's Social Security plan <57u6s0tmu6hrfdaqsjirlk290r7u5p3v4l@4ax.com> > Put it like this. Right now you get ~ 1.75% annual return on your > FICA investment, paid out of the Treasury when you retire. If you > invested that money privately you'd get at least that and probably > much more, paid out of the growth in the GDP between now and I figure it differently. If this applied to one or two individuals, it would indeed work the way you say. But when applied to entire populations, I think it would work another way. All this money is going to be put in one particular area of the economy - the stock market. > or the real estate market, or the municipal bond market, or any of a > hundred other investments. Historically all of these have > outperformed Social Security over time. Most pension plans don't put > all their eggs in one basket, like Social Security does. SS is > REQUIRED BY LAW to invest in the infusion of new workers over time and > we know that ain't gonna happen; just the opposite is happening. Think you missed the point here. What do you think is going to happen to interest rates when massive amounts of money are pulled out of the government bond market? No doubt this would have a big impact on the stock market as interest rates go way up which could result in a negative return for the stock market and a detrimental impact on the economy. And why do you assume that real estate is a safe investment? Have you ever heard of people making a bad investment? What will happen to those who invest in riskier investments and see their retirement fund wiped out? > -- > Iraq was a brilliant campaign fought with minimal > casualties, 11 September was a humiliating failure > by government to fulfill its primary role of > national defence. But Democrats who complained that > Bush was too slow to act on doubtful intelligence > re 9/11 now profess to be horrified that he was too > quick to act on doubtful intelligence re Iraq. This > is not a serious party. I am amazed that a president who outright lied over the reason to go to war in Iraq, WMD that did not exist was reelected. Iraq and 9/11 are unrelated events or at least thats the conclusion of the 9/11 commission === Subject: Re: Question about President's Social Security plan >by the amount of $2.7 trillion in 75 years. >Bush administration has a plan. The plan is >to privatize some parts of social security. Bush has a notion, not a plan. The actual plan, following the precedent of Cheney's energy plan, will be prepared by brokerages and mutual funds, with their recommendations being weighted according to their contributions to the GOP. === Subject: Re: Question about President's Social Security plan by the amount of $2.7 trillion in 75 years. Bush administration has a plan. The plan is to privatize some parts of social security. > Bush has a notion, not a plan. > The actual plan, following the precedent of Cheney's energy plan, > will be prepared by brokerages and mutual funds, with their > recommendations being weighted according to their contributions to > the GOP. Bush does have something in it for him, too. It will leave the stock market jumping for joy, and Bush's legacy will be to leave the economy booming. Whether it's a false or temporary boom, won't be his concern. And in any case, he will have lots of defenders who will claim the followup bubble-bursting and social-security money getting swallowed up by the smart crowd, was caused by the evil lefties and had nothing to do with the original mighty grand plan... === Subject: Re: Question about President's Social Security plan This [privatize some parts of social security] will cost $2 trillion to set up. So my question is, wouldn't it make more sense to just GIVE that $2 trillion to social security, which is guaranteed to fix things by exactly $2 trillion, just leaving a small $0.7 trillion shortfall after 75 years? The counter-argument would be that in doing so you'd simply be perpetuating a poor system that by your own words remains in deficit, still will be unsustainable because of its design, and you are also erroneous (if not deliberately misleading) by referring to $0.7 trillion as small. There are other counter-arguments as well (you fail to view the entire exchange of revenues and benefits, for example). You were on stronger ground elsewhere on this thread when you rightly noted that Wall Street would love to get its hands and and make money off all that money. Government would indirectly be giving money (taxing it and seeing that it was redirected) to Wall Street. A more legitimate problem with Bush's proposal is this would be nominally private but still would be a government-program set of accounts, and there would be temptations by Democrats, especially liberal Democrats, to engage in evil, anti-American lefty-fascist follies that Ralph Nader only could dream of decades ago, such as making the federal government the largest-by-far institutional investor -- and with that would come government influence and social responsibility, faddish far-left idiocy such as Israeli divestiture gimmicks, maybe government shareholder influence on politically disfavored industries like guns and automobiles, and so on. No normal American wants any threat of that. === Subject: Re: Question about President's Social Security plan > A more legitimate problem with Bush's proposal > is this would be nominally private but still would be > a government-program set of accounts, and there > would be temptations by Democrats, especially > liberal Democrats, to engage in evil, anti-American > lefty-fascist follies that Ralph Nader only could > dream of decades ago, such as making the > federal government the largest-by-far institutional > investor -- and with that would come government > influence and social responsibility, faddish far-left > idiocy such as Israeli divestiture gimmicks, maybe > government shareholder influence on politically > disfavored industries like guns and automobiles, and > so on. No normal American wants any threat of that. I can't imagine that political agitation for restriction of these investments would be confined to the left. Would the backers of this plan permit any of the money to be invested in ,for instance, manufacturers of contraceptives, particularly the so-called morning after pill? I have a plan that would make SS solvent for the rest of this century. Just increase the interest rate that the general fund pays to the Social Security fund by one percentage point. The same amount of cash that changes hands would be the same as present, but the SS fund has immediately become much more solvent due to the increase in the future value of its reserves. Just bookkeeping, but that is all claims of SS insolvency are anyway. -- To e-mail me get rid of the cats and dogs. === Subject: Re: Question about President's Social Security plan >> A more legitimate problem with Bush's proposal >> is this would be nominally private but still would be >> a government-program set of accounts, and there >> would be temptations by Democrats, especially >> liberal Democrats, to engage in evil, anti-American >> lefty-fascist follies that Ralph Nader only could >> dream of decades ago, such as making the >> federal government the largest-by-far institutional >> investor -- and with that would come government >> influence and social responsibility, faddish far-left >> idiocy such as Israeli divestiture gimmicks, maybe >> government shareholder influence on politically >> disfavored industries like guns and automobiles, and >> so on. No normal American wants any threat of that. > I can't imagine that political agitation for restriction of these >investments would be confined to the left. >Would the backers of this plan permit any of the money to be invested >in ,for instance, manufacturers of contraceptives, particularly the >so-called morning after pill? >I have a plan that would make SS solvent for the rest of this century. >Just increase the interest rate that the general fund pays to the Social >Security fund by one percentage point. The same amount of cash that >changes hands would be the same as present, but the SS fund has >immediately become much more solvent due to the increase in the future >value of its reserves. The SS tax was dramatically increased under Reagan and he used the money to fund the tax breaks for the wealthy and his monstrous increase in military spending to go to war against Russia and the Middle East (Iraq and Iran). Charlie >Just bookkeeping, but that is all claims of SS insolvency are anyway. >-- >To e-mail me get rid of the cats and dogs. === Subject: Re: Question about President's Social Security plan > The counter-argument would be that in doing so > you'd simply be perpetuating a poor system that > by your own words remains in deficit, still will be > unsustainable because of its design, and you are I thought the problem was not the design, but baby boomers. Reverse booms should put money into social security, which could then be used up for the next boom. > also erroneous (if not deliberately misleading) by > referring to $0.7 trillion as small. Not looked at the budget figures lately, have we? Maybe you have a point -- perhaps it's erroneous in this context to call 0.7 trillion over 75 years as small. It's better called negligible or trivial or irrelevant. Any single one of the next presidents over the next 75 years could fix it when the need became apparent. In the worse case, by borrowing (just like Bush plans to do), in the best case by using up some surplus. === Subject: Re: Question about President's Social Security plan > I thought the problem was not the design, but > baby boomers. Reverse booms should put money > into social security, which could then be > used up for the next boom. Reverse booms??? The problem is both with the design and with demographics (not merely the Baby Boomers but lower fertility rates and longer lifespans). The program, if kept (which is most likely), should be converted to fully funded rather than as pay-as-you go. Also, it is irresponsible (and given our lives are involved, illogical) to count on recovery of the program after the Baby Boomers are dead, much less to be superficial in our approach and simply tax more during low-beneficiary-number years or decades to better finance benefits during high-beneficiary-number years or decades that follow them. > Not looked at the budget figures lately, have we? I'm fully aware of them, as well as what Social Security and Medicare will do to the entire federal budget (not just those two programs) long before either of these two programs go bankrupt. > Maybe you have a point -- perhaps it's erroneous > in this context to call 0.7 trillion over 75 years > as small. It's better called negligible or trivial > or irrelevant. That's even more erroneous. > Any single one of the next presidents over the next > 75 years could fix it when the need became > apparent. In the worse case, by borrowing (just like Bush > plans to do), in the best case by using up some surplus. Only a fool would count on a surplus then, much later, the realistic time when politicians will take action -- which is only when forced to, or in other words -- eventually. (They avoid doing anything unpleasant now even though it solves worse problems for us later.) There will not be an easy fix later. Note that you say that at any time, it can be fixed. That is the admission of error of all those who deny there is anything wrong with the system now. (There is; it is unsustainable and will cause many other fiscal problems for the federal government long before it goes broke.) If it were up to me and we had to keep Social Security, I would take the roughly ten years we have left before the Baby Boomers start their retirements to convert from our pay-as-you-go (Ponzi scheme) system to a fully funded system. As that causes some pain to all, perhaps it is that one time conversion that might, might justify borrowing, because it would solve the problem with not only S.S. but with the federal finances, and so justify the extra cost of interest in the long run. === Subject: Re: Question about President's Social Security plan Maybe you have a point -- perhaps it's erroneous in this context to call 0.7 trillion over 75 years as small. It's better called negligible or trivial or irrelevant. > That's even more erroneous. I think I was not clear enough. I will simplify. It works out to less than 10 billion per year. How big a deal in the federal budget do you want to make that? > There will not be an easy fix later. Sure there will be. Note that Bush's fix involves borrowing $2 trillion. What's to stop a future president, who is facing the problem here and now instead of in the future, to simply borrow the much smaller amounts needed to keep it solvent, as and when needed? And it is indeed possible that when needed, the budget will happen to have a surplus, at least some of the times. === Subject: Re: Question about President's Social Security plan > by the amount of $2.7 trillion in 75 years. > Bush administration has a plan. The plan is > to privatize some parts of social security. > This will cost $2 trillion to set up. > It is not guaranteed to fix things, but > is only one part of an integrated plan. > So my question is, wouldn't it make more > sense to just GIVE that $2 trillion to > social security, which is guaranteed > to fix things by exactly $2 trillion, just > leaving a small $0.7 trillion shortfall after > 75 years? Your solution makes too much sense. Forget about it in this administration. === Subject: Re: Question about President's Social Security plan John Deere says... >by the amount of $2.7 trillion in 75 years. >Bush administration has a plan. The plan is >to privatize some parts of social security. >This will cost $2 trillion to set up. >It is not guaranteed to fix things, but >is only one part of an integrated plan. >So my question is, wouldn't it make more >sense to just GIVE that $2 trillion to >social security, which is guaranteed >to fix things by exactly $2 trillion, just >leaving a small $0.7 trillion shortfall after >75 years? You're assuming that Bush *wants* to fix social security. Nothing could be farther from the truth. He wants to make sure that it goes bankrupt, and his privatization plan is a crucial step. -- Daryl McCullough Ithaca, NY === Subject: Re: Question about President's Social Security plan > You're assuming that Bush *wants* to fix social security. > Nothing could be farther from the truth. He wants to make > sure that it goes bankrupt, and his privatization plan is > a crucial step. I don't think he particularly cares to see social security bankrupt -- I suspect the motivation is entirely different. Wall Street was helped strongly during the internet bubble by 401K funding. The major players made tons of money. But then the bubble burst, and the wall street honchos felt they weren't rich enough. They saw a ray of light -- the next possible bubble could come from social security. So they are pulling all kinds of strings. And of course, our venerable president is a sucker for anything the rich guys say -- it must be the right thing to do if the havemores say so. He just doesn't believe the nice havemores would advise something that could have bad results. It's kind of Darwinian. Remeber, the folks who would be most exploited are mostly staunch Bush supporters, and would jump at anybody with both feet for suggesting he is doing something wrong. I don't know how much is really wrong if they are left -- by the actions of their favorite president -- with nothing in their golden days. Besides, they will blame the lib dem boogiemen anyways. And I suppose if it does go through, the smart folks could make some money at the expense of these non-havemore Bush supporters, because the stock market will have some irrational moments. Still, it just doesn't seem right, or American, or a good thing at all for the long term future. === Subject: Re: Question about President's Social Security plan >by the amount of $2.7 trillion in 75 years. >Bush administration has a plan. The plan is >to privatize some parts of social security. >This will cost $2 trillion to set up. >It is not guaranteed to fix things, but >is only one part of an integrated plan. >So my question is, wouldn't it make more >sense to just GIVE that $2 trillion to >social security, which is guaranteed >to fix things by exactly $2 trillion, just >leaving a small $0.7 trillion shortfall after >75 years? Yes, except that the government doesn't have an account with $2 trillion in it. So it would have to sell bonds in that amount. The simpler method is to directly credit the SS trust fund with whatever it needs to cover the shortfall, if and when it occurs. === Subject: Re: Question about President's Social Security plan >Yes, except that the government doesn't have an account with $2 >trillion in it. So it would have to sell bonds in that amount. The >simpler method is to directly credit the SS trust fund with whatever >it needs to cover the shortfall, if and when it occurs. Except of course the government would have to print money to cover the shortfall which would quickly lead to hyperinflation. You can issue bonds but there has to be some expectation that you can pay off those bonds when they come due, you see, and with 95% or whatever percent of the federal budget dedicated to paying pension checks that doesn't leave much room to pay any other bills. Ideally you take care of this by increasing the contributions slightly well in advance and restraining the growth of the program to inflation. If we had done this back in, oh, 1983 the system would be solvent. Unfortunately we only did *half* of this: We raised FICA taxes and declared we had a surplus. But Congress never reined in entitlements and then spent the FICA surplus on other things. Bottom line is the surplus disappears from the books around 2011 and from that point on the system runs in the red. And it only gets *redder* the further out you go, with too few workers paying into the system to cover benefits going out. -- Iraq was a brilliant campaign fought with minimal casualties, 11 September was a humiliating failure by government to fulfill its primary role of national defence. But Democrats who complained that Bush was too slow to act on doubtful intelligence re 9/11 now profess to be horrified that he was too quick to act on doubtful intelligence re Iraq. This is not a serious party. === Subject: Re: Question about President's Social Security plan OrionCA says... >Iraq was a brilliant campaign fought with minimal >casualties, 11 September was a humiliating failure >by government to fulfill its primary role of >national defence. But Democrats who complained that >Bush was too slow to act on doubtful intelligence >re 9/11 now profess to be horrified that he was too >quick to act on doubtful intelligence re Iraq. This >is not a serious party. Whoever said that is not a serious commentator. The comparison is stupid. -- Daryl McCullough Ithaca, NY === Subject: Re: Question about President's Social Security plan >OrionCA says... >>Iraq was a brilliant campaign fought with minimal >>casualties, 11 September was a humiliating failure >>by government to fulfill its primary role of >>national defence. But Democrats who complained that >>Bush was too slow to act on doubtful intelligence >>re 9/11 now profess to be horrified that he was too >>quick to act on doubtful intelligence re Iraq. This >>is not a serious party. >Whoever said that is not a serious commentator. The >comparison is stupid. Brilliant counterargument there. Mark Stein, btw, is a respected political commentator both in the UK and the United States. And he was absolutely correct in his statement. you are not a serious political party anymore. -- Iraq was a brilliant campaign fought with minimal casualties, 11 September was a humiliating failure by government to fulfill its primary role of national defence. But Democrats who complained that Bush was too slow to act on doubtful intelligence re 9/11 now profess to be horrified that he was too quick to act on doubtful intelligence re Iraq. This is not a serious party. === Subject: Re: Question about President's Social Security plan >>Yes, except that the government doesn't have an account with $2 >>trillion in it. So it would have to sell bonds in that amount. The >>simpler method is to directly credit the SS trust fund with whatever >>it needs to cover the shortfall, if and when it occurs. >Except of course the government would have to print money to cover the >shortfall which would quickly lead to hyperinflation. The government prints money, i.e. monetizes the debt, for only two reasons: (1) to meet the public's demand for wallet money in lieu of bank deposits, and (2) to provide the reserves banks need to meet their reserve ratio requirements. Future payments to social security beneficiaries will not require the printing of money. It will involve deficit spending, all of which will be covered by the sale of bonds. >You can issue >bonds but there has to be some expectation that you can pay off those >bonds when they come due, you see, and with 95% or whatever percent of >the federal budget dedicated to paying pension checks that doesn't >leave much room to pay any other bills. The Treasury has no problem redeeming its bonds, and never will. The amount of government spending going to social security and medicare is currently about 35% of the total spending. That will increase as baby boomers retire, but it will never come close to 95% of total spending. >Ideally you take care of this by increasing the contributions slightly >well in advance and restraining the growth of the program to >inflation. If we had done this back in, oh, 1983 the system would be >solvent. The program is solvent and will be for at least forty years. If and when the trust fund runs out, the shortage can be made up by government borrowing. That will increase the deficit, but only during the peak years of benefits for baby boomers. They too will die. >Unfortunately we only did *half* of this: We raised FICA >taxes and declared we had a surplus. But Congress never reined in >entitlements and then spent the FICA surplus on other things. There is no way the FICA surplus can be kept in a lock box. Those funds would be spent even if the on-budget were in balance. >Bottom line is the surplus disappears from the books around 2011 and >from that point on the system runs in the red. And it only gets >*redder* the further out you go, with too few workers paying into the >system to cover benefits going out. The surplus FICA revenues are expected to disappear about 2018, which simply means the so-called trust fund will stop increasing in value at that time. It is conservatively projected to remain in the black decades longer. === Subject: Re: Question about President's Social Security plan : by the amount of $2.7 trillion in 75 years. : : Bush administration has a plan. The plan is : to privatize some parts of social security. : : This will cost $2 trillion to set up. : : It is not guaranteed to fix things, but : is only one part of an integrated plan. : : So my question is, wouldn't it make more : sense to just GIVE that $2 trillion to : social security, which is guaranteed : to fix things by exactly $2 trillion, just : leaving a small $0.7 trillion shortfall after : 75 years? : Why not just level with the people and tell them Social Security is an unconstitutional use of their money and in 25 years that will cease? So, all you nice folks that have been letting the government have some walking around cash better start saving now for that rainy day coming up 25 years from now. -- Who are these guys? If the world were a logical place, men would ride horses sidesaddle Smith or Jones === Subject: Re: Question about President's Social Security plan Why not just level with the people and tell them Social Security is an unconstitutional use of their money and in 25 years that will cease? Liberals and those they've made dependent on the federal government don't care about Social Security's unconstitutionality; as the childish people they are, they're delighted that Government can and will do all kinds of things to meet human needs. It is for this reason that for decades Social Security has been treated by politicians in DC as something that people hold sacred. (The dopiest of Dems even take sacredness further: they worship their God, FDR.) === Subject: Re: Question about President's Social Security plan Alias: Smith or Jones says... >Why not just level with the people and tell them Social Security is an >unconstitutional use of their money and in 25 years that will cease? So, all >you nice folks that have been letting the government have some walking >around cash better start saving now for that rainy day coming up 25 years >from now. The way I see it, Bush's social security plan is like one of those paradoxical science fiction stories about predicting the future. The hero foresees some great catastrophe, and so takes steps to prepare for it. Ironically, those steps turn out to be exactly what *causes* the catastrophe. Except that it's not haplessness, it is intentional. Bush is actively taking steps to *insure* that social security goes bankrupt. -- Daryl McCullough Ithaca, NY === Subject: Re: Question about President's Social Security plan : Alias: Smith or Jones says... : : >Why not just level with the people and tell them Social Security is an : >unconstitutional use of their money and in 25 years that will cease? So, all : >you nice folks that have been letting the government have some walking : >around cash better start saving now for that rainy day coming up 25 years : >from now. : : The way I see it, Bush's social security plan is like one of those : paradoxical science fiction stories about predicting the future. The : hero foresees some great catastrophe, and so takes steps to prepare : for it. Ironically, those steps turn out to be exactly what *causes* : the catastrophe. : : Except that it's not haplessness, it is intentional. Bush is actively : taking steps to *insure* that social security goes bankrupt. : : -- : Daryl McCullough : Ithaca, NY : Bush Smush, who cares. They're all in it together. Scrap it and move on. -- Who are these guys? If the world were a logical place, men would ride horses sidesaddle Smith or Jones === Subject: A Quantum Poem for Xmas A Quantum Poem for Xmas --------------- I wonder if science shall ever see, a quantum discrete as a tree, Alas, the answer no must be, For what science views is energy, Where, as discrete, Quantum Mystics see in their symmetric reverie the integral over r of nm0c. === Subject: Re: A Quantum Poem for Xmas > A Quantum Poem for Xmas > --------------- > I wonder if science shall ever see, > a quantum discrete as a tree, Hopeless ignorant idiot. SCANSION! Zick Limerick ------------- There once was an old idiot who had some poor idea of how mum Went to bed With dread Because Lester suckled her left bum cheek in the night. -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz.pdf === Subject: Re: A Quantum Poem for Xmas