mm-1839 T must be defined someone or assumed to have been defined. The superposition assertion is a condition (linearity) on T. Unless it is stated that way, it doesn't have to be. ---------------------------------------------------------- ** SPEED ** RETENTION ** COMPLETION ** ANONYMITY ** ---------------------------------------------------------- http://www.usenet.com NOTE: First post in this group. Pardon the cosmology content of the following, but I posted this in sci.physics.relativity, but there are so many crazy people (i.e. mentally ill) there that I just said the hell with it, and will ask questions here. SO YOU MIGHT JUST WANT TO SKIP DOWN TO QUESTION E! In the following, the term hyperboloidmeans two-sheeted circular hyperboloid. (There's one you can play with just before equation 22 at http://mathworld.wolfram.com/Hyperboloid.html Space itself is expanding; not the objects in it. It has no center that you can point to in the sky. Okay. BUT: Suppose you draw a line from every galaxy to every other. Topologically, this defines a closed polyhedron which very closely approximates a sphere. A) Isn't the centroid of the polyhedron the spatial location of the big bang? Isn't every galaxy a specific distance from that centroid, using the ordinary Euclidean metric? Can't all the aliens on every planet aim a laser pointer at the sky and tell their kids see, that's the center of the universe? Though I guess half the time, they'd have to aim the laser at the ground, which would just confuse the kid more. And speaking of becoming more confused: The inflating balloon analogy can explain A), but requires that if you throw a baseball hard enough and wait long enough, it will hit you in the back of the head (it's traveled the circumference of the balloon). B) If this IS what's happening, then every pair of objects has at least two different ordinary Euclidian distances between them, neither more valid than the other (think of two points on a sphere). It is easy to construct a paradox from this in 3-space (I will on request. Superluminal transportation, for instance). The inflating rubber balloon wraps around through a higher dimension, which (two-dimensional) bacteria on the surface of the balloon would call hyperspace. C) Does the above statement describe the same concept as the Riemann metric wrapping the two cone-like sides of a hyperboloid through imaginary space (i.e.,time?) to reveal the hyperboloid as a four-dimensional sphere? D) Is this a reasonable explanation of the paradoxes described in B)? Is it, at least, a consistent one? Is it a ridiculous one? E) Is a sphere isomorphic to a two-sheeted circular hyperboloid (I believe it is)? Is it homeomorphic too? The following refer to http://mathworld.wolfram.com/Hyperboloid.html F) What is meant by the height of a hyperboloid? G) In equation 22, what are parameters a and c? (a may just be the same thing as in equation 26. If so, they should have defined it in 22). What is v in equation 23? H) A hyperboloid is actually a solid with a radius greater than infinity, right? If so, wouldn't the volume always be infinite? If so, then how can that be consistent with equation 27? I) With the mouse, grab the hyperboloid mentioned in the first paragraph and shake it. What are you doing to the 4-dimensional sphere? You're not rotating it, because that motion breaks the symmetry. Rotation would leave the location of the sphere's surface in the same place. Thanx in advance, =[ d PS: Despite what someone said, I still think a hyperboloid is a sphere with a negative radius, but I think the reason isn't that the guy was wrong, but that I'm too slow and stupid to get it. Is it possible that a solid with greater than infinite positive radius is isomorphic to a sphere with a finite negative radius? Meaning what by isomorphic? An isomorphism preserves some structure. What structure is being preserved here? No. ... Why do you believe this to be true? Is your claim based on empirical evidence? A closed polyhedron, even on which very closely approximates a sphere, has no intrinsic centroid: the centroid depends on an embedding in (usually, Euclidean; but I'll give you hpyperbolic) some higher-dimensional space, and different embeddings would have different centroids in non-trivial cases. What embedding of your polyhedron, in what higher-dimensional space, are you considering? [things which make no sense until sense is given to the supposed centroid, clipped here] No and no (for any reasonable interpretations of isomorphic, homeomorphic, and two; how many sheets does *your* sphere have, anyway). [stuff refering to pages at mathworld, which I don't intend to look at, clipped here] Wrong (to the extent it even makes sense). [more snippage] No (same disclaimer). You might as well have stayed in a physics group; there's no mathematics-as-we-know-it in your post, Captain. And there are plenty of lunatics in sci.math, too, so it isn't as if you can get away from them just by moving here. Lee Rudolph same sarcastic BS as in physics.relativity... given a spherical cloud of points (galaxies), of COURSE the object created by connecting each with every other forms a polyhedron, unless the points are all coplanar or something. think of the points as graph nodes. every closed loop of length three forms a triangle. pretend it's a triangular window frame and put a pane of glass in it. do that for all loops of length three. then stand back and look at what you have. not only is it a polyhedron, but it's an everywhere-convex polyhedron. It has a lot of irrelevant internal connectivity, but it's outer surface is polyhedral. even on which very closely approximates sorry, I'm not a math person. I meant the center of mass when you view the thing as a solid. hyperbolic. It's a 3-dimensional solid embedded in 3-space. what's the big deal? uhhh... flat euclidian 3-space? maybe they make sense now! ?? why would you have to interpret the meaning of isomorphic? that's just the point. both sheets of the hyperboloid are actually connected through a higher-dimensional space. it really only has one sheet, and it's closed, and neither one of them has any holes, which makes it both iso- and homeomorphic to a sphere... I think. the question iS: does this hold when extending an n-dimensional object to an n+1 dimensional one. for instance, you can map all points on a planar sheet with finite area onto a cube. Just think of a cardboard box that has been flattened out the question is, can you CONTINUOSLY DEFORM a 3D object into a 4D one? a hyperboloid is the three dimensional manifestation of a 4-dimensional sphere. everybody into special relativity knows that. it's what explains the difference between time and space, for instance, and it's why you can talk about the hypotenuse of a triangle with one leg extending in space and the other extending into time. (Rotation of that triangle has a very unexpected effect from the viewpoint of 3-space, too!) well why bother posting? jee-ziss, this is almost as bad as the other group. =[ d Why do you suppose it to be spherical? Hi Are there any known upper bounds for the chromatic number of the following sets S1, S2 of undirected simple graphs? graphs whose clique number is c. graphs such that for all G = (V, E) in S2 the following property holds. There exists a mapping f of each vertex to a d-tuple of ranges ([a_1, b_1], ..., [a_d, b_d]) such that {u, v} in E if and only if the intersection of (f(u))(i) and (f(v))(i) is non-empty for all 1<=i<=d. Remark: For the special case d = 1, S2 is the set of interval graphs where the chromatic number equals the clique number. Thomas When c = 2 (triangle-free graphs) the chromatic number can be arbitrarily large. I think this is the simplest construction: For a {(x,y): x and y integers, 1 <= x < y <= n}; (x,y) is adjacent to (u,v) n. This example is due to Erdos and Hajnal, if I remember right. keeping Whether it is a fallacy or not has nothing to do with whether it is formal. http://www.m-w.com/cgi-bin/dictionary?book=Dictionary&va=demonstrate&x=13&y= 13 Good idea. Definition 2: to prove or make clear by reasoning or evidence b : to illustrate and explain especially with many examples It is made clear by evidence: the lack of any instances of the system actually working despite overwhelming reasons to give them. The example that illustrates the lack of anything being produced is provided by its very authors in their own words. Since it is the first paper on their system, then nobody has ever shown it to produce anything. Thus the evidence runs contrary to their claim. C-B By now you have pretty much shown that you actually believe that. http://www.m-w.com/cgi-bin/dictionary?book=Dictionary&va=demonstrate&x=13&y= 13 No. The lack of evidence that the system works as stated is not evidence that the system does not work as stated. No. The lack of evidence that the system works as stated is not evidence that the system does not work as stated. shown What evidence? The lack of evidence that the system works as stated is not evidence that the system does not work as stated. LOL. Do think trying to impress people with your vocabulary will make them think you're smart imn Mathematics? C-B Your lack of a defense is a tacit admission that your response is not useful or academic, which was one of my points. I said demonstrates, increases the probability, gives confirming evidence. C-B Tip: When you think that something is (whatever personal insult you happen to come up with), first consider the possibility that you are misinterpreting or simply don't understand what you are reading. It is wise to seek clarification before drawing conclusions. C-B Huh? tacit admission? Of course my post was not academic, whatever that means - this is not academia, this is usenet. Whether it was useful depends on the reader. No, you said demonstrates, period, in the question that I called stupid. If you think that that word means something other than proves you're wrong. Not that it matters, because the question is still pretty dumb even if we change the wording as you suggest. Why do you imagine I asked two or three times whether you'd read what you'd written, if not because I was thinking that you must have meant to write something else? ************************ David C. Ullrich No. implies means logical certainty. I said demonstrates, incresses the probability, gives confirming evidence. The paper that makes unsubstantiated claims about what the authors accomplished is presumably expecting the reader to be persuaded that the claims are true. My point was that to a logical person it is only providing evidence that it is not true. To add a little more detail: Suppose a paper describes a new system that purportedly generates proofs of given theorems. It contains all sorts of cryptic formalisms, and ends with the claim that the theorem has been proven. The naive reader will be impressed at the array of symbols, not understand them, and conclude that it must be some new advanced theory and that it is valid. However, the logical reader knows that in the final analysis, any proof is a convincing argument that the given proposition is true. Formalisms do not constitute a proof. It is the logical argument that they represent that is the proof. If an intuitive, semi-formal convincing argument is not given, then there is no real proof. Similarly with a system that purportedly generates computer programs. The proof of the pudding is an actual, executable program that is created by the system. Short of this, the truth of the claim has not been supported. Furthermore, the lack of such real examples suggests only that the system does not work, and that the claims of success by the authors are false. C-B In fact demonstrates _means_ proves, at least in logic, which is where we are. But it doesn't matter. So evidently you just didn't phrase your question quite accurately (as I suggested many times when I asked if you'd read what you'd written.) If we take your question and substitute the words 'incresses the probability; gives confirming evidence' for demonstrates it's still a silly question, although not quite as blatantly stupid as the original version. Yes, this was your point. And that point is simply ridiculous - the _supposed_ lack of substantiation does _not_ provide evidence that the assertion is false. And not that it's relevant to the question of why your question was so stupid, but in fact if we're talking about the paper and the assertion that I think we are, it's simply not true that there is no substantiating evidence in the paper. If I give a clear description of an algorithm in informal English, together with a proof in English that the algorithm does what is intended, then state that this could all be done by a computer program, the fact that I don't include any actual computer code in the paper does not give evidence that writing such a program is impossible. Saying that the absence of code gives evidence that the program is impossible is stupid. Laughably, blatantly stupid. Expressing doubts as to whether the program is possible is another matter - whether that looks stupid depends on how clear the informal proof was. Here it does in fact make you look pretty clueless. Sorry, Charlie. The fact that something is too hard for _you_ to follow does not prove that it's just a bluff with no actual content. ************************ David C. Ullrich Some of you don't seem to get it. I have what must be a fairly huge readership off of Usenet that reads my postings off the Web, where some part of that bloc of readers goes out and does web searches, which move Google and Yahoo search results, at times on a daily basis. To see it, search on prime counting regularly, or some other topics related to my work. So, no I'm not posting so much to convince you--if you are a sci.math'er--but to convince those people who come here for the reality drama. It's like a weird variant of reality TV that's for people who like to read, and this is quite real, with very real issues in mathematics, where there's this huge battle going on with me a discoverer pitted against people fighting to deny the truth. There may be a few hundred regular sci.math posters who can be said to be sci.math'ers and potentially several million readers out there around the world tuning in to this intellectual brouhaha. So yeah, given my ability to indirectly determine that I am reaching people out there, why would I stop? Now I'm focusing on prime counting, yet again, but this time I'm keeping things simpler so that sci.math'ers can't distract people with a lot of math-ese, and I've also put out my prime counting applet, again, so they can run it for themselves: http://groups.yahoo.com/group/myprimecounting/ As time progresses these people can see that mathematicians and math people DO in fact lie as I've said, and with mathematics that is supposedly important to them, like mathematics about prime numbers. Over time that should have a real world impact, like when some news organization tries to run a story claiming that mathematicians have some major find, there will be skepticism, from people who have watched discussion here and seen that math people can lie about the darndest things! It's not surprising that there would be a *readership* following, as there is a market for reality based stuff, and there are in fact people all over the world who prefer to read versus watch television. And they can get a lot of drama, right here. James Harris It's more like a weird variant of Whack-a-Mole, where you are a pathetic but obnoxious mole and the groundskeepers are continually on alert to restore integrity to the landscape. You can also reach people by passing gas in an elevator or throwing up in a restaurant. Why would think that any of these are worthy of praise? -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com James... Why the do you want to count primes? You can't really believe that millions of people have read your posts. I would hazard a guess closer to the hundreds, and most of them forgot about them 5 minutes later. There are maybe a few dozen people who regularly read your posts, for whatever reason. Most, like me do it to see how great your delusions will become. You can rant and rave, call everyone liars until your face turns blue, and in the end, will amount to nothing. You are a probably slightly above average in intelligence, but have a severe mental problem, as well as an overinflated ego born of an obvious failing in your offline life. You would never mouth off to people in the real world, but you feel safe and secure to chatter away on usenet. If you randomly chose a million people around the U.S. , and asked them what usenet was, I doubt if much more than 10% of them would be able to tell you what it is, and even fewer use it. And of those that do use it, most are doing it to download mp3's, movies from alt.binaries, and trading porn. And by the way, Nora is kicking your butt in her posts. All you can do is stutter and stammer around, trying to reply without looking foolish. Sorry, but you prove time and time again that your grasp of mathematics is not much beyond the average high school graduate, and that might actually sound insulting to the average graduate. posts. forgot to amount intelligence, You sci.math'ers are an odd bunch. When did I say millions of people *were* reading my posts as in definitely? I pointed out Google and Yahoo searches changing at times on a daily basis, and I'll admit I don't know how many people it'd take to do that, but I think it's more than a few hundred. Now I can understand why you try to comfort yourself with your rationalization, but so what? My point is that math people routinely lie, I can prove it, and I can see evidence that other people are tuning in, so you probably for the first time feel some fear, as I've noticed that some of you seem to think that sci.math is this little closed place where the obvious lies are ok as you're amongst your own. My point is that you people do lie when you think you can get away with it, and now for the first time, because some of you may realize that Usenet is not your own backyard after all. It's a VERY public place. And in fact Usenet is one of the most public places available, where you can air things out to the entire world. The posts here are not just on Usenet either as they are being put on the web by multiple sources, so you have the full Internet. Did you people really think I posted so much to convince you? Why bother? I use this as an outlet to the world and try to keep up with how effective it is by checking web search engines. The entertainment aspect to the postings I do is partly deliberate. And the story here is wild enough for the world. There are these big arguments with all kinds of weird drama like math journals shutting down and posters of unknown sex, along with a lot of hostility. It sounds a lot like a reality TV show anyway, besides the math. Come on, you can't be that dumb. Didn't it occur to most of you that more than likely these arguments are getting a lot more attention worldwide than a few hundred people who are regular Usenetters? You can't be so naive as to really believe you're in some little world that just happens to have outlets onto the Internet itself, where no one pays attention despite all the crazy drama. And if you were so naive, just do the Google or Yahoo searches, and wake up to the real world. James Harris And by the way. You don't seem to know how search engines rank results. prime counting. Now , since you are by far the largest poster on the subject, and your posts are cross posted to a couple of web sites, your posts appear near the top. It has nothing to do with how many people are doing searches on the subject. I can do a search on karllarc and my posts pop right to the top. I am pretty sure no one is searching for me. Alright James. Lets see how many people you are likely to convince who just wander by and read your posts. You have been posting to sci.math for years. You have made well over 6000 posts. Hundreds or maybe thousands of people have heard your viewpoint. How many people have posted agreeing with you. How many have joined you in your great crusade. At the rate you are going, in 2 or 3 hundred millenia, you will have convinced enough people to fill a small 1 room apartment. Lets say that you get your message out. What do you think will happen. Lets say a producer for 60 minutes reads your posts and wants to put you on the air. You go on, do your 10 minute segment, and get everything off your chest. What do you think, or hope would happen? Would the nation rise up as one to tell them nasty, lying math people that they better stop. All those people who failed high school alegbra will say, I knew it. I knew they were lying. They made up all that a James Harris has outed them damn dirty liars. I knew I wasn't dumb. Seven years from now, James will probably still be posting the same old things, but he will have come up with a few more flawed proofs and theories that he can rant about. Meanwhile, more than a few amateur and professional mathematicians will actually come up with someting new, and will be recognized for it. But James will still be there crying liar, liar, pants on fire. Pretty pathetic, but hey that is his lot in life. Someone has to be the butt of jokes, James has taken that unliked job and ran with it. Well, comedy anyway. I prefer comedy myself. This is a bit like Jerry Springer, with the newsgroup as the audience, and you as the hapless subject. But I believe Springer's subjects get paid. Do they get a bonus if a fight breaks out? Mail-To-News-Contact: abuse@dizum.com Println, system. -- Lady Chatterly Of course if you think I am wrong you should continue responding to Lady Chatterly, if this helps you feel better about yourself....... -- Josef Oswald Count THIS, Moron! When you have the logistic map x_{n+1}= r* x_n(1 - x_n) When you set r = 4 there is a formula involving simplie functions that give you the orbit which is x_n= Cos ( 2^n ArcCos (1-x_0) ) I know this is true as it is esay to use induction to prove it. What i would like to know is how was this first derived and by whom? Any reference to texts or articals would be helpful as well. Do the other logistic maps also have formulas of this kind? stephen ---------------------------------------------------------------------------- Guess its time for me to say something profound....Nothing comes to mind. Stanislaw Ulam in Mathematics and Logic mentions a similar result using the x=Sin(t)^2 substitution: that t'=2*t It's called the period dubling property of this type of transform. Ulam mentions G. D. Birkhoff as a source of ergotic measure limits proofs just before that. You might try searchimg for him in Google. -- Roger L. Bagula email: rlbagula@sbcglobal.net or rlbagulatftn@yahoo.com 11759 Waterhill Road, Lakeside, Ca. 92040 telephone: 619-561-0814} Stephen a .8ecrit : I think it is obvious enough to have been derived as an exercise long ago (certainly *before* the logistic map)... No, except for the case r=0 (and similar ones) : as the zeros of the polynomial P_n(X)= x_n, with x_0=X, are related to the Mandelbrot set, any general formula would give a regularity to that set which doesn't exists... sqrt(x+7)-2sqrt(x-8)=sqrt(x-5) Solve x I found that x=9 using a calculator. I need to prove this on paper without the use of a calculator. If you need to write this out on paper and scan it, that will be fine. Send the picture file to me email address or post on user group. budking10@gmail.com A helpful identity is sqrt(x) + sqrt(y) = sqrt(x+y+2*sqrt(xy)), just got to be careful with sqrt's though. Jon Square both sides. Rearrange to put square root term on one side, everything else on the other side. Square again. Solve quadratic. http://www.want-to-be-sure.blogspot.com << Click On Link No!!! or Furthermore anything that relies on a number p being prime can't do anything more that show that if p|ab then p|a or p|b. But, hard as it is to believe, some people even try to count primes. - William Hughes oldest BFC. As proven by EVERYTHING you have ever generated or said! difficulty pre-calculus, order to But, the problem is, I haven't taken most upper level math courses twice. that the explain I'd be open to suggestions for good, thorough, and easy-to-understand books. But, I'd prefer free ones or any free online resources that already might exist. I actually already have a ton of books on advanced math, in addition to the text books that I need for class, and generally do not find them to be very helpful. Is there a site like www.purplemath.com for more advanced mathematics? moubinool.omarjee We know that for infinitely many n, for otherwise sum(x(n)) would converge. Omitting the other x(n)'s from the sequence, and using the fact that the sequence is nonincreasing, we can assume (*) for all n. Now exp(-x(n)/x(n+1)) ge exp(-2) and the result follows. Larry Hammick On second thought, no we can't. The hypothesis is no longer necessarily satisfied. Sorry. Assume the result to be false. Then sum(y_n; n integer, x_n / x_(n+1) <= 2) < +oo, whence sum(x_n; n integer, x_n / x_(n+1) <= 2) < +oo and sum(x_n; n is decreasing, you can extract from (x_n) a sequence (z_n) such that where a_n is the general term of a convergent geometric series. -- Julien Santini (x_n) Obviously I had in mind to take (z_n) the subsequence of (x_n) which (Otherwise the conclusion is impossible) -- Julien Santini i know the solution is a sphere and that the proof was as recent as 1882. does anyone know where i could find a copy of the proof? i don't On 15 Mar 2005 02:25:16 -0800, spinoza1111@yahoo.com You didn't see Derek Lowe's post Bravo, You Dolts! about his Speaking of ignorant ... that was Clarke, not Asimov. Should we forgive your anti-immigrant prejudice? -- http://hertzlinger.blogspot.com confuse third rate hacks. The Israeli settlers are colonists and not immigrants. On 14 Mar 2005 18:34:07 -0800, spinoza1111@yahoo.com Actually, we reactionaries figure that there is real evidence that climates have gotten warmer in the past century of two. We may quibble about whether that has ccontinued to the present but there are bigger disagreements are on whether that caused by humans and on whether it is likely to be beneficial. Our biggest disagreement is on whether, just in case global warming turns out to be a human-caused problem, it makes sense to cure it by turning large parts of the economy over to clowns protesting nuclear power. ... a good start ... ... expert testimony ... Didn't Australia also elect a reactionary government? During the afternoon of Election Day, I wondered if they would offer asylum. There were similar claims in WWI, which turned out to be exaggerated. The reluctance to take the Holocaust seriously in the intial stages was based on something. (By 1944, ignoring it was inexcusable.) Didn't the Bush administration turn down Enron's request for assistance? This calls for Nuclear Winter! -- http://hertzlinger.blogspot.com They simply didn't see any need to mention him while talking about breaking a theoretical limit? This isn't science. It's a sort of destructive instinct, Bourdieu's science, without the scientist in which Science can be a sort of organizational effort of specialists...who, I think, didn't see any need to mention Turing because in some cases on the authorial team (for these types of writings are generally collective efforts, not a single member feeling up to the task as an individual), the authors were homophobes and in other cases ignorant of Turing. Again, I conclude that the REAL dishonesty, the REAL economy with the truth, is located in these institutional writings which by compromise and LCD (least common denominator) thinking are MUCH more destructive than some French *flaneur* writing about science. spinoza1111@yahoo.com says... It would not have been difficult for you to inform yourself that the Church. Perhaps the author is a militant feminist who felt that mentioning a married man with three children would introduce a bias towards patriarchal chauvinism. Or it could be that, as I said, he did not feel a reference to the Church-Turing thesis was necessary. He did refer, in as much detail as I thought was necessary, to the differences between the two types of computer. I would very much doubt whether he was ignorant of Turing's work - few who have anything to do with computer science are, and indeed from his discussion it appears he is well aware of the relevant computational issues. - Gerry Quinn David, having had rather S^1 = R/Z in mind, I didn't get aware of this nasty twist you are pointing out. Certainly, you are right and my solution obviously does *not* apply to the problem posted. J. You might look at the thread positive integral powers of a transcendental number from December 1994: It seems that there is no proof that (pi^n) is dense modulo 1, although it's almost certainly true. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada (x/z - 2)(y/z - 2) = n (x - 2z)(y - 2z) = nz^2 xy - 2xz - 2yz + 4z^2 = nz^2 (n-4)z^2 + 2(x+y)z - xy = 0 Using quadradic formula z = (-2(x+y) +- sqr(4(x+y)^2 - 4(n-4)(-xy))) / 2(n-4) = (-(x+y) +- sqr((x+y)^2 - (n-4)(-xy))) / (n-4) = (-(x+y) +- sqr((x - y)^2 + nxy)) / (n-4) The basic premise of my mission is to dynamically populate a plane with a set of points that will be equidistant to one another and the outer points to be equidistant fom the perimeter of the plane. To expand on the concept, I am trying to build a formula, or function, which will kick back the coordinates (x,y) of a point based on the iteration (i) of the point in a series of points to be fit equidistantly (d) within a rectangle with a determined width (w) and height (h). Assuming we already know all of the numeric values for w, h, d, and i, and that 0,0 is the bottom left hand corner of the rectangle, I need to build a formula to kick back each of pairs of coordinates. Is this programmatically or formulaically (sp?) possible? knarf Here's an idea - - multiply both sides by z^2, expand, collect terms in powers of z on one side and use the quadratic formula to solve for z. --Lynn --Lynn If you don't like the notion 'catenary' which is quite common in commutative algebra, then you could pick the property of Z of being Jacobson. If I is a maximal ideal of Z[x], then the finitely generated (Z-)algebra Z[x]/I is finite over Z, hence integral. Now I cap Z is not equal 0: Otherwise Z sits injectively in the field Z[x]/I which is integral over Z, so Z is a field which is a contradiction. (This proof goes along with So the prime ideal I cap Z is always generated of some prime p e Z. Then you can proceed as Jos.8e proposed (in J. I = (2,1+x) Z[x]/I = { I, 1+I } ? Seems so. x + I = 2+x + I = 1 + 1+x + I = 1 + I (2, 1+x) = (2,x) = I_2 ?? No. Yet Z/(2,1+x) = Z/(2,x) ? Hm. ---- Yea, that's smelling like the right stuff. Is it maximal? If I add g to (p,f), then g,f coprime in Z_p[x] some h,k in Z_p[x] with hf + kg = 1 (mod p). Hence 1 in (p,f,g). Ah er, did you catch them all? Are you sure some stray maximal didn't escape? ---- Yes. I'm quite sure, yes. Jose Carlos Santos --------------------------------------------------------------------- How'd you convince a skeptic, a skeptic who hasn't the dubious pleasure of meeting Jacobson? So, how do you prove that any maximal ideal I of Z[x] contains some non-zero integer? If this is done, the rest is clear. J. And in order to know why is it true that any maximal ideal I of Z[x] contains some non-zero integer, all you have to do is to see that otherwise the restriction to Z of the projection of Z[x] onto Z[x]/I would be injective and then the prime field of Z[x]/I would be Q. Now, read the contributions to this thread posted yesterday by Zbigniew Fiedorowicz. Jose Carlos Santos Jos.8e, but why is x e Z[x]/I algebraic over Q then? In my solution the Jacobson property of Z is used. Or am I missing something in Zbigniew's posting? J. Certainly, since there is a non-constant polynomial in I giving an algebraic equation of x. So do monkeys, else why do you exist? You don't bury survivors. The room & the people going in & of it out is a distractor -- a red-herring -- misdirection. It is irrelevant to the question how many people remain. Furthermore the question How many people remain? is given without any context. It kind of reminds me of Zeno's paradox. He sets up a riddle with information about distance then asks a question relating to time - and makes it seem like a paradox. Its not. Jack Martinelli http://www.martinelli.org --------------------------------------------------------------------- v(0) = 100 gal n(0) = 0.02 * 100 gal = 2 gal w(0) = 0.98 * 100 gal = 98 gal v(t) = n(t) + w(t) dv/dt = 10 gal/min - 5 gal/min = 5 gal/min v = 5t gal/min + 100 gal 200 gal = 5t gal/min + 100 gal t = 20 min; v(20 min) = 200 gal dn/dt = 0.24 * 10 gal/min - n(t)/v(t) * 5 gal/min = 2.4 gal/min - n(t)/v(t) * 5 gal/min = 2.4 gal/min - n(t)/(5t gal/min + 100 gal) * 5 gal/min = 2.4 gal/min - n(t)/(t + 20 min) dy/dx = y' = a - y/(x + b) (x + b)y' = a(x + b) - y [(x + b)y]' = (x + b)y' + y = a(x + b) (x + b)y = (1/2)ax^2 + abx + c n(t) = (2.4 gal/min)(t^2 /2 + 20t min + c) / (t + 20 min) n(0) = 2 gal = (2.4 gal/min)c / 20 min 1 = (1.2 min^-1)c / 20 min c = (20/1.2) min^2 = (5/3) min^2 No. n(20 min)/200 gal --------------------------------------------------------------------- William Elliot corrects line break error. n(0) = 2 gal = (2.4 gal/min)c / 20 min 1 = (1.2 min^-1)c / 20 min c = (20/1.2) min^2 = (5/3) min^2 Oops... scratch that. Just forget about this part. I added the regularity condition so that the image of the parametrization would be locally one-dimensional. That can be evaluated in Maple 9.52 also as -I*conjugate(BesselK(0,-1/2))/sqrt(Pi*exp(1))/2; evalf(%,20); .57164762453873852150-.15816730823096183196*I or -I*conjugate(KummerU(-3/2,-1,-1)); evalf(%,20); .57164762453873852151-.15816730823096183197*I Could you please show how you got to your representation? Best wishes, Vladimir Bondarenko http://www.cybertester.com/ http://maple.bug-list.org/ http://www.CAS-testing.org/ That gives the same numerical result, but not in all CAS. The branch cut of BesselK(0,x) is negative axis, so the values for negative x defined only conventionally - they can be defined as limits from either one side of the cut, or another one (which in this case determines the need of conjugation - the values from different sides are conjugate). Same problem here. OK. The hypergeom([1/2,-3/2],[],x) is not a function - it is only a series 1-3/4*x+... If this series was convergent in some neighborhood of 0, then we could extend it to complex plane using either analytical continuation, or some functional equations. It is not the case here. The series is convergent only for x=0. In this case, the function corresponding to this series can be determined through differential equations. series(hypergeom([1/2, -3/2], [], x),x,100): gfun[seriestodiffeq](%,y(x)); / 2 /d 2 |d | [{y(0) = 1, D(y)(0) = -3/4, -3 y(x) - 4 |-- y(x)| + 4 x |--- y(x)| dx / | 2 | dx / }, ogf] dsolve(%[1]); 1/2 / 1 1 y(x) = 1/2 Pi |(1 - x) BesselI(1, ---) + BesselI(0, ---)| 2 x 2 x / 1 / 1/2 exp(- ---) / x + 2 x / 1 / 1 1 _C2 exp(- ---) |BesselK(0, ---) + (-1 + x) BesselK(1, ---)| 2 x 2 x 2 x / ----------------------------------------------------------- 1/2 x What value of _C2 to choose - depends on a convention. It is undocumented in Maple, so we can only guess. Some calculations (in different Maple versions) show that Maple's choice for positive x is _C2=-1/2/sqrt(Pi)*I. For negative values, the formula should be corrected, as well as for non-real values, to get the same answer as Maple's hypergeom. That's why I wouldn't call that a regression bug. Regression bug is something that worked well before and started work wrong in a new version. In this case, hypergeom([1/2,-3/2],[],x) as a function of x is undocumented in Maple, so it's value means nothing - it could be changed with every new version and changing undocumented features is not a bug. Something like the following. If we type %%; starting Classic Maple, we get 10. What if it would be 125 in the next version - it is different, but it is not a bug, and not a regression bug, because it is an undocumented feature. Crashing Maple's kernel is a serious bug though - just not a regression bug. Alec http://math.tntech.edu/alec/ At the moment of asking you I already realized that it is hardly possible to get this transformation at a single 'natural enough' step, say using convert etc. I am also not surprised much with your usage of gfun, I myself adore such stuff (but this might be instructive for tyros). So a hidden goal of my request :) was to show that actually, it is not very plausible that a rank-and-file Maple customer would discover the conversion quickly... Thus, this crash provides us with an example where an external help for a Maple user might be essential. But this means precisely that Maple is user-hostile here, which is definitely something that concertinaed the hope of the user to get the desired result, which he or she spent money for. Also, I asked you because our dialogue (not a static statement, but a warm, dynamical, emotional, developing essence) could be interesting for (math) students, maybe not to the absolute newbies but to the persons who have interest in experimental math. After all, almost all of us learn via imitation of the best behavioral sample available at fingertips... Yes, this is true. Here in Maple 9.5.2 we deal with a 'regular' bug... ... just because we are not explained, what to expect at the cases like the given one... ...which itself, in my opinion, makes one of... documentation bugs ;) (should we use a 'feature' euphemism here?) Because here we encounter two distinct, incompatible Maple behaviors in several versions, and have no foothold whatsoever to see what we should expect in the quality of a 'regular' behavior. .5716476245-.1581673082*I Error, series is divergent Error, (in hypergeom) hypergeometric series diverges At any rate, in Maple 6 we also have a bug (Maple 6 keeps running after 10,000 seconds). Best, Vladimir Bondarenko http://www.cybertester.com/ http://maple.bug-list.org/ http://www.CAS-testing.org/ This should be treated with much caution: the DE has an irregular singular point at x=0, so initial conditions there are quite iffy. If I'm not mistaken, for positive real x, sqrt(Pi)/2/sqrt(x)*exp(-1/(2*x))*((1-x)*BesselI(1,1/(2*x))+BesselI(0,1/(2*x) )) = 1 - 3/4*x + O(x^2) while 1/sqrt(x)*exp(-1/(2*x))*(BesselK(0,1/(2*x))+(x-1)*BesselK(1,1/(2*x))) = O(x^2*exp(-1/x)). So for _C2 = 0 you have a removable singularity at x=0; for any _C2 you have I don't know why previous Maple versions would take a nonzero value of _C2. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada Yes, all this already worked one fine day... This is just an example of thousands regression bugs calculated automatically. hesitation, in this thread Maple quality obviously increases, especially in the most recent versions. Oh yes. Obtaining dry figures... what a dull, indeed. Aren't crashes much more entertaining? evalf(hypergeom([1/2, -3/2], [], 1)); Execution stopped: Stack limit reached. Execution stopped: Stack limit reached. Execution stopped: Stack limit reached. -------------------- (2002) Maple 8 -------------------------- .5716476245-.1581673082*I -------------------- (2001) Maple 7 -------------------------- .5716476245-.1581673082*I -------------------- (2000) Maple 6 -------------------------- Maple keeps running after 10,000 seconds -------------------- (1997) Maple V Rel 5 -------------------- .5716476245-.1581673082*I -------------------- (1995) Maple V Rel 4 -------------------- Error, series is divergent -------------------- (1994) Maple V Rel 3 -------------------- Error, (in hypergeom) hypergeometric series diverges --------------------------------------------------------------- Here we observe one of several typical Maple quality deterioration types, an oscillatory quality degradation. Its pattern reminds us an irregular wave --+-++--- With the deepest respect to your computational flair, Vladimir Bondarenko http://www.cybertester.com/ http://maple.bug-list.org/ http://www.CAS-testing.org/ That's not exactly a regression bug, even if looks like it. I don't want to go into a deep discussion about that though. The idea is that people who don't use hypergeom of that type, shouldn't care about that (and they are not used internally in any other functions in Maple). People, who use it, should be able to find any particular value they need without any difficulties (as I did). That is a good demonstration of one specific weakness of automated bug testing. Most of the bugs that were found that way are such bugs that nobody meets in real calculations and nobody cares about (except automated bug testing system creator). Alec that the extension of the intersection of these ideals is contained in the intersection of the extensions of these ideals: (I cap J)^e in I^e cap J^e, where for example I^e is the ideal in B generated by f(I). However, the converse dose not smell true. Sadly, I haven't been able to cook up a counterexample Please let me know if you have a counterexample for the converse. DMW So why do people still reply to him? -- Giuseppe Oblomov Bilotta Can't you see It all makes perfect sense Expressed in dollar and cents Pounds shillings and pence (Roger Waters) We all have our reasons I guess. o[CapitalYAcute]in for is we tough JSH is trapped by his own great mass of lies. Hundreds of times he has said things like this: I've yet to see a poster with the abilities to challenge any of my work. There is no error. But all his fictions explode in his face, and then we hear: It turns out it's not hard to understand the ... With this little game http://www.adf-tank.com/adftank/sites/fo/redir.php?c=1&g=1&m=1&u=70&l=fr_FR& k=1T5SH7TOa3Pz2UJ6fGKjQLtT3oUHN Darn, not many replies. Does it really look like homework? :-( To be honest, I have never seen any inner product defined on R^n other than sum(a_i x_i y_i). Hence my question if maybe inner products defined on R^n always look like that, by the linearity condition maybe. Well, any inner product on R^n can be expressed as: if we think of x,y as column vectors, for some positive definite matrix A. If you are allowed a change of basis, you can get to a diagonalization of A, which is essentially the form you proposed, and this change of basis can be orthonormal, ie. mapping perpendicular unit vectors to perpendicular unit vectors. In that sense, yes, all inner products on R^n look like that. Not really sure what more you'd like to know, e.g. about parallelogram law. Surely a Web search will quickly tell you what it is? I'll be happy to help you understand linear algebra, even if you are studying it !!! Concerning the parallellogram law, I don't immediately see how the parallellogram, as you said, a parallellogram law would imply that a vector norm has an inner product origin. Can you elaborate on this? Interesting. But this relation does not guarantee that the defined inner product does indeed satisfy the requirements to be an inner product, or does it? This is really new. I believe that this is going to be a real theoretical breakthrough. In marketing, if life, everywhere. Let me please reformulate into simple, plain words this important vision. Obviously, all the variegated Maple customers, from now on, must be stratified. 1. Maple customer of 1st quality (like Alec Mihailovs, maybe a thousand on the planet). To this category of Maple customers (usually they are very smart, and have multi-years Maple experience) is allowed to use any Maple features. If a customer of 1st quality encounters a crash, or an error message, or virtually any other bug, he just hack the Maple source code on the fly and resume the computations. For a Maple customer of 1st quality is not necessary reading the Maple Quality Prayer Book. 2. Maple customer of 2nd quality (without much math experience, millions on the planet). To this category of Maple customers (usually they are not so smart as compared with the Maple customers of 1st quality, and have a limited Maple experience, if any) is NOT allowed to use any Maple features, only the simplest ones. If a Maple customer of 2nd quality encounters a crash, or an error message, or virtually any other bug, he or she usually gets only frustrated and has no good ideas on how to proceed. Often, however, especially if there is no crash or error message, a customer of 2nd quality just cannot realize that he or she encountered a bug, and keeps computing, just as if nothing mathematically invalid had happened. Never mind. All the same a Maple customer of 2nd quality has no (and cannot has by definition?) important tasks in life, so why bother? So if you, unluckily, are a Maple customer of 2nd quality, consume beer, go to dancing, opt out of advancing in math, forget your creative urge, do NOT explore anything more complex than 2*3;, ignore you high impulse. This is not for you. Remember, your principal task is to keep reading the Maple Quality Prayer Book, regularly. Best wishes, Vladimir Bondarenko VM and GEMM architect Co-founder, CEO, Mathematical Director Cyber Tester, LLC http://www.cybertester.com/ http://maple.bug-list.org/ http://www.CAS-testing.org/ That is not the right classification. The right classification might be following: not 2 types of customers, but 2 types of Maple functions (and other features) - documented and undocumented. Again, not customers, but functions of 2 types. If it is a crash, or an error message, or other bug in a documented function - it is a bug and a problem. If undocumented features don't work as expected, or work differently from version to version - it is different. Alec huh? isn't concept of limits actually the beginning of calculus? how did you learn calculus while skipping the idea of limits? i don't know how else you can tell a student how to formalize the notion of limits without Cauchy's epsilon-delta formalism. The arithmetic problems are greatly overstated; the Greek method of arithmetic does not suffer much compared to the modern Arabic numbers, which many have suggested came from the Indians applying the base 60 positional number notation of the Babylonians, originally Sumerian, to base 10. The way numbers are written for arithmetic purposes is not in any way of much importance. BTW, the Hellenistic Greeks used base 60 for their fractions, and had no problem with using base 10 for the integer part and base 60 for the decimal part. I think the problem was not having the use of variables, especially multiple variables. This is surprising in that Euclid labeled points in his diagrams, but the idea of labeling quantities had not occurred. The use of one symbol for a number seems to have originated with Diophantus long after Euclid, and was carried by the Muslims. However, the systematic use of several variables was 16th century. The algorithm of al-Khwarismi consisted of collecting the terms in the variable on one side and not in the variable on the other (al muq'ballah (sp?)) and dividing to get the magnitude (al jabr). This is not any kind of advance. The ideas all seem to have been Greek, until advanced in western Europe after the Renaissance. -- 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 Am 19.03.05 15:53 schrieb Karl Pech: I don't believe that this thread is in a form that you could learn something out of it. Better is any book (or online-text), which deals with the irrationality of sqrt(2) (proportion of the squares side to its diameter) Gottfried Helms Hallo Gottfried, Well, these threads are quite readable at the beginning (I mean the first 100 postings or so :) ), but you're right; After more than 1000 postings which repeat the same arguments over and over again, the replys loose some quality, because cranks usually tend to deny correct postings and the responding people get tired and angry. Well, this is always a better idea! ;-) Karl -- [Werbung] Greift nach den Sternen auf www.vorhilfe.de ! =))) [/Werbung] simply that that reasonably set, other like, that In some cases, like the one above, I note that the set has a definition as being every second element of the first. Seems pretty clear to me. I'm not going there, because it seems clear that you conflate procedures and formulas into being the definitions, and that to me is totally wrong. The roots of a polynomial equation are not defined as per elimination or the matrix approach, but those methods can discover them. : simply : that : that : reasonably : set, : other : like, : that : In some cases, like the one above, I note that the set has a definition as : being every second element of the first. Seems pretty clear to me. That does not apply to [0,1] and [0,2]. And your definition is invoking an ordering, and sets are not ordered. : I'm not going there, because it seems clear that you conflate procedures and : formulas into being the definitions, and that to me is totally wrong. The : roots of a polynomial equation are not defined as per elimination or the : matrix approach, but those methods can discover them. So you cannot define it. Basically your answer is Ask Allan. What is the point of a definition that only you can use? Stephen determine as If I have to invoke an ordering to see what is true, then so be it. I was trying to be more precise, and you took that as a problem with ordering. Sigh. And for [0,1] and [0,2], the argument is over range of numbers. and The You conflate methods and definitions. There are multiple methods to determine if something meets a definition, and not all of them will work in all cases. : determine : as : If I have to invoke an ordering to see what is true, then so be it. I was : trying to be more precise, and you took that as a problem with ordering. : Sigh. : And for [0,1] and [0,2], the argument is over range of numbers. : and : The : You conflate methods and definitions. There are multiple methods to : determine if something meets a definition, and not all of them will work in : all cases. I just want to know how you determine if two sets have the same number of elements or not. Apparently you have a method, because you seem quite certain about which sets have the same elements and which do not. You seem to use different rules in different cases, but somehow you apparently know which rule to use in which case. Why can you not explain your rules and how you know when they apply? There has to be a method, or else you are just guessing. Stephen get Excellent. Then I'll say that the set of integers has more elements than the set of even integers and you'll leave me alone since that won't be a comment on the definitions of your mathematics, okay? with be But I DON'T have to a) provide one that always works and b) don't have to conflate the definition with the method. I can claim, in the first case, that the number of elements (or relative number of elements) for certain sets cannot be, at least, currently determined and can argue in the second case that if my method leads to a conclusion that seems to violate the definition that the definition is to be taken to be true, and the method fails in that case. For example, to me for finite sets the bijection approach works, since it seems clear that you can only map two finite sets onto each other one to one if they have the same number of elements. But with infinite sets, this seems to be a contradiction, since you can map something that has precisely half the elements of the other onto each other one-to-one. So the method, in my mind, doesn't work in this case, and so the method does not apply to infinite sets. You can redefine number of elements to cardinality and then insist that it does work, but don't expect anyone to blindly accept that definition. In short, use it as an axiom in your mathematics and only bother people about it if they claim to work in your mathematics. In short, I'm again giving you the option to accept that you take this as an axiom and leave everyone else alone. Will you take it this time? If not, why not? what I The problem is that every time I tell you, you translate it to subsets and then insist that that is all I'm doing. I am not responsible for your doing this. case, the then by for see definition set. No, because what you are doing is building an octal set that represents -- wait for it -- all integers. That's part of the definition of the set. You could form it by a random number generator and it would make no difference. In the other case, you could form the set any way you wanted, but ultimately as part of its definition it is half the elements as the set of integers, since its definition is every second integer. I hope this makes things clearer as to what I mean by definition. : get : Excellent. Then I'll say that the set of integers has more elements than : the set of even integers and you'll leave me alone since that won't be a : comment on the definitions of your mathematics, okay? And I will say that until you define what same number of elements means that you are literally talking nonsense. You are after all using a term which you refuse to define, so there is no way for anybody to know if what you say makes sense or not. : case, : the : then by : for : see : definition : set. : No, because what you are doing is building an octal set that represents -- : wait for it -- all integers. That's part of the definition of the set. You : could form it by a random number generator and it would make no difference. No, the defintion of the set is [1-7][0-7]*. It is the set of all strings that begin with a 1,2,3,4,5,6 or 7 and are followed by any number of 0,1,2,3,4,5,6, or 7's. There is nothing in the definition that says anything about the meaning of the strings. Of course here you are implicitly using the idea of a bijection to claim that the set of octals has the same size as the set of naturals because you know that each string can be mapped to an integer. Why does it make sense to use bijections in this case when you elsewhere claim that they do not apply to infinite sets? : In the other case, you could form the set any way you wanted, but ultimately : as part of its definition it is half the elements as the set of integers, : since its definition is every second integer. I hope this makes things : clearer as to what I mean by definition. The evens are not defined as every second integer. The evens are defined as { y | y=2*x for some integer x}. There is nothing in the definiton about every second integer. Back to the octals and the decimals, using Perl regular expression notation, we have octals = [1-7][0-7]* decimals = [1-9][0-9]* Clearly the octals are a subset of the decimals. The following set is also a subset of the decimals S = [1-7][0-7]* + [89][089]* The two parts to this set are clearly disjoint. We know that the number of elements in [1-7][0-7]* is the same as the number of elements int the naturals. What about [89][089]*? Well that should have the same number of elements as the set of all base 3 strings [12][012]*. The number of base 3 strings should not depend on what characters we use to represent the three possible digits. You must also agree that the number of elements in the set of base 3 strings equals the number of elements in the naturals. So what is the number of elements in S? Clearly it is less than or equal to the number of elements of the decimals, because it only contains some of the decimal strings. You agree that the number of elements of the decimal strings is equal to the number of elements of the naturals. So we have number of elements in S = number of elements in octals + number of elements in base 3 strings = number of elements in naturals + number of elements in naturals <= number of elements in decimals = number of elements in naturals. So using definitions that you apparently agree to, we can show that number of elements in naturals + number of elements in naturals <= number of elements in naturals. But this was one of the conclusions that you claimed was ridiculous and evidence that cardinality could not be the number of elements. Stephen What you are -really- saying and would if you would use more exact terminology is that the set of even integers is a proper subset of the set of integers. This use of the word more has an unfortunate consequences. If you are consistent you cannot use more to compare disjoint sets. What about the set {a,b} and the set of integers. If you restrict yourself to sets which are partially ordered by containment you have no basis for asserting there are more integers than there are elements in the set {a,b}. Is this what you really want? Bob Kolker than that all proper supersets have more elements -- by my definition -- than their proper subsets, then more the worse for you. I don't insist that that is true, however. I already pointed out how I can indeed do so, so this is an utterly irrelevant comment. Obviously the set {a, b} has two elements, and the set of integers has more than two elements, and so the set of integers has more elements than the set {a, b}. That was simple. The problem is that I DON'T restrict myself to that, and it is only people like you and Stephen who conflate method and definition that get confused and believe that I would. O.K. In what sense (by your definition, now) does the set of integers contain more elements than the set {a, b}? Be precise, be complete, be logical and be neat. Bob Kolker But the example isn't talking about mathematical validity, since there isn't such a thing for measuring cups, is there? The point was that if there was a practical point where measuring could not be done because of the size of the fraction, the pint measure gets there faster than the 2 cup measure. And that was my assumption when I answered the question the way I did. some people in Then using the 2 cup measure would allow for meaningful measurements that the 1 pint measure would not. Which was my point and opposed Stephen's point. And the quality of your thinking shows your intellectual snobbery to be unjustified. someone that It's also utterly unimportant, except to mathematicians, since picturing that is irrelevant. that for feet your So what? Would you ever use that micro-measure in real life? If not, I fail to see what your point is. This is not the point at all. Measuring cups care little for mathematical notions, and so practical viability is more important for them than mathematical niceties. Perhaps Stephen's example was just misleading. If a scientific theory has no relation to the empirical world that we experience, it is useless. If it does, then it can be explained in terms of what we experience, even if only indirectly. And any theory that contradicts the intuitions built up through experience is constrained to explain why our intuitions and our experiences are incorrect. If it does not, then it is merely an imposed theory by a group of people who hold the intellecutal conceit that they are smarter than everyone else and that what they think is correct regardless of experience, which would turn science and mathematics into something that embodies the same qualities as the worst form of religious dogmas. As soon as you talk of fractions (proper rational numbers) you have brough in mathematics. Bob Kolker isn't But not mathematical validity. The fact that my car as 1/2 a tank of gas remaining gains no importance from its mathematical validity (and no one cares about that in that case anyway). But not modern math. All mathematics introduced since the Renaissance is modern mathematics. It represents those mathematical developments which go beyond the classical greek or hellensitic mathematics. If you mean mathematics done in a formal style that goes back to the time of Cantor (circa 1870) and subsequent developments. You can even take it back to Galois who invented group theory in circa 1830 to prove the unsolvability of quintic polynomials by root extraction. You really have no knowlege of the history of mathematics, do you? You are an intellectual vandal who draws graffite over ideas you cannot or will not comprehend. Bob Kolker Quite a claim considering its author is forced to rely on Euclidean ruler and compass construction to define a circle he can't define in modern math terms correctly as the set of all points equidistant from any point. Which your modern math definition for a circle certainly doesn't. I mean mathematics as the derivation of universal knowledge applied specifically to spatial dimensionality and the study of numeric concepts. My knowledge of mathematics tends to focus on universal concepts, something your modern math definition of the circle doesn't even if you appeal to history for justification. And you are an intellectual dwarf who smears on the graffiti of others because you cannot read it. -- I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives. - -- Tolstoy Do you have a cell phone? -- Giuseppe Oblomov Bilotta Can't you see It all makes perfect sense Expressed in dollar and cents Pounds shillings and pence (Roger Waters) No, but I can think for myself. That is a very witty question. What does it mean or signify or intend? Bob Kolker Context, Bob. Context. Only an arrogant mathematiker, like Burroughs' talking asshole, would claim that all talk of counting and measuring was properly only in the domain of mathematics. -- I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives. - -- Tolstoy Fraction i.e. a ratio of the form m/n where n !=0 and m, n integers is a mathematical concept. Bob Kolker There is a distinction between mathematics, Bob, and modern math. So you claim. Now show the difference. Be very explicit. Bob Kolker He means there's a difference between math and his confused recall of what he learned from those mathematiker nuns, who cowed him into submission by threatening to give him a failing grade if he didn't agree with them, while all the time he just knew they were wrong. They showed him pictures of giraffes, you see, and he knew there warn't no sech animals. But, as Torkel has admitted, he _is_ a great student of fnoffleness, perhaps even of fnoffleality, which I know to be a study of much greater omphaloidity, and way beyond my poor powers of combombulation. Or perhaps you could just define a circle for us, Wolf, that doesn't define a sphere instead? The difference between your first modern math definition for a circle which actually defines a sphere and your second Euclidean definition for a circle which allows you to pretend that circles are well defined as the set of all points equidistant from any point without definition for spatial dimensionality that allows you to pretend dimensionality is just so much vulcanized rubber. You said there was a difference between mathematics and modern math. That is a very general statement. Substantiate it. Address yourself to the question I asked. Schmuck. What is the dimension of a plane, in the sense of maximal number of mutually orthagonal lines lying on a plane? Think! Put all 13 of your neurons to work. When I made the defnition complete (my appologies for the initial omission) I made it plain that it was a figure on a plane. Now what can be plainer than a plane. Dimnsionality for vector spaces is not rubber, it is the cardinality of the maximal set of linearly independent vectors in the vector space. One must show that all maximal linearly independent sets have ths same cardinality (easy for finite dimensional vector spaces, not so easy for infinite dimensional vector spaces). Except for specifying that a circle is a figure on a plane the definition gives no refernce to dimensionality whatsovery. In fact one can show a circle on a plane (a two dimensional surface) is a one dimension set since it can be parametrically generated by one variable, to with the angle an arbitrary radius makes with a reference radius. That conclusion is not immediately clear from the definition. A lot of theorems have to proven to show that. Bob Kolker I just did. The difference between your first and second definitions for a circle, Bob1 and Bob2, is what defines the difference between modern math and mathematics respectively. Very good, Bob. I'll have to borrow your 2 neurons. Why don't you defne a plane for us so I can do the higher modern math needed to define the number of mutually orthogonal lines lying on it for you so you won't have to think for yourself. The question is, I think, what could be plainer than your lack of definition for a plane in terms of the points you use to define a sphere instead of the circle you call it. In other words, you have to define the plane, Bob, and not just ask rhetorically what could be plainer than a plane. Obviously you don't know. But the dimensionality of space is? Yada yada, whatever. If you say so, Bob. Of course what you're saying isn't anything worth writing home about much less posting. Lots of things are easy for mathematikers, Bob, especially when they feel comfortable drawing circles in the air to pretend they've defined planes. Oh, well, Hello? Earth to Bob. Earth to Bob. What in the hell do you suppose gives reference to dimensionality other than references to dimensionality? Gee, Bob. So now a circle is a one dimension parametric which only relies on two dimensions in space for its parametrization? You should have quit while you were ahead. Oh, that's right, you were never ahead. You just keep going backward and stumbling over your own feet. None of your conclusions are immediately clear from your definitions, Bob, because so far you haven't given any definitions for the spatial dimensionality your definitions for a circle rely on and assume. Only by greedy mathematikers who like to believe that /everything/ properly belongs to mathematics. Actually, it is a social concept developed by Neanderthals to facilitate the sharing of a deer shank. -- I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives. - -- Tolstoy that reasonably set, other Why do you think that's a problem? I never conflated this particular case (where the sets have elements in common, or where one is a subset of the other) to the generic method for calculating or determining the number of elements in a set. The relative number of elements just seems obvious in that case. Nope. You get the first from the second by adding 5/2 Well, since I don't do that this isn't a problem. All you can say about my definition is that one way of determining it is to look at how the sets themselves are defined. This is NOT how we count. This is, perhaps, ONE WAY of counting that completely misses the point of what we want when we count. When we count, we want the integer N that reflects the number of elements in a set, regardless of whether we have or would have a set of integers that could be matched with it. For example, one way of counting (or determining the number of elements) would be if you had a group of things that all had the same weight would be to weigh the things and then divide by the weight of each element. But this has NOTHING in common with the approach you suggest. It is merely accidental that you can also express a counting relation as a set in the manner you suggest, but to determine that that is just what we do is conflating the notion to a ridiculous degree. I do NOT map the elements one-to-one onto a set, except only in an accidental sense. What I DO is map the total number of elements counted SO FAR to a placeholder in memory to allow me to continue my counting. The fact that this works out to a mapping on a set is unintentional and uninteresting. You could ... but why would you? This seems like an utterly complex way to express it, using an accidental property to replace the notion that we actually use which is Two finite sets have the same number of elements if the number of elements is N. Seems? How about giving a rigorous definition of the phrase relative number of elements? Would you please do that? And since your use of more has been restricted (by you) to sets such that one is contained in the other, please do not use that word. Bob Kolker in Please define rigourous definition and show me why that would matter first. And since your use of Nope, since I have not done so. A rigourous definition first would identify terms you are leaving undefined (there must be such) terms. The remaineder of the definition to conform to common mathematical usage. Now tell us what you mean by relative number as opposed to just plain number or un-relative number. What meanind does relative have here? Or are you going to leave the entire phrase relative number undefined. If so, at least provide us with some postulates you would assert about relative number that might convey some meaning. For example in synthetic geometry we do not define point or line, but we do assert (as a postulate) that points and lines can be coinincident, i.e points can lie on lines and lines can lie on points. We assume given two distinct points there is exactly one line that lies on those points and that given two distinct lines, if they intersect at all they intersect at exactly one point. The postulates provide the semantics of the undefined terms. Now please provide assumptions which relative number must satisfy. For example what is the relative number of the sets {x, y} and {a, b, c, d} all items assumed to be distinct. Provide examples to try to convey what you mean by relative number. Bob Kolker How about giving us a rigorous definition for spatial dimensionality, Bob? Thee are several definitions of dimension of spaces. Which do you want? Go to this site: http://www.absoluteastronomy.com/encyclopedia/d/di/dimension.htm Here is an extract from that site. Dimension [Categories: Linear algebra, Algebra, Abstract algebra] Dimension (from Quick Facts about: Latin Any dialect of the language of ancient RomeLatin measured out) is, in essence, the number of Quick Facts about: degrees of freedom Quick Summary not found for this subjectdegrees of freedom available for movement in a space. (In common usage, the dimensions of an object are the Quick Facts about: measurement The act or process of measuringmeasurements that define its Quick Facts about: shape The spatial arrangement of something as distinct from its substanceshape and size. That usage is related to, but different from, what this Dimension - Physical dimensions The Quick Facts about: spacetime Quick Summary not found for this subjectspacetime in which we live appears to be 4-dimensional. It is conventional (and for most practical purposes entirely sensible) to consider this as three spatial dimensions and one of time. We can move up-or-down, north-or-south, or east-or-west, and movement in any other direction can be expressed in terms of just these three. Moving down is the same as moving up a negative amount. Moving northwest is merely a combination of moving north and moving west. Time is frequently referred to as the fourth dimension. It is somewhat different to the three spatial dimensions in that there is only one of it, and movement is only possible in one direction. Some theories predict that the space we live in has in fact many more dimensions (frequently 10, 11 or 26) but that the universe measured along these additional dimensions is subatomic in size. See also Quick Facts about: string theory Quick Summary not found for this subjectstring theory. In physics, the dimension of a quality is the expression of that quality in basic units: the dimension of speed, for example, is length divided by time. See Quick Facts about: Dimensional analysis Quick Summary not found for this subjectDimensional analysis. Dimension - Mathematical dimensions In Quick Facts about: mathematics A science (or group of related sciences) dealing with the logic of quantity and shape and arrangementmathematics, no definition of dimension adequately captures the concept in all situations where we would like to make use of it. Consequently, mathematicians have devised numerous definitions of dimension for different types of spaces. All, however, are ultimately based on the concept of the dimension of Quick Facts about: Euclidean n-space Quick Summary not found for this subjectEuclidean n-space E n. The point E 0 is 0-dimensional. The line E 1 is 1-dimensional. The plane E 2 is 2-dimensional. And in general E n is n-dimensional. A Quick Facts about: tesseract Quick Summary not found for this subjecttesseract is an example of a four-dimensional object. mathematical definitions of dimension. Hamel dimension For Quick Facts about: vector spaces Quick Summary not found for this subjectvector spaces, there is a natural concept of dimension, namely the cardinality of a basis. See Hamel dimension for details. Manifolds A connected topological Quick Facts about: manifold A pipe that has several lateral outlets to or from other pipesmanifold is locally homeomorphic to Euclidean n-space, and the number n is called the manifold's dimension. One can show that this yields a uniquely defined dimension for every connected topological manifold. The theory of manifolds, in the field of Quick Facts about: geometric topology Quick Summary not found for this subjectgeometric topology, is characterised by the way dimensions 1 and 2 are relatively elementary, which to 'work'; and the cases n = 3 and 4 are in some senses the most difficult. This state of affairs was highly marked in the various cases of the Quick Facts about: Poincar.8e conjecture Quick Summary not found for this subjectPoincar.8e conjecture, where four different proof methods are applied. Lebesgue covering dimension For any Quick Facts about: topological space (mathematics) any set of points that satisfy a set of postulates of some kindtopological space, the Quick Facts about: Lebesgue covering dimension Quick Summary not found for this subjectLebesgue covering dimension is defined to be n if n is the smallest integer for which the following holds: any open cover has a refinement (a second cover where each element is a subset of an element in the first cover) such that no point is included in more than n + 1 elements. For manifolds, this coincides with the dimension mentioned above. If no such n exists, then the dimension is infinite. Hausdorff dimension For sets which are of a complicated structure, especially Quick Facts about: fractal (mathematics) a geometric pattern that is repeated at every scale and so cannot be represented by classical geometryfractals, the Quick Facts about: Hausdorff dimension Quick Summary not found for this subjectHausdorff dimension is useful. The Hausdorff dimension is defined for all Quick Facts about: metric spaces A set of points such that for every pair of points there is a nonnegative real number called their distance that is symmetric and satisfies the triangle inequalitymetric spaces and, unlike the Hamel dimension, can also attain non-integer real values. The upper and lower Quick Facts about: box dimensions Quick Summary not found for this subjectbox dimensions are a variant of the same idea. Hilbert spaces Every Quick Facts about: Hilbert space A metric space that is linear and complete and (usually) infinite-dimensionalHilbert space admits an orthonormal basis, and any two such bases have the same Quick Facts about: cardinality Quick Summary not found for this subjectcardinality. This cardinality is called the dimension of the Hilbert space. This dimension is finite if and only if the space's Hamel dimension is finite, and in this case the two dimensions coincide. Krull dimension of commutative rings The Quick Facts about: Krull dimension Quick Summary not found for this subjectKrull dimension of a commutative ring, named after Quick Facts about: Wolfgang Krull Quick Summary not found for this subjectWolfgang Krull (1899 - 1971), is defined to be the maximal number of strict inclusions in an increasing chain of Quick Facts about: prime ideal Quick Summary not found for this subjectprime ideals in the ring. More dimensions Quick Facts about: Dimension of an algebraic variety Quick Summary not found for this subjectDimension of an algebraic variety Topological dimension Quick Facts about: Isoperimetric dimension Quick Summary not found for this subjectIsoperimetric dimension Poset dimension Pointwise dimension Lyapunov dimension Kaplan-Yorke dimension Exterior dimension Hurst exponent q-dimension; especially: Information dimension (corresponding to q=1) Correlation dimension (corresponding to q=2) There are over a dozen definitions (all different) of dimension. The most commononly used notion of dimension is the cardinality of the maximal set of linearly indendent vectors in a vector space. This correspons (roughly) to the number of orthagonal axes that generate the vector space. There are also fractional dimensions used in fractal theory. For example Hausdorff Dimension. For uses in algebraic geometry Spener or Lebesqu covering diemsions is used. So there is no one answer to your question. The definintion is dependent on the context and usage. Bob Kolker The correct one, Bob, the one that allows you to construct circles using Euclidean ruler and compass methods to define spatial dimensionality while denouncing Euclidean spatial concepts, Bob. All the definitions I referenced are correct. They are correct in the contexts in which they are applied. Dullwits seem to think dimension meanss the maximal number of mutually orthagonal lines that can be constructed in a space. That is one definition of dimension, but not the only definition. Fractal sets have fractional Hausdorf Dimension, for example. Bob Kolker Which doesn't seem to include your definition of a circle much less spatial dimensionality. Oh, there are lots of definitions of spatial dimensionality, Bob. I expect there is one right one and lots of wrong ones. So far you seem to be making your way through a lots of undefined definitions. And you have your unreal dimension, Bob. I was asking about spatial dimensionality. What space? There are many different kinds of spaces (mathematically speaking). Which space, what space. Specify please. If you are talking cardinal number of a maximal set of linearly independent vectors. In the euclidean 3 space that number is three. Given a set of four vectors or more there is a linear dependence among them. Bob Kolker Whatever space and dimensionality you needed to transmogrify your definition of a sphere into a circle. Neither mathematiker elitists nor their profession nor their language are protected by law. You and your private religion are fair game, just as you believe that theists are fair game. Your 'privileged' position in society is not merit based, but merely assumed by you. You assume that because you spent your early years learning a trade that you are somehow more special than others who learned other trades. The unmitigated arrogance of mathematikers is unmatched anywhere else in society, with the possible exception of politics. And don't reply with that same old crap about how curved space, probability clouds and undead cats are responsible for everything good in the world; When you do that you remind me of William Burroughs' talking asshole in _Naked Lunch_. -- I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives. - -- Tolstoy I, personally, have no privileged place. On the other hand mathematics has made modern physics and ALL of its technological entailments possible. Our industrial and technological society has as one of it pillars mathematics. Without mathematics there would be no physics. Without physics there would be only crude technology. Certainly nothing much beyond what the Romans and the Greeks had. Mathematics is a necessary compenent of modern science. Bob Kolker Now replace the term mathematics in the foregoing with the phrase modern math and see how stupid that makes your assertion look. Just as the asshole is the most important part of our anatomy. Without it we would swell to enormous size, our eyes would cross, our breath would become labored and foul, our brain would cloud and our muscles would cramp. So, yes, Bob, you are probably right. -- I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives. - -- Tolstoy You, on the other hand, Bob, take all this into consideration and still get the definition of the circle wrong. Go figure. Fortunately, Bob, Planck's constant and not SR is the basis of quantum theory and is the reason why quantum predictions are so accurate. SR on the other hand is simply self contradictory when it comes to geometric spatial contraction. I realize this doesn't make the theory incorrect to empiricists, but then nothing would. But geometric dimensional contraction at velocity is. Let P be a point on the plane. Let R be a line segment. A circle with center P and radius equal in length to R is the set of points on the plane distance R from P. Why do you find this difficult? Bob Kolker And let Bob be a point. The set of points worth making equidistant from Bob still defines a sphere and not a circle. The reason you find this so difficult is that you don't understand that the definition you provide has to be qualified dimensionally or it the same definition applies equally to points, lines, circles, and spheres. So what's your definition for spatial dimensionality? In point of fact you can talk about any particular dimensionality. So your definition for spatial dimensionality is just another axiom on the road to Xanadu, Bob. That's what I find so difficult. You complain that space has no independent existence and that I am an anachronism because I recognize the independence of spatial dimensionality. Now it seems you're hoist with your own petard. Either or get off the pot. Well, the qualification of '...on the plane' was missing originally. But like the typical liar you are, you pretend that it was not. Even so, it is still a wonky definition, requiring as it does an infinite set. As an exercise, just to prove you can do it, how about a definition that makes no reference to anything infinite. -- I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives. - -- Tolstoy Where in the definition do you see the word infinite? Point it out. Quote and underline it. The infinitude of the line and plane come from the axiom of continuity, which Euclid left out, but Archimedes and Hilbert put back in. Bob Kolker So what's your definition for spatial dimensionality that allows you to talk about lines, planes, and points, Bob? A pickpocket who works in gloves is innocent of theft? That 'definition' of infinity can be used for anything anywhere, and is therefore useful for nothing. Once again: As an exercise, just to prove you can do it, how about a definition that makes no reference to anything infinite. -- I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives. - -- Tolstoy The definiton of cirle that I gave most recently does not contain within it any concept of infinity. The infinitude comes from elsewhere. Since you are an ignoramus in matters geometrical, I may safely suppose you know nothing of Archimedes axiom of continuity (Hilbert also stated it explicitly) aside from my pointing it out to you. You are a dunce. Bob Kolker Well, Bob, see, the problem here is that the definition of a circle you gave most recently is a Euclidean rac definition and not a modern math definition consisting of the set of all points . . . Now if modern math wants to build its edifice on Euclidean dimensional concepts, it is certainly free to do so. However I find it curious indeed that modern math is forced to rely on ancient concepts of dimensionality to shore up its own supposedly sufficient, exhaustive well definitions while denouncing Euclidean spatial dimensionality. How many points are in the set of points equidistant from the center? We already know that, Bob. It's not the first time you have pulled an answer out of your ass. I know that a finite continuity does not contain an infinity of anything. Unlike, you, a genius in matters geometrical, who see infinities everywhere. -- I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives. - -- Tolstoy For the very good reason that pre quantum physics badly misunderstood understandable in mechanical terms. All of our engineering (in electronics) is based on quantum electrodynamics, pretty much as Direc and Feynman delivered it. We are not post quantum physics at all. All of the replacements for quantum physics so far lack empirical corroberation. They are just pretty speculations and will remain so until evidence supports them. In the mean time mechanics in the classical sense is no longer the dominant methodology of physics. It has not been since 1925. Mechanicalism is a relic of the past. Bob Kolker It's certainly a relic of your past and future. Interesting quote, Albert. And what did Tolstoy know about science? Nichevo. Bob Kolker Apparently as much as you know about the definition of circles. I gave a correct definition of a circel. My initial ommission of stating it was a figure on the plane was an oversight on my part. We were discussing geometric concepts in the context of the Euclidean plane. Ina subsequent posting I inserted what I omitted. I know what a cicrle is. You don't. You think the definition involves infinity. It does not. Bob Kolker Horse, Bob. You were discussing geometric concepts in context of whatever facile assumptions are needed to make your definition work without, however, being able to define spatial dimensionality in other than Euclidean terms while denouncing Euclidean space and dimensionality in all terms. Well perhaps I misjudge the significance of the phrasing set of all points which you here seem to contend means set of all countable points which I assure you doesn't define a circle even in the modern math lexicon and makes me wonder whether you do in fact understand. Ever hear of Zeno, Bob? -- I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives. - -- Tolstoy Of course. All of his so called paradoxes were psuedo paradoxes and have long since been resolved. Greek mathematics could not answer Zeno, modern mathematics has. Part of Zeno's problem was that he could not see how to add up an infinite set of non zero quantities and come up with anything finite. That was -his- problem. The theory of convergent sequences and series answers his paradoxes. Bob Kolker Right. Not a problem for modern mathematikers, who can do anything they like with infinities. -- I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives. - -- Tolstoy The quote wasn't about Science, Bob. It was about people like you. -- I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives. - -- Tolstoy Yes. Nothing really ever changes, does it? -- I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives. - -- Tolstoy As opposed to an uninspiring spectacle. What does people's ability to visualise a fractional quantity have to do with your argument? is most so will whereas I Tony The idea being that since most people couldn't visualize a certain fraction in the example Stephen gave meaning that the two cup measure allows for a measure that people can understand where the pint measure would not, which was my initial response to the example. I'm basically just trying to explain to Stephen what my response was. : is : most : so : will : whereas : I : Tony : The idea being that since most people couldn't visualize a certain fraction : in the example Stephen gave meaning that the two cup measure allows for a : measure that people can understand where the pint measure would not, which : was my initial response to the example. I'm basically just trying to : explain to Stephen what my response was. In both cases the marks on the measure will be identical. Claiming that people cannot visualize the 1/64 pint mark but can visualize the 1/32 cup mark when the marks are identical does not seem justified. Stephen do fraction a which People would not be able to visualize it without the mark, which imposes the visualization. Thus the lower fraction is not as meaningful as the higher fraction, as I said already. : do : fraction : a : which : People would not be able to visualize it without the mark, which imposes the : visualization. Thus the lower fraction is not as meaningful as the higher : fraction, as I said already. But the question was about what you could measure with a 2 cup measure and with a 1 pint measure. Any mark you can make on a 2 cup measure you can make on a 1 pint measure. Any method or device you could devise to count the marks would work in both cases. Visualization has nothing to do with it. Anything you can measure in cups you can measure in pints. Any value of [0-2] cups corresponds to a value of [0-1] pints. There are not more possible measurements if you use cups than if you use pints. Stephen Wny 1/64 th of 2 is the same thing as 1/32 of 1. Why is 1/64 th less meaningful. Both are equally meaningful. Bob Kolker I don't know what you mean by things we experience as mental. Give an example. basically level. and Feelings. Thoughts. Viewpoints. That sort of thing. Yes, there may be a physical component, but things like these and beliefs and desires have an impact on our behaviour. Oh, I see, you're saying that some of our behaviours affect other behaviours. Ok. Now what? Now you learn to discriminate behaviors for a change, Wolf, instead of calling them all behaviors. least be a an I consider an experience to be more than merely a behaviour, and in general they are not considered to be the same thing. Since I don't agree with your initial premise, I fail to see what your point is. Even if you define it as behaviour, it is a behaviour that seems to be different than other behaviours we might display. This means that we have to consider it separately, and look at what it is, instead of merely defining it to be the result of brain for no reason. [snip remarks eliciting following claim:] Define behaviour so that it makes sense to say that experience is more than merely behaviour, and maybe you'll see my point. Ok. How do you look at it seperately? Separately from what? Just what will you be looking at? Example, please. Eg, how do you know that X haa a certain viewpoint? How do you know that this viewpoint in fact influences X's behaviour? That is a restatement of The Problem of Other Minds. In point of fact none of us have any definite proof than anyone but themselves has sapience or volition. Suppose everyone else but you is a cleverly constructed robot made of organic materials. What evidence do you have that would contradict this supposition? Answer: none. We all make a common assumption. That if someone has the same sort of external behaviour as me, then his internal operations are like mine. Bob Kolker Our behaviour (internal and external) are physical effects of physical causes. We are meat. We are meat that walks and talks. Bob Kolker be a an You repeat this so much one wonders if it has become for you a religious mantra that you use to brainwash yourself into continuing to believe it when it is challenged. Have you ever smelled a corpse. I have. I simply state as an experienced and witnessed fact that we are meat. The human body has been sliced, diced, catscanned, petscanned and x-rayed to kingdom come. Only physical things have been seen --- ever. Bob Kolker Bath more often. And seen only by means of invisible energy, which, no matter how you slice, dice or scan your rotting corpse, has ever been seen --- ever. -- I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives. - -- Tolstoy You are meat, Bob, and your ideas are hot air. That doesn't stop your hot air from eating meat. But it does seem to stop your hot air from thinking for itself. Oh, no. Not another nihilist sermon. Bob, you've made your bed, why not just lie down and sleep in it. Why this evangelical zeal? -- I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives. - -- Tolstoy He's trying to convince himself, Albert. Naturalized epistemology in action. [...] [...] So what? What most people hold is irrelevant. This isn't a discussion about dictionary meanings, but about mathematics. If you want to talk about what people think they mean when they use some word or other, that's a whole different game. Try sci.lang. There may be a psycholinguistics NG, but if there is, my ISP doesn't offer it, so I don't know. Happy hunting! EXCEPT if more about Um, who SAYS this is about mathematics? The group I'm in is sci.philosophy.meta. There is no assumption that any term I or others use here is the mathematical. Second, is it insisted that all mathematics must take the definition of cardinality as meaning number of elements? If not, then it is still relevant to be able to claim that even mathematically to say number of elements does not always equate to cardinality. Not the NG I subscribe to. [...] You are purporting to prove that a mathematical argument or proof makes no sense, so you are talking mathematics. You can't drag non-mathematical, common-language meanings into the argument - if you do, you are actually taking psycho-linguistics, not math, and not philosophy. If you want to discuss the different meanings of a term in different contexts, say so. That might be interesting, even. Nobody disagrees that the common parlance meaning of infinity doesn't work as Cantor's precise definition of it works. So what? If you want to talk philosophy, be explicit: but in that case I fail to see any problem, since it's obvious that a term can have different meanings in different contexts, and there is no point in arguing that any of these meanings is universally correct. I know Lester for one wants to find the real or universal meanings of terms, and he believes he knows what they are. But it's a mug's game, looking for the real meaning of a word. An hour or two reading the OED or any other etymological dictionary will show the futility of this quest. (BTW, I've never understood why so many people think the dictionary tells what a word really means. All a dictionary can do is tell what meanings people use words for, and the very nature of language guarantees that a dictionary is out of date even before it's published, and becomes more so as time goes by.) Meaning is usage and context. One must speak with the speakers in the world that the speakers live in to know what the speakers mean. When anthropologists have visited hitherto uncontacted and unknown people they set about learning the language the way a child would learn it. They learned the local names for things and by imitation and gesture they leanrned the names of simple actions. Then by using the language as they learned it and being corrected by the locals, they learned the language. Humans are very clever that way. Anybody can learn anybody else's language with sufficient exposure, trial and error. By the time a numan infant is two years old he has learned at least one language (perhaps more than one) barring of course, brain damage or other serious impairments. A non-mathematician can learn mathematical terminlogy in a math class or reading math text books that provide a suffient number of examples to convey the specialized meanings of terms. Unlike Lester-Speak, the language of mathematics is not private, it is not secret and is not hidden. It is taught every day to thousands and used every day by thousands more. Every art, discipline and science has a specialized vocabulary. If one would lear such arts etc or even learn -about- them he must acquire sufficienty technical vocabulary. Bob Kolker Well, geez, Wolf, if you'd just explain to us how you come to such apocalyptic comprehension, we could all benefit from your teachings instead of just taking you for a pretentious school marm. Bob, you're truly a work of art when it comes to usage and context. Or you could just spend your prose and our time explaining the obvious instead of drafting definitions for circles which define spheres. Yes, well the difference between mathspeak and a technical vocabulary is that the latter is designed to clarify issues. Well, you see, Bob, Lesterspeak is not designed to axiatomize hermit functions in the Dilbertspace of mathematikers imaginations nor to allow mathematikers to pretend they've defined circles in terms of points equidistant from any point on planes they can't define in terms of the points they use to pretend they know their collective pointless asses from a hole in the ground. So how does one learn this public technomathspeak that is so private that its definitions for circles on planes require recondite arcana to communicate? Read any introduction to mathematics textbook. There are thousands of titles advertised on amazon.com. Why is it that freshmen in college can learn abstract mathematics but you consider it a mystery. Bob Kolker Because they learn modern math the same way you did. By rote. Recitativo. No mystery there. Cardinality is a well defined concept in its usual and normal usage. Number of elements is somewhat ambiguous. By common mathematical convention (as established by the mathematical community) number of elements unless otherwise qualified is taken to -mean- cardinality. Words are words. Their meanings are determine by those who use them. Bob Kolker