mm-299 Subject: Re: The Cerebrals Society - is their math test bunkum?> The test measures one aspect of intelligence, namely number pattern> recognition. It requires very fast ability to test hypotheses about> sequences of integers.I'm sure this is useful in some mathematical> endeavors, but is neither a necessary nor sufficient condition for> mathematics nor a substitute for a strong foundation in analysis.> Save your money.But look at the later questions! Surely the answers aren't welldefined?One may be able to confidently continue 111, 222, 333, 444, ... for afew more terms, but are the (intended) answers to the later questionsreally simpler than polynomial interpolation?HJ === Subject: Re: The Cerebrals Society - is their math test bunkum?> The test measures one aspect of intelligence, namely number pattern> recognition. It requires very fast ability to test hypotheses about> sequences of integers.I'm sure this is useful in some mathematical> endeavors, but is neither a necessary nor sufficient condition for> mathematics nor a substitute for a strong foundation in analysis.> Save your money.> But look at the later questions! Surely the answers aren't well> defined?> One may be able to confidently continue 111, 222, 333, 444, ... for a> few more terms, but are the (intended) answers to the later questions> really simpler than polynomial interpolation?> HJWhat they ask you to do is to select the simplest of all reasonablecandidates.Quite often, especially on simpler tests, a really bright person might seetwo or more candidates which appear equally simple, and then has to guess atthe answer the test designer had in mind. Sometimes the questions requirespecialized cultural Knowledge as well...A typical example is the sequence 3,3,5,4,5,3,... the Correct, obvious, and only true answer is ..TA-DA ... 5The number of letters in the names of integers . *******How about 7, 2, 6, 3, 2 ,4 3, ?Correct answer..10Introibo ad altare Dei ,ad deum qui laetificat...Doesn't EVERYBODY know THIS??exceedingly complica.Also some of the questions require a peculiar vocabulary and context notgenerally familiar to Americans..I also suggest the test makes claims to percentile placement of high scorerswhich they cannot justify statistically.Bob Pease === Subject: Re: The Cerebrals Society - is their math test bunkum?> http://www.cerebrals.com/tests/default.htm> Other PhD mathematicians, what do you think?> HJI didn't look at the test, but I did notice that http://www.cerebrals.com/links.htmhas this link: megafoundation.org. Member site.This is the foundation bold enough to publish Harris's revolutionarywork.-- | Jim Ferry | Center for Simulation |+------------------------------------+ of Advanced Rockets || http://www.uiuc.edu/ph/www/jferry/ +------------------------+| jferry@[delete_this]uiuc.edu | University of Illinois | === Subject: Re: Geodetic sphereswell, they're both big promoters of iffy sortsof 4D math (if not just homogenous coords.),just like Bucky was. however,he actually made several dyscoveries, alongwith his many unattribu rediscoveries. see this-- http://www.rwgrayprojects.com/synergetics/plates/plates.html --for a nuts-and-bolts go at what he revolutionized the study of.his most essential teaching amounts to,Why do They call the tetragon, Skware, andthe hexahedron, Qyoob -- what's the etymology of that?oops; that's the nice plates. here's the tensegrity thing:http://www.channel1.com/users/bobwb/tenseg/book/ cover.html> I made no assertions about Fuller. Rather I made fair comment> on the extracts from his books pos in this group by> such as Urner and Neslon.--Dec.2000 'WAND' Chairman Paul O'Neill, reelecto Board. Newsish?http://www.rand.org/publications/randreview/issues/rr .12.00/http://members.tripod.com/~american_almanac === Subject: Re: trunca normal - aargh!A few weeks ago, Ohad Kadan asked for a proof of the following:Let random variable X have standard normal distribution, and let a be a positive number. Show thatE[X | |X-a| 0.To be precise, this is not more general, since OK wan strictly decreasing (I think). However, one should be able modify the proof to get strict monotonicity when as long as P{|X-a|=g(t2) when t1<=a and t1 0 for any subinterval I of positive length contained in [-a, a], and thatP{X in I} / P{-X in I} >= P{Y in I} / P{-Y in I} for any subinterval I of positive length contained in [0,a].Then EX >= EYTo prove this, first prove a discrete version, then discretize the general version and take limits.To prove the proposition: If 0=g(t2). (You need some fussiness to deal with the case where ess sup X < a+t2.) It is then simple show that t2>a implies g(t2)<=g(a). But what about the case where t1>a?-- === Subject: A diophantine equationI'm studying the following diophantine equation:a! - b! = n^2Of course, a, b and n are positive integers. It's easy to show that a < 2band, more strictly (assuming b > 3): if p is the least prime between b/2 andb, then a < 2b. (with b>3 is certain that there is such a prime as aconsequence of the Bertrand's postulate).I think the only two solutions are:2! - 1! = 1^23! - 2! = 2^2Can anyone proof it?Spider === Subject: Re: A diophantine equationSupersedes: Originator: erick@sfu.ca (Erick Bryce Wong)>I'm studying the following diophantine equation:>a! - b! = n^2>Of course, a, b and n are positive integers. It's easy to show that a < 2b>and, more strictly (assuming b > 3): if p is the least prime between b/2 and>b, then a < 2b. (with b>3 is certain that there is such a prime as aI assume you mean a < 2p here :).>consequence of the Bertrand's postulate).>I think the only two solutions are:>2! - 1! = 1^2>3! - 2! = 2^2For sure there are only finitely many solutions. It is known that forsufficiently large n there is always a prime between n and n + o(n^0.6)(I believe the best known exponent is 0.525 by a result of Baker, Harmanand Pintz). So for sufficiently large b, we must have a < b + b^0.6.This means that log(a!/b!) < b^0.6 log(b+b^0.6) = b^0.6 log b + o(b^0.6).On the other hand, as you observed, if p is a prime such that b/2 < p <= b,then a!/b! - 1 must be divisible by p. So a!/b! is greater than the productof all the primes in the interval (b/2,b]. The log of this product is atleast b/2 - o(b), which easily exceeds the upper bound above. Thereforethe equation has no solutions for sufficiently large b.I'm not sure how easy it is to get an effective upper bound using thisline of argument. We actually don't need anything quite as strong as then^0.6 result; I think any prime gap smaller than, say, 0.3 n/log n willprobably do. The primes in (b/2,b] are handled quite well by the Rosser-Schoenfeld inequalities. -- Erick === Subject: Re: Little Quantum Mechanics Question>> One of the postulates of quantum mechanics is that the state of the system>> is represen by a unit vector in Hilbert space. There are no >> eigenstates of the position operator.By the way, the Hilbert space of which I was thinking was L_2(R) orL_2(R^3).>Thanks for your comments. Well, the connection between the first>sentence and the second sentence is not transparent to me. I know about>the postulate, but it seems a big jump to infer from it that there is no>such thing as an eigenstate of position. There have been debates here>previously about whether there is such a thing as an eigenstate of>position and, on the whole, I'd say that the no's have it (and clearly>you agree with that). Please could you be more clear about why you think>this postulate (or any postulate) of quantum mechanics rules out the>existence of eigenstates operators with continuous spectra. If I can>get it clear in my mind why such things are not admissible then I would>feel I've made a step forward in my understanding.>>David.The spectrum of a self-adjoint operator A is the set of all real numberslambda such that A-lambda does not have a bounded inverse. The spectrumof A is closed. lambda is in the discrete spectrum iff A-lambda does nothave an inverse. All other lambda in the spectrum are in the continuousspectrum. If lambda is in the spectrum but not the discrete spectrum,then A-lambda has an inverse, but that inverse is unbounded. lambda is inthe discrete spectrum of A iff lambda is an eigenvalue of A.For any lambda, the range of X-lambda is dense in the Hilbert space.Let psi(x) be an arbitrary element of the Hilbert space, then for anydelta > 0, then psi_delta(x), defined by psi_delta(x) = psi(x) if |x-lambda| > delta and psi_delta(x) = 0 for |x-lambda| < 0, is in the range of X-lambda, and ||psi-psi_delta||^2 = int_{lambda-delta}^{lambda+delta} |psi(x)|^2 dx,which can be made as small as you like by making delta small.Other options to show that the range of X-lambda is dense in the Hilbert space are also available.Suppose, contrary to assertion, that X has eigenvalue lambda in L_2(R), and let f(x) be the corresponding eigenfunction, so thatf(x) is square-integrable, and int_{-infinity}^infinity |f(x)|^2 dxis positive. Then (x-lambda)f(x) = 0 for almost all real x. So for almost all real x, f(x) = 0. Since f(x) = 0 for almost allreal x, then int_{-infinity}^infinity |f(x)|^2 dx = 0, contraryto the requirement that int_{-infinity}^infinity |f(x)|^2 dx > 0.Therefore the assumption that lambda is an eigenvalue of X leads to a contradiction, and so lambda is not an eigenvalue of X.Therefore X has no eigenvalues and no eigenstates.David McAnally-------------- === Subject: Re: Little Quantum Mechanics Question>>Thanks for your comments. Well, the connection between the first>>sentence and the second sentence is not transparent to me. I know about>>the postulate, but it seems a big jump to infer from it that there is no>>such thing as an eigenstate of position. There have been debates here>>previously about whether there is such a thing as an eigenstate of>>position and, on the whole, I'd say that the no's have it (and clearly>>you agree with that). Please could you be more clear about why you think>>this postulate (or any postulate) of quantum mechanics rules out the>>existence of eigenstates operators with continuous spectra. If I can>>get it clear in my mind why such things are not admissible then I would>>feel I've made a step forward in my understanding.>>David.>The spectrum of a self-adjoint operator A is the set of all real numbers>lambda such that A-lambda does not have a bounded inverse. The spectrum>of A is closed. lambda is in the discrete spectrum iff A-lambda does not>have an inverse. All other lambda in the spectrum are in the continuous>spectrum. If lambda is in the spectrum but not the discrete spectrum,>then A-lambda has an inverse, but that inverse is unbounded. lambda is in>the discrete spectrum of A iff lambda is an eigenvalue of A.As an additional note, the continuous spectrum of A is the set ofall real lambda such that the range of A-lambda is not closed. Itis possible for some self-adjoint operators that the discrete spectrumand the continuous spectrum are not disjoint.David McAnally-------------- === Subject: Set theory is seriously order collection> That should be {a, {a,b}}. Sorry abut that.> Where do you see order explicitly invoked?> There is no circularity in the definition.> Bob KolkerI can't see how an 'unordered collection' is supposed to look when it isexactly defined. For if it was presen to us exactly defined, we wouldhave no lesser or greater images of an unordered collection than we wouldhave of an ordered collection.JJ>>The ordered pair (a,b) = {{a}, {a,b}} where curly brackets {, } mean an>>unordered collection.>No, not unordered, but order not considered.> JJ> That should be {a, {a,b}}. Sorry abut that.> Where do you see order explicitly invoked?> There is no circularity in the definition.> Bob Kolker === Subject: Set theory is seriously flawed 3fSets don't form unions. There is only the named set.> How can you have a 'union' of two singletons?> {a} union {b} = {a, b}Sticking them in a formula don't help.Thats no answer to me.>> For each particular case of a relation is always an> object. oo I forgot what I was going to say there.I remember. It was that sets consider objects and not relations, except asobjects.JJ> Formally, in set-theory, a pair is not a function but a set> p = {a, {a, b}}> that is the union of two singletons ({a} and {{a, b}}) such that> 1. the only element in one of them (a in {a}) is an element of the> only element of the other singleton (a is an element of {a,b} which> is the only element of {{a,b}}).That can't be right.> How can you have a 'union' of two singletons?> {a} union {b} = {a, b}> And are you saying that the formation of sets is through 'unions'?> One of the many ways. In particular, the union of sets is a set.> (Formally, in pure set theory, it's somewhat more complica because> you take the union of ONE set (of sets))> This is strange indeed. It presupposes that the properties of theobjects of> the set are conferred upon the set. But does the 'set of boilingliquids'> evaporate?> We're talking maths, troll.> Also, formally, a function is not a pair but a relation between two> sets, that is a particular subset of the cartesian product of the two> sets.How can you have a set of one of a pair?> There is no glue here that set theory can come up with to say that it> considers 'relations'. For each particular case of a relation is alwaysan> object. oo I forgot what I was going to say there.> Yes, I figured that out.> --> Giuseppe Oblomov Bilotta> Can't you see> It all makes perfect sense> Expressed in dollar and cents> Pounds shillings and pence> (Roger Waters) === Subject: Multitude of infinities. 2Infinity is not a limitless horizon, but an unplaced one. A horizon notconsidered falls anywhere. Where the horizon falls cannot be an undertakingof mathematics, but represents the contingency of applying mathematics. Thecontingency of consideration of applying mathematics is mathematics itself,therefore, infinity is not a construct of mathematics.JJ>Take the universal set. There is no set greater than that.That set of all subsets of the universal set has more elements than the> universal set. By the way, there is not such set. The assumption that> there is a set which includes all other sets as elements leads to a> contradiction. Consider the set of all sets that are not elements of> themselves. See what you get.>> What if we define a 'cardinality' (I hate that word, you know. There> has GOT to be a better way to describe an infinity) of infinity for> which this is not the case? For which, even though it containes> infinities within infinities, it remains the same. It would> essentially be an infinity of infinite cardinality.>Therefore, the amount of information it contains is not only infinity,> but it must be the highest infinity possible. There could not be an> infinity higher, since all the numbers contained within that would be> contained within the universal set, thus contradiction the notion that> something could be a higher form of infinity.This leads to contradictions. The history of set theory is essentially> the elimination of contradictions caused by overreaching. The paradox I> sta above is one of the earliest of the set theoretical paradoxes.>> Yes, but how is that a contradiction? It would also contain all its> subsets, and those subsets would be bigger, et cetara. It is only a> contradiction if we limit ourselves to finite cardinalities of> infinity.> What about the set of all universal sets. Is it a universal set or not.Just because there is no highest set that can be represen using the> outda method of using cardinalities to represent infinities, IE, an> 'algebraic representation' of infinities, doesn't mean that such a set> does not exist.There is nothing outda in set theory. After the contradictions were> removed, about 90 years ago, it is as good as the day it was first> inven. Unlike physics, sound mathematics never becomes outda. That> is why Euclid's geometry (properly tightened up with Hilbert.s axioms)> is as good now as it was 2500 years ago. Physics theories come and go,> but mathematics is forever.>> Alright, but how many of those contradictions were actually meant to> be removed, and how many should we have actually learned something> from and revised rather than thrown away?> Why don't you learn some math instead of spewing ax ano?Bob Kolker> I do know a lot of math. I also believe that our current understanding> of infinity is laughable.> (...Starblade Riven Darksquall...) === Subject: Re: Multitude of infinities. 2> Infinity is not a limitless horizon, but an unplaced one. A horizon not> considered falls anywhere. Where the horizon falls cannot be an undertaking> of mathematics, but represents the contingency of applying mathematics. The> contingency of consideration of applying mathematics is mathematics itself,> therefore, infinity is not a construct of mathematics.There is no point introducing an adult discussion of Georg Cantor andaleph because you obviously do not possess the intelligence to doanything with the information. You are a lout, a spewing moronwithout education in the sciences who knows directly from god thatperfect ignorance is tantamount to perfect knowledge. We willtherefore descend to your level,http://www.asa3.org/ASA/PSCF/1993/PSCF3- 93Hedman.htmlhttp://www.wikipedia.org/wiki/Infinityhttp:// www.amazon.com/exec/obidos/tg/detail/-/0679406883/102-5425845- 4496125?v=glancewhich are undoubly still too tightly connec with reality to betolera by your perseverative ineducable stupidity, even the firstURL.If it isn't math it is only opinion - and the universe doesn't carewhat you believe as it offhandedly crushes you.--Uncle Alhttp://www.mazepath.com/uncleal/eotvos.htm (Do something naughty to physics) === Subject: Re: Multitude of infinities. 2> Infinity is not a limitless horizon, but an unplaced one. A horizon not> considered falls anywhere. Where the horizon falls cannot be an undertaking> of mathematics, but represents the contingency of applying mathematics. The> contingency of consideration of applying mathematics is mathematics itself,> therefore, infinity is not a construct of mathematics.> There is no point introducing an adult discussion of Georg Cantor and> aleph because you obviously do not possess the intelligence to do> anything with the information. You are a lout, a spewing moron> without education in the sciences who knows directly from god that> perfect ignorance is tantamount to perfect knowledge. We will> therefore descend to your level,> http://www.asa3.org/ASA/PSCF/1993/PSCF3-93Hedman.html> http://www.wikipedia.org/wiki/Infinity> http://www.amazon.com/exec/obidos/tg/detail/-/0679406883/102- 5425845-4496125?v=glance> which are undoubly still too tightly connec with reality to be> tolera by your perseverative ineducable stupidity, even the first> URL.> If it isn't math it is only opinion - and the universe doesn't care> what you believe as it offhandedly crushes you.Hi Al, Hey, what's happenin', man. I like to write to Al, I call him a serious chemist. Al willsuggest premier potent potables, and he knows how they work!Hey Al, OK, I know almost nothing, but I got a book the other daycalled Magnetochemistry and I really think it's cool, Carlin'sMagnetochemistry, Springer-Verlag, 1986. I've read five or sixpages and think it's really good. I want you to spiel aboutmagnetochemistry, I know you like talking about gravity, it's notgravitochemistry, nor magentochemistry, nor meganotchemistry, normenotchristmegay, it's magnetochemistry. Lay on, MacDuff.Hey, about infinity, Cantorian infinity is simply not the end-allbe-all. The transfinite cardinals aleph_0, 2^aleph_0=aleph_1, ...,are... something, they're the transfinite cardinals, but theyplainly aren't adequate for some purposes. Heh, duff. Heh, planely.Kid, people have been thinking about and theorizing upon infinity formany thousands of years. You're probably right, and so are lots ofother people. Now, to actually make a statement about infinity, ameaningful statement that advances the field, takes some finesse, andas well some knowledge of what comes before, barring having consistenttheories about the infinite. That said, I wholehearly encourageyou to develop your own theories of the infinite, be prepared toexplain them if you're going to defend them or promote them asprogress.Here's a good place to start in considering things infinite: ZdislavKovarik's list of classifications of things infinite, he pos one tosci.math.Keep in mind that sci.math readership comprises many hundreds ofPh.D.'s in Mathematics, and thick skulls like myself, phrenologically. I'm not a doctor, cribbage is a game of chance, mostly, it's aquestion of maximization. Alas, poor Yorick, I knew him, Horatio. Myquotes are shallow.Back to prior art, what you should do is check out surreal numbers,and also non-standard analysis, this is after you are using theinfinite summation of the integral calculus. Look at the integralbar, it's a big S for summation, the evaluation as summation, overreals, is, in effect, what it does, the sum of 1/2^x for integer xfrom zero to infinity is two, that kind of thing. UnderstandCantorian theories of the infinite. Learn about asymptotic density ofnumber theory, and Schnirelmann density. Learn about binary, and feelfree to represent a real number as 0.0101000..., with the bar over therepeating terminal sequence, a rational number, but do it right.Consider 2/x + 2/x, it's 4/x. Multiply the result by x. Consider lg2^x.Offhanded crush! Offhanded crush! Offhanded crush! Yay! Offhandedcrush!Anyways, Al, as usual reading your writings is a slum-bunny Buickthrill, you're volatile. Buick: a fine automobile.Frere Al: If it isn't math it is only opinion ...Do you have a problem with paradoxoi? There might be a single one. Then again, I say there are none.Thanks for selecting those resources. That's what I like about Al: utter lack of cynicism.Anything is a mathematical construct, except perhaps theanti-prerogative.Anyways, about magnetochemistry, I'm beginning to think that EM fieldshave a lot more to do with reaction than I thought. Yet, it's stillmostly just ionization, stochastic chemistry.I went to a library the other day and walked through the shelves andwas overwhelmed and awestruck. Then I selec some books and leftwith them. I love books. I read them.Anyways, besides that sap, we're talking about infinity here, mostlyin theory potential and actual infinity are equivalent. This threadis on sci.physics. I'm trying to think of something that's infinitein the physical universe. My opinion is that the universe isinfinite, although I saw an MWI scenario in a comic book when I wasyoung, multiple worlds intepretation, it's turtles all the way down. The zero point projects to all infinite-dimensional polyhedra. WhatI'm trying to say with that is that that doesn't matter from ourmedio-scale perspective. About alternative osophy, I understandthere are various meanings of the term objectivist, one of which isnot Randian. Maybe you've heard of that, too, that might make ussemi- or pseudo-objectivists, non-Randian objectivism, and perhapsanti-Randian objectivism. In terms of logic, effect => cause for allintents and purposes, although infinitely many consistent theories areaxiomatizable.The Infinity of Points on a Line: Non-Standard Measure Theory.Burp.Thanks for reading my post, have a nice day, Ross FinlaysonFinlayson Consulting === Subject: Re: Multitude of infinities. 2> Infinity is not a limitless horizon, but an unplaced one. A horizon not> considered falls anywhere. Where the horizon falls cannot be anundertaking> of mathematics, but represents the contingency of applying mathematics.The> contingency of consideration of applying mathematics is mathematicsitself,> therefore, infinity is not a construct of mathematics.Please tell me the difference between your statement and a line is not a limitless horizon, but an unplaced one. A line notconsidered falls anywhere. Where the line falls cannot be an undertaking ofmathematics, but represents the contingency of applying mathematics. Thecontingency of consideration of applying mathematics is mathematicsitself,therefore, line is not a construct of mathematicsRJ P === Subject: Re: Multitude of infinities. 2> a line is not a limitless horizon, but an unplaced one. A line not> considered falls anywhere. Where the line falls cannot be an undertaking of> mathematics, but represents the contingency of applying mathematics. The> contingency of consideration of applying mathematics is mathematics> itself,therefore, line is not a construct of mathematicsJohn Jones spews randomly genera sequences of words which are fashioned into syntactically correct tommyrot.Bob Kolker === Subject: Re: Multitude of infinities. 2> Infinity is not a limitless horizon, but an unplaced one. A horizon not> considered falls anywhere. Where the horizon falls cannot be an undertaking> of mathematics, but represents the contingency of applying mathematics. The> contingency of consideration of applying mathematics is mathematics itself,> therefore, infinity is not a construct of mathematics.Perhaps Jones would benefit from review of what a function is http://mathworld.wolfram.com/Function.html === Subject: Re: Multitude of infinities. 2> Infinity is not a limitless horizon, but an unplaced one.Infinity is a property of a set that can be bijectively mapped onto a proper subset of itself. Why do you persist in nonsense?Bob Kolker === Subject: Re: Multitude of infinities. 2>> Infinity is not a limitless horizon, but an unplaced one.>Infinity is a property of a set that can be bijectively mapped onto a >proper subset of itself. Why do you persist in nonsense?>Bob Kolkerpipe dream, infinity is ineffable, and nonesense is a virtue.-- ---------------------------| BBB b barbara minus knox at iname stop com| B B aa rrr b || BBB a a r bbb | | B B a a r b b | | BBB aa a r bbb | ----------------------------- === Subject: Re: Multitude of infinities. 2John Jones top-pos:> Infinity is not a limitless horizon, but an unplaced one. A horizon not> considered falls anywhere. Where the horizon falls cannot be an> undertaking of mathematics, but represents the contingency of applying> mathematics. The contingency of consideration of applying mathematics is> mathematics itself, therefore, infinity is not a construct of mathematics.To sound even more impressive, replace therefore by ergo.-- His mind has been corrup by colours, sounds and shapes. The League of Gentlemen === Subject: contingent application count infinity 2> What? What does fill the infinite limit mean and where is> one supposing that numbers fill the infinite limit?If something is 'larger', it must be measured.> What? How is not countable different from infinite?'Not countable' I said, meant 'no count to be made', as opposed to infinitysidea of 'not countable', as according to an 'endless count'.JJ> I don't necessarily need to be able to count to infinity to show that> one infinity is larger than another.I had to plough through twenty bucket loads of old posts to scrawl downto> your one-liner. I hope you are sorry.> Back to your post -I don't necessarily need to be able to count to infinity to show that> one infinity is larger than another.You don't show that one infinity is larger than another unless yousuppose> that numbers fill the infinite limit.> What? What does fill the infinite limit mean and where is> one supposing that numbers fill the infinite limit?> But the infinite limit is a limit not> applied,> What? What is a limit not applied and how does it differ from> other limits?> so you can have no conception of a limit except the limit to which> you count.> What?> The one to one mapping we do when we compare functions, is not> 'infinite', it is not countable at all.> What? How is not countable different from infinite?> but I can't find any meaning when they are strung together.> [snip a mind-boggling amount of unedi verbiage]> - Randy === Subject: contingent application count infinity 3Infinity has no scale. It has 'sizes' now. Thats a laugh in itself.JJ> I don't necessarily need to be able to count to infinity to showthat> one infinity is larger than another.I had to plough through twenty bucket loads of old posts to scrawldown> to> your one-liner. I hope you are sorry.> Back to your post -I don't necessarily need to be able to count to infinity to showthat> one infinity is larger than another.You don't show that one infinity is larger than another unless you> suppose> that numbers fill the infinite limit.What? What does fill the infinite limit mean and where is> one supposing that numbers fill the infinite limit?But the infinite limit is a limit not> applied,What? What is a limit not applied and how does it differ from> other limits? so you can have no conception of a limit except the limit to which> you count.What?The one to one mapping we do when we compare functions, is not> 'infinite', it is not countable at all.What? How is not countable different from infinite?but I can't find any meaning when they are strung together.[snip a mind-boggling amount of unedi verbiage] - Randy> Infinity is but the illusion that occurs when mathematical> constructs fail due to scale... There are no infinities.... === Subject: Re: contingent application count infinity 3> Infinity has no scale. It has 'sizes' now. Thats a laugh in itself.What is so funny. The hierarchy of infinite cardinalities is well defined conceptually.Bob Kolker === Subject: contingent application count infinity 4I found a couple of faults with Mays axioms.JJ> I don't necessarily need to be able to count to infinity to show> that> one infinity is larger than another.I had to plough through twenty bucket loads of old posts to scrawl> down> to> your one-liner. I hope you are sorry.> Back to your post -> I don't necessarily need to be able to count to infinity to show> that> one infinity is larger than another.You don't show that one infinity is larger than another unless you> suppose> that numbers fill the infinite limit.What? What does fill the infinite limit mean and where is> one supposing that numbers fill the infinite limit?But the infinite limit is a limit not> applied,What? What is a limit not applied and how does it differ from> other limits? so you can have no conception of a limit except the limit to which> you count.What?The one to one mapping we do when we compare functions, is not> 'infinite', it is not countable at all.What? How is not countable different from infinite?but I can't find any meaning when they are strung together.[snip a mind-boggling amount of unedi verbiage] - Randy>Infinity is but the illusion that occurs when mathematical> constructs fail due to scale... There are no infinities....> For clearification.. and just to be confrontive...> Mays's Axiom's> 1) There are no infinities... are but illusion that occurs whenmathematical> constructs fail due to scale...> 2) There are no paradox's ... Are but a mental construct in the absence> of all known rules....> 3) All observed constants in nature are variable... but on a scales as> to be undetectable as varing from the limi scale of observation of> the observer...> 4) Man knows far less than he knows he knows....> 5) Physical Laws apply whether or not man has symbolically defined> it.....> a) Mathmatical symbolisim is not the event its describes...> b) All physical aspects will occure in the universe wheather we> reconize it or modle it... or even if we cease to exist...> 6) Vines will not grow out your butt if you swallow water mellon seeds...> Paul R. Mays> -------------------------------------------------------------- --------------> -> Some where within the Quantum State> Http://Paul.Mays.Com> http://paul.mays.com/resume.html> Almost all really new ideas have a certain aspect of foolishness when> they are first produced. - Alfred North Whitehead>> === Subject: Homogeneous, Isotropic, CompactAre there any compact spaces of dimension 3 other than S^3 that arehomogeneous and isotropic about every point?Eugene Shuberthttp://www.everythingimportant.org === Subject: Re: Homogeneous, Isotropic, Compact> Are there any compact spaces of dimension 3 other than S^3 that are> homogeneous and isotropic about every point?> Eugene Shubert> http://www.everythingimportant.orgRP^3, real projective 3-space.John Mitchell === Subject: Re: Homogeneous, Isotropic, Compact> Are there any compact spaces of dimension 3 other than S^3 that are> homogeneous and isotropic about every point?What about (S^1)^3?Best regards === Subject: Is there a function?Is there a continuous function y=f(x) where y and all it's derivatives,y', y, y''', etc. are equal to zero, at x=0, and yet the function y hasreal values for all other values of x? === Subject: Re: Is there a function?> Is there a continuous function y=f(x) where y and all it's derivatives,> y', y, y''', etc. are equal to zero, at x=0, and yet the function y has> real values for all other values of x?Yes. A slight modification of example 10 in chapter 3 of Counterexamples in Analysis, by Gelbaum and Olmstead, produces such a function.If f(x) is the function of example 10, theng(x) = f(x) + f(-x) has the desired properties. === Subject: Re: Is there a function?> Is there a continuous function y=f(x) where y and all it'sderivatives,> y', y, y''', etc. are equal to zero, at x=0, and yet the function yhas> real values for all other values of x?> Yes. A slight modification of example 10 in chapter 3 of> Counterexamples in Analysis, by Gelbaum and Olmstead, produces> such a function.> If f(x) is the function of example 10, then> g(x) = f(x) + f(-x) has the desired properties.... which is the function defined in chapter 6, example 23.-- Clive Toothhttp://www.clivetooth.dk === Subject: Re: Is there a function?X-ID: XcWfy8ZSge2YySo4RBgL+3XMQHuV75pzqlSHC+f9xIbVYnil7hJ2wM> Is there a continuous function y=f(x) where y and all it's derivatives,> y', y, y''', etc. are equal to zero, at x=0, and yet the function y has> real values for all other values of x?... real x values?f(x) = exp( -x^(-2)) === Subject: Re: Is there a function?> Is there a continuous function y=f(x) where y and all it'sderivatives,> y', y, y''', etc. are equal to zero, at x=0, and yet the function yhas> real values for all other values of x?... real x values?> f(x) = exp( -x^(-2))... if x<>0, and =0 if x=0. === Subject: Re: Is there a function?> Is there a continuous function y=f(x) where y and all it's> derivatives, y', y, y''', etc. are equal to zero, at x=0, and> yet the function y has real values for all other values of x?... real x values?f(x) = exp( -x^(-2))> ... if x<>0, and =0 if x=0.That may well be the sort of thing the OP was wanting. But takinghis question at face value, a much simpler answer would be, say,f(x) = 0 for _all_ x.David === Subject: Re: Is there a function?Is there a continuous function y=f(x) where y and all it's>> derivatives, y', y, y''', etc. are equal to zero, at x=0, and>> yet the function y has real values for all other values of x?... real x values?f(x) = exp( -x^(-2))... if x<>0, and =0 if x=0.>That may well be the sort of thing the OP was wanting. But taking>his question at face value, a much simpler answer would be, say,>f(x) = 0 for _all_ x.I was gonna say that. You pos this while I was still in bed -that's cheating.>David************************ === Subject: Re: Is there a function?> Is there a continuous function y=f(x) where y and> all it's derivatives, y', y, y''', etc. are equal to zero,> at x=0, and yet the function y has real values for all> other values of x?I saw this sign at a troll booth:EZPass, why, y = x^2. === Subject: Re: Is there a function?> Is there a continuous function y=f(x) where y and> all it's derivatives, y', y, y''', etc. are equal to zero,> at x=0, and yet the function y has real values for all> other values of x?> I saw this sign at a troll booth:> EZPass, why, y = x^2.y'' = 2 for y=x^2 === Subject: Re: Obfuscation NumbersInjector-Info: news.mailgate.org; posting-host=adsl-67-119-172-150.dsl.frsn01.pacbell.net; posting-account=48257; posting-date=1064043124X-URL: http://mygate.mailgate.org/mynews/sci/sci.math/ 329e1672061dcc8b5d5a2801c195e7be.48257%40mygate.mailgate.org> let posterity decide if I> am ... or just plain nuts.hold that thought.xanthian.-- Pos via Mailgate.ORG Server - http://www.Mailgate.ORG === Subject: Re: Obfuscation NumbersMail-To-News-Contact: abuse@dizum.com>The complex number i is defined as the positive square-root of -1.>This simple concept helped to identify roots of polynomials that were>in a number of differing fields of human endeavours... though>electrical engineers still prefer to use j instead of i.We use j to represent the imaginary unit, because i is alreadyused to represent current. If we used i to represent the imaginaryunit, as well, beginning EE students would find themselves having towrite things like i = 3i, to the confusion of all concerned.We can't use c for current (or charge) because it's already beingused for capacitance.>The idea:>I define i to be the positive square-root of -1, and j to be the>negative square-root of -1.>Identities:>i = sqrt(-1)>j = - sqrt(-1)>i = -j>j = -i>b) It eliminates the need to use a +/- symbol, such that consider>expressions like sqrt(-4) = +/-2i can instead be represen as 2i+2j.Well, no, that turns out not to be the case. By your definition, j isthe additive inverse of what mathematicians normally call i, so2i+2j is the same as 2i-2i, also known as 0.-- Michael F. Stemper#include This message contains at least 95% recycled bytes. === Subject: Re: Obfuscation Numbers>> The complex number i is defined as the positive square-root of -1.>> This simple concept helped to identify roots of polynomials that were>> in a number of differing fields of human endeavours... though>> electrical engineers still prefer to use j instead of i.> We use j to represent the imaginary unit, because i is already> used to represent current. If we used i to represent the imaginary> unit, as well, beginning EE students would find themselves having to> write things like i = 3i, to the confusion of all concerned.Beginning students, indeed. When, pray, is current imaginary?> We can't use c for current (or charge) because it's already being> used for capacitance.>> The idea:I define i to be the positive square-root of -1, and j to be the>> negative square-root of -1.>> Identities:i = sqrt(-1)>> j = - sqrt(-1)>> i = -j>> j = -i>> b) It eliminates the need to use a +/- symbol, such that consider>> expressions like sqrt(-4) = +/-2i can instead be represen as>> 2i+2j.> Well, no, that turns out not to be the case. By your definition, j> is the additive inverse of what mathematicians normally call i, so> 2i+2j is the same as 2i-2i, also known as 0. === Subject: Re: Obfuscation Numbers> The complex number i is defined as the positive square-root of -1.>> This simple concept helped to identify roots of polynomials that were>> in a number of differing fields of human endeavours... though>> electrical engineers still prefer to use j instead of i.We use j to represent the imaginary unit, because i is already> used to represent current. If we used i to represent the imaginary> unit, as well, beginning EE students would find themselves having to> write things like i = 3i, to the confusion of all concerned.> Beginning students, indeed. When, pray, is current imaginary?Answer: when working with alternating current. If I is a (constant)complex number, then the current is given by I(t) = Re(I * exp(j*omega * t)), where omega is the frequency of the current.We can't use c for current (or charge) because it's already being> used for capacitance.>> The idea:I define i to be the positive square-root of -1, and j to be the>> negative square-root of -1.>> Identities:i = sqrt(-1)>> j = - sqrt(-1)>> i = -j>> j = -i>> b) It eliminates the need to use a +/- symbol, such that consider>> expressions like sqrt(-4) = +/-2i can instead be represen as>> 2i+2j.Well, no, that turns out not to be the case. By your definition, j> is the additive inverse of what mathematicians normally call i, so> 2i+2j is the same as 2i-2i, also known as 0. === Subject: Re: Obfuscation Numbers> The complex number i is defined as the positive square-root of -1.>> This simple concept helped to identify roots of polynomials that>> been used in a number of differing fields of human endeavours...>> though electrical engineers still prefer to use j instead of i.We use j to represent the imaginary unit, because i is already> used to represent current. If we used i to represent the imaginary> unit, as well, beginning EE students would find themselves having to> write things like i = 3i, to the confusion of all concerned.Beginning students, indeed. When, pray, is current imaginary?> Answer: when working with alternating current. If I is a (constant)> complex number, then the current is given by I(t) = Re(I * exp(j*> omega * t)), where omega is the frequency of the current.Yes, I know all that. Do you *really* believe there could be a confusion inmore than one case in 1000 (especially with sqrt(-1) always deno i, inopposition to I), i.e. the case when there is pi/2 phase...? And what aboutIodine or Italy?We can't use c for current (or charge) because it's already being> used for capacitance.>> The idea:I define i to be the positive square-root of -1, and j to be the>> negative square-root of -1.Identities:i = sqrt(-1)>> j = - sqrt(-1)>> i = -j>> j = -ib) It eliminates the need to use a +/- symbol, such that consider>> expressions like sqrt(-4) = +/-2i can instead be represen as>> 2i+2j.Well, no, that turns out not to be the case. By your definition, j> is the additive inverse of what mathematicians normally call i, so> 2i+2j is the same as 2i-2i, also known as 0. === Subject: Re: Obfuscation NumbersWe use j to represent the imaginary unit, because i is already>> used to represent current. If we used i to represent the imaginary>> unit, as well, beginning EE students would find themselves having to>> write things like i = 3i, to the confusion of all concerned.> Beginning students, indeed. When, pray, is current imaginary?When supplied by certain power companiesAnyway, before I gave up on physics etc., I always deno currentby I (capital) not by i. Does this mean that electrical engineershave some problem with the identity matrix too?-- His mind has been corrup by colours, sounds and shapes. The League of Gentlemen === Subject: Re: Obfuscation Numbers>The complex number i is defined as the positive square-root of -1.>This simple concept helped to identify roots of polynomials that were>in a number of differing fields of human endeavours... though>electrical engineers still prefer to use j instead of i.> We use j to represent the imaginary unit, because i is already> used to represent current. If we used i to represent the imaginary> unit, as well, beginning EE students would find themselves having to> write things like i = 3i, to the confusion of all concerned.> We can't use c for current (or charge) because it's already being> used for capacitance.>The idea:>>I define i to be the positive square-root of -1, and j to be the>negative square-root of -1.>Identities:>>i = sqrt(-1)>j = - sqrt(-1)>i = -j>j = -i>b) It eliminates the need to use a +/- symbol, such that consider>expressions like sqrt(-4) = +/-2i can instead be represen as 2i+2j.> Well, no, that turns out not to be the case. By your definition, j is> the additive inverse of what mathematicians normally call i, so> 2i+2j is the same as 2i-2i, also known as 0.I had a theory 1/0 = j, 2/0 = 2j.... it was meant to eliminate referring to limits of oo.wouldn't j = +/- i be a better theory, sqrt(-4) = 2j.or more accurately since sqrt is a positive function by defn,x^2 = -4x = 2jThis is the supposition principle you referred to as motive for the numbers.Now all you need is some elegant axioms that look tidier than CN notation andyou just inven the number system.Obfuscation has to go...... super numbers, double numbers, ...Herc === Subject: Re: Is ...9999.9999... = 0 ?> Is ...9999.9999... = 0 ?Consider the Manipulation:> ...9999.9999... = ...9999.0000... + ...0000.9999...> = ...9999.0 +(1.0-1.0)+ 0.9999...> = ...9999.0 +1.0 -(1.0 - 0.999...)= ...00000.0 - 0.0000... = 0.0 = 0> since> ...9999.0 +1.0 = ...0000.0 by carrying to the left (try it!)> and> 1.0 - 0.9999... = 0.0000... by borrowing to the right.Woops, I may be getting my topologies confused(*),> But can anybody obtain a real contradiction(**) from assuming...9999.9999... = 0and the things it would clearly imply like...3333.0 = - 0.3333... ?* It's kind of like trying to keep your feet in seperate boats, that> are being pulled apart.> ** Such as 0=1, but no fair dividing by zero or nothing like that.(: I hope this note isn't offensive to those trying to teach kids> that 1/3 = .3333.... :)> Okay... but the thing is, you see... if you carry to the left, you> just get a number like lim n->inf 10^(n), not 0. Even this I> understand.> If you keep on carrying to the left, then where do you think it goes?> Apparantly you think that it simply 'falls off the edge of the> mathematical universe'. Kind of like what happens on the outda> ideas of a flat earth if you sail too far. ;Pthe newsgroups I've been to its hardly outda!we've had infinity, 0, nothing, trunc error,...999.999....with 2s complement ...9999 is -1, and .9999.... is 1 which makes 0.but recurring decimals converge, recurring leading numbers diverge.still I see no reason ...999 can't be an alternative to represent -ve numbers....9998.0 = -2Test it out :...9998+ 3______....0001Herc === Subject: Re: Is ...9999.9999... = 0 ?I don't know if there's a proof of the fact that*any* digital sequence repeating across the point is zero, butthis is a good introduction by Madore (took me a whileto find it, the 4th or 5th page of a search).http://www.eleves.ens.fr:8080/home/madore/math/ padics.pdf > Is ...9999.9999... = 0 ?> The lefthand side looks suspiciously like infinity.--UN HYDROGEN (sic; Methanex (TM) reformanteurs) ECONOMIE?...La Troi Phases d'Exploitation de la Protocols des Grises de Kyoto:(FOSSILISATION [McCainanites?] (TM/sic))/BORE/GUSH/NADIR @ http://www.tarpley.net/aobook.htm.Http://www.tarpley.net/ bushb.htm (content partiale, below): 17 -- L'ATTEMPTER de COUP D'ETAT, 3/30/81 === Subject: Re: Is ...9999.9999... = 0 ?This reminds me of Ord is less than nothing and in a way, Ordequals negative one', and one, in a way, equals negative Ord, aboutnegative ordinals and dual values of ordinals.Yet, in a way, Ord equals i, and Ord is very similar to zero. Ord isthe order type of all ordinals, in a way, infinity. What's theopposite of a projection?Speaking of ordinals, sets, I wonder what is the representation of arational in terms of the empty set and sets containing the empty set. A rational is the ratio p/q for integer p and non-zero integer q. Iguess one way to consider that in terms of set is in terms of setdivision.Set subtraction, set division, positive, negative, and rationalordinals; what's next, density of sets?There really is a lot of the behavior of infinitely many digitsembodied in the finite length binary word.I'm reading this Graphs and Hypergraphs, it's a pretty good graphtreatise.Ross === Subject: Re: 3 elements generate C_n> What is the number of 3-tuples (a,b,c) in the cyclic group C_n such> that a,b,c generete C_n ?> ThanksIsn't this the Jordan function J_3(n) ?Bill === Subject: Re: Mays' First Axiom .>>The same applies to constants.. We cannot verify the value>of c if the mechanism of causation is of a universal scale we>would always experimentally observe it as constant due to>any variation taking longer that the life span of our solar system>to detect. So we perceive c as a constant. We experimentally>observe it as a constant... and will be only observable as a>constant within the limits of experimental error for say the>next 100,000 years... but actually varies at a rate... While>it gives us a model that the mathematical constructs we use>can work to explain observation on our scale, if it did vary>it would give a different vision of the universe around us...> To see the value of c change we would have to redefine our units,> of time (seconds), space (meters, feet or whatever) or both. In> the current system (SI), c = 299792458 m/s exactly, by definition,> just as 1 foot = 0.3048 m exactly (again, by definition of the> international foot -- the US survey foot is slightly different,> but equally precisely defined as 1200/3937 meter).No we would not... the value of c is a calcula then experimentallyverified value..... Someone didn't just wake up one morning andsay a foot is 12s and the speed of light is 299792458 m/sNow consider that if the value of c varied 1 m/s per 100,000 yearswould you be able to verify the variance? Nope, to you it wouldseem a constant due to any experimental verification givingthe same value today, tomorrow and for the next 50 generations.There's no need to redefine anything except that c varies ata rate as to be undetectable to a sub scale observer. Toalmost all calculation applied to the observed on our scalewould remain intact... but there would be outcome changesin the equations that are applied to sub scale and super scaleevents being described...> What could happen is that two distances construc to be equal> could be juxtaposed so as to differ visibly in length, or that> of two sequences star at the same time, and construc to> last the same number of seconds, one ends before the other in> an observable way. THEN we'd know that we have to revise our> standards.> Astronomical measurements could be used, but it can be very> difficult to tease out the various effects. We may observe> drift between atomic time (based on quantum mechanics) and> dynamical time (based on General Relativity) -- but there> are always confounding factors, since dynamical systems (for> example, binary star systems) dissipate energy at a rate that> is usually not well known. Never mind that unification of QM> and GR is still unsolved (i.e. mathematically the two theories> are relatively inconsistent, though each agrees extremely well> with observations in its respective domain of applicability).> Michel. === Subject: transitive closures uniquely generaa transitive closure of a binary relation is uniquely genera by a base relation iff it is antisymetric.i think i can prove this using properties of antimatriod closure spaces.is there a much more simpler proof (may be using partial orders)? === Subject: Maximal determinant of matrix with elements 1..n^2In the On-Line Encyclopedia of Integer Sequences we have A085000:http://www.research.att.com/projects/OEIS?Anum= A085000Maximal determinant of an n X n matrix using the integers 1 to n^2.1,10,412,40800The value 40800 has been found just a few days ago.Examples for the corresponding matrices are:n=24 21 3(3 others with same determinant)n=39 3 54 8 12 6 7(36 with same determinant)n=416 7 3 8 4 15 5 11 9 10 14 1 6 2 12 13(576=(n!)^2 with same determinant? -- I'm not sure if this is ok)Can we find a continuation to n=5 (and further) by better methodsthan brute force (or random) search? The best result I've foundafter trying more than 10^10 random matrices (don't tell me storiesabout insufficient period of RNGs, I used the Mersenne twister ;-)is det A_5 = 6553521 for the matrix25 15 10 8 7 5 24 3 17 16 6 13 23 18 214 1 9 22 1912 11 20 4 21Brute force is hopeless, since 25!~=1.55*10^25, even taking intoaccount the possible reduction by a factor of (5!)^2=14400Can we prove that the best result can always be written with then largest matrix elements in the main diagonal?Any comments or improved results are welcome.Hugo Pfoertner === Subject: ellipse constructionis there any *practical* solution to construction an ellipse? somethingwhich gives you the compass needle precision you get when construction acircle. i mean, ok, using a pen and a piece of rope fixed in the foci works,but if i want to draw an ellipse in a 3 by 4 inch rectangle, this is farfrom accurate. any tips? === Subject: Re: ellipse constructionIn sci.math, <%System%><><3f6c2d3c$0$24174$ba620e4c@reader0.news.skynet.be> :> is there any *practical* solution to construction an ellipse? something> which gives you the compass needle precision you get when construction a> circle. i mean, ok, using a pen and a piece of rope fixed in the foci works,> but if i want to draw an ellipse in a 3 by 4 inch rectangle, this is far> from accurate. any tips?Well, it's mostly an issue in the placement of the foci, methinks.In the case of the 3 x 4, one can do the following computations,first dissecting the rectangle in the usual way into 4 subrectangles,with (0,0) at the center. We orient the 4-edge with the X axis.[1] We assume the foci are placed on the 4-edge (x axis) and need merely compute their position thereon. We further assume symmetricity, for rather obvious reasons. We'll call this position (a,0) (with (-a,0) being the other).[2] We need the length of the string loop. There are two constraints on the string loop length: (a) it must touch (2,0). This means the loop length must be 4+2a. (b) it must touch (0,1.5). The loop in this configuration is an isosceles triangle, with sides 2a, sqrt(1.5^2 + a^2),sqrt(1.5^2 + a^2). The loop length is therefore 2a+2*sqrt(1.5^2 + a^2). This means 2*sqrt(1.5^2 + a^2) = 4 sqrt(1.5^2 + a^2) = 2 1.5^2 + a^2 = 4 a^2 = 4 - 2.25 = 1.75 a = 1.32288... and of course l = 4 + 2a = 6.64575...[3] This can of course be generalized. If we have a rectangle of size A x B with A > B, we do similar computations and get a loop length of A + 2a = 2a + 2*sqrt((B/2)^2 + a^2), or a = sqrt( (A/2)^2 - (B/2)^2) and l = A+2a. This suggests the following reasonably simple construction. Let the rectangle A x B be on a sheet of paper, with points enumera clockwise from upper left CDEF. Construct the perpendicular bisectors to the sides; they will intersect at O and hit the sides clockwise from the left GHJK. N----C------H------D | | | G-L----O----M-J | | | F------K------E Take your compass and draw an arc of center H, radius CH down to the line parallel to the longer side, intersecting at L and M. OL is a; we have both foci. Extend the top edge leftward sufficiently, then adjust your compass to the radius OL and describe an arc around C, and call the point of intersection N on the extended line (the other intersection point isn't of interest). Take two pushpins and place them on HN, and affix a loop of string such that it tautly loops around them. Now move the pushpins to LM and draw your ellipse with the string. It works reasonably well mathematically, though the usual physical issues apply -- for starters, strings aren't usually equipped with turnbuckles. However, the width of the string shouldn't contribute to any error here. A slightly simpler alternative might be to use three pushpins loca at L, M, and H; the string can then be tied around them. One can then remove the third pushpin and draw the ellipse.-- #191, ewill3@earthlink.netIt's still legal to go .sigless. === Subject: Re: ellipse construction> is there any *practical* solution to construction an ellipse? something> which gives you the compass needle precision you get when construction a> circle. i mean, ok, using a pen and a piece of rope fixed in the foci works,> but if i want to draw an ellipse in a 3 by 4 inch rectangle, this is far> from accurate. any tips?How about using graphics software and a printer?John Mitchell === Subject: Re: ellipse construction> is there any *practical* solution to construction an ellipse?> something which gives you the compass needle precision you get when> construction a circle. i mean, ok, using a pen and a piece of rope> fixed in the foci works, but if i want to draw an ellipse in a 3 by 4> inch rectangle, this is far from accurate. any tips?1. They do (or used to) make gadgets for drawing ellipses. Imet a very old geometer about 20 years ago who owned such things.2. My grandpa used to make a kids' toy called a nothing grinderwhich had two sliding pieces of wood running perpendicularly insidea block of wood. Turning a handle ran the pieces of wood back andforth. The knob on the handle descibed an ellipse. I imaginethe design could be (and probably has been) adap to ellipsedrawing.Bart === Subject: Re: ellipse construction>Subject: Re: ellipse construction>Message-id: is there any *practical* solution to construction an ellipse?>> something which gives you the compass needle precision you get when>> construction a circle. i mean, ok, using a pen and a piece of rope>> fixed in the foci works, but if i want to draw an ellipse in a 3 by 4>> inch rectangle, this is far from accurate. any tips?>1. They do (or used to) make gadgets for drawing ellipses. I>met a very old geometer about 20 years ago who owned such things.>2. My grandpa used to make a kids' toy called a nothing grinderMy dad used to call it a bull grinder.>which had two sliding pieces of wood running perpendicularly inside>a block of wood. Turning a handle ran the pieces of wood back and>forth. The knob on the handle descibed an ellipse. Maybe that's why I was never able to figure out how to build a computerizedversion. I never got past the single axis type:http://members.aol.com/rotanasnem/lego.htm> I imagine>the design could be (and probably has been) adap to ellipse>drawing.>Bart--Mensanator2 of Clubs http://members.aol.com/mensanator666/2ofclubs/2ofclubs.htm === Subject: Re: David Ullrich on IdentityInjector-Info: news.mailgate.org; posting-host=adsl-67-119-172-150.dsl.frsn01.pacbell.net; posting-account=48257; posting-date=1064043124X-URL: http://mygate.mailgate.org/mynews/sci/sci.math/ c010e5b54ca97545d77ee9f27243668b.48257%40mygate.mailgate.org> Of course, my ability to admit my mistakes and correct them is a> trait that many of you seem to never have properly apprecia.> -- JSH, discussing his 1463rd proof of Fermat's Last Theorem.Yeah, don't you just hate it when people split infinitives?xanthian.-- Pos via Mailgate.ORG Server - http://www.Mailgate.ORG === Subject: Re: David Ullrich on Identity <8765jve4zr.fsf@phiwumbda.org> linux)>> Of course, my ability to admit my mistakes and correct them is a>> trait that many of you seem to never have properly apprecia.>> -- JSH, discussing his 1463rd proof of Fermat's Last Theorem.> Yeah, don't you just hate it when people split infinitives?No.That's a rule that to which I never much cottoned.-- Jesse HughesCertainly he who can digest a second or third fluxion neednot, methinks, be squeamish about any point in divinity. George Berkeley, 1734Subject: Re: David Ullrich on Identity === > David Ullrich says:> And yes, identity is in _fact_ reflexive. To> refute that statement you need to give an> example of something which is not identical> to itself.You never meet a person who was 'just beside himself'or 'not feeling himself'?> The idea that there is something> which is _not_ identical to itself is simply> ludicrous: That's what identity _means_: A> thing is identical to itself and to nothingOnce upon a modern neotime, in secrete of course, a politician clonedhimself. This went unnoticed for a few decades until..., but just ananosec, that's a digression. The morrow of the story is political clonescan be literately beside themselves. === Subject: Re: David Ullrich on Identity> David Ullrich says:And yes, identity is in _fact_ reflexive. To> refute that statement you need to give an> example of something which is not identical> to itself. The idea that there is something> which is _not_ identical to itself is simply> ludicrous: That's what identity _means_: A> thing is identical to itself and to nothingFor a contrasting standpoint, see> ************************************************************** > David Ullrich asks:What's an example of something that's not identical****************************************************** ********> David Ullrich dares:Exhibit of proof of Ex~(x=x) from> C1-C4 and someone will point out the error.>C1 AxAy[x=y -> Az(z in x <-> z in y)] LL1>>C2 AxAy[Az(z in x <-> z in y) -> Az(x in z <-> y in z)] LL2>>C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in A)>>Classification>>C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}]> WeakWould someone be kind enough help David out with a proof?> *************************************************************> David Ullrich remonstrates:I 'blunder' by saying that equality is reflexive by definition?> Huh. Do you have any idea what the word definition means?Homework for David Ullrich:1) What osopher said: ...definitions are available only for transforming> truths, not for founding them.2) In your own words, explain why (or why not) you think> this is true.--John> Because I'm stupid, can anyone tell me what's the point of this discussion?> Because you're stupid, why would anyone want to bother?Because, under certain situations, like the one you are involved,stupidity suffices in providing the answer.True, the provability of Ex~(x=x) will vary from one system tothe next. Nevertheless, there is more to truth than'truth-in-a-model';and just as the truth of Ex~(x is red) depends on how things standwith individuals and redness so the truth of Ex~(x=x) depends onhow things stand with individuals and identity.Because I'm stupid, can you offer an example where Ex~(x=x) is true? === Subject: Re: David Ullrich on Identity> David Ullrich says:And yes, identity is in _fact_ reflexive. To> refute that statement you need to give an> example of something which is not identical> to itself. The idea that there is something> which is _not_ identical to itself is simply> ludicrous: That's what identity _means_: A> thing is identical to itself and to nothingFor a contrasting standpoint, see> ************************************************************** > David Ullrich asks:What's an example of something that's not identical****************************************************** ********> David Ullrich dares:Exhibit of proof of Ex~(x=x) from> C1-C4 and someone will point out the error.>>C1 AxAy[x=y -> Az(z in x <-> z in y)] LL1>>C2 AxAy[Az(z in x <-> z in y) -> Az(x in z <-> y in z)] LL2>>C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in A)>>Classification>>C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}]> WeakWould someone be kind enough help David out with a proof?> *************************************************************> David Ullrich remonstrates:I 'blunder' by saying that equality is reflexive by definition?> Huh. Do you have any idea what the word definition means?Homework for David Ullrich:1) What osopher said: ...definitions are available only for transforming> truths, not for founding them.2) In your own words, explain why (or why not) you think> this is true.--JohnBecause I'm stupid, can anyone tell me what's the point of this discussion?> Because you're stupid, why would anyone want to bother?> Because, under certain situations, like the one you are involved,> stupidity suffices in providing the answer.> True, the provability of Ex~(x=x) will vary from one system to> the next. Nevertheless, there is more to truth than> 'truth-in-a-model';> and just as the truth of Ex~(x is red) depends on how things stand> with individuals and redness so the truth of Ex~(x=x) depends on> how things stand with individuals and identity.> Because I'm stupid, can you offer an example where Ex~(x=x) is true?I'll provide not one but THREE examples. The firstdepends on a (very simple) proof that DAVID ULLRICH COULDN'T HACK!A) Take any 'pure' first order logic, that is, any FOL with nosingular terms other than variables (just to keep things simple).B) To your FOL add this formation rule; If a and b are variables, a=b is an atomic formula.and this axiom scheme: Let be C and D be wff's which differ only in that a occurs free in C where b occurs free in D. If a=b, then C <-> D.C) Call the resulting logic FOL+.In FOL+, identity is symmetric and transitive and identicals areindiscernible, but neither Ax(x=x) nor ~Ax(x=x) is a thesis. So,FOL+ is a subtheory of FOL= (because every thesis of FOL+ isalso a thesis of FOL=), but not conversely (because Ax(x=x)is a thesis of FOL= but not of FOL+).Example 1: Set TheoryAdd to FOL+ the NBG axiom scheme, C3, and Correy's principle ofextensionality, C4:C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in A) Classification C4 AyAx[Az(z in y <-> z in x) -> {(set y & set x) <-> y=x}] (Equi-membered classes are identical iff these are sets.)1) EyAx(x in y <-> Et(x in t) & ~(x in x)) C3Hence2) Ax(x in r <-> Et(x in t) & ~(x in x)) 1,EIand3) r in r <-> (Et(r in t) & ~(r in r)) 2,UIso that4) ~Et(r in t) 3and 5) ~(set r) 4and 6) ~(r=r) 5,C4so that7) Ex~(x=x) 6,EGTo be continued.--John === Subject: Re: David Ullrich on Identity> David Ullrich says:> And yes, identity is in _fact_ reflexive. To> refute that statement you need to give an> example of something which is not identical> to itself. The idea that there is something> which is _not_ identical to itself is simply> ludicrous: That's what identity _means_: A> thing is identical to itself and to nothing> For a contrasting standpoint, see> ************************************************************** > David Ullrich asks:> What's an example of something that's not identical> ************************************************************** > David Ullrich dares:> Exhibit of proof of Ex~(x=x) from C1-C4 and someone will point out the error.>>C1 AxAy[x=y -> Az(z in x <-> z in y)] LL1>>C2 AxAy[Az(z in x <-> z in y) -> Az(x in z <-> y in z)] LL2>>C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in A)>>Classification>>C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}]> Weak> Would someone be kind enough help David out with a proof?> ************************************************************* David Ullrich remonstrates:> I 'blunder' by saying that equality is reflexive by definition?> Huh. Do you have any idea what the word definition means?> Homework for David Ullrich:> 1) What osopher said: > ...definitions are available only for transforming> truths, not for founding them.> 2) In your own words, explain why (or why not) you think> this is true.> --JohnBecause I'm stupid, can anyone tell me what's the point of this discussion?Because you're stupid, why would anyone want to bother?> Because, under certain situations, like the one you are involved,> stupidity suffices in providing the answer.True, the provability of Ex~(x=x) will vary from one system to> the next. Nevertheless, there is more to truth than> 'truth-in-a-model';> and just as the truth of Ex~(x is red) depends on how things stand> with individuals and redness so the truth of Ex~(x=x) depends on> how things stand with individuals and identity.> Because I'm stupid, can you offer an example where Ex~(x=x) is true?> I'll provide not one but THREE examples. The first> depends on a (very simple) proof that DAVID ULLRICH COULDN'T HACK!> A) Take any 'pure' first order logic, that is, any FOL with no> singular terms other than variables (just to keep things simple).> B) To your FOL add this formation rule;> If a and b are variables, a=b is an atomic formula.> and this axiom scheme:> Let be C and D be wff's which differ only in that a occurs free> in C where b occurs free in D. If a=b, then C <-> D.> C) Call the resulting logic FOL+.> In FOL+, identity is symmetric and transitive and identicals are> indiscernible, but neither Ax(x=x) nor ~Ax(x=x) is a thesis. So,> FOL+ is a subtheory of FOL= (because every thesis of FOL+ is> also a thesis of FOL=), but not conversely (because Ax(x=x)> is a thesis of FOL= but not of FOL+).> Example 1: Set Theory> Add to FOL+ the NBG axiom scheme, C3, and Correy's principle of> extensionality, C4:> C3 EyAx[x in y <-> Et(x in t) & A] (with y not free in A) Classification> C4 AyAx[Az(z in y <-> z in x) -> {(set y & set x) <-> y=x}]> (Equi-membered classes are identical iff these are sets.)> 1) EyAx(x in y <-> Et(x in t) & ~(x in x)) C3> Hence> 2) Ax(x in r <-> Et(x in t) & ~(x in x)) 1,EI> and> 3) r in r <-> (Et(r in t) & ~(r in r)) 2,UI> so that> 4) ~Et(r in t) 3> and > 5) ~(set r) 4> and > 6) ~(r=r) 5,C4> so that> 7) Ex~(x=x) 6,EG> To be continued.> --JohnHi John,Your very interesting proof leads to some questions.1. What is the domain of your variables ? I assume you admitsets and non-sets. Are physical objects and proper classes and setsvalues of the variable x ?2. Do you claim: Et(x in t & y in t) <-> (set y & set x), or,Et(x in t)& Et(y in t) .<->. set x & set y. ?C4 AyAx[Az(z in y <-> z in x) -> {(set y & set x) <-> y=x}] (Equi-membered classes are identical iff these are sets.)C4 AxAy[Az(z in x <-> z in y) -> {Et(x in t & y in t) <-> x=y}] Weak3. Is C3 <-> EyAx[x in y <->. Et(x in t) & set y & A] (with y not freein A) ?4. It seems your C4 entails x=x <-> set x. That is, x=x fails forempirical values (non sets). Where does this leave us with respect toRussell's descriptions?5. How do you distinguish: E!x, x=x, set x, Ey(x in y), Ez(z in x) ?Thanks in advance,Witt