Galileo's Paradox
in [Math]
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From: Virgil on 9 Dec 2006 17:33 In article <MPG.1fe4fcfbe5d995e3989a2d(a)news.rcn.com>, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > Virgil wrote: > > In article <457af8dd(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > > David Marcus wrote: > > > > Tony Orlow wrote: > > > >> David Marcus wrote: > > > >>> Tony Orlow wrote: > > > > > > > >>>> Isn't the purpose of math be to quantify? > > > >>> No. > > > >> What,in mathematics, has a solution which is neither a real measure, > > > >> or > > > >> the measure of truth of a statement, 0, 1, or somewhere in between? > > > >> Measure=maths. > > > > > > > > You have a very limited view of what mathematics is. A better > > > > description is, Mathematics is the study of patterns. I believe > > > > Saunders > > > > MacLane presented this view in his book "Mathematics, Form and > > > > Function". You might like this book. MacLane believed set theory is not > > > > the best foundation for mathematics. > > > > > > > > > > I have a simplistic view of it. There is measure, the language of > > > measure, and the operations within the language. To me, logic is a form > > > of mathematics where all values lie within the first unit interval. I > > > try to stick to the basics - Occam's Razor, eh? Thanks for the > > > reference, by the way. > > > > > > > > > TOny > > > > Saunders Mac Lane preferred to have his family name spelled with a space > > between "Mac" and "Lane". > > E.g., See http://en.wikipedia.org/wiki/Saunders_Mac_Lane > > > > It is a preference often ignored. > > My apologies. I looked at his and Birkhoff's book "Algebra" to check the > spelling. The title page spells it "SAUNDERS MacLANE". On the other > hand, the copyright page does spell it "Mac Lane". If the text book you looked at made that mistake, you can hardly be held accountable.
From: Virgil on 9 Dec 2006 17:38 In article <457b3353(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > I also, do not regard the set {(2,4), (4,2)} as a"real measure", nor a > > "measure of the truth of a statement", but that does not mean that I > > regard 2 or 4 as anything other than natural number quantities. > > > > {x,y}| xeN ^ yeN ^ x^y=y^x = {(2,4),(4,2)} > > That's a logical statement with a truth value of 1. It is, logically speaking, not even a WFF, so cannot be either true or false. { (x,y) | xeN ^ yeN ^ x^y=y^x} = {(2,4),(4,2)} is a logically WFF.
From: Tony Orlow on 9 Dec 2006 19:28 David Marcus wrote: > Tony Orlow wrote: >> Lester Zick wrote: >>> On Fri, 08 Dec 2006 14:20:32 -0500, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>>>> Now we're all aware of phase transition rules for imaginary numbers. >>>>> c+ni is no greater or less than c alone unless n is even in which case >>>>> there is a transition from the i phase to the c phase and c+ni becomes >>>>> larger than c alone. >>>> I am not aware of that. Please give me a good link that explains it. It >>>> might prove quite important. Thanks. >>> Just complex algebra, Tony, where powers of i contribute to cardinal >>> magnitudes according to whether they're even or not. >>> >> Uh, why don't I see any powers of i in c+ni? Do you mean c+i^n? Yes, >> there's a funny interaction between the real and imaginary components of >> complex numbers which maps to angular and linear operations on the >> radius of a circle. What does this have to do with infinitesimals? They >> are not produced by calculus, but used to form the basis of it. A number >> infinitesimally close to x is considered equal to x in standard math. > > I think Lester would like to avoid having a real number plus an > infinitesimal be a real number. So, he wants to keep them separate, like > we do when we add a real number and an imaginary number. > Yes, but I'm not sure why he wants that. Like I said in another post, I think it has to do with his infinitesimal change in radius as dr/dt or something, uh, nonstandard like that. Tony
From: Tony Orlow on 9 Dec 2006 19:31 David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> Countably infinite means potentially infinite. >>> If you wanted to say "potentially infinite" means countably infinite, >>> why didn't you just say so? I don't think anyone else who has been using >>> the phrase "potentially infinite" recently shares your meaning. >> No? Why don't you ask them. It seems pretty clear to me that that's a >> pretty common conception. > > Are you using the standard meaning for "countably infinite"? Yes, an unbounded set (no last element), with all elements finitely close to the beginning of the set, in whatever order is chosen. In other words, a set bijectible with the naturals. > > As for asking WM, EB, etc., they rarely give coherent answers when they > are asked questions. I'm sure I've asked both WM and EB several times to > explain what they mean by "potential infinity". I'm afraid I still have > only the vaguest idea what they mean. > I think "countable infinity" is what they mean. Endless, and yet, not requiring infinite strings to index any of its elements, using a normal finite digital system. I personally see a conflict between countable infinity and what I consider actually infinite.
From: Tony Orlow on 9 Dec 2006 19:34
David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> There is nothing wrong with saying E and N have the same cardinality. >>>> It's a fact. Six Letters is essentially suggesting IFR, my Inverse >>>> Function Rule, which indeed can parametrically compare sets mapped onto >>>> the real line, using real valued functions, as a generalization to set >>>> density. It works for finite and infinite sets. So, what is wrong with >>>> trying to form a more cohesive theory of infinite set size, which >>>> distinguishes set sizes that cardinality cannot? There certainly seems >>>> to be intuitive impetus for such a theory. >>> Fine. Please state your rule. Let's take a look. >> I've been over this a lot, but hey, what's one more time. Practice makes >> perfect. >> >> We start with the notion of infinite-case induction, such that a >> equation proven true for all n greater than some finite k holds also for >> any positive infinite n. > > What is a "positive infinite n"? > A value greater than any finite value. >> Inequalities can also be proven true for >> infinite n, but only provided that the difference between expressions >> which forms the inequality does not have a limit of 0 as n grows without >> bound. If it does, the inequality holds only for finite n. Now, given >> this extension of classical inductive proof, we can easily prove such >> facts as, say, 2<n <-> 2 < n < 2n < n^2 < 2^n < n^n, and this ordering >> will be true for any infinite n. Thus we have a full spectrum of >> infinite expressions which can be ordered, provided we have some common >> infinite n with which to express them. Okay so far? > > No. See above. > Why do you have a problem with the mere suggestion of an infinite value? Surely you must have guessed enough what I meant to follow the paragraph? (sigh) >> Now, where we are bijecting the naturals with a subset of the reals >> through a mapping formula, if that function is monotonically increasing >> or decreasing, then there is a formulaic relation between the count and >> the value range of each set. For instance, we map the naturals to the >> evens using e=2n. For every set of naturals up to n, there is a >> corresponding set of evens up to 2n. For every set of evens up to e, >> there is a corresponding set of naturals up to e/2. In other words, the >> COUNT of the set up through the value e is e/2, because that's the upper >> bound on the naturals which map to it. So, the inverse of the formula >> describes the size of the set. That is the Inverse Function Rule. >> >> This works for finite sets mapped from the naturals as well as infinite, >> but in order to accommodate any value range that one might plug in, >> whether the values map to naturals or not, we have to employ the floor >> function. Where N maps to S using f(n), and f(g(n))=g(f(n))=n, within >> the value range [x,y] we have floor(|g(y)-g(x)|+1) elements. For >> infinite sets we can dispense with the floor function, and consider the >> interval [0,n], for some assumed infinite range of n, such that the >> count if g(n)-g(0). >> >> I think this might be appealing to Six Letters. > > Let's hope not. > I get the feeling you didn't read past the first sentence. Let's hope Six Letters isn't so lazy. Tony |