Galileo's Paradox
in [Math]
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From: Tony Orlow on 9 Dec 2006 13:03 Virgil wrote: > In article <MPG.1fe42308b91485ee989a0b(a)news.rcn.com>, > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > >> Virgil wrote: >>> In article <4579d1c7(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>>> I chose to work within computer science, after having planned to become >>>> a mathematician, for the obvious reasons.... >>> Couldn't cut it as a mathematician? >> Probably couldn't understand any of his math courses. > > I think TO once said that he disagreed with one of his math teachers so > strongly that he decided to give up on math as a major. It was never my major. > > I suppose pat of it might have been that the teacher was a graduate > assistant with a bad attitude, but I rather think that a large part of > it was the student's bad attitude. I understood my math courses and aced them, thanks. I did fine with transfinitology, even though I didn't believe it. I disagreed with the teacher on it, but we didn't have a conflict over it. It certainly made me think that pure mathematics had a great potential for nonsense, and that going into an applied math area, like the math machines we call computers, would be more rewarding and productive. Part of it had to do with the binary code upon which they are built, which I had been developing myself in conjunction with a dualistic model of the universe, and part had to do with the computer being a simple brain, and the model describing the structure of mind, which might be implemented on a computer. I never ran away from math. I mixed it with reality. :)
From: Tony Orlow on 9 Dec 2006 13:05 David Marcus wrote: > Tonico wrote: >> Tony Orlow ha escrito: >> >>> You can call me a troll if that makes you feel better. You seem to need >>> to bolster your ego by piling it on top of others. Hopefully you'll work >>> that out eventually. I won't concern myself with your spiritual >>> development too deeply, but I do have some questions. >> ***************************************************** >> Lemme see: troll....yup, it makes me feel better....ah. > > If a troll has to know they are trolling, then I don't think Tony is a > troll. He thinks he is merely posting what is correct. Of course, it > would be better if he was a troll. He's more a crank. > Thank you, David. I'm definitely more of a crank than a troll. :) Tony
From: Tony Orlow on 9 Dec 2006 13:14 imaginatorium(a)despammed.com wrote: > Tony Orlow wrote: >> Lester Zick wrote: >>> On Tue, 05 Dec 2006 11:53:13 -0500, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>> [. . .] >>> >>> You know, Tony, I got to thinking last night... > > Oh dear, that was Lester... <snip> > >>> Now there is actually a precedent in conventional mathematics for this >>> situation. With complex numbers you actually have two component >>> numbers: one conventional algebraic and one imaginary. Thus we can't >>> say that r+ni is actually larger than r unless n is even. >> That's true,... > > Is it? One can't say that r+i is larger than r, but we can say that > r+2i _is_ larger than r. I'm totally baffled by this, and wonder if you > can explain to us mathematikers what Lester is on about now? > > Actually I wasn't going to nitpick that "even" statement, since the point is that the 2D plane isn't inherently ordered linearly, and that imaginary numbers don't lie on the same metric line as reals. The "even" condition doesn't make sense to me either. >>> ~v~~ > >> 01oo > > Woof! > > Brian Chandler > http://imaginatorium.org > Mrraaaowww!!
From: Tony Orlow on 9 Dec 2006 13:23 cbrown(a)cbrownsystems.com wrote: > Tony Orlow wrote: > >> 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? > > "Find all pairs of distinct naturals (x,y) such that x^y = y^x". The > solution to which is the set {(2,4), (4,2)}, which doesn't appear to be > a "real measure", nor a "measure of the truth of a statement" (as far > as I can understand your meaning of the terms). Those are specifications for two points in a 2D array of naturals, the values within each pair denoting the distance of each point in each of the two directions, from the origin. Are you saying x and y and 2 and 4 are not taken to be quantities? That's a rather strange position to take. > > I assume that you offer more than the trivial observation that all > mathematical statements P, including the statement "2^4=4^2", are > examples of the (boolean) truth valued statement "it can be proved that > P". If so, I would claim that the mathematical question is "Find a > proof of P, or proof of not P". And the solution is not "0" or "1"; the > solution is either a proof of P, or a proof of not P. First of all, that trivial observation is plenty. All logic is subsumed under math as a calculation of truth values between 0 and 1. Further, a proof is precisely this calculation of truth value for a given statement. To say "find a proof" is to say "define a sequence of operations on the given statements to produce another such that its value is 1". Truth is a form of quantity. > > "Find all finite groups G having a maximal subgroup S, and having a > subgroup T which is isomorphic to S but not maximal". This question was > asked in sci.math a few days ago. The groups in question are not > "numbers" at all; and a set of them possesses no particularly natural > total orderings. They can be partially ordered by "size" (number of > elements); but there are, in general, multiple distinct groups on a set > of any given size. Assign each element a bit, and every group corresponds to a binary string, which corresponds to a value. These letters on your screen are numbers. > > You might look at the following threads currently active in sci.math > within the last 24 hours (believe it or not, not everyone posts to > argue about Cantor/infinity): > > "some advance algebra q's. please help" > "Group on arbitrary ordinal" > "Another two universal algebra questions" > "? how subspaces are like" > "large ordinals, help!" > "Continuous injection from a subset of R^n to R" > "Smith normal form in a binary field (F2), symbolic" > "Infinite width Moebius strip" > "notation of fields and rings" > "(Universal Algebra) The function determined by a term..." > "Intermediate Fields in Galois Theory" > "Mapping of integer functions into reals" > "Sets of quaternions with no repeats in infinite set of products:" > "please help me explain: Counter-Example for Riesz representation > Theorem" > "how to deduce the algebra structure" > "Infinite simple groups and their proper subgroups" > "Right cosets intersection" > "Valuations in field extensions" > "Groups and commutators" > "Compactness" > > ... and so on. > > There are many mathematical questions of the forms: "does there exist X > such that P(X)?", "characterize all X such that P(X)", and "find an X > such that P(X)" that do not use a "real measure" to describe X, and are > not about "measuring" things with "numbers" in the sense I'm guessing > you mean. In particular on sci.math, there's topology, number theory, > abstract algebra, linear algebra, and graph theory. > > These areas exist because of a combination of two factors: they are > sometimes useful, and sometimes interesting in their own right to its > practitioners. > > As far as /I'm/ concerened, these are the only actually /interesting/ > parts of mathematics. Face it - Calculus is boring! I don't like adding > up columns of numbers, so I have a calculator. Nor do I like plodding > through many certainly quite well-known transformations; so I look it > up in a table if I need it. > > On the other hand, given my interests, I don't need it that often; so I > also don't personally find it very "useful" :). > > Cheers - Chas > That's fine. I am still of the opinion that math boils down to measure, the language of measure, and the operations allowed on that language. Tony
From: Tony Orlow on 9 Dec 2006 13:28
MoeBlee wrote: > Tony Orlow wrote: >> You might want to look into Internal Set Theory, a partial >> axiomatization of Nonstandard Analysis. > > Why do you say 'parital'? I said "partial", and said that because that's what I've read. I am not sure what parts of nonstandard analysis are not included in the axioms of IST. > >> Both infinitesimal and infinite >> values are "nonstandard", and no reference to "standard" values is >> allowed in the definition of any set. > > Not ANY set. IST includes standard sets too. You do realize that IST is > an EXTENSION of ZF, right? > > MoeBlee > Sorry, "standard" is not allowed in the definition of any *internal* set. |