Obections to Cantor's Theory (Wikipedia article)
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From: David Kastrup on 23 Jul 2005 16:17 Han.deBruijn(a)DTO.TUDelft.NL writes: > Martin Shobe wrote: > >> Yet your response is that *mathematics* is what must change. Not >> the physicists. Most mathematicians do not believe that >> mathematics has more than an accidental connection to reality. If >> the physicists believe otherwise, then the phsycists need to have >> their training adjusted to emphisize that mathematics is not >> reality. > > That sounds reasonable. Alas. Seems that we have become such > separatists that almost nobody of us still believes in that great > ideal: > > One World or No World > > Doesn't anybody agree with me that a Unified Science would be > desirable? Uh, no. I don't want to recreate the whole building of mathematics whenever physics discovers something new. The whole point of mathematics is that it is not subject to external influences. New math models don't invalidate previous ones, new physics models _do_ invalidate previous ones. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Han.deBruijn on 23 Jul 2005 16:26 Daryl McCullough wrote: > Give an example of a calculation in physics in which using modern > ("Cantorian") mathematics gives you the wrong answer, and using > some other kind of mathematics gives you the right answer. That's actually not difficult. Many mainstream mathematicians find that physicists are quite sloppy when they are doing calculus: they blindly assume, for example, that every function which is _defined_ on the reals is automatically continuous there as well. This is not a valid assumption within common mathematics. But, in Intuitionism this result is true, and known as Brouwer's Continuity Theorem: http://huizen.dto.tudelft.nl/deBruijn/grondig/science.htm#bt Han de Bruijn
From: Han.deBruijn on 23 Jul 2005 16:33 Jesse F. Hughes schreef: > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > > > Robert Low wrote: > > > >> stephen(a)nomail.com wrote: > >> > >>> Are you claiming that set theory is directly applied in General > >>> Relativity? Do you actually have evidence of that? > >> > >> I have seen a paper in which transfinite induction was > >> used. And in the existence of maximal solutions to the > >> initial value problem, appeal is (at least sometimes) > >> made to Zorn's lemma. > > > > I'm not surprised. Whether people like it or not, mathematics cannot be > > separated from its applications. > > That conclusion is not justified from what Low said. All he said was: > this particular part of mathematics has been applied. So what? Am I not allowed to draw my own conclusions? Han de Bruijn
From: Virgil on 23 Jul 2005 17:05 In article <1122127909.708336.250430(a)g47g2000cwa.googlegroups.com>, malbrain(a)yahoo.com wrote: > Peter Webb wrote: > > > > > > e.g. we agree on the basis of our experience with the axiom's veracity > > > and viability. > > > > > > > > > You seem to think that somehow mathematics is a physical science, and the > > axioms are like physical laws, which can be true or false. You think that > > you can observe that zero does not have a suucessor just as you observe that > > every action has an equal and opposite reaction. > > When we talk about natural numbers we AGREE that zero has no > predecessor. We get to say with complete VERACITY that minus one is > not a natural number. > > karl m Not everyone agrees that zero is a natural number. And the issue of whether it has a predecessor depends on whether it is an integer (in which case it does) or a natural (in which case it does not). It depends on which game one is playing.
From: Virgil on 23 Jul 2005 17:07
In article <1122128108.244029.116020(a)g47g2000cwa.googlegroups.com>, malbrain(a)yahoo.com wrote: > David Kastrup wrote: > > malbrain(a)yahoo.com writes: > > > > > David Kastrup wrote: > > >> malbrain(a)yahoo.com writes: > > >> > > >> > David Kastrup wrote: > > >> >> Robert Kolker <nowhere(a)nowhere.com> writes: > > >> >> > > >> >> > Peter Webb wrote: > > >> >> >> How do you disagree with an axiom? > > >> >> > > > >> >> > By assuming a contrary axiom, as is done in non-Euclidean > > >> >> > geometry. > > >> >> > > >> >> Oh, but that is not disagreeing with it. In fact, it is expressing > > >> >> faith that the axiom indeed _is_ an axiom. > > >> > > > >> > Axioms are agreements -- an shared expression of faith. > > >> > > >> Uh, no. Axioms have nothing to with faith at all. If you are playing > > >> chess, you don't have _faith_ that a knight moves always two squares > > >> and then one perpendicular. If it moves differently, that does not > > >> cause you to lose faith in the knight, but rather in your opponent's > > >> mental sanity. > > > > > >>From webster (1913): > > > > > > "Faith (?), n. [OE. feith, fayth, fay, OF. feid, feit, fei, F. foi, fr. > > > L. fides; akin to fidere to trust, Gr. to persuade." > > > > > > "1. Belief; the assent of the mind to the truth of what is declared by > > > another, resting solely and implicitly on his authority and veracity; > > > reliance on testimony." > > > > > > e.g. we agree on the basis of our experience with the axiom's veracity > > > and viability. > > > > Axioms don't have "authority" outside of the game, and certainly not > > "veracity". And I don't see why you drag in the dictionary here. The > > meaning of the word "faith" was not at all in question. > > Natural numbers are not a game. They are a part of our language. To the pure mathematician they are a game in the sense of being systems of rules (axioms) within which one must move. We > have a set of agreements that define them and their properties. karl m |