Obections to Cantor's Theory (Wikipedia article)
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From: malbrain on 22 Jul 2005 19:54 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. karl m
From: David Kastrup on 23 Jul 2005 02:42 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. It seems like you did not understand a single word of what I wrote. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Peter Webb on 23 Jul 2005 03:22 > > 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. Well, they are not like that. As David said, they are just the rules of the game. Mathematicians just pick the rules in order to make interesting games. The example I gave before - the axioms of group theory - make for a really interesting game. The axioms of set theory (ZFC) make a really, really interesting game. There is no question of "veracity". Is it true that there is an element of the group e such that for all g, g*e=g ? Yes, because that is an axiom of group theory. It is true *by definition*. It is part of the rules of the game called group theory. As David said, arguing about whether axioms are true makes about as much sense as arguing about how knights move in chess - if you think you can invent axioms which make better games, go for it. But you can't say - "that isn't how a knight moves" because we defined a knight in chess as moving that way.
From: Jesse F. Hughes on 23 Jul 2005 03:27 "david petry" <david_lawrence_petry(a)yahoo.com> writes: > Han de Bruijn wrote: > >> Indeed, idealizations >> don't necessarily obey the same laws of logic as the directly observable >> objects. But, at least, there should exist a path from the abstractions >> back to the observable objects. Directly observable sets are idealized - >> and that's a good thing - but idealized sets must also be "materialized" >> again, for the sake of _applications_. The latter tends to be forgotten. > > That hit's the nail on the head. Thanks Han. > > For those who missed it, the key sentence is: > > "But, at least, there should exist a path from the abstractions > back to the observable objects." Seems like nonsense to me. You are still claiming that an abstraction is useful, but applying deductive logic to the abstraction may yield false results unless we check nature. But this has two obvious problems: (1) The abstraction doesn't exist in nature. Now, maybe that means (to you) that there is no "path from the abstractions back to the observable objects", but then why did you call infinity a *useful* abstraction? Or what path do you have in mind and how does it contradict Cantor's theorem? (2) How can any useful abstraction fail to satisfy the basic laws of deductive logic? You have still not explained that. If our useful abstraction justifies the presumption of A and A -> B, then it damn well better justify the presumption of B and there's no need to check nature to see. But that is all that Cantor's theory uses, too. Basic axioms prove a perfectly simple result. You agree that the proof is correct, but want to claim that normal logic doesn't apply to infinite sets (which are nonetheless "useful". Of course, you don't give any principle to determine *what* derivations involving infinite sets are acceptable, aside from "look out the window and see how infinite sets really behave." Your position is really incoherent. You can't simply say that infinite sets are legitimate mathematical objects but that normal logic doesn't apply to them. You must have a principled objection (not "Cantor violates my intuitions." or "Everything before Cantor was about computation.") and you must have a solution. Which axioms are wrong? Do you really suppose that deductive logic is not the right logic for set theory? Then what is? Anyway, viva la revolution and all that. -- "No feeling sympathy for mathematicians who start marching with signs like 'Will work for food' in the future... I will not show mercy going forward. I was trained as a soldier in the United States Army after all... We play to win." --James Harris, feel his wrath!
From: Jesse F. Hughes on 23 Jul 2005 08:01
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. -- Jesse F. Hughes "We will run this with the same kind of openness that we've run Windows." Steve Ballmer, speaking about MS's new ".Net" project. |