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Date: Tue, 4 May 93 22:49:35 EDT From: sowa <sowa@turing.pacss.binghamton.edu> Message-id: <9305050249.AA05043@turing.pacss.binghamton.edu> To: sowa@turing.pacss.binghamton.edu, phayes@cs.uiuc.edu Subject: Re: 20th Century Mathematics Cc: cg@cs.umn.edu, interlingua@ISI.EDU

Pat, > John, I don't want to keep arguing with you about the history of mathematics, > but I can't let your last message go unanswered. You will have the last > word on this as I won't spend any more email time on it. Thanks, I agree that it has gone on far beyond its original usefulness. > >I just want to emphasize how totally disjoint Cantor's diagonal is from > >anything in mathematics that has any useful applicability to anything. > >NOTHING, absolutely NOTHING, in analysis depends in any way on Cantor's > >diagonal proof. All the work on epsilon's and delta's, not to mention > >infinitesimals, was completed long before by Cauchy, Riemann, Weierstrass, > >etc. They never made any assumptions about orders of inifinity, and in > >fact, they never even talked about infinity as a completed whole. All > >of modern analysis follows directly in their footsteps and does not in > >any way depend on anything that Cantor did or said about uncountable sets. > > But you omit a vital aspect of the historical story. While these ideas were > being developed, serious worries were arising about whether any of it made > sense, in particular whether or not it was consistent. Arguments about > whether series converged or not were often felt to be only marginally > coherent, and intuitions often failed or disagreed. Yes, of course. I certainly agree about the importance of such foundational issues. But the issues of convergence of series were well worked out by Cauchy, Riemann, and Weierstrass without any recourse to Cantor's work. > These worries largely > motivated the development of set theory and the famous attempts to reconstruct > mathematics in set-theoretical terms. And 'Cantor's paradise' is a conceptual > framework which provides a consistent foundation for all that machinery. I agree that set theory was developed for that purpose and that it was applied to it very successfully. My only claim is that the constructions beyond the countable sets were unnecessary to establish any of the results in analysis or computer science. > Cantor thought there was only one kind of infinity - countable infinity - > and spent years trying to prove it: he was as surprised as anyone by his > argument. I agree, analysis does not explicitly depend on Cantor's > construction, but it does rely on notions of infinity - infinite series, > for example - which seem to clearly allow it. I am willing to admit countable sets, although I have a preference for avoiding talk about completed infinities unless absolutely necessary. As for uncountables, I prefer to remain agnostic, but I won't deny anyone else the joy of discussing them, exploring them, or "proving" theorems about them if they please. > If one decides to expel oneself from paradise then one is obliged to > explain how these ideas make sense. Or perhaps I should say, that I > won't follow you until I hear such an explanation. I'm not asking anyone to do anything. I'm only stating my own doubts, which I share with a number of respectable mathematicians. But as long as there are doubts, it seems best to state clearly which theorems depend or do not depend on the disputed "proofs". > The real problem is that the basic idea of Cantor's construction is > very clear and can apparently be used with any other attempt to define > limits. For example, take Whitehead's 'extensive abstraction' concept, > which identifies a point with a suitably convergent infinite nested > collection of spatial patches. It is easy to give a Cantorian argument > for this kind of definition. The slender thread you talk about seems > to be remarkably tough, in fact. Yes, I have as much admiration for it as I would have for a well-played game of chess. But at least the chessplayers aren't making any claims about the nature of reality. > But anyway, we can argue about this for ever. All I want to do here > is make it clear that ordinary working mathematics, these days, accepts > these ideas without fuss over foundational issues. I don't dispute that point. Most mathematicians are Platonists, which is a philosophical position that I ardently held in my youth. But I got over it. > While > we are playing citation games, try these, from the Encylopedia of > Philosophy, 1967. > > First, Mostowski's article on Tarski: > > "It should be noted that in these papers, Tarski has not criticised > the assumptions of set theory. Like most mathematicians, he has simply > accepted them as true." Yes. And Tarski also used Lesniewski's mereology, which he accepted as true. In the opening section of his famous paper that introduced model theory, Tarski has a very generous acknowledgment of his discussions with Lesniewski and how they helped him to clarify many of the points. > Second, James Thomson on Infinity: > > "Much of Cantor's theory is now almost universally accepted. There > are philosphers who still ask whether there really are infinite numbers, > or whether (a very slight improvement) whether what are called infinte > numbers deserve to be called numbers,....Such questions do not seem > very profitable or interesting. As a mere fact of anthropology (but > nonetheless interesting for that), one may mention that it is now > virtually impossible to instil a general skepticism about infinite > numbers among freshmen who have had a good high-school education." I read George Gamow's _One, Two, Three... Infinity_, when I was in high school, and I was fascinated by it. And I didn't acquire my skepticism until many years later. You can blame Wittgenstein for perverting me. But I also learned about language games from W., and I have no objection to letting people play language games with the word "infinity" or "uncountable" if they like. John