From larsen@math.upenn.edu Fri Mar 8 21:12:05 1996 Received: from hans.math.upenn.edu (root@HANS.MATH.UPENN.EDU [130.91.49.156]) by euclid.math.temple.edu (8.6.12/8.6.12) with ESMTP id VAA15651 for ; Fri, 8 Mar 1996 21:12:05 -0500 From: larsen@math.upenn.edu Received-Date: Fri, 8 Mar 1996 21:12:05 -0500 Received: from diophantus.math.upenn.edu (DIOPHANTUS.MATH.UPENN.EDU [130.91.48.118]) by hans.math.upenn.edu (8.7.4/SAS 8.03) with SMTP id VAA16952 for ; Fri, 8 Mar 1996 21:12:03 -0500 (EST) Received: by diophantus.math.upenn.edu id AA16157; Fri, 8 Mar 96 21:12:02 -0500 Date: Fri, 8 Mar 96 21:12:02 -0500 Posted-Date: Fri, 8 Mar 96 21:12:02 -0500 Message-Id: <9603090212.AA16157@diophantus.math.upenn.edu> To: zeilberg@euclid.math.temple.edu Subject: Marilyn vos Savant Status: RO Dear Doron, I just read the Opinions column on your home page with interest. Though I am generally sympathetic to your point of view, I think that with Opinion 4 (Fermat's Last Theorem) you go too far. You say that Wiles' proof is not convincing. I strongly disagree. Indeed, it is so convincing that the experts believed it even when it was wrong. You say that Wiles' proof is not "psychologically satisfying, since it uses many concepts that are far removed from the statement." This is, of course, a matter of taste. In my opinion, transforming a problem so that new and unexpected methods can be brought to bear is one of the best things we do in mathematics. It is also the style of mathematical reasoning which is furthest from what computers can be made to do and therefore, as I see it, the most characteristically human. You say that some ingredients in the proof are "extremely dubious" on logical grounds, especially with reference to the axiom of choice. Though I do not understand Wiles' proof well enough to make a categorial claim, I strongly doubt that it uses the axiom of choice in an essential way. Usually in such cases AC is merely a convenience; to avoid making a finite number of choices, the author makes all possible choices at the outset and never considers the matter again. But even if I am mistaken, yes, even if the usual axioms of set theory turn out to be inconsistent, I would not worry more about Wiles' proof than about any other piece of mathematics. I believe that the conceptual structure is sound enough to survive a complete collapse of the foundations, just as 18th century analysis survived the 19th century. Finally, you complain that Wiles uses "esoteric results". One of the great strengths of mathematics is that we build solidly enough that we can safely add floors to existing buildings. We could not build so high nor see so far if everyone had to start from bedrock. Besides, these results are not all that esoteric. To people who use them every day, they are quite homely and familiar. I am reminded of Rutherford's reply to the fellow who suggested that electrons were a merely theoretical construct. "Not real? I can practically see the little buggers." Sincerely, Michael Larsen From andrew@sophie.math.uga.edu Thu May 4 17:57:32 1995 Return-Path: Received: from dns1.uga.edu by euclid.math.temple.edu (8.6.10/Math-HUB-v8) id RAA13440; Thu, 4 May 1995 17:57:31 -0400 Received: from sophie.math.uga.edu (sophie.math.uga.edu [128.192.3.59]) by dns1.uga.edu (8.6.10/8.6.10) with SMTP id RAA20102 for ; Thu, 4 May 1995 17:58:00 -0400 Received: by sophie.math.uga.edu id AA14081 (5.67b/IDA-1.5 for zeilberg@euclid.math.temple.edu); Thu, 4 May 1995 17:55:21 -0400 Date: Thu, 4 May 1995 17:55:21 -0400 From: Andrew Granville Message-Id: <199505042155.AA14081@sophie.math.uga.edu> To: zeilberg@euclid.math.temple.edu Subject: Re: Don't Burn your Thesis Yet! Cc: andrew@sophie.math.uga.edu X-Sun-Charset: US-ASCII Status: RO Dear Doron, I like your opinion. However I back a wealth of ideas over nitpicking about axioms, existential quantifiers and the meaning of proof any day! The more I try to understand what Wiles did, the more mind-blowing it is. I find myself thinking -- how can anyone have employed so many different and fascinating ideas in such a complicated yet accurate approach? It is our `dogma', but it is a dogma which appreciates elegance, charm, concepts and ideas. The destruction of a `I refuse to accept the paradigm in which you work' leaves me cold, because it lacks those ingredients that make it fun to do math. On the other hand I am with you on `proof by computation', and even argument by heuristic. The pedants who disagree with this seem not to recognize that this usually is what one does when `learning how to think'. We don't learn mathematical tools by learning every detail, but rather learning the way to think about them --- often it turns out to be too hard to turn that into a proof. I really learnt about thinking through examples when I was at the IAS and seeing how people like Bombieri understand so much through one judiciously chosen example/computation. Of course, then I read Gauss, and learnt more about quadratic forms from the examples he gives than any of the many books I have read containing the theory. I enjoyed your analogy of Thonas Aquinas with Marilyn vS. Best wishes, Andrew