Written: May 4, 1995.
Andrew Granville and Nigel Boston just came out with a very well-written and well thought-out book review on Marilyn vos Savant's little paperback on FLTs [AMM, May 1995]. They should be commended for praising what they consider the strengths of the book, and for being tactful on what they consider its weaknesses. After reading their review, and reading again vos Savant's book, however, I am not so sure that vos Savant's critique is so groundless.
Sure, (some of) the specific details are wrong. It is OK to use hyperbolic geometry etc., even though one can square the circle there. But the spirit of her critique is more than valid. Wiles's proof is not convincing. In the good old days `when air was clean and sex was dirty', the raison d'etre of a proof was to explain why the theorem is true. So, Euclid and his buddies started out with some self-evident truths, called them axioms, and tried to deduce everything else from them, by a step by step deduction called `proof'. This paradigm at arriving at `truths' is the sine qua non of our mathematical religion.
We all know, but choose to suppress, Godel's discovery that this methodology is bankrupt. Before Euclid, many people believed that you can `explain' (read factorize) every number in terms of a finite number of `prime numbers'. Euclid proved that you need an infinite number of them. Godel did the same for axioms. But mathematical life goes on.
Wiles's proof is certainly not psychologically satisfying, since it uses many concepts that are far removed from the statement of FLT. Some of them are extremely dubious, to say the least. For example, using continuous notions that assume the axiom of choice. Let me remind you that the consistency of set theory is still not proved. It may be inconsistent, or undecidable.
So all that Wiles did (and that we all do, with somewhat less dramatic outcome), is play a highly artificial game, that we call `proving', that presumably shows, that under many philosophically dubious assumptions, lots of esoteric results proved before `imply' FLT.
Of course, this is our GAME, and our DOGMA, so we have a right to rejoice. But let's not be so narrow-minded and fanatical as to fool ourselves that people outside our faith should rejoice as well. For them, including some, like Marilyn, that outsmart all of us (with the possible exception of Andrew Wiles, who was too busy to take an IQ test), this is as meaningless as the question of how many angels can stand on a pinhead, that occupied Thomas Aquinas, who probably had a comparable IQ to vos Savant's.
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