The book is: Marilyn vos Savant, "The World's Most Famous Math Problem", St.
Martin's Press, NY, 1993, paperback. Original price: $7.95. (I looked for
it just now on Amazon and found it listed at $8.95, but temporarily (?)
unavailable; it's not clear whether the publisher plans a new printing.)
ISBN 0-312-10657-2
I am the "lurking but reputable private communicator" to whom John
Mallinckrodt referred a few days ago. I'll accept the "lurking" (I try to
find time to look at this list, but I do not have time to participate; I'm
making an exception for 350-year old theorems); "reputable" was John's
adjective.
I don't see Marilyn's column often, but I have been favorably impressed when
I have. In particular, several years ago she did far better on the doors,
goats, and cars puzzler than did a lot of arrogant (and wrong)
mathematicians who wrote to her on the subject. More recently, she seems to
do very well at M/F birth probabilities.
But this book is not good (understatement). Only 80 pages but filled with
what are (in my opinion) serious misunderstandings of the real nature of
mathematics and what mathematical proofs are all about.
In the context of F's theorem, her basic problem seems to be that because
Wiles, in his proof, used some theorems that were originally developed for
hyperbolic geometry, and because our physical space is Euclidean, (at any
rate, not hyperbolic), she thinks the whole proof is suspect. (pp. 18-19).
(Along the way, she appears to condemn as nonsense most or all of 20th
century mathematics and mathematicians.)
Is our space really Euclidean? How does she know? And what does that have to
with anything in mathematics? (She quotes approvingly, p.16, Lobachevsky's
description of hyperbolic geometry as "imaginary geometry"; she also
appears to think (p.61) that any math using i (SQRT(-1)) is suspect.) She
says (p.61) that "the square root of -1 is imaginary because -1 times -1
would equal +1, instead of -1. This appears to be a contradiction." (Thus
she gives yet another illustration of the well known theorem that for any
correct result there exist numerous fallacious arguments.)
She does not seem to understand that a proof that is supposed to be valid
for all n>2 cannot be proved by a long list of n values for which it holds.
(p.62). (Though at various places in the book, she does seem to understand
that no finite list will suffice; the book is not strong on internal
consistency.)
Because she (and most of the rest of us, including most mathematicians) have
not worked through the proof, she suspects that whatever Wiles' proof is, it
could probably be used to "prove" that there exist no n=2 examples, either.
(p.62). (Here she manages to mangle a statement of the Pythagorean theorem,
for good measure.)
She suggests that because Wiles uses lots of math that has been developed
since Fermat's time, then his proof can hardly be considered a proof of
Fermat's theorem. (pp 63-64, where she says that "few would deny that
[Wiles'] proof would be totally unacceptable to [Fermat]"). (In this
contentious age, F's heirs might take her to court.)
She does not seem to accept a mathematical proof by contradiction. Nor one
by complete mathematical induction (which she does not seem to be able to
distinguish from the ordinary inductive arguments that underlie physics and
that we use to predict that the sun will probably rise tomorrow).
I could go on, but enough already. Marilyn is very good at some things, but
I guess she should restrict her public discussion of mathematics to
non-negative real numbers and go think about what math and mathematical
reasoning & proof are all about.
Incidentally, I wrote an AJP editorial (Oct.'93) called "Fermat's Last
Theorem", but you won't find me - not there, anyhow - making any egregious
mathematical errors, because the editorial is not about Fermat's last
theorem at all; rather, it's about my sadness that this particular Mt.
Everest of my childhood had been climbed and my wondering whether there
exist similar things in physics. I also have written an editorial titled
"Cold Fusion" (Dec. '92), which was not about cold fusion but brought lots
of postcard reprint requests. (Moral: the only way to find out what an AJP
editorial is about is to read it.)
I am not alone in thinking unkind thoughts about this book. In decreasing
order of clout (with me) are the opinions of:
1. My oldest son, Evan Romer, a super math teacher at Susquehanna Valley
Highschool in Binghamton, NY. He and I looked at this book in the bookstore
and were shocked. We only plunked down our $7.95 because, as I recall, of
the apparent endorsement on the back cover by one of his and my intellectual
heroes, Martin Gardner. See next item.
2. The book carries a back cover blurb by MG: "..delightful, informative....
highly recommended...". Some time later, I wrote MG, chastising him for
those opinions. (One of the fun things about being AJP editor is having
occasional correspondence with MG, who has written two Guest Comments for
AJP: 3/89 and 4/97.) MG replied that he had seen some of her discussion of
the pre-Wiles history of Fermat's theorem, had been favorably impressed, and
had written her to that effect, but that what he saw had none of the
silliness and errors that are so prominent in the published book; he
requested that she publish a retraction in Parade, but as far as he and I
know, that has not been done. (Some day I hope to meet MG in the flesh; he
received an award from the APS Forum on Physics & Society at Washington last
April, I went hoping to meet him, but he had sent James Randi to receive it
on his behalf. Is MG a real person? As with Bourbaki, one wonders how one
person could write so much good stuff. Is Marilyn a real person or a
collaboration?)
3. A devastating review by Nigel Boston and Andrew Granville in American
Mathematical Monthly (the "AJP of Math", more or less) 102 (5), 470-473
(May, 1995)), who refer to the book as "drivel" and back this up with 4
pages of details.