June 24, 1993: At Last, Shout of 'Eureka!' In Age-Old Math Mystery by Gina Kolata.
(Note that the NYT doesn't seem to render superscripts correctly; I trust this won't be a problem for you!) They're interesting articles, but they don't say anything that hasn't been rehashed a zillion times in the last fourteen years; I bring them up mostly as historical curiosities.
From the June 24 article:
Dr. Ribet said that 20th-century work on the problem had begun to grow ever more divorced from Fermat's equations. "Over the last 60 years, people in number theory have forged an incredible number of tools to deal with simple problems like this," he said. Eventually, "people lost day-to-day contact with the old problems and were preoccupied with the objects they created," he said.I've definitely had this feeling about number theory before; a lot of what goes under the name "number theory" doesn't seem to have all that much to do with, well, numbers: 1, 2, 3, 4, and so on. Somehow understanding the structure of those numbers seems more important to me, in some aesthetic sense, than understanding the structure of some crazy set of "numbers" one invents; should these auxiliary structures be anything more than scaffolding? Perhaps this is why I'm not a number theorist.
A column by James Gleick from October of 1993, which includes the following:
What a peculiar bit of knowledge! Yet if anyone is out there judging the galaxy's civilizations according to some handbook of "What Every Bright Species Ought to Know," it seems reasonable to suppose that somewhere on the list will be the fact of arithmetic that we call Fermat's Last Theorem. It must be a universal truth -- true here, true everywhere, true now, true always.
I don't know about that. Fermat's Last Theorem somehow seems accidental, even though it's so simple, and it's not used to prove other results. It wouldn't have nearly the same renown if Fermat hadn't scribbled that note in the margin. It's a "cute" problem, nothing more, one that's become important by historical accident. Gleick points out that whole branches of mathematics have arisen from failed attempts to solve FLT -- but wouldn't they have arisen in some other way? It's important to have big unsolved problems that will inspire people. But I don't know if it matters what problems, exactly, they are. If there are alien mathematicians, I wouldn't be surprised if to them FLT is just a footnote, if they've actually proved it at all. (I think they'll have at least thought of the problem, though, because the Pythagorean theorem seems inevitable, and any mathematical community that never thinks to attempt the generalization to higher exponents isn't a mathematical community worthy of the name.)
But if the Riemann Hypothesis turns out to be true (and I don't doubt that it is) I'd put it above Fermat's Last Theorem on such a list. (Not for the form in terms of the zeta function, but in one of the equivalent forms in terms of strengthenngs of the Prime Number Theorem.)
Also from the Gleick column:
It's an impressively economical way of speaking, really -- a picture's-worth-a-thousand-words approach wherein one word is worth a thousand words. Never mind that the published version of Wiles's proof will run 200 pages. To expand the shorthand, flesh it out so that, say, Fermat could understand it, would require many times more.That's an interesting point. I've heard it claimed that the true "difficulty" of a piece of mathematics is not its length, but the length of all the pieces of mathematics that had to come before it. And perhaps if we couldn't explain it to Fermat, that means we don't really understand it. Feynman said, when he was teaching the courses that became the Feynman Lectures on Physics, that if he couldn't express something in a way that could be understood by freshmen, it meant he didn't really understand it. Fermat, one might say, could be considered one of these freshmen.
Here are some other things dragged out of the archives:
June 27, 1993: But How Did Fermat Do It?
June 27, 1993: After 350 Years, A "Q. E. D." (This is a very short item; it appears to be from some sort of "week in review".
June 29, 1993: SCIENTIST AT WORK: Andrew Wiles; Math Whiz Who Battled 350-Year-Old Problem by Gina Kolata.
A review of Amir Aczel's book "Fermat's Last Theorem"
And a challenge: find the error in this "parody proof" of FLT, which I found from the Wikipedia article. (I don't think this is particularly difficult. But I'm lazy, so I want you to tell me what it is.)
I think I'll stop here.
3 comments:
The issue in the "proof" is the classic problem of taking square roots and not considering sign. After step 8, we have the exponents reading
((p+1) - (1/2)(2p+1))^2 = (p - (1/2)(2p+1))^2
which you can see is true if you expand it out into one polynomial (and, as an aside, is kind of cool). After step 9, we have the exponents reading
(p+1) - (1/2)(2p+1) = p - (1/2)(2p+1)
which is trivially false. However,
(p+1) - (1/2)(2p+1) = -(p - (1/2)(2p+1))
is true, so the contradiction doesn't follow.
Flooey,
that's one of two problems that I saw.
the other one is that we're meant to assume that all the exponents are integers. But in step 8, all the exponents are actually 1/4, and in step 9 they're 1/2 or -1/2.
Step 5 is a pretty blatant error. You can't just "subtract exponents from both sides" like that! If you could, then because 3^2 + 4^2 = 5^2, 3^(2-1) + 4^(2-1) would = 5^(2-1)!
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