Wednesday, February 24, 2010

The Last Theorem: Pro and Con

As I'm sure you all know, back when Arthur C. Clarke died two years ago, he left an unfinished novel entitled The Last Theorem, about a Sri Lankan mathematician who finds a three-page proof of Fermat's Last Theorem and, as one would expect, gets recruited by the CIA to fight aliens. All in all it sounds like a perfect idea for a book. I mean, aliens! Math! Sri Lanka! What's not to like?

Making matters even more awesome, none other than Frederik Pohl was hired to finish the book and it was published in July of 2008. Yet when looked it up on it averaged three stars. This shattered my confidence.

Of course, there are plenty of good books that get mediocre scores. Maybe some people who don't normally like science fiction gave it a read because it was his last book and, not liking sci-fi, didn't like it. Maybe those reviews were canceled out by other reviews that gave it unnecessarily high scores because it just didn't feel right for Clarke to go out on a low note. The only way to know for sure was to read the reviews and see for myself.

On the one hand, the following one-star review entitled "major disappointment" by one S. E. Greer completely reinvigorated my enthusiasm for the book:
If you want to read about anti-American views and homosexual glorification that has nothing at all to do with the story being told, this is it. This book is a total joke. Clarke's grand finale just shows how bad he hated America, what a strong socialist if not communist he was, and how pro-homo he was. If you're looking for a sci-fi story, all you'll get is an opiniated, bitter, hateful, old man, with a sorry excuse for a book, lashing out at anything American.

On the other hand, the following five-star review entitled "Pleasant Surprise Finding it Pans Wiles-Ribet-Fry" by one Ronald L. Wooden suggests maybe the book isn't so good after all:
This book has leveled a critique at the touted "proof" of Fermat's Last Theorem from two authors who knew and know more than a little about math and how proofs are done.

At least the plot would seem to be recycling Hollywood in a new way, Sneakers and Mercury Rising, for example. The alien subplot is similar to Childhood's End or The War of the Worlds, still that's pretty much what you get all the time with science fiction.

To suggest Ranjit's proof takes only three pages might seem absurd or impossible given the hundreds of years so many mathematicians have labored away at it, then the disappointing claim that basically is 300 pages of still further and obscure journal references unavailable in most American university libraries. Pohl says it would take a computer to assess the satisfiability of Wiles-Ribet-Fry, or that that "no biological human could read it"--which is a strange way only a science fiction author could put it I guess. Ignorance then is bliss.

Completion of the n-square actually follows directly from the standard technique in algebra for deriving the quadratic equation, that and some awareness of simple fractal renormalization approaches. Nested binomial expansion is a little tricky but a good exercise with indexed sums (a + (b + c))^n or group inside the other way. What should result is a statement in the presentation of the Fundamental Theorem of Algebra demonstrating reducio ad absurdum. My son accomplished this in his Modern Algebra course he took as independent study, and indeed, even expanded in places it takes only three pages:


The covering dimension is the span between like powers of natural numbers, as a difference. And you get the usual recursion relation, a simple mapping to the rationals, cancellation and substitution and using triangular matrix representation as the Pascal Triangle. See for yourself.

All in all, I am left just as uncertain as I was before I read the reviews.

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