Beautiful Mathematics

January 20, 2012

“Beauty is the first test: there is no permanent place in the world for ugly mathematics.”

G. H. Hardy

I’ve been reading Proofs from THE BOOK  by Aigner and Ziegler, a paean to beautiful mathematics. The Book refers to a conceit of the mathematician Paul Erdős. From wikipedia:

[Erdős] spoke of “The Book”, an imaginary book in which God had written down the best and most elegant proofs for mathematical theorems. Lecturing in 1985 he said, “You don’t have to believe in God, but you should believe in The Book.”

Proofs from THE BOOK is a collection to these “book proofs,” some of the most elegant arguments in mathematics. One thing I like about this book is that it samples multiple, very diverse ideas to prove the same, often classic, result. For example, the book opens with six different proofs that there are infinitely many prime numbers. A number of these are classics, or variations on a classic: assume that there are only finitely many, and then find a contradiction by constructing a new prime, or showing one must exist. But some of the constructions are incredibly novel.

Now here’s the thing: this is a book that is decidedly not for the layperson. To read it, you need to be extremely comfortable with calculus and limits, infinite sums, and some serious subtleties, both conceptual and notational. But if you are, the surprises keep coming. The results on primes are nothing short of astonishing (especially to me, as a non-number theorist who’s always loved the subject).

For example, there are these incredible inequalities throughout the book. Bertrand’s Postulate states that there is always a prime number between any number n and its double 2n. I’ve always thought that this was a great result. But the proof is magical: a series of inequalities that leads to a contradiction that a quantity ends up growing larger than it should, unless there’s a prime between n and 2n. On the way, there’s this elegant proof of the remarkable fact that the product of all primes less than a number x is less than or equal to 4^{x-1}. The proof is too involved to get into here, but consider the depth of that statement: on one side, you’re going up number by number, multiplying primes when you come to them, ignoring nonprimes; on the other side, you’re putting in a factor of 4 for each number. No matter how large the primes get, the product of fours is always bigger.

These are the kind of details that at once make math rich and, sadly, inaccessible. For me, I read the book blown away, exclaiming aloud when a new idea is dropped into the mix, all the time with that sense of “how did anyone think of this?”
I suppose that’s what happens whenever you’re looking at great art. How did Stravinsky compose The Rite of Spring? How did Picasso make Guernica? It takes a huge amount of work to get a sense of how they thought of it (though there’s always a path). It’s inspiring, and also leads to the despair that comes from idealism. There’s a prime between n and 2n! Wow! And no one knows if there’s always a prime between n^2 and (n+1)^2? I’ll solve that! And then, of course, you realize just how hard these unsolved problems are. It’s easy to go from surveying the best work to deciding that you don’t have what it takes. Still, these works of art are worth seeing.

There’s another way to appreciate the wonder of it all, and that’s with the results that connect things that seem like they shouldn’t even be remotely connected. Consider Stirling’s Approximation. We could ask, how different are N! and N^N? If you need to adjust one of them, what is the adjustment? The answer: the adjustment factor is roughly \sqrt{2\pi N}/e^N. What in the world is \sqrt{2}, e, and \sqrt{\pi} doing there?! It seems like magic that these terms are all connected. If you look at the proof, it makes more sense. Nevertheless, there’s a way that I continue to carry a sense of wonder that all these pieces fit together, again and again, unexpectedly and inevitably.


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