Every Mathematician Has Only a Few Tricks

A long time ago an older and well-known number theorist made some disparaging remarks about Paul Erdos’s work. You admire Erdos’s contributions to mathematics as much as I do, and I felt annoyed when the older mathematician flatly and definitively stated that all of Erdos’s work could be “reduced” to a few tricks which Erdos repeatedly relied on in his proofs.

What the number theorist did not realize is that other mathematicians, even the very best, also rely on a few tricks which they use over and over.

Take Hilbert. The second volume of Hilbert’s collected papers contains Hilbert’s papers in invariant theory. I have made a point of reading some of these papers with care. It is sad to note that some of Hilbert’s beautiful results have been completely forgotten. But on reading the proofs of Hilbert’s striking and deep theorems in invariant theory, it was surprising to verify that Hilbert’s proofs relied on the same few tricks. Even Hilbert had only a few tricks!

— Ten Lessons I Wish I Had Been Taught

— Gian-Carlo Rota

2012.05.13 Sunday ACHK