Mathematical alchemy and wrestling

David Mumford wrote a blog post a few weeks ago in which he identified four tribes of mathematicians. Here’s a summary of his description of the four tribes.

  • Explorers are people who ask — are there objects with such and such properties and if so, how many? …
  • Alchemists … are those whose greatest excitement comes from finding connections between two areas of math that no one had previously seen as having anything to do with each other.
  • Wrestlers … thrive not on equalities between numbers but on inequalities, what quantity can be estimated or bounded by what other quantity, and on asymptotic estimates of size or rate of growth. This tribe consists chiefly of analysts …
  • Detectives … doggedly pursue the most difficult, deep questions, seeking clues here and there, sure there is a trail somewhere, often searching for years or decades. …

I’m some combination of alchemist and wrestler. I suppose most applied mathematicians are. Applications usually require taking ideas developed in one context and using them in another. They take often complex things then estimate and bound them by things easier to understand.

One of my favorite proofs is Bernstein’s proof of the Weierstrass approximation theorem. It appeals to both alchemists and wrestlers. It takes an inequality from probability and uses it in an entirely different context, one with no randomness in sight, and uses it to explicitly construct an approximation satisfying the theorem.

I thought of David Mumford’s tribes when I got an email a couple days ago from someone who wrote to tell me he found in one of my tech reports a function that he’d studied in his own research. My tech report was motivated by a problem in biostatistics, while he was looking at material structural fatigue. The connection between remote fields was a bit of alchemy, while the content of the tech report, an upper bound on an integral, was a bit of wrestling.

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