Mathematical coincidences

“In the early 1970s the mathematician John McKay made a simple observation. He remarked that

“196,884   =      1     +     196,883

“What is peculiar about this formula is that the left-hand side of the equation, the number 196,884, is well known to most practitioners of a certain branch of mathematics (complex analysis, and the theory of modular forms), while 196,883, which appears on the right, is well known to most practitioners of what was in the 1970s quite a different branch of mathematics (the theory of finite simple groups). McKay took this “coincidence” — the closeness of those two numbers — as evidence that there had to be a very close relationship between these two disparate branches of pure mathematics, and he was right! Sheer coincidences in math are often not merely sheer; they’re often clues — evidence of something missing, yet to be discovered.”

From Shadows of Evidence.

How proofs happen

When you see a proof in print it’s like an Old Master painting – everything in its place and polished and perfect. You get the feeling that no human being could produce such a thing, and you despair. This is why Old Master drawings are so encouraging. Repeated attempts at getting the same thing. There is a lovely Guercino drawing where the hero has three left arms: one because Guercino got the proportions wrong, one because he changed his mind about where to put it.

The description of how something comes about is inevitably far longer than the act of doing it. Try describing a BJJ move in terms of what happens, why it happens, and what would have happened if you’d done something else. Try describing walking, for that matter.

So this description of how to prove the Fundamental Theorem of Arithmetic (that every number has a unique factorization into prime numbers) is invaluable. It describes how one actually thinks when proving things, not merely the result of doing the thinking.