organization:simons foundation

• Essay on Science: Shadows of Evidence | Simons Foundation
(texte de février 2013)

The Evidence of Coincidence

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, i.e., the number 196,884, is well known to most practitioners of a certain branch of mathematics (complex analysis, and the theory of modular forms), (3) 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). (4) McKay took this “coincidence” — the closeness of those two numbers (5) — 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.
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(3) 196,884 is the first interesting coefficient of a basic function in that branch of mathematics: the elliptic modular function.

(4) 196,883 is the smallest dimension of a Euclidean space that has the largest sporadic simple group (the monster group) as a subgroup of its symmetries.

(5) McKay gave a convincing interpretation of the 1 in the formula as well.

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