[Math] When Have Numerology and Computational Experimentation Been Successful

Question

When has numerology been successfully used in math and science? The Monstrous Moonshine conjecture led to a Fields medal for Borcherds. Balmer's formula for hydrogen spectra led to the Bohr model of the atom.

We could extend this to general computational experimentation. For example, the Birch-Swinnerton-Dyer conjecture was originally formulated based on sketchy computational results. Gauss guessed the law of quadratic reciprocity and the prime number theorem from his calculations too. Are there other interesting or instructive examples?

Answer 1

Three examples from the theory of dynamical systems (in a broad sense).

• Edward Lorenz' almost accidental discovery of the first example of a strange attractor and the butterfly effect.

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• The Fermi-Pasta-Ulam numerical experiments which for the first time exhibited the soliton-like dynamics in a nonlinear system.

• Mitchell Feigenbaum's experiments with iterations of the logistic map which led to the discovery of universality in nonlinear systems.