David Lindley, The End of Physics. The Myth of a Unified Theory, 1993. From Prologe, The Lure of Numbers.
Scientific history, however, tends not to record the zigzags. History is written by the winners, and failed attemnts to explain this or that phenomena are soon forgotten, though the faild theory may have provided a useful stimulus. Kepler spent years working with the observational data collected by the Danish astronomer Tycho Brahe, and trying who knows how many geometrical shapes for the planetary orbits before he hit on ellipses. But when he published his results, ellipses were all that he offered the reader. In the same way, Isaac Newton, though he presumably reached his theories of motion and gravity and his theorems on orbiral shapes by some sort of trial and error (albeit guided by general principles), wrote up his researches in the Philosophiae naturalis principia mathematica as if he had been fortunate enough to glimpse directly the divine rules that govern the universe and, thus blessed, had worked out the consequences.
Quantum mechanics was born when Max Planck, struggling to understand the relationship between the temperature of a glowing body and the color and intensity of the light it emitted, played at fitting mathematical formulas to experimentally derived results until he found one that worked.
In the same vein, Niels Bohr began to understand how the quantum idea could explain the characteristics frequencies of light emitted by hydrogen atoms because he was familiar with a simple numerical formula, stumbled accros many years before by the Swiss mathematician Johann Jakob Balmer, which expressed all the known wavelength of hydrogen light in terms of the difference between the reciprocal of the squares of whole numbers. Balmer’s formula had been the result of arithmetical quesswork, but it turned out to be of profound significance.
There are many cases, however, where an excessive devotion to the search for mathematical simplicity can mislead. For example, there is a venerable numerical formula that claims to explain the distances of the planets from the Sun by simple arithmetic.
Good and bad numerology are easiliy distingquished in retrospect.
What forms does numerology take today, and how are we to judge, whether they are good or bad?