Surprises in numerical expressions of physical constants

TL;DR

This paper investigates the likelihood that simple mathematical expressions approximating physical constants are coincidental, emphasizing the importance of null models to distinguish genuine insights from chance.

Contribution

It introduces a naive probabilistic approach to assess whether simple expressions of physical constants are likely coincidences or meaningful approximations.

Findings

Provides a framework to estimate coincidence probabilities

Highlights the role of null models in interpreting simple expressions

Suggests many simple expressions may occur by chance

Abstract

In science, as in life, `surprises' can be adequately appreciated only in the presence of a null model, what we expect a priori. In physics, theories sometimes express the values of dimensionless physical constants as combinations of mathematical constants like pi or e. The inverse problem also arises, whereby the measured value of a physical constant admits a `surprisingly' simple approximation in terms of well-known mathematical constants. Can we estimate the probability for this to be a mere coincidence, rather than an inkling of some theory? We answer the question in the most naive form.