Bayesian Evidence Accumulation in Experimental Mathematics: A Case Study of Four Irrational Numbers

TL;DR

This paper applies Bayesian inference to evaluate the evidence supporting the normality of four fundamental irrational constants, demonstrating its effectiveness in experimental mathematics for updating beliefs based on accumulating data.

Contribution

It introduces a Bayesian approach to quantify evidence for mathematical conjectures, specifically testing the normality of constants like π, e, √2, and ln(2).

Findings

Overwhelming evidence supports the normality hypothesis for each constant.

Bayesian inference effectively quantifies evidence in experimental mathematics.

The approach is suitable for continually updating beliefs with increasing data.

Abstract

Many questions in experimental mathematics are fundamentally inductive in nature. Here we demonstrate how Bayesian inference --the logic of partial beliefs-- can be used to quantify the evidence that finite data provide in favor of a general law. As a concrete example we focus on the general law which posits that certain fundamental constants (i.e., the irrational numbers $\pi$, $e$, $\sqrt2$, and $\ln{2}$) are normal; specifically, we consider the more restricted hypothesis that each digit in the constant's decimal expansion occurs equally often. Our analysis indicates that for each of the four constants, the evidence in favor of the general law is overwhelming. We argue that the Bayesian paradigm is particularly apt for applications in experimental mathematics, a field in which the plausibility of a general law is in need of constant revision in light of data sets whose size is increasing continually and indefinitely.