  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.
