Bayesianism in Mathematics

Abstract

I shall begin by giving an overview of the research programme named in the title of this paper. The term ‘research programme’ suggests perhaps a concerted effort by a group of researchers, so I should admit straight away that since I have started looking investigating the idea that plausible mathematical reasoning is illuminated by Bayesian ideas, I have not encountered in the literature anyone else who has thought to develop the views of the programme’s founder, the Hungarian mathematician, George Pólya. I should further admit that Pólya never termed himself a Bayesian as such. Motivation for the programme may, therefore, be felt sorely necessary. Let us begin, then, with three reasons as to why one might want to explore the possibility of a Bayesian reconstruction of plausible mathematical reasoning:

• (a) To acquire insight into a discipline one needs to understand how its practitioners reason plausibly. Understanding how mathematicians choose which problems to work on, how they formulate conjectures and the strategies they adopt to tackle them requires considerations of plausibility. Since Bayesianism is widely considered to offer a model of plausible reasoning, it provides a natural starting point. Furthermore, Pólya has already done much of the spadework with his informal, qualitative type of Bayesianism.

• (b) The computer has only recently begun to make a serious impact on the way some branches of mathematics are conducted. A precise modelling of plausibility considerations might be expected to help in automated theorem proving and automated conjecture formation, by providing heuristics to guide the search and so prevent combinatorial explosion. Elsewhere, computers are used to provide enormous quantities of data. This raises the question of what sort of confirmation is provided by a vast number of verifications of a universal statement in an infinite domain. It also suggests that statistical treatments of data will become more important, and since the Bayesian approach to statistics is becoming increasingly popular, we might expect a Bayesian treatment of mathematical data, especially in view of its construal of probability in terms of states of knowledge, rather than random variables.

• (c) The plausibility of scientific theories often depends on the plausibility of mathematical results. This has always been the case, but now we live in an era where for some physical theories the only testable predictions are mathematical ones. If we are to understand how physicists decide on the plausibility of their theories, this must involve paying due consideration to the effect of verifying mathematical predictions.