Tim Hodgetts – A “Bayesian” or “Newmanian” approach to proof in mathematics

I have delivered several talks to the Circle in the last few years, exploring the concepts of “proof” (or “demonstration”) exemplified quantitatively by Rev Thomas Bayes and qualitatively by Cardinal St John Newman. The talks have been well received (somewhat to my surprise!), so I am encouraged to offer another one.

I begin with a very brief review of the essential previous material. Bayes created the modern theory of “probability of propositions”, showing how the probability that a compound proposition is true can be built up by successively including the probabilities of the simple propositions on which it depends. Newman discussed in his Grammar of Assent what he called “the accumulation of converging probabilities”, as a qualitative tool for increasing the credibility of complex theological propositions by clarifying their dependence on elementary propositions of “natural theology” used as axioms. Both writers aimed to take logical reasoning beyond the rigidity of syllogistic logic, in which the only possible conclusions are “true”, “false” or “it doesn’t follow”.

Can we apply such methods to proof in mathematics or mathematical physics? Non-mathematicians instinctively say we cannot, because they implicitly view mathematics as syllogistic logic in excelsis, so that to them a mathematical proof is “exactly and perfectly true” (Lewis Carroll, The Hunting of the Snark) or it is not a mathematical proof at all. Mathematicians and mathematical physicists are rarely so dogmatic. The work of Kurt Gödel in the 1930s demolished the idea that all mathematical reasoning can be reduced to symbolic manipulation, by proving that symbolic systems such as Bertrand Russell’s Principia Mathematica are necessarily incomplete. Even before Gödel, Russell himself had already realised that his system could at most be complete in pure mathematics. His comment that “Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true” neatly sums up the essential distinction between mathematics and the mathematical sciences: mathematics may give us a convincing method of solving some collection of equations, but only the appropriate mathematical science can lead us to the preliminary scientific conclusion that the collection of equations is a usefully-accurate description of some real-world problem we want to solve.

Since it is always possible that the preliminary scientific conclusion is unsound and so the mathematical description of our problem is inadequate, is it pointful to strive for a mathematically-perfect solution to a possibly-inadequate mathematical model? If not, what is the status of an imperfect solution; can solutions in the mathematical sciences be “good enough” even if imperfect? I will explore these questions, with the aid of a problem set in a recent student mathematics Olympiad which is suitable for non-mathematical philosophers (it requires deep thought but very little symbolic mathematics).