Non-Deductive Logic in Mathematics: The Probability of Conjectures

Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, as Fermat’s Last Theorem did and the Riemann Hypothesis still does, have had to be considered in terms of the evidence for and against them. It is not adequate to describe the relation of evidence to hypothesis as “subjective”, “heuristic” or “pragmatic” there must be an element of what it is objectively rational to believe on the evidence, that is, of non-deductive logic. Mathematics is therefore (among other things) an experimental science. The occurrence of non-deductive logic, or logical probability, or the rational support for unproved conjectures, in mathematics is however an embarrassment. It is embarrassing to mathematicians, used to regarding deductive logic as the only real logic. It is embarrassing for those statisticians who wish to see probability as solely about random processes or relative frequencies: surely there is nothing probabilistic about the truths of mathematics? It is a problem for philosophers who believe that induction is justified not by logic but by natural laws or the “uniformity of nature”: mathematics is the same no matter how lawless nature may be. It does not fit well with most philosophies of mathematics. It is awkward even for proponents of non-deductive logic. If non-deductive logic deals with logical relations weaker than entailment, how can such relations hold between the necessary truths of mathematics? Work on this topic was therefore rare in the mid-twentieth century “classical” period in the philosophy of science and mathematics. The recent turning of attention in philosophy of mathematics towards mathematical practice has produced a number