The Argument of Mathematics

Volume 30 of the series Logic, Epistemology, and the Unity of Science pp 11-29


Non-deductive Logic in Mathematics: The Probability of Conjectures

  • James FranklinAffiliated withSchool of Mathematics and Statistics, University of New South Wales Email author 

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Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or objective Bayesianism or non-deductive logic), and some detailed examples of its use in mathematics surveyed. Examples of inductive reasoning in experimental mathematics are given and it is argued that the problem of induction is best appreciated in the mathematical case.