Non-deductive logic in mathematics

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, such as Fermat's Last Theorem and the Riemann Hypothesis, have had to be considered in terms of the evidence for and against them. It is argued here that it is not adequate to describe the relation of evidence to hypothesis as `subjective', `heuristic' or `pragmatic', but that there must be an element of what it is rational to believe on the evidence, that is, of non-deductive logic.
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DOI 10.1093/bjps/38.1.1
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Alexander Paseau (2015). Knowledge of Mathematics Without Proof. British Journal for the Philosophy of Science 66 (4):775-799.
John R. Welch (2013). New Tools for Theory Choice and Theory Diagosis. Studies in History and Philosophy of Science 44 (3):318-329.
Ian J. Dove (2013). Towards a Theory of Mathematical Argument. In Andrew Aberdein & Ian J. Dove (eds.), Foundations of Science. Springer 291--308.

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