In Andrew Aberdein & Ian J. Dove (eds.), The Argument of Mathematics. Springer. pp. 11--29 (2013)

Authors
James Franklin
University of New South Wales
Abstract
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.
Keywords nondeductive logic  mathematical conjectures
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Laws and Symmetry.Bas C. Van Fraassen - 1989 - Oxford, England: Oxford University Press.
The Uses of Argument.Stephen E. Toulmin - 1958 - Cambridge University Press.
Argumentation Schemes.Douglas Walton, Chris Reed & Fabrizio Macagno - 2008 - Cambridge and New York: Cambridge University Press.
The Structure of Science.Ernest Nagel - 1961 - Les Etudes Philosophiques 17 (2):275-275.

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