Why has there been such enormous resistance to the idea that there should be an account of rational belief in mathematics for propositions not known to be true or false?
I took this up in my chapter in Foundations of Bayesianism, and revised it as Chap. 5 of Towards a Philosophy of Real Mathematics, having noticed Polya’s neglected account in Mathematics and Plausible Reasoning. Only afterwards did I discover James Franklin had done some very useful work in ‘Non-Deductive Logic in Mathematics’, The British Journal for the Philosophy of Science, Vol. 38, No. 1 (Mar., 1987), pp. 1-18. He writes
Previous work on this topic has therefore been rare. There is one notable exception, the pair of books by the mathematician George Polya on Mathematics and Plausible Reasoning . Despite their excellence, they have been little noticed by mathematicians, and even less by philosophers. Undoubtedly this is largely because of Polya’s unfortunate choice of the word ‘plausible’ in his title–‘plausible’ has a subjective, psychological ring to it, so that the word is almost equivalent to ‘convincing’ or ‘rhetorically persuasive’. Arguments that happen to persuade, for psychological reasons, are rightly regarded as of little interest in mathematics and philosophy. Polya in fact made it clear, however, that he was not concerned with subjective impressions, but with what degree of belief was justified by the evidence ( I p. 68).
Once you are made aware of the idea of changing degrees of belief in mathematical statements, you see references to them everywhere:
…it is my view that before Thurston’s work on hyperbolic 3-manifolds and his formulation of the general Geometrization Conjecture there was no consensus amongst experts as to whether the Poincaré Conjecture was true or false. After Thurston’s work, notwithstanding the fact that it has no direct bearing on the Poincare Conjecture, a consensus developed that the Poincaré Conjecture (and the Geometrization Conjecture) were true. Paradoxically, subsuming the Poincare Conjecture into a broader conjecture and then giving evidence, independent from the Poincare Conjecture, for the broader conjecture led to a firmer belief in the Poincaré Conjecture.(John W. Morgan, ‘Recent Progress on the Poincare Conjecture and the Classification of 3-Manifolds’, Bulletin of the American Mathematical Society 2004, 42(1): 57-78)
For a simple case of plausible reasoning in mathematics, consider that you are set the task of finding a formula for the curved surface area of a frustrum of a cone. With your shaky and rudimentary calculus you arrive at the conjecture that the surface area is . You are moderately confident that this is correct, but you accept that you may have erred.
You then consider 1. The dimensions look right. 1. gives annulus area. 1. gives cylinder area. 1. gives cone area. 1. symmetrical under exchange of and .
The probability is now very high, apparently resulting by conditioning on old evidence. Which checks give most support? Why?
It looks as though, rather than
we are using