Gunnar_Zarncke comments on A Limited But Better Than Nothing Way To Assign Probabilities to Statements of Logic, Arithmetic, etc.

[Gunnar_Zarncke]

For those wanting to read the Haim Gaifman paper: It took me about one hour to read it.

Notable quotes:

Rationality, let us agree, implies that if I know that something is deductively implied by what I believe, then I should believe it, or revise my other beliefs.

Now the fact that mathematical activity is truth-revealing should not blind us to the a priori nature, or to the necessity, of mathematical truth; to its being constitutive of our conceptual organization, without which thought looses its coherence.

[The] deductive shortsightedness that makes [Mersennes] error possible isolates it to a large extent from the main body of other beliefs. The holding of it does not prevent one from developing other sound arithmetical beliefs about prime numbers. On the mathematical background of Mersenne’s time, believing that $2^67–1$ is prime was like having a false belief about some far away island. This is not due merely to the number’s size, but to the scarcity of known deductive chains between that belief and others. When the known territories of the deductive web expand, the contradiction between that belief and others emerges; at that stage revision takes place.