Abstract: Accounts of the validity of mathematical proofs traditionally base them in the validity of their underlying deductive steps. However, in a skeptical argument going back to Hume, this should make even weak belief in the final claim of the theorem unjustified. To understand how and why mathematical arguments appear, by contrast, to be paradigms of certainty, we undertake a data science study of the epistemic structure of actual proofs, ranging from Euclid’s Geometry and Apollonius’ Conics to fifty computer-assisted contemporary proofs including Godel Incompleteness and the Four Color Theorem. Our analysis shows that these proofs share an underlying network structure. This structure enables the emergence of certainty even in the presence of skepticism of the validity of any particular step. This emergence is explosive, and has an isomorphism to phase transitions in material objects. I finish with some remarks from practicing mathematicians that serve to validate the model of mathematical belief formation we propose.
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