  Mathematical proofs are both paradigms of certainty and some of the most
explicitly-justified arguments that we have in the cultural record. Their very
explicitness, however, leads to a paradox, because the probability of error
grows exponentially as the argument expands. When a mathematician encounters a
proof, how does she come to believe it? Here we show that, under a
cognitively-plausible belief formation mechanism combining deductive and
abductive reasoning, belief in mathematical arguments can undergo what we call
an epistemic phase transition: a dramatic and rapidly-propagating jump from
uncertainty to near-complete confidence at reasonable levels of claim-to-claim
error rates. To show this, we analyze an unusual dataset of forty-eight
machine-aided proofs from the formalized reasoning system Coq, including major
theorems ranging from ancient to 21st Century mathematics, along with five
hand-constructed cases including Euclid, Apollonius, Hernstein's Topics in
Algebra, and Andrew Wiles's proof of Fermat's Last Theorem. Our results bear
both on recent work in the history and philosophy of mathematics on how we
understand proofs, and on a question, basic to cognitive science, of how we
justify complex beliefs.
