Theorems by Consensus: How Belief Becomes Mathematical Truth

The mythology of mathematics holds that theorems are proven or not proven̶ abinary determined by logical validity. History tels a different story. Theorems havebeen accepted without valid proof, proofs have been accepted despite containingerrors, and the boundary between "proven" and "accepted" has been policed by socialconsensus, not logic alone. This paper documents historical cases where mathematical"truth" was determined by who believed it, who had authority, and what thecommunity was wiling to accept. Cases include: the Italian algebraists' secret methodsaccepted without proof; Euler's manipulations of divergent series; the century ofacceptance of flawed proofs of the four-color theorem; the social negotiation aroundcomputer-assisted proofs; and the ongoing acceptance of results "too long to check"by human verification. The conclusion is not that mathematics is arbitrary, but thatthe social process of acceptance is not the purely logical process it is claimed to be.Mathematical truth, in practice, is what mathematicians agree is true.