Mathematical Proof

A logically airtight chain of reasoning from agreed starting assumptions to a conclusion, the thing that distinguishes a mathematical claim, once proved, from every other kind of well supported belief: it is not merely well evidenced, it is meant to be certain, given the assumptions. Euclid's Elements (c. 300 BCE) is the founding model still followed today, building an entire body of knowledge from a small set of definitions and postulates by strict logical deduction rather than by observation or authority. The idea of proof itself became an object of mathematical study, not only a method, in the twentieth century foundations crisis: David Hilbert's program sought to prove mathematics' own consistency using finite methods, and Kurt Godel's 1931 incompleteness theorems showed a fundamental limit on what any such self-verification can achieve, without diminishing the ordinary, working certainty an individual proof provides.