Introduction Mathematicians spend most of their time thinking about what’s knowable. But the unknowable can be just as compelling. Perhaps the most famous example comes from a theorem by the logician Kurt Gödel. Gödel’s celebrated result — one of two “ incompleteness theorems ” he published in 1931 — established that for any reasonable set of basic mathematical assumptions, called axioms, it’s impossible to prove that the axioms won’t eventually lead to contradictions. Though mathematicians continued their research much as they had before, they would never again be certain that their rules were self-consistent. More than 50 years after Gödel’s theorem, cryptographers devised a radical new proof method in which unknowability played a very different role. Proofs based on this technique, called zero-knowledge proofs , can convince even the most skeptical audience that a statement is true without revealing why it’s true.…