Proof at a roll of the dice

摘要

The PCP theorem encapsulates the idea that randomization allows the immediate verification of any mathematical proof. A simple route to this striking result was proposed earlier this year. Many branches of mathematics have their signature numbers: geometry has the transcendental π; analysis Eulers exponential e; and algebra the imaginary unit, the square-root of-1, i. These numbers are the stars of exotic — but seemingly quite separate — worlds scattered across the vast expanse of the mathematical cosmos. Until, that is, one day a master stargazer spots a stellar alignment of breathtaking beauty, and nothing ever looks quite the same again. For these worlds, that alignment is Euler's identity, e~(iπ) = -1, and it is proof, if proof were needed, that mathematics is about more than shape (π), change (e) and structure (i): it is also about magic.