Proofs

Mathematical proofs based on a technique called diagonalization can be relentlessly contrarian, but they help reveal the limits of algorithms.

Number theorist Andrew Granville on what mathematics really is — and why objectivity is never quite within reach.

How hard is it to prove that problems are hard to solve? Meta-complexity theorists have been asking questions like this for decades. A string of recent results has started to deliver answers.

Transcendental numbers include famous examples like e and π, but it took mathematicians centuries to understand them.

The four-color problem is simple to explain, but its complex proof continues to be both celebrated and despised.

Zero-knowledge proofs allow researchers to prove their knowledge without divulging the knowledge itself.

Three computer scientists have posted a proof of the NLTS conjecture, showing that systems of entangled particles can remain difficult to analyze even away from extremes.

What makes a proof stronger than a guess? What does evidence look like in the realm of mathematical abstraction? Hear the mathematician Melanie Matchett Wood explain how probability helps to guide number theorists toward certainty.

Artificial intelligence has bested humans at problem-solving challenges like chess and Go. Is mathematics research next? Steven Strogatz speaks with mathematician Kevin Buzzard to learn about the effort to translate math into language that computers understand.