Proofs

The math, which combines chaos, quantum theory, and infinitely complex fractal structures, has been called a “foundational result.”

AI’s greatest mathematical successes have come from answers to problems posed by a mid-20th century iconoclast. By examining what makes the Erdős problems unique, mathematicians are trying to understand how AI might change the rest of math.

When checking that solutions to certain problems are correct, it turns out, you can’t get around the inherent complexity of the quantum world.

With automated proof-checkers, a problem can be broken up into small chunks, solved bit-by-bit, then reassembled with confidence that every piece is correct. For some, this heralds a new area in mathematical research.

At 25, Kurt Gödel proved there can never be a mathematical “theory of everything.” Columnist Natalie Wolchover explores the implications.

A graduate student recently harnessed the complexity of mathematical proofs to create a powerful new tool in cryptography.

AI is being used to prove new results at a rapid pace. Mathematicians think this is just the beginning.

The quest to make mathematics rigorous has a long and spotty history — one mathematicians can learn from as they push to formalize everything in the computer program Lean.

Researchers have used metamathematical techniques to show that certain theorems that look superficially distinct are in fact logically equivalent.