Foundations of mathematics

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

Ultrafinitism, a philosophy that rejects the infinite, has long been dismissed as mathematical heresy. But it is also producing new insights in math and beyond.

Zermelo-Fraenkel set theory is so widely accepted that modern mathematicians hardly think about it. But believing in its core principles didn’t come easily.

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.

David E. Dunning explores how mathematical notation is a social, world-building technology.

In an 1874 paper, Georg Cantor proved that there are different sizes of infinity and changed math forever. A trove of newly unearthed letters shows that it was also an act of plagiarism.

Intuition breaks down once we’re dealing with the endless. To begin with: Some infinities are bigger than others.

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