foundations of mathematics

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.

One of the strangest results in mathematics explains how it’s possible to turn one sphere into two identical copies, simply by rearranging its pieces.

For 50 years, mathematicians have believed that the total number of real numbers is unknowable. A new proof suggests otherwise.

For almost a century, the anonymous members of Nicolas Bourbaki have written books intended as pure expressions of mathematical thought.

A small community of mathematicians is using a software program called Lean to build a new digital repository. They hope it represents the future of their field.

Emily Riehl is rewriting the foundations of higher category theory while also working to make mathematics more inclusive.

His incompleteness theorems destroyed the search for a mathematical theory of everything. Nearly a century later, we’re still coming to grips with the consequences.