Foundations of mathematics

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.

The laws of physics imply that the passage of time is an illusion. To avoid this conclusion, we might have to rethink the reality of infinitely precise numbers.

Two monumental works have led many mathematicians to avoid the equal sign. The process has not always gone smoothly.

Decades after the landmark proof of Fermat’s Last Theorem, ideas abound for how to make it even more reliable. But such efforts reflect a deep misunderstanding of what makes the proof so important.

Two mathematicians have proved that two different infinities are equal in size, settling a long-standing question. Their proof rests on a surprising link between the sizes of infinities and the complexity of mathematical theories.

When two mathematicians raised pointed questions about a classic proof that no one really understood, they ignited a years-long debate about how much could be trusted in a new kind of geometry.

A surprising new proof is helping to connect the mathematics of infinity to the physical world.