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Foundations of mathematics

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Mathematical Beauty, Truth and Proof in the Age of AI

April 30, 2025

Mathematicians have started to prepare for a profound shift in what it means to do math.

New Proofs Probe the Limits of Mathematical Truth

February 3, 2025

By proving a broader version of Hilbert’s famous 10th problem, two groups of mathematicians have expanded the realm of mathematical unknowability.

The Jagged, Monstrous Function That Broke Calculus

January 23, 2025

In the late 19th century, Karl Weierstrass invented a fractal-like function that was decried as nothing less than a “deplorable evil.” In time, it would transform the foundations of mathematics.

How the Square Root of 2 Became a Number

June 21, 2024

Useful mathematical concepts, like the number line, can linger for millennia before they are rigorously defined.

The Year in Math

December 22, 2023

Landmark results in Ramsey theory and a remarkably simple aperiodic tile capped a year of mathematical delight and discovery.

The Deep Link Equating Math Proofs and Computer Programs

October 11, 2023

Mathematical logic and the code of computer programs are, in an exact way, mirror images of each other.

Alan Turing and the Power of Negative Thinking

September 5, 2023

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

Q&A

Why Mathematical Proof Is a Social Compact

August 31, 2023

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

Complexity Theory’s 50-Year Journey to the Limits of Knowledge

August 17, 2023

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.

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