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# foundations of mathematics

## Latest Articles

### Alan Turing and the Power of Negative Thinking

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

### Why Mathematical Proof Is a Social Compact

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

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.

### Banach-Tarski and the Paradox of Infinite Cloning

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.

### How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer.

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

### Inside the Secret Math Society Known Simply as Nicolas Bourbaki

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

### Building the Mathematical Library of the Future

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.

### Conducting the Mathematical Orchestra From the Middle

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

### How Gödel’s Proof Works

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.