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# foundations of mathematics

## Latest Articles

### 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.

### Does Time Really Flow? New Clues Come From a Century-Old Approach to Math.

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.

### With Category Theory, Mathematics Escapes From Equality

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

### Why the Proof of Fermat’s Last Theorem Doesn’t Need to Be Enhanced

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.