## 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. ..

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

A deceptively simple math proposition known as the Kakeya conjecture underpins a tower of other questions in physics, number theory, and harmonic analysis. ..

Mathematicians can often figure out what happens as quantities grow infinitely large. What about when they are just a little big? ..

There’s a surprisingly straightforward answer to how many numbers are needed to fill an infinite grid so that identical numbers never get too close together. ..

By imbuing enormous vectors with semantic meaning, we can get machines to reason more abstractly—and efficiently—than before. ..

In 50 years of searching, mathematicians found only one example of a “subspace design” that fit their criteria. A new proof reveals that there are infinitely ..

The "Monty Hall problem" is a classic example of how games of chance can have surprising results. Here’s a fun way to model the problem. ..

A decades-old conjecture about the best way to minimize the surface area of a three-bubble cluster seemed unprovable—until a breakthrough result. ..

Richard Feynman’s path integral is a powerful prediction machine and a philosophy. Physicists still struggle to figure out how to use it, and what it means. ..

In three-dimensional space, the surface of a black hole must be a sphere. But in higher dimensions, an infinite number of configurations are possible. ..