Proofs

When pigeons outnumber pigeonholes, some birds must double up. This obvious statement — and its inverse — have deep connections to many areas of math and computer science.

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.

Three researchers have figured out how to craft a proof that spreads out information while keeping it perfectly secret.

After decades of uncertainty, a motley team of programmers has proved precisely how complicated simple computer programs can get.

How are scientists able to crack fundamental questions about nature and life? How does math make the complex cosmos understandable? In this episode, the physicist Nigel Goldenfeld and co-host Steven Strogatz explore the deep foundations of the scientific process.

Terence Tao, who has been called the “Mozart of Mathematics,” wrote an essay in 2007 about the common ingredients in “good” mathematical research. In this episode, the Fields Medalist joins Steven Strogatz to revisit the topic.

A team of four prominent mathematicians, including two Fields medalists, proved a conjecture described as a “holy grail of additive combinatorics.”

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

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