Logic

Researchers have used metamathematical techniques to show that certain theorems that look superficially distinct are in fact logically equivalent.

Descriptive set theorists study the niche mathematics of infinity. Now, they’ve shown that their problems can be rewritten in the concrete language of algorithms.

Marijn Heule turns mathematical statements into something like Sudoku puzzles, then has computers go to work on them. His proofs have been called “disgusting,” but they go beyond what any human can do.

Two new notions of infinity challenge a long-standing plan to define the mathematical universe.

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

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.

Two mathematicians have shown that origami can, in principle, be used to perform any possible computation.

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.