Logic

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.

Number theorist Andrew Granville on what mathematics really is — and why objectivity is never quite within reach.

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.

The numbers count a variety of seemingly unrelated mathematical structures.