Logic

Ultrafinitism, a philosophy that rejects the infinite, has long been dismissed as mathematical heresy. But it is also producing new insights in math and beyond.

Zermelo-Fraenkel set theory is so widely accepted that modern mathematicians hardly think about it. But believing in its core principles didn’t come easily.

David E. Dunning explores how mathematical notation is a social, world-building technology.

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.