Foundations of mathematics

The quest to make mathematics rigorous has a long and spotty history — one mathematicians can learn from as they push to formalize everything in the computer program Lean.

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

In an 1874 paper, Georg Cantor proved that there are different sizes of infinity and changed math forever. A trove of newly unearthed letters shows that it was also an act of plagiarism.

Intuition breaks down once we’re dealing with the endless. To begin with: Some infinities are bigger than others.

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.

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

Mathematicians have started to prepare for a profound shift in what it means to do math.

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