Foundations of mathematics

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.

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.

Useful mathematical concepts, like the number line, can linger for millennia before they are rigorously defined.

Landmark results in Ramsey theory and a remarkably simple aperiodic tile capped a year of mathematical delight and discovery.

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