Foundations of mathematics

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.

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.