When I tell someone I am a mathematician, one of the most curious common reactions is: “I really liked math class because everything was either right or wrong. There is no ambiguity or doubt.” I ...

The uncompromising ethos of pure mathematics in the early post-war period was that any theorem should be provided with a proof which the reader could and should check. Two things have made this no ...

In 1931, the Austrian logician Kurt Gödel pulled off arguably one of the most stunning intellectual achievements in history. Mathematicians of the era sought a solid foundation for mathematics: a set ...

Kurt Gödel's incompleteness theorem demonstrates that mathematics contains true statements that cannot be proved. His proof achieves this by constructing paradoxical mathematical statements. To see ...

Some mathematical statements feel undeniably true the moment you hear them. Yet proving them can be impossible. This theorem ...

Lashi Bandara is affiliated with Australian National University. These are questions I’m often confronted with when people discover I do pure mathematics. I always manage to provide an answer but it ...

Talia Ringer is in the Siebel School of Computing and Data Science, University of Illinois at Urbana–Champaign, Urbana, Illinois, 61801, USA. Read the paper: Olympiad-level formal mathematical ...

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 ...

Computers are extremely good with numbers, but they haven’t gotten many human mathematicians fired. Until recently, they could barely hold their own in high school-level math competitions. But now ...

Black holes exist in our universe. That’s widely accepted today. Physicists have detected the X-rays emitted when black holes feed, analyzed the gravitational waves from black hole collisions and even ...