"All Cretans are liars", said Epimenides, a Cretan. But this means that his statement must be a lie too. But then it is false that Cretans are liars and the statement must be true. So what now?

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 ...

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 ...

In 1931, the Austrian logician Kurt Gödel published his incompleteness theorem, a result widely considered one of the greatest intellectual achievements of modern times. The theorem states that in any ...

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 ...

The starting point for rigorous reasoning in mathematics is a system of axioms. An axiom is a statement that is assumed, without demonstration, to be true. It is usually self-evident, for example, ...

What is pure mathematics? What do pure mathematicians do? Why is pure mathematics important? These are questions I’m often confronted with when people discover I do pure mathematics. I always manage ...

This article is from Proof Positive, our friendly math newsletter that's delivered to your inbox every Tuesday afternoon. Sign up today and read it first. Last week I explained how a then 25-year-old ...