What is Mathematics?

William P. Thurston, from Proof and Progress in Mathematics, 1994:

Mathematicians generally feel that they know what mathematics is, but find it difficult to give a good direct definition. It is interesting to try. For me, “the theory of formal patterns” has come the closest, but to discuss this would be a whole essay in itself.

Could the difficulty in giving a good direct definition of mathematics be an essential one, indicating that mathematics has an essential recursive quality? Along these lines we might say that mathematics is the smallest subject satisfying the following:

• Mathematics includes the natural numbers and plane and solid geometry.
• Mathematics is that which mathematicians study.
• Mathematicians are those humans who advance human understanding of mathematics.

This definition is overlarge, since mathematicians study many things that are not mathematics. I do like the idea of a recursive definition of mathematics, and maybe this definition can be improved to meet my objection.

David Spivak (personal communication, 2024) picks up on the other possibility, of a good direct definition:

Math is the collection of all the thought-patterns that are stable enough that we can work with them collectively.

I like this definition as it emphasizes the social character of mathematics.