Is there a mathematical definition of mathematics?

Question

First of all, I apologize if this question is inappropriate for math SE. In mathematics textbooks, there are defined all sorts of things, like groups, fields, boolean algebras, turing machines, etc. However, there is no definition of mathematics itself. I know some dictionaries define math as the study of this or that, but that is simply the study of math, not math itself. So, what is a mathematical definition of mathematics? Is there even such a thing at all?

Answer 1

You may be interested in the paper "On Proof and Progress in Mathematics" by Bill Thurston, in particular the recursive definition of mathematics given on page 2. Here is the relevant passage:

It may sound almost circular to say that what mathematicians are accomplishing is to advance human understanding of mathematics. I will not try to resolve this by discussing what mathematics is, because it would take us far afield. 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.

In other words, as mathematics advances, we incorporate it into our thinking. As our thinking becomes more sophisticated, we generate new mathematical concepts and new mathematical structures: the subject matter of mathematics changes to reflect how we think.

This definition appeals to me, but like any recursive definition, it is "bottom up", so its extent is not so obvious. Here is an alternative (the phrasing is mine, but the idea is hardly original to me):

Mathematics is the domain of inquiry where logical reasoning is the sole methodology. That is, a question is a mathematical question if and only if it can (in principle) be settled by logical reasoning alone.