Solving the Mystery of Mathematics

Jared Warren says, think of a number. What exactly are you thinking of?

- JARED WARREN

Mathematics is mysterious. By this I don't just mean that there are many difficult, unsolved mathematical questions. I mean instead that mathematics itself, as a subject, and as an activity, is mysterious. Our familiarity with elementary mathematics sometimes lets us lose sight of this. This is probably a good thing most of the time. Reflecting on the nature of mathematics is no aid to mathematical problem-solving. But doing mathematics is one thing, understanding the nature of mathematics is another. The first task is mathematical, the second philosophical.

Why is mathematics mysterious? Because it is difficult to see how mathematics fits into our picture of the world. Mathematics concern specific truths and falsehoods: it is true that 1+1=2 and it is false that 1+1=3. It is also true that there is an empty set (a set of things with nothing in it), and false that some sets are members of themselves. These are truths and falsehoods in arithmetic and set theory respectively, which are branches of pure mathematics. Pure mathematics contrasts with applied mathematics, as in applied to the physical world. There are questions about how mathematics manages to apply to the world in all of the unexpected ways it does. The physicist Eugene Wigner famously said that mathematics was 'unreasonably' effective at applying to the world. However, the puzzles I am raising here are even more basic; they concern the most fundamental aspects of pure mathematics itself.