H ow do mathematicians decide that something is true? They write a proof. Often they start with proofs that already exist, building on or drawing connections between proven claims. Each of these proofs, in turn, has relied on other proofs to make its point, and so on. Proofs upon proofs. Truths upon truths. But eventually this process must come to an end. At some point, things are true simply because they are. These truths are the axioms, the ground rules. And it is tempting to stop there — to declare, as Penelope Maddy , a philosopher of mathematics at the University of California, Irvine, put it, “that axioms are obvious or intuitive or conceptual truths.” After all, most mathematicians simply accept that their work relies on an axiomatic system — namely, “Zermelo-Fraenkel set theory with the axiom of choice,” or ZFC — if they bother to acknowledge the axioms at all. ZFC is a list of 10 basic principles that together form the foundation on which nearly all of modern mathematics is built.…