Morals

Abstract

This concluding chapter draws a pair of morals. First, what's gone before shows that there's a form of objectivity in mathematics that doesn't depend on the existence of mathematical objects or the truth of mathematical statements. The second moral concerns the distinction between intrinsic justifications for set-theoretic claims—what's asserted is implicit in the concept of set—and extrinsic justifications—what's asserted is effective, fruitful, produces theories with important virtues. Intrinsic considerations are standardly regarded as the best, sometimes the only legitimate justifications. After examining more set-theoretic examples, the Second Philosopher concludes that this evaluation gets things backwards: intrinsic justifications are only legitimate insofar as they lead to extrinsic benefits.