"One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed case that one would be wrong. In particular, I argue that computational complexity theory---the field that studies the resources (such as time, space, and randomness) needed to solve computational problems---leads to new perspectives on the nature of mathematical knowledge, the strong AI debate, computationalism, the problem of logical omniscience, Hume's problem of induction, Goodman's grue riddle, the foundations of quantum mechanics, economic rationality, closed timelike curves, and several other topics of philosophical interest. I end by discussing aspects of complexity theory itself that could benefit from philosophical analysis."
A math professor was explaining a particularly complicated calculus concept to his class when a frustrated pre-med student interrupts him. "Why do we have to learn this stuff?" the pre-med blurts out. The professor pauses, and answers matter-of-factly: "Because math saves lives." "How?" demanded the student. "How on Earth does calculus save lives?" "Because," replied the professor, "it keeps certain people out of medical school."