On the Relation Between Mathematics and Physics: How Not to ‘Factor’ a Miracle

On the Relation Between Mathematics and Physics: How Not to ‘Factor’ a Miracle: Mathematics is a bit like Zen, in that its greatest masters are likely to deny there being any succinct expression of what it is. It may seem ironic that the one subject which demands absolute precision in its definitions would itself defy definition, but the truth is, we are still figuring out what mathematics is. And the only real way to figure this out is to do mathematics. Mastering any subject takes years of dedication, but mathematics takes this a step further: it takes years before one even begins to see what it is that one has spent so long mastering. I say
“begins to see” because so far I have no reason to suspect this process terminates. Neither do wiser and more experienced mathematicians I have talked to. In this spirit, for example, The Princeton Companion to Mathematics [PCM], expressly renounces any tidy answer to the question “What is mathematics?” Instead, the book replies to this question with 1000 pages of expositions of topics within mathematics, all written by top experts in
their own subfields. This is a wise approach: a shorter answer would be not just incomplete, but necessarily misleading. Unfortunately, while mathematicians are often reluctant to define mathematics, others are not. In 1960, despite having made his own mathematically significant contributions, physicist Eugene Wigner defined mathematics as “the science of skillful operations with concepts and rules invented just for this purpose” [W]. This rather negative characterization of mathematics may have been partly tongue-in-cheek, but he took it seriously enough to build upon it an argument that mathematics is “unreasonably effective” in the natural sciences—an argument which has been unreasonably
influential among scientists ever since. What weight we attach to Wigner’s claim, and the view of mathematics it promotes, has both metaphysical and practical implications for the progress of mathematics and physics. If the effectiveness of mathematics in physics is a ‘miracle,’ then this miracle may well run out. In this case, we are justified in keeping the two subjects ‘separate’ and hoping our luck continues. If, on the other hand, they are deeply and rationally related, then this surely has consequences for how we should do research at the interface. In fact, I shall argue that what has so far been unreasonably effective is not mathematics but reductionism—the practice of inferring behavior of a complex problem by isolating and solving manageable ‘subproblems’—and that physics may be reaching the limits of effectiveness of the reductionist approach. In this case, mathematics will remain our best hope for progress in physics, by finding precise ways to go beyond reductionist tactics.