At the dinner, I was seated next to a 18-month-old. She slid her hand under a napkin and pulled it out again. Strangely, her hand was still there. She repeated this action many times, puzzled as to why her hand wasn’t vanishing. A few times, I pressed her hand from above the napkin to demonstrate that, not only did the hand still exist, but it retained all tactile functionality. This mystery of object permanence was baffling, and she concluded that future research was going to be necessary. First, though, there were cookies to eat.
Now, people sometimes ask why mathematics is so unreasonably effective at describing the physical universe we occupy. I start off with the object permanence example because all joking aside, the infant was engaged in scientific experimentation of the purest kind. We can imagine a viable universe in which only things that are seen possess existence or, equivalently, a universe in which we see all that is. There is nothing fundamentally illogical or self-contradictory about this idea, it just doesn’t happen to describe the world as it actually is, and experimentation is the only way of sorting such things out. Smart baby.
So, math can describe things that do not match the universe. There are also things in the universe that are not amenable to mathematical description. Godel’s Theorem, for example, exists beyond the grasp of finite axiomatic systems, and the concept of now has no physical interpretation. Moral judgements are also highly problematic, and appear to defy quantization. The Venn diagram is looks this:
Speaking of Godel, on the occasion of Albert Einstein’s 70th birthday, mathematician Kurt Godel presented to Einstein an alternate solution to the Einstein field equations. Godel’s solution described a universe of rotating, uniformly distributed dust. Though thoroughly consistent with Relativity, this is nothing like the universe we inhabit, and Einstein was delighted. Smart mathematician.
Another smart mathematician, Cornelius Lanczos, pointed out one obvious-in-retrospect reason why math and physics get along together so happily: “Newton would not have been interested in this tremendous discovery of calculus if it wouldn’t have been directly applicable to the laws of nature.” He makes an excellent point. A lot of mathematics was motivated by physics and would not look as it does had physics been something different. Math fits so beautifully with physics because we wrote it that way.
For my money, I think there is one other reason, though I am speculating here, and on very shaky ground. But I suspect that both math and physics are both derivative of yet a third thing, more general than both of them: geometry. From geometrical considerations alone, without calculating a single number, the ancient Greeks knew that the proportions a parabola were given by the square of a certain magnitude. (For details, see Conics, by Apollonius of Perga. How many smart mathematicians are going to turn up in this post?) Mathematics, it seems to me, has to yield to geometry, and not the other way around.
Now, perhaps I am wrong. Perhaps considerations of Gaussian curvature and Hilbert spaces demolish geometry as generalized truth. I’m not sure. But if math is subsumed by geometery, then physics is too, and this would explain why the one is so unreasonably effective at explaining the other. They are both, I suspect, manifestations of geometrical truth.
Of a Sphinx Athwart the Evening Sky 