Why is mathematics so apt for describing the physical world? Why do our local descriptions turn out to be pretty generalizable (like the law of gravity, first developed for objects around here, then applied to planets, then expanded by Einstein). I have always thought that this problem had an easy solution but this morning I started to reconsider.
I remember back in 2014 Max Tegmark gave a talk at the Graduate Center (and again at NYU) where he argued that the universe itself is a mathematical structure and part of his argument was based on the unreasonable effectiveness argument. At the time I raised an objection along the following lines: there are an infinite number of mathematical structures so one of those will turn out to be a pretty good description of reality. It is not magic, it is just that the number of mathematical structures out there covers all the possibilities so it is no mystery why one (or several) map onto the world. At the time he responded that there were only a finite number of Platonic solids not an infinite number (or at least this is how I remember his response) and of course that is true of our kind of word with three spatial dimensions (at least three macroscopic ones 😉 but there are higher-dimensional Platonic solids and if we lived in an 8 dimensional world we would have those mapping onto physical reality rather than the 5 we know and love. At the time I felt like that settled the issue, and a couple of people remarked afterwards that they had basically agreed with my point.
But now I think that perhaps this kind of response misses Tegmark’s point. Perhaps the question isn’t ‘why does this equation (rather than that one) so accurately describe physical reality?’ but rather ‘why does any equation at all (whatever it is) so accurately describe physical reality?’…the kind of answer I gave before seems like a good answer to the former question but it is not a good answer to the latter question. I think I can conceive of a world, much like ours, but which no mathematical description sufficiently describes. If that is conceivable then, if it is also possible that there be such a world, then it is a genuine question whether our world has any non-mathematically describable properties. And it would also be a genuine further question why mathematics does accurately describe the part it does (since it could have failed to do so) not to mention the question of why mathematical speculation can lead to fruitful empirical discoveries.
On the other hand, is it really conceivable that there be world like ours that no mathematics could describe?
One thought on “The Unreasonable Effectiveness of Mathematics in Natural Science”
I tracked down this blog after listening to some of your YouTube lectures (Hume/Kant/Utilitarianism/Egoism) to keep me awake on an overnight, 700 mile drive from down south. Riveting stuff!
I may be misunderstanding everything here, but I’ll take a few stabs:
(1) “Why does any equation at all (whatever it is) so accurately describe physical reality?”
This sounds like a reframing of the problem of induction, i.e., why do past and future events resemble each other so closely to the extent their relationship can be expressed by a fixed mathematical formula? It seems to me that all events that repeat, and whose results are unfailingly reproduceable, would be subject to being described by an mathematical equation that captures that repetition. Can you conceive of a repeatable event that wouldn’t be?
(2) I can certainly conceive of a universe in which the physical laws change from day to day or instant to instant, being so inconstant and chaotic that no law could ever be derived from the observation of two or more events. Is that the world you are conceiving, “which no mathematical description (can) sufficiently describe? A world in which balls fall at different speeds each time you drop them (despite all surrounding conditions being the same), or the billard balls go in different directions each time they’re hit, or sometimes explode, or morph into elephants?
(3) I’m having trouble conceiving of a universe in which math cannot be applied to anything in it at all. Let’s suppose a universe containing only four motionless, indivisible marbles. Wouldn’t simply counting them qualify as employing math? Or, if they were arranged in the shape of a square (each being a point representing one of the four corners) wouldn’t expressing the shape with geometry count? Even if they were just randomly placed, I assume you could construct a single equation that would generate a line passing through each of the points. While I understand that equations expressing physical laws generally track some sort of movement or change over time, it seems to me that math could be applied to describe the position of things in a completely “frozen” universe.
(4) I recall that in one of your lectures (I forget which, I think you were making the point that even 7 + 5 = 12 could be viewed as merely an a posteriori claim), you said something to the effect that math wouldn’t exist in a world without sentient beings capable of doing math. Is that observer part of the non-mathematically-describable world you’re postulating?