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?