Thoughts the Unreasonable Effectiveness of Maths

Half a year or so ago I stumbled across Eugene Wigner's 1960's article "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". It asks a fairly simple question. Why does mathematics generalize so well to the real world? Even in cases where the relevant math was discovered (created?) hundreds of years before the physics problems we apply it to were known. In it he gives a few examples. Lifted from wikipedia:

Wigner's first example is the law of gravitation formulated by Isaac Newton. Originally used to model freely falling bodies on the surface of the Earth, this law was extended based on what Wigner terms "very scanty observations" to describe the motion of the planets, where it "has proved accurate beyond all reasonable expectations." Wigner says that "Newton ... noted that the parabola of the thrown rock's path on the earth and the circle of the moon's path in the sky are particular cases of the same mathematical object of an ellipse, and postulated the universal law of gravitation on the basis of a single, and at that time very approximate, numerical coincidence."

Wigner's second example comes from quantum mechanics: Max Born "noticed that some rules of computation, given by Heisenberg, were formally identical with the rules of computation with matrices, established a long time before by mathematicians. Born, Jordan, and Heisenberg then proposed to replace by matrices the position and momentum variables of the equations of classical mechanics. They applied the rules of matrix mechanics to a few highly idealized problems and the results were quite satisfactory. However, there was, at that time, no rational evidence that their matrix mechanics would prove correct under more realistic conditions." But Wolfgang Pauli found their work accurately described the hydrogen atom: "This application gave results in agreement with experience." The helium atom, with two electrons, is more complex, but "nevertheless, the calculation of the lowest energy level of helium, as carried out a few months ago by Kinoshita at Cornell and by Bazley at the Bureau of Standards, agrees with the experimental data within the accuracy of the observations, which is one part in ten million. Surely in this case we 'got something out' of the equations that we did not put in." The same is true of the atomic spectra of heavier elements.

Wigner's last example comes from quantum electrodynamics: "Whereas Newton's theory of gravitation still had obvious connections with experience, experience entered the formulation of matrix mechanics only in the refined or sublimated form of Heisenberg's prescriptions. The quantum theory of the Lamb shift, as conceived by Bethe and established by Schwinger, is a purely mathematical theory and the only direct contribution of experiment was to show the existence of a measurable effect. The agreement with calculation is better than one part in a thousand."

The puzzle here seems real to me. My conception of mathematics is that you start with a set of axioms and then explore the implications of them. There are infinitely many possible sets of starting axioms you can use^[1]. Many of those sets are non-contradictory and internally consistent. Only a tiny subset correspond to our physical universe. Why is it the case that the specific set of axioms we've chosen, some of which were established in antiquity or the middle ages, corresponds so well to extremely strange physical phenomenon that exist at levels of reality we would not have access to until the 20th century?

Let's distinguish between basic maths, things like addition and multiplication, and advance math. I think it's unsurprising that basic maths reflects physical reality. A question like "why does maths addition and adding objects to a pile in reality work in the same way" seems answered by some combination of two effects. The first is social selection for the kinds of maths we work on to be pragmatically useful. e.g: maths based on axioms where addition works differently would not have spread/been popular/had interested students for most of history. The second is evolution contouring our minds to be receptive to the kind of ordered thinking that predicts our immediate environment. Even if not formalized, humans have a tendency towards thinking about their environment, reasoning and abstract thought. Our ancestors who were prone to modes of abstract reasoning that correlated less well with reality were probably selected against.

As for advance math, I do think it's more surprising. The fact that maths works for natural phenomenon which are extremely strange, distant from our day to day experience or evolutionary environment and often where the natural phenomenon were discovered centuries after the maths in question seems surprising. Why does this happen? A few possible explanations spring to mind:

1. Most advance math is useless and unrelated to the real world. A tiny proportion is relevant. When we encounter novel problems we search for mathematical tools to deal with them and sometimes find a relevant piece of advance math. Looking back, we see a string of novel discoveries matched with relevant math and assume this means advance math is super well correlated with reality. In reality most math is irrelevant and it's just a selection effect where physicists only choose/use the relevant parts.
2. Our starting maths axioms are already very well aligned with physical reality. Anything built on top of them, even (at the time) highly abstract things still is applicable to our universe in some way.

Hmmmmm. I think 2 kind of begs the question. The core weird thing here is that our maths universe is so well correlated with our physical universe. The two answers here seem to be

1. It's actually not that correlated because
   1. it's just a selection effect where we ignore the uncorrelated parts of maths and only pick the correlated/useful parts to use
2. It is correlated. This is explained by
   1. evolution priming with deep structures so we choose or care about maths that is correlated
   2. selection effects over the centuries as humans study and fund only correlated maths
   3. something else weird like all math even based on weird unconnected axioms leading to common methods that are applicable in very different universes. So basically it could be the case that any sufficiently developed mathematical tradition will eventually generate tools applicable to any sufficiently regular universe.

I don't have a good answer here. This is a problem I should think about more.

1. Maybe. Possibly at some point you cease being able to add non contradictory axioms that are also cannot be collapsed/simplified. ↩︎