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:
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.
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
It’s actually not that correlated because
it’s just a selection effect where we ignore the uncorrelated parts of maths and only pick the correlated/useful parts to use
It is correlated. This is explained by
evolution priming with deep structures so we choose or care about maths that is correlated
selection effects over the centuries as humans study and fund only correlated maths
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.
- ↩︎
Maybe. Possibly at some point you cease being able to add non contradictory axioms that are also cannot be collapsed/simplified.
From this twitter thread by Jonathan Gorard, lightly edited by me https://x.com/getjonwithit/status/2009602836997505255?s=20
He also describes one such system a bit in https://x.com/getjonwithit/status/2010422931583860860?s=20
I think this is evidence that humans have a tendency to invent mathematical axioms that generate useful mathematics for the natural sciences, somehow.
Here is the claimed Gorard’s “alien logic system” in the linked tweet
I had Claude chew on this for a bit, and Claude determined that this proof system was the trivial one (ie ∀a,b:a=b) by writing a script in SPASS (an automatic theorem prover) to try to prove this statement.
Here is the SPASS proof
Here is Claude’s proof summary
Proof sketch (from SPASS, depth 3, length 23):
Axioms 1 & 2 ⟹ a ⊕ (a ⊕ b) = b ⊕ (b ⊕ a) [step 15: “symmetrized absorption”]
Substitutions into Axiom 3 + rewrites ⟹ a ⊕ (a ⊕ b) = b ⊕ a [step 173: “double apply = flip”]
(1) + (2) ⟹ a ⊕ (a ⊕ b) = a ⊕ b [step 174: idempotent-like]
(2) + (3) ⟹ a ⊕ b = b ⊕ a [step 209: commutativity]
(4) + earlier lemmas ⟹ a ⊕ b = a [step 228: left absorption]
Rewrite Axiom 2 with (5) ⟹ a ⊕ b = b [step 229: right absorption]
(5) + (6) ⟹ a = b ∀ a,b ∎
Therefore I think its quite likely that many of the supposedly rich alien axiom systems Gorard found are actually just trivial almost-contradictory systems which are hard to prove the triviality of. It also explains why he couldn’t “make sense” of the system. There is nothing to make sense of.
Commenting on the footnote:
Your original statement was correct. There are infinitely many non-isomorphic assemblies of axioms and inference rules.
For many systems (e.g. all that include some pretty simple starting rules), you even have a choice of infinitely many axioms or even axiom schemas to add, each of which results in a different non-contradicting system, and for which the same is equally true of all subsequent choices.
This is a popular view but in my opinion it is wrong. My conception of math is that you start with a set of definitions and the axioms only come after that, as an attempt to formalize the definitions. For example:
The natural numbers are defined as the objects that you get by starting with a base object “zero” and iterating a “successor operation” arbitrarily many times. Addition and multiplication on the natural numbers are defined recursively according to certain basic formulas. The axioms of Peano arithmetic can then be viewed as simply a way of formalizing these definitions: most of the axioms are just the recursive definitions of addition and multiplication, and the induction schema is an attempt to formalize the fact that all natural numbers result from repeatedly applying the successor operation to 0.
The universe of sets is defined as the collection you get by starting with nothing, and repeatedly growing the collection by at each stage replacing it with the set of all its subsets (i.e. its powerset). The axioms of Zermelo-Fraenkel set theory are an attempt to state true facts about this universe of sets.
Of course, it’s possible to claim that the definitions in question are not valid—they are not “rigorous” in the sense of modern mathematics, i.e. they do not follow from axioms because they are logically prior to axioms. This is particularly true for the definition of the universe of sets, which in addition to being vague has the issues that it presupposes the notion of a “subset” of a collection while we are currently trying to define the notion of a set, and that it’s not clear when we are supposed to “stop” growing the collection (it’s not at “infinity”, because the axiom of infinity implies that we are supposed to continue on past infinity). But Peano arithmetic doesn’t have those problems, and in my opinion is therefore on an epistemologically sound basis. And to be honest much (most?) of modern mathematics can be translated into Peano arithmetic; people use ZFC for convenience but it’s actually not necessary much of the time.
Richard Hamming provided a (formalist) response to Wigner’s argument about the “unreasonable effectiveness of mathematics” in this 1980 paper, which was helpfully expanded upon by Derek Abbott in 2013. The core point is that despite the appearances, mathematized science answers comparatively little about the world, and that both the mathematics we use and the phenomena we apply those mathematics to are probably parochial to our cognitive processes. Mathematics is, in this view, a powerful tool rather than the underlying truth of the universe, and we shouldn’t be surprised it managed to drive a few nails really well.