The Unreasonable Effectiveness of Mathematics in the Natural Sciences [pdf] (1960)

I believe there’s a reasonable explanation for this effectiveness: mathematics is created by brains. Brains have evolved to track and entrain to various physical phenomena, and the internal dynamics of the brain resonate with the dynamics of the world around that is perceptible through sensory channels.

It hardly seems odd that the mathematics created by the physics and chemistry of the brain is very good at describing the physics and chemistry of the world the brain’s physics and chemistry is evolved to track. It would be unreasonable if human math were somehow complete useless at describing the world. That would indeed be a mystery.

It is only unreasonable if you have the unreasonable assumption that this is not a simulation.

This is an interesting piece and it is kind of an outstanding puzzle, but I feel like it was also indicative of the zeitgeist. The 60s were a more optimistic time for physics. We had succeeded in Quantum Electrodynamics, Quantum Chromodynamics were underway. The standard model was being developed. To say nothing of the radical success of earlier advances applying fundamental insights to industry. We had not yet spent 40 years trying to develop a theory of quantum gravity. Or gotten into the immense computational challenges of solid state physics. We are now sifting through a huge number theories which all describe the universe reasonably well. To a reasonable approximation anything can be modeled mathematically and it can be modeled in multiple ways. But that doesn't really tell us too much. Plenty of mathematical frameworks could be described as universal function approximators. And an SVM and a Transformer NN can, in principle, both model language to an arbitrary degree of accuracy. One is computationally tractable and achieves good results. But it feels realistic to say that that neither model captures to actual structure of language as it is generated in the human brain. This is all to say that this essay might be reading too much into the success of various mathematical models.

The thing that's the real mindfuck for me is group theory. It makes a bit of sense when doing crystallography in the 1830s, because there's obviously symmetry in crystals. Fine.

But go ahead 90 years, and you get to Noether's theorem: the laws of conservation come down to symmetry. And then it just completely infests quantum theory. What the hell. How is this possible? God must be a mathematician.

It's just that the universe is ordered, and math is the study of order.

Never seemed unreasonable to me - math is “rule-based” and so is nature. The later is perhaps mysterious, but Occom’s Razor makes it more likely than lawless.

(1960)

Nine pages without subheadlines, long lines of text with heavy paragraphs.

Could someone be so kind and make tldr; version? Please.