I studied complexity, evolution, emergent behavior, etc. quite a bit, and I think this is a mirage. More accurately, it's confirmation bias. Mathematics is onl

I studied complexity, evolution, emergent behavior, etc. quite a bit, and I think this is a mirage. More accurately, it's confirmation bias. Mathematics is only unreasonably effective at describing the subset of physical systems for which it's unreasonably effective. These are systems that are deterministic (or statistically so), linear, and generally well behaved. Things like the laws of motion, thermodynamics, heat transfer, and of course basic arithmetic all qualify as well behaved systems and lend themselves to simple straightforward reductionistic mathematical reasoning. These systems however are only a subset of the much larger set of all natural systems. This larger set includes living systems, chaotic systems with feedback loops, and systems like those in QM that exhibit what seems to be non-deterministic behavior. Mathematics does have some content around chaotic systems. It's possible to write equations for systems that exhibit chaotic and emergent behavior. Yet what you can't do easily with math is write equations that correctly model real instances of these systems in the same way that you can for, say, billiard balls on a table or the motion of bodies in the solar system. You can try, but what you end up with is a system of mathematical relationships whose complexity approaches that of the source data you are trying to describe. You do not get the kind of miraculous conceptual "data compression" you achieve with inorganic physics. I am not arguing for some kind of supernatural underlying principle or that these things could never be described mathematically. I'm just saying that this should be a fertile area for new math. The language as it stands is not up to the challenge of describing stuff like the N^N^N^N^N^N^N^N^... causal interaction combinatorics of genetic regulatory networks or the behavior of real economies at scale. I'm still very much a fan of Stephen Wolfram's core thesis in A New Kind of Science. The book as a whole is a mixed bag and I understand why some had a negative reaction to it, but what he's basically saying if you cut away some of the hype is that CS may offer new mathematical primitives that can describe some of these systems. IMHO the biggest problem with the book is the title-- it's not a "new kind of science," just a new domain of math. But Wolfram also oversells this emerging area of math a little... there is still a ton of work to be done here. In studying this subject I really got the impression that there are monumental mathematical (and possibly physical) discoveries hiding in there.