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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.
It's just that the universe is ordered, and math is the study of order.
But why? Why are the rules of the universe consistent? What orders them? These are some of the most baffling questions in philosophy. This is even more of mystery if you take non-theist interpretation of the universe. If all possible universes (including ones with changing rules) are equally probable, then what are the odds that we would be in both a universe with simple rules such that the elegant mathematical models referred to the in the paper have such predictive power and the rule are not changing over time. This is one of those papers that is massively under appreciated by people who do not study philosophy.
> then what are the odds that we would be in both a universe with simple rules such that the elegant mathematical models referred to the in the paper have such predictive power and the rule are not changing over time.

Could conscious life capable of asking such a question evolve in a world with volatile rules?

Also, math spots patterns. It makes sense that we would spot patterns with it. But these patterns could either be just a tiny slice or an extremely high abstraction of reality.

Supposs Earth is visited by a bunch of friendly aliens. They're eager to explore our oceans, and for some reason a magnetic crane is their favorite tool for scientific inquiry. They drop the magnes into the ocean and pull out heaps of metal junk. They conclude that the ocean floor is made up exclusively of metal junk, and then they use the distribution patterns of that junk to come up with coherent explanations for all sorts of observable phenomena.

I guess what I'm saying is: you can only see what you can see. There could be loads that we won't discover any time soon because it's undiscoverable with a math mind.

A sibling comment to my reply made the point already about deterministic not being simple, so let me address your first point that there could be a filter causing it.

First, a disclaimer. I'm a theist. I like arguments like the so-called "modal ontological argument" and find them convincing. That said, I love explore philosophy especially non-theistic positions because I'm already convinced of theism.

Second, your arguing that there is a filter, and, yes, I agree that one of the possible explanations is a filter. But just asserting that filter with a simple claim as the original commenter did dismisses what is a fascinating exploration of why? How do we know that's a sufficient filter. How do we know that volatility prevents life from existing and asking the question? Is it not possible that there is a volatility that preserves our existence. When your dealing with infinite possible "worlds" this paper becomes more interesting.

Third, if I could put on my non-theist hat (and do it without my biases affecting me), it seems that discoveries such as those made in the paper require non-theist explanations to depend on the existence of a multiverse.

Fourth, there is something else that always get's missed. In a non-theist philosophical framework that forces us to consider all logical possible worlds and universes there is no restriction that a universe begin in a simple state such as a singularity that causes a big bang. A universe can logically begin to exists in an infinite set of states. And again, if every possible universe is equally probable. Why should we be in a universe where what we see can be logically extrapolated backwards to imply that there was a history that follows the mathematical models that we construct. Even a life filter like what you suggested is not capable of answering why we should end up in such a universe. That's a deep and fascinating mystery to be explored.

> Also, math spots patterns. > ... you can only see what you can see. ... undiscoverable with a math mind.

It's not that we believe that we have a math mind. It's that we believe that we have logical minds. Math is just one part of the abstract domain of logic. I agree that there are limitations to our ability to observe and limitations with the capacity (storage depth and processing depth) of our brains. If our brains can be trusted then they are logical, and nothing illogical can exists. It seems to me like this is a very hard argument for you to make.

The article makes a more subtle point. You can still have deterministic physics but not necessarily explained by simple math equations. For instance, physics could have worked through small algorithms with results that can't be captured in elegant math equations like E=mc2.
You don't have enough information to simulate and predict what goes on in an arbitrary 10 cubic cm part of your tissues for an arbitrary purpose (things are different if you had sensing instruments and are satisfied measuring something easily measurable for that purpose). Doctors can't predict, either. Yet many lives could be greatly improved it we were able to do this cheaply.

But you and I have ignored all that and are blind to all these things that are part of reality that you still don't have knowledge of, or abide by mathematics. If we simply put our attention to the parts of experience that have order that can be described by maths and blind ourselves to everything else, then sure, maths seems very powerful.

I think your conflating emergent behavior that results from complex physical states / arrangements of matter with the surprisingly simple laws of physics which can be modelled with mathematics and are the foundations that such complex emergent behavior operate according to. Discoveries since the paper was created have not changed that.
Let me ask you this:

1. what makes you confident that the rules of particle physics alone can lead to a perfect simulation of macroscopic phenomena? (as you call it "emergent behavior")

2. what makes you confident that our universe is not "turtles all the way down", that as soon as we develop more precise and powerful instruments we will discover new "fundamental phenomena" that requires new theory? This has happened before where neat Newtonian physics was found to fail at edge cases detectable by more precise instruments.

3. considering that we have no feasible way of snapshotting a 1 cubic inch of the world, and even if we have done that, no feasible way of simulating its dynamics at the qft level for one second, and yet we have "approximate" physical theories with less invasive instruments that can do that (for specific purposes), is the ideology of calling macroscopic phenomena an emergent behavior of particle physics still relevant? In a similar bent, there exists a Starbucks in certain places in the world, and there does not exist a Starbucks in other places. This is a fundamental truth of the present world, and many peoples' lives are oriented around that and affected by it. Nevertheless, it cannot be precisely predicted by mathematics where there exist Starbucks and where there does not exist Starbucks one year into the future.

4. I hope you can see that just as Newtonian physics was useful for describing an approximation to the world but still fails at giving knowledge about sufficiently complex systems and sufficiently small scales, you now see that likewise particle physics is just an "effective lens" for a particular genre of human activity.

Interesting questions. I'll try to take a shot.

> 1. what makes you confident that the rules of particle physics alone can lead to a perfect simulation of macroscopic phenomena? (as you call it "emergent behavior")

Given the general axiom of science that the same experiments under the same conditions will yield the same results, I do think we can trust the standard model because of how successful it has been at matching real world tests. That doesn't mean that the standard model is "correct" or complete--we know that we still don't have a quantum theory of gravity--only that model is amazingly predictive, which is all that we could ask of a mathematical model. There are also philosophical explanation of quantum physics that need to be factored in (Copenhagen, many-worlds, pilot wave).

All that said, I think your mixing up the need for a perfect simulation with the fact that the currently understood quantum level interactions of matter and space produce produce the lions share of influence at the larger macroscopic levels. The fact that medical science has been massively successful at using our understanding of chemistry and physics to explain so much of what we know about the details of the body is evidence of how amazing our understanding already it, even if it's not perfect.

> 2. what makes you confident that our universe is not "turtles all the way down", that as soon as we develop more precise and powerful instruments we will discover new "fundamental phenomena" that requires new theory? This has happened before where neat Newtonian physics was found to fail at edge cases detectable by more precise instruments.

All of physics is dying to be the one who discoveries the next level or foundation or theory of everything in physics. No disagreement there, but if that next level turns out to be indescribable with simple mathematical models as is the case with the currently understood quantum level, then that make the effectiveness of relatively simple mathematical models at the quantum level and in relativity/Newtonian physics all the more astounding. How amazing it would be that the complexity would produce such elegance. That would answer one of the whys that we wish to answer!

> 3. considering that we have no feasible way of snapshotting a 1 cubic inch of the world, and even if we have done that, no feasible way of simulating its dynamics at the qft level for one second, and yet we have "approximate" physical theories with less invasive instruments that can do that (for specific purposes), is the ideology of calling macroscopic phenomena an emergent behavior of particle physics still relevant? In a similar bent, there exists a Starbucks in certain places in the world, and there does not exist a Starbucks in other places. This is a fundamental truth of the present world, and many peoples' lives are oriented around that and affected by it. Nevertheless, it cannot be precisely predicted by mathematics where there exist Starbucks and where there does not exist Starbucks one year into the future.

"...is the ideology of calling macroscopic phenomena an emergent behavior of particle physics still relevant?" How did this become an ideology? That said, yes, even with all that said, the effectiveness of mathematics in the physical sciences is still amazing. How the universe came to have it's current arrangement is a fascinating question with it's own mysteries, but that doesn't take away from the points about the effectiveness of mathematics.

"Nevertheless, it cannot be precisely predicted by mathematics..." The ontological naturalists would push back on that claim. Although, I do my best to explore naturalism under it's own terms and conditions, I'm a theist who believes that humans have a soul. For that reason I would agree that we cannot predict such a scenario because there are inputs from outside the physical universe. That said, if we are just talking about physics only, it would be an assumption on your part that it cannot be precisely predicted by mathematics.

> 4. I hope you can see that just as Newtonian physics was useful for describing an approximation to the world but still fails at giving knowledge about sufficiently complex systems and sufficiently small scales, you now see that likewise particle physics is just an "effective lens" for a particular genre of human activity.

I think my other comments have addressed this. Thank you. I enjoy these discussions & debates.

I'm sorry, I cannot continue this conversation. To do so we need an intense back-and-forth that this format is too asynchronous to provide. Either that, or I will need to spend hours trying to anticipate your responses and address them, and I am not sufficiently motivated to do that. I can state my position succinctly in that my view on science and knowledge is very close to philosophical pragmatism.
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 is only unreasonable if you have the unreasonable assumption that this is not a simulation.
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 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.

You just assigned intent to evolution. Evolutionary theory emphatically avoids, even precludes, intent.

Furthermore, the "mathematics created by the physics and chemistry of the brain" is only our modeled attempt to describe the actual "physics and chemistry of the world". The two ideas are separated by so many layers of complexity that work should be shown to establish that there is, in fact, circularity in the process. (Water molecules are bound by electromagnetism, but this did not cause us to build Maxwell's Laws with our watery brains.)

Where did I assign intent? The brain is good at tracking the physics and chemistry of the world not because it was designed to, but because being able to is an evolutionary advantage. I’m not clear why that needs intent. Good fortune through a random process doesn’t meant there was a teleological purpose to that random process.
In pre scientific times people mostly thought it was driven by spirits. God told things to exist, the devil made bad things happen, the rain god made rain and so on. It was a bit of a surprise when experimental investigation found mathematical laws, starting mostly with Newton. Also human brains are pretty bad at some of the maths. I can't figure out renormalization.
I would say a mathematical mind makes a huge difference in natural science whether or not you are doing any equations at the time or not.

Probably since before the cave men or maybe even homo sapiens.

Of course when equations do come up over the course of eons it could be even more helpful.

And whenever there is a shortage of equations there's quite often been a need for somebody to step up to the plate and make some up ;)

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.
Interesting typo: math (the study) is "later" than nature.

Also, Occam's Razor, but here I risk Muphry's Law.

(1960)
Timeless
Nine pages without subheadlines, long lines of text with heavy paragraphs.

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

Math good, strangely effective, much science, reality deeply weird, needs more stochastic approximation of real valued functions.

Edit: em dashes available on request

It's that it's surprising physical behavior fits with mathematical equations so well.

My guess is it's because the universe is maths in some sense.

Max Tegmark is chatting elsewhere
Yeah he wrote a lot about that.