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.
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.
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.
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 left over by our civilization. They believe that the metal junk is all there is to the ocean, and then they use the distribution 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.
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.
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.
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.
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.
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.
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 ;)
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 left over by our civilization. They believe that the metal junk is all there is to the ocean, and then they use the distribution 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.
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.
It is only unreasonable if you have the unreasonable assumption that this is not a simulation.
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.
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)
Timeless
Nine pages without subheadlines, long lines of text with heavy paragraphs.
Could someone be so kind and make tldr; version? Please.
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
Math good, strangely effective, much science, reality deeply weird, needs more stochastic approximation of real valued functions.
Edit: em dashes available on request