Interesting post ! But I wonder: do we have any good reason to think that a non-mathematical world or an entirely chaotic world is even possible? Because if this is pure speculation without any model behind it, it would be like asking the question: why are mathematics so effective compared to magic? It would be an even stronger argument than an anthropic filter.
Maybe you would say that the shrimp’s world is a serious example. But to me, the shrimp’s world is a mathematical world that is not entirely chaotic. It’s essentially the same mathematical world as ours, exhibiting chaotic evolution in some cases, like meteorology, fluid mechanics, and the three-body problem, for instance. But not everything is chaotic. Our dependence on weather didn’t prevent us from mastering mathematics, so a smart shrimp could arguably end up with the same mathematics and physics. However I admit that the shrimp could have more practical difficulties than we do, like aliens in the Three-body series.
Because if this is pure speculation without any model behind it, it would be like asking the question: why are mathematics so effective compared to magic? It would be an even stronger argument than an anthropic filter.
I don’t think “why is mathematics unreasonably effective?” is the same as “why are mathematics so effective compared to magic?”
The fundamental question is not why mathematics, in general, works. It’s also not why our universe has[1] a deep fundamental mathematical structure. These are both interesting questions in their own right, but they are importantly different from what Wigner and Hamming, respectively, were discussing. The critical question, as I see is, is why is mathematics effective in the particular ways it is in our world right now? Meaning:
not only does the physical world appear to be well-modeled by mathematics, it actually seems well-modeled by simple, compact, elegant, human-comprehensible mathematics.[2]
not only do areas of mathematics have naturally-emerging constants/structures/etc, but seemingly disparate areas of math, designed/discovered for entirely unrelated purposes, often share the same natural entities.[3]
not only do we have the ability to design mathematical structures to help us in modeling the physical world, but sometimes mathematical objects initially created solely for the aesthetic enjoyment and study by pure mathematicians eventually end up as critically important pieces of physics, applied math, or engineering.[4]
Even if it’s the case that we can’t design or conceive of a non-mathematical world,[5] it doesn’t follow that all our important confusions about why our world is mathematical in this specific way go away. Math being effective doesn’t explain why it’s unreasonably effective, after all.
As an illustration, the fact that Newton’s three laws (which are such a simple and compact set of mathematical principles) can by themselves generate such accurate predictions in a macroscopic, significantly-slower-than-light environment is remarkable. Why are three laws sufficient, and we don’t need a trillion of them? See also Science in a High-Dimensional World for related ideas
Pi was originally defined as the ratio between a circle’s circumference and its diameter. A fundamentally geometrical constant. And yet it also appears when you invoke the Central Limit Theorem to study the distribution of IID coin flips in probability theory, as a critical term in the normal distribution. The fact that such a deep and fundamental connection between geometry and probability theory/chance exists certainly wasn’t obvious to mathematicians and scientists in the past.
As examples, consider Hilbert spaces in Quantum Mechanics, Fourier series and Lebesgue integration in signal analysis, and Radon transformations in Computerized Tomography
I admit my objection was more specifically addressed to the question of conceiving alternative worlds that would be non-mathematical or entirely chaotic.
However, you are right that, without going that far, following the cosmological landscape hypothesis, we could seriously conceive of alternative universes where the physical laws are different. We could arguably model worlds governed by less simple and elegant physical principles.
That said, today’s standard model of physics is arguably less simple and elegant than Newtonian physics was. Simplicity, elegance, and symmetry are sometimes good guides and sometimes misleading lures. The ancient Greeks were attracted by these ideals and imagined a world with Earth at the center, perfect spheres orbiting in perfect circles, corresponding to musical harmony. Reality proved less elegant after all. We also once hoped to live in a supersymmetrical world, but unfortunately, we find ourselves in a world where symmetries are broken.
It seems that in the distribution of all possible physical worlds, we probably occupy a middle position regarding mathematical simplicity, elegance, and symmetry. This is what we might expect given the general principle that we should not postulate ourselves to occupy a privileged position. I acknowledge that a form of the anthropic principle could also explain such a position: extremely simple (crystal) or extremely complex (noise) universes might be incompatible with the existence of intelligent observers.
Regarding the intriguing fact that certain mathematical curiosities turn out to be necessary components of our physical theories, my insight is that mathematicians have, from the very beginning (Pythagoras, Euclid), been attracted to and interested in patterns exhibiting strong regularities (elegance, symmetry). The heuristic instincts of mathematicians naturally guide them toward fundamental formal truths that are more likely to be involved in the fundamental physical laws common to our world and many possible worlds (but only more likely).
Interesting post ! But I wonder: do we have any good reason to think that a non-mathematical world or an entirely chaotic world is even possible? Because if this is pure speculation without any model behind it, it would be like asking the question: why are mathematics so effective compared to magic? It would be an even stronger argument than an anthropic filter.
Maybe you would say that the shrimp’s world is a serious example. But to me, the shrimp’s world is a mathematical world that is not entirely chaotic. It’s essentially the same mathematical world as ours, exhibiting chaotic evolution in some cases, like meteorology, fluid mechanics, and the three-body problem, for instance. But not everything is chaotic. Our dependence on weather didn’t prevent us from mastering mathematics, so a smart shrimp could arguably end up with the same mathematics and physics. However I admit that the shrimp could have more practical difficulties than we do, like aliens in the Three-body series.
I don’t think “why is mathematics unreasonably effective?” is the same as “why are mathematics so effective compared to magic?”
The fundamental question is not why mathematics, in general, works. It’s also not why our universe has[1] a deep fundamental mathematical structure. These are both interesting questions in their own right, but they are importantly different from what Wigner and Hamming, respectively, were discussing. The critical question, as I see is, is why is mathematics effective in the particular ways it is in our world right now? Meaning:
not only does the physical world appear to be well-modeled by mathematics, it actually seems well-modeled by simple, compact, elegant, human-comprehensible mathematics.[2]
not only do areas of mathematics have naturally-emerging constants/structures/etc, but seemingly disparate areas of math, designed/discovered for entirely unrelated purposes, often share the same natural entities.[3]
not only do we have the ability to design mathematical structures to help us in modeling the physical world, but sometimes mathematical objects initially created solely for the aesthetic enjoyment and study by pure mathematicians eventually end up as critically important pieces of physics, applied math, or engineering.[4]
Even if it’s the case that we can’t design or conceive of a non-mathematical world,[5] it doesn’t follow that all our important confusions about why our world is mathematical in this specific way go away. Math being effective doesn’t explain why it’s unreasonably effective, after all.
Or at least can be modeled, in powerfully predictive ways, as having
As an illustration, the fact that Newton’s three laws (which are such a simple and compact set of mathematical principles) can by themselves generate such accurate predictions in a macroscopic, significantly-slower-than-light environment is remarkable. Why are three laws sufficient, and we don’t need a trillion of them? See also Science in a High-Dimensional World for related ideas
Pi was originally defined as the ratio between a circle’s circumference and its diameter. A fundamentally geometrical constant. And yet it also appears when you invoke the Central Limit Theorem to study the distribution of IID coin flips in probability theory, as a critical term in the normal distribution. The fact that such a deep and fundamental connection between geometry and probability theory/chance exists certainly wasn’t obvious to mathematicians and scientists in the past.
As examples, consider Hilbert spaces in Quantum Mechanics, Fourier series and Lebesgue integration in signal analysis, and Radon transformations in Computerized Tomography
A claim I am highly skeptical of
I admit my objection was more specifically addressed to the question of conceiving alternative worlds that would be non-mathematical or entirely chaotic.
However, you are right that, without going that far, following the cosmological landscape hypothesis, we could seriously conceive of alternative universes where the physical laws are different. We could arguably model worlds governed by less simple and elegant physical principles.
That said, today’s standard model of physics is arguably less simple and elegant than Newtonian physics was. Simplicity, elegance, and symmetry are sometimes good guides and sometimes misleading lures. The ancient Greeks were attracted by these ideals and imagined a world with Earth at the center, perfect spheres orbiting in perfect circles, corresponding to musical harmony. Reality proved less elegant after all. We also once hoped to live in a supersymmetrical world, but unfortunately, we find ourselves in a world where symmetries are broken.
It seems that in the distribution of all possible physical worlds, we probably occupy a middle position regarding mathematical simplicity, elegance, and symmetry. This is what we might expect given the general principle that we should not postulate ourselves to occupy a privileged position. I acknowledge that a form of the anthropic principle could also explain such a position: extremely simple (crystal) or extremely complex (noise) universes might be incompatible with the existence of intelligent observers.
Regarding the intriguing fact that certain mathematical curiosities turn out to be necessary components of our physical theories, my insight is that mathematicians have, from the very beginning (Pythagoras, Euclid), been attracted to and interested in patterns exhibiting strong regularities (elegance, symmetry). The heuristic instincts of mathematicians naturally guide them toward fundamental formal truths that are more likely to be involved in the fundamental physical laws common to our world and many possible worlds (but only more likely).