This is a linkpost for https://linch.substack.com/p/why-reality-has-a-well-known-math
Why Reality Has A Well-Known Math Bias
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But it has been discussed here:
https://www.lesswrong.com/posts/3LcyoqNTJuCZ65MbL/mo-putera-s-shortform?commentId=Lif85TC2zJgmQieZH
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1Hm, nice rabbit hole.
Though the ordinary "semi-formalization" of UDASSA seems to rely on Turing machines, which are already mathematical objects. I find the idea that there is some universal Turing machine endlessly running all programs in parallel (because to be clear, many never halt) that exists somewhere in the ether... a little unintuitive, personally, even as an AIT researcher.
Noted! Will check out the comments later.
(I did discuss earlier versions of this argument at least a few years ago, so at least I'm confident I couldn't have been subconsciously influenced by that thread specifically!)
(EDIT: Upon a skim they discuss a pretty different set of considerations here, closer to what Hamming calls "You see what you look for")
I'm honestly surprised that other people haven't covered this question on LW before, since it feels like very centrally in the space of questions LW folks tend to be interested in.
Your core argument was familiar to me, but I don't recall immediately if or where I encountered it on LW before. I strongly associate it with the sequences on induction and Occam's razor, and with List of Lethality, but I doubt that they mention an anthropic filter argument about what environment minds exist in, and probably it's just that I read those documents around the time I contemplated Wigner's puzzle myself.
1I think that the anthropic argument fails.
If there were 2 options, mathematical realities like this one, and realities of total chaos, the argument would work. Evolution can't happen in total chaos.
But, this anthropic argument is talking about the evolution of intelligence.
For intelligence to evolve, there must be patterns that can be spotted by the unusually smart caveman, but not by the dumber cavemen. And knowledge of these patterns must increase evolutionary fitness. Sometimes these patterns will be how stones chip into flint axes. Often they will be the behavior of other members of the tribe.
The unusually smart caveman isn't deducing all existence from the first principles of quantum field theory.
Thus the evolutionary anthropic arguments can't distinguish between.
- There are some subtle effects that only appear at very high speeds. (Ie, unobservable to the caveman). These effects can be understood as a beautifully simple system of 4-vectors.
- There are some subtle effects that only appear at very high speeds. These effects are totally ad hoc, and have no compact mathematical description. Moving near the speed of light makes apples ripen faster, oranges ripen slower, and bananas turn purple. Looking for a deep consistent mathematical pattern, there isn't one.
For examples of environments where intelligence is useful, consider.
- Basically any fantasy magic system.
- How humans used to think reality worked.
- Most computer games. If you take the minecraft source code as the fundamental laws of physics, it's a lot more ad hoc than real physics. But intelligence is still useful.
I agree that the anthropic filter may be sufficient to explain the unreasonable effective of mathematics, but I don't think it's a necessary explanation. I doubt that universes like ours are vastly outnumbered by alternative universes where:
- Math isn't unreasonably effective
- Life still evolves
- Yet intelligence fails to evolve because there are no patterns to predict
The anthropic filter is necessary for explaining why Earth has water (when most planets don't), and may be necessary for explaining why the universe seems fine tuned for life. But probably isn't necessary for explaining the unreasonably effectiveness of mathematics.
Wigner argues this is "unreasonable" because there is no logical reason why the universe should obey laws that conform to man-made mathematical structures.
I think Eugene Wigner is misunderstanding something. The causality isn't:
"The math which humans make/discover" --determines--> "The structure of mathematics" --determines--> "How the world works"
Instead, the causality is:
"The math which humans make/discover" <--determines-- "The structure of mathematics" --determines--> "How the world works"
It's true that "there is no logical reason why the universe should obey laws that conform to man-made mathematical structures," but there's a very logical reason why the universe should obey laws that man-made mathematical structures also obey.
For example, math and logic can simply be defined as the rules which many different phenomena (regardless of origin) follow. The difference between math and logic is that math is very complex logic (e.g. a large number can be represented as a binary string of TRUE and FALSE, math operations are made out of AND OR and NOT logical statements, and so math statements are essentially just complex logical statements).
As for why the same logic applies to different things, the answer is probably as elusive as "why does the universe exist? Why does math and logic exist?" Not every explanation has an explanation.
The problem with arguments like this is that they are typically circular. At the end of the day you are using math to try to show why math is necessary for reasoning or whatever.
Best to just take a few unjustified axioms so that you're honest about the uncertainty at the bottom of any worldview
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.
- ^
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
1I 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).
I can't help myself but I am compelled to link this comic because of the superficial similarity
https://www.smbc-comics.com/comic/precise
Thanks! Yeah "Wigner's puzzle isn't a puzzle after all" appears to be a common response by scientists and philosophers upon first hearing it. Hamming discusses some of them here, under "some partial explanations." Though he also ultimately concludes:
Conclusion. From all of this I am forced to conclude both that mathematics is unreasonably effective and that all of the explanations I have given when added together simply are not enough to explain what I set out to account for. I think that we - meaning you, mainly - must continue to try to explain why the logical side of science-meaning mathematics, mainly - is the proper tool for exploring the universe as we perceive it at present. I suspect that my explanations are hardly as good as those of the early Greeks, who said for the material side of the question that the nature of the universe is earth, fire, water, and air. The logical side of the nature of the universe requires further exploration.
