(sorry, for some reason substack won't let me post a comment on your blog, so I'll have to respond to your essay here.)
The anthropic principle is an interesting and fresh answer to the problem of the unreasonable effectiveness of mathematics. You are a very creative thinker.
Here are a few things which you might want to think about:
1. Is Math really that effective? When I was a geeky teenager, I knew a lot of math, but no matter how much I learned, I couldn't get cheerleaders to date me. There is a lot in our lives which we have no way to model mathematically.
2. As an engineer, sure, I use math. But I almost never use, say, math from the laws of physics on up. Much more frequently, I use formulas which I know *for sure* are wrong, but are nevertheless close enough approximations.
3. As a computer scientist, I did a lot of programming finding approximate solutions to NP-hard problems. There are even computer science problems we can prove have no mathematical solution. Knuth famously ended one of his papers with "I don't know whether this algorithm works or not; I haven't implemented it, I've only proved it correct." 🙂 Even in computer science, we can trust our models only insofar as they bear out in practice.
Now for a few more probing questions:
1. Suppose we lived in a universe which was a physicist's nightmare, but an engineer's heaven 🙂 I.e. there really were no laws of nature. Nothing really held true at all times and places. Nevertheless, engineers could gradually accumulate a grab bag of techniques, some resembling mathematics, some resembling poetry, some which resembled worship of a god or gods, etc, which eventually lead them to possess a civilization not unlike our own. What would your counterpart say in such a universe? Would he be able to advert to the anthropic principle as well to justify how his engineers were able to do their thing?
2. Imagine a universe which was created by a benevolent deity, ordered such that there were mathematical laws, and populated by creatures who could do math and find out these laws. What would your counterpart say to justify the existence of that universe? Could you even prove that *this* universe is not such a universe?
Now for some brief observations of my own as to why math is so effective.
1. Let's start with what we mean by a law of nature. Best answer I could find is something like a) it hold at all times in all places, under all circumstances, i.e. it is utterly exceptionalness, and b) any two observers will be able to agree that those laws hold.
2. Unpacking the agreement among agents, at the very least, if two agents are to agree on something, that something has to be *communicatable* from one agent to the other. Which means that it has to be expressible in a language which both agents can understand and agree on the meaning of the messages expressed in that language--in particular, they need to be able to agree that they can refer to the same object or situation.
3. Unpacking communicable: what are the conditions under which a language is communicatcble: the best answer to this question was given IMHO by Turing, in his original paper where he introduces the Turing machine. For space reasons I can't put his full argument here (I do recommend the paper to you if you haven't read it! It's marvelous). Turing had to formalize a notion of communicatability in order to argue that his machines were indeed could do any possible computation. Namely, the symbols on the tape, and the symbols used to represent the states of the machine, must be communicatable because, really, when a machine goes from state to state, it is communicating with itself as to where it is in the computation.
Bad description--pleae read Turing's paper if that isn't clear.
4. So lets telescope back: For something to be a law, any two observers must agree that it holds everywhere excepitionlessly. For them to agree, they have to communicate. For them to communicate, they must share a language which is computable. And that's where the math comes in: computation is just math in action, as it were.
5. So that's why math is all over the place: anything we can communicate with each other, anything we can agree on--or even specify with enough exactness to be able to disagree about, must be expressed in a computable language.
I hypothesise that, actually, everything can be modelled mathematically but at levels of precision and complexity that are beyond human comprehension, maybe.
Higher kolmogorov complexity is difficult to model but I doubt that it’s a physical property as much as a human limitation. Thoughts?
Also, I’m skeptical that a law of nature requires two observers who can communicate. I think it’s very often a property that tends to naturally follow but I’m unsure if it’s a requirement. I posit the existence of several laws of nature that we cannot observe, and thus have no observers. Unless I’m misunderstanding any of the commenter’s arguments.
This is a fascinating post. I am a huge believer in a deep biological origin of mathematics. Specifically, the emergent complexity of biological systems is one of many ways in which our mathematical system of logic seems to build itself.
Life started out in small havens of predictable sources of chemical complexity (primordial deep sea vents). From there, life began to colonize the rest of the world through a process of biological evolution and ecosystem engineering — e.g. photosynthesizers stabilizing our atmosphere over time, forests creating their own ideal growing conditions.
This process is pretty analogous to the semantic colonization of our world by rationalist societies. Math started out in controlled, predictable corners (like Euclidean geometry) and was built and expanded to cover any practicable area. You are right that math proved surprisingly able to predict many unrelated aspects of our universe, just as life proved capable of colonization many surprising niches.
One thing you are missing though is that math, like organic life, creates many of its own areas of application. For example, our financial system emerged and dominated in its current form because that is the form that allows clear and simple analysis by mathematics. However, not every aspect of our universe is equal in this regard. Quantum mechanics is starkly concise in mathematics, and biological life has no real effect on that (quantum Darwinism is an interesting theory related to that though). On the flip side, day to day weather events are still entirely unpredictable and mathematically ugly. The process of biological and financial evolution is mainly the expansion of semantics into these frontiers, and the logical corrections needed to maintain systemic consistency.
In short, complex life and mathematical reason both originate in small havens of predictability and reason. They then spread into available areas of logical consistency through evolution, or create new pockets through semantic colonization.
I rly like how comprehensive this is! Will defo be giving this a re-read or few.
The bit about anthropic thinking seems pretty accurate and likely satisfies several other questions but I’d be wary of how quickly it could slide into circular reasoning territory.
Say, if you’re particularly interested in the question of how we came to evolve minds that can do mathematics, you might like Dennett’s book From Bacteria to Bach and Back. It attempts to answer this very question and it’s got a disproportionately high S/N ratio.
Also I’m intrigued by any attempts to formalise this question (or other questions within this area) further. If you come across anything like that I’d be interested.
In any case, I don’t know if answering the question of “why is mathematics so unreasonably effective” is possible, really, or if the question makes sense to begin with. Maybe the very question is faulty. I don’t know. It gets really messy, really quickly.
And because of that, I know writing this essay wasn’t easy, so honestly mad respect for pulling it off so well. Thanks.
You posit that our universe necessarily possesses a "gradient of mathematical tractability" and this explains the effectiveness of mathematics. But it's not obvious to me that a gradient of mathematical tractability is more likely to continue far beyond the patterns that the mathematical abilities evolved to handle than it is to stop at that level. As a kind of example: if we set up artificial universes with a small number of simple rules, then we can often apply very deep mathematics to analysing them, so we could get this gradient from "merely" from the combination of a few elementary tractable patterns, and it's not obvious that tractability needs to generalise beyond the elementary patterns.
I do find the anthropic argument compelling, though, and it suggests to me that it must be the case that it's actually hard to construct universes that feature tractable patterns at all without it having more tractable patterns than you intended.
Thanks! As a partial answer to both you and a subpoint of Shlomo’s, to be clear my argument doesn’t presuppose a form of “mathematical realism” (probably not the right technical jargon) where simple mathematics correctly describe literally everything in the universe. As alluded to in the article, we should be less surprised if things very far from normal human range (like very big or very small things) are less well-captured by simple mathematics. And my argument doesn’t give us additional reason to be surprised if we see a specific subspace in the universe that strangely doesn’t obey the normal mathematical laws, except for all the usual reasons (consistent with other theories) you might find that surprising.
“You posit that our universe necessarily possesses a "gradient of mathematical tractability" I searched for “necessarily” and I don’t think I used it in this way in my post. Can you point to where I said it? (Not being difficult, I edited the article in several rounds so it’s very likely that a quantifier skipped past me). I did mean for my argument to be probabilisitic.
True that your argument doesnt assume that simple mathematics governs everything...but science does assume that...sort of...like we obseve mathematical laws by looking at the solar system and then by default expect them to apply outside the solor system as well. So my point is we need a general theory of why we expect that and my answer is that it is just a law of epistemology that simple things are more likely to exist than complicated ones.
I see Linch’s argument as trying to explain why that epistemological law is helpful. Consider a world where we assume it a priori but it’s also completely unhelpful. The assumption doesn’t explain the helpfulness.
You didn’t say necessarily, I misspoke. Though you say you don’t really know how to construct priors, so we’re both speaking a bit loosely. In any case my argument about gradients was probabilistic.
One possible solution to the issue I pose: maybe the surprising part is the tractability of micro-scale stuff. Macro scale “has” (with high probability) to have some tractable features, and maybe via “complex universe from simple rules” type principles this almost always ends up being many tractable features, but micro scale stuff which is not directly observed doesn’t seem to have the same anthropic constraints.
One possibility is that an ordered macroscale requires an ordered microscale unconditionally. Another is that “macro scale tractability” means that macro scale stuff is predictable via micro scale processes (i.e. brains), so the micro scale has to be ordered enough to do the prediction.
I think that scale separation between the predictor and the predicted could be needed for “intelligent life”, because if the machine to predict the phenomenon is the same size and speed as the phenomenon it’s not clear that there’s any way to use the prediction in order to grow. So at least one of (smaller, faster) seems needed.
(actually I don’t think any of this explains gravity)
I have no idea what this universe would look like so I can't confidently say that minds can't evolve in it to seek patterns. Like I also don't even now what this universe is…or how minds would look.
Is it really true that patterns don't "exist at every scale and abstraction level" in such a universe? Also, maybe other properties of the Weierstrass function would remove other constraints we have to developing minds. Like sure, in our universe we can develop minds evolutionarily because as you say simple pattern recognition helps you get to advanced pattern recognition. But also in our universe we are constrained by the fact that our lives are finite and short as are our sense capacities. As we have no idea what beings in Weierstrass-Universe even look like I can't confidently say that that's true about them.
2. Chaotic Dynamics
To some extent this is our universe. So I'm not sure what you mean. Unless you are using "dominated by" to mean that it is dominated by Chaotic Dynamics so much so that there are no observable patterns…in which case, … sure … it's tautological that if there are no noticeable patterns then there are no noticeable patterns
3. not governed by any mathematical rules at all.
This is incoherent. Like, if there are no rules to the universe than such a universe might have brains that are capable of doing math. There are no rules against such brains popping up. You might say "well they could pop up but they wouldn't be very likely to" but that means you are asserting a pattern to such a universe… namely the pattern of consistently not having brains pop up. And this is why the notion is incoherent… any universe where stuff happens…you can count the frequency of that stuff happening and boom! you have math. The intuitive-notion of a "non-mathamatical" universe could possibly be made more precise into something that is coherent but until that's done I don't know what this means.
But more importantly, this argument misses a bigger point…
You are assuming mathematical reasoning does work in our universe. I agree with this point, but it's important to consider why we think this is indeed true.
For instance, suppose it was the case that the general laws of gravity exist everywhere in the universe, except for one tiny rock which has a slightly smaller gravitational constant than everywhere else. The slightly smaller constant for this rock is not noticeably different from the main gravitational constant so we wouldn't notice the difference.
(maybe if the rock were the size of Jupiter we could tell the difference, but since it's the size of a pebble and its gravitational force is so small anyway, we can't).
So our experimental results couldn't tell us it's not the case. And it's not logically impossible for that to be the case either. Yet we don't think that's the case and we assume that our universe does in fact obey mathematical laws.
The reason we do this is because we a priori would consider a universe where all rocks to obey the same laws of gravity more likely than a universe with one exception. We express this by saying "the law of gravity has been tested so many times…what are the chances this one rock is different?" Indeed. What are the chances? Low. Because we apriori don't expect the universe to work that way.
We a priori expect the universe to be simple (hence Occum's razor). Simple things follow mathematical rules (by definition) so there is an apriori expectation that our universe would work this way.
This is an excellent post which I really enjoyed reading. I'm currently doing an undergrad in mathematical physics and hoping to start a PhD soon so it's very refreshing to see original takes on Wigner's problem. A few points though:
1) I agree that Kolmogorov complexity is a good starting point, but it seems to me like the defining feature of mathematical simplicity is very often beauty, or, as you mentioned, symmetry. So you could make the case that complexity could be, instead, measured by the order of a group, or more specifically its representation, given that the group (on the scales you're considering) could be infinite. In general I feel like making a link between aesthetic property and simplicity could give the specific measure you're looking for, although this inevitably leads you down the cascade of defining aesthetics. But the main supporters of my argument would be Dirac and Einstein, since they both said that it was beauty which guided them to simplicity.
2) What about the possibility that a mathematically 'complicated' world (e.g. the Weierstrass world) evolves a brain which adapts to this? You mention that the gradient would be flat, but I don't think this is the case - it doesn't seem impossible to me that a universe governed by things which seem like random discontinuities in this world would not be able to give rise to minds which interpret the patterns differently, and begin seeing deep patterns, albeit still tailored to a more intricate system. But for them the baseline of simplicity would be the upper bound of our current perception; i.e. the baseline of what's considered 'simple' in the Weierstrass world is just much higher. In that case the evolution gradient would be the same if not steeper than in our world, for the same reason (I'm assuming laws of physics still exist). The gradient would still exist, even though this is not a mathematically 'simple' universe. What seems like turbulence to us would be intuitively easy to the inhabitants of that universe.
3) Point already mentioned in the comments below and once considered by Taleb - it very much doesn't seem to me like evolution selected for mathematical thinking at all. This is why every single human being, including mathematicians, find the subject very difficult. We evolved to recognise patterns, of course; but my claim is that human minds are at the lowest minimum bound to even recognise that mathematical patterns or truth exists. I think this is more of a byproduct of a more relevant 'module' in our minds (recognising patterns) than a special ability selected for by evolution. The natural thing to compare with here is language; sure, not everyone can write like Shakespeare, but every single human can speak fluently before the age of six. The same is not true for math; our brains are not adapted to be fluent with it, and this takes up a lot of compute. If this was selected for, then we'd have the same fluency in math as we do with natural language; and the level of abstraction in something like calculus is quite far above any abstraction you need in the daily world. The gradient exists, definitely, but evolution-wise, you quite quickly begin hitting diminishing returns once the abstraction level is for something like mathematics. Evolution did not strongly select for mathematical ability, and I think humans are just barely smart enough to do any kind of math (I mean this in the sense that even our greatest discoveries are, in contrast to the population of a hypothetical mathematically-selected universe in which everyone is completely fluent in mathematics, quite trivial).
Thank you for writing this, I'd love to see more of your work!
I like this post and the idea. Though as with the response you pinned, I don’t actually think mathematics is as good as we think. We approximate all the time. A lot of math is unsolvable. Pythagoras thought that a complete musical tuning system could be created with perfect math. It still hasn’t happened. We instead just use approximations.
It appears to me that what we are really trying to do is find better ways of understanding the universe with language. Mathematics is a better language than our culturally dependent spoken languages when dealing with objective “non human” problems/ideas, (anything cultural and anthropological, ie, “human”, should be left to spoken languages) but the main thing is that mathematics is still just a language. This is why it seems both made up and yet very accurate, the same way our other languages are.
The power is not in math itself, it’s in the language. I think at the root of these questions is a question of linguistics, and perhaps the universe itself is actually fundamentally linguistic. We just need to find better and better languages to use.
Detailed comment by someone on Facebook:
(sorry, for some reason substack won't let me post a comment on your blog, so I'll have to respond to your essay here.)
The anthropic principle is an interesting and fresh answer to the problem of the unreasonable effectiveness of mathematics. You are a very creative thinker.
Here are a few things which you might want to think about:
1. Is Math really that effective? When I was a geeky teenager, I knew a lot of math, but no matter how much I learned, I couldn't get cheerleaders to date me. There is a lot in our lives which we have no way to model mathematically.
2. As an engineer, sure, I use math. But I almost never use, say, math from the laws of physics on up. Much more frequently, I use formulas which I know *for sure* are wrong, but are nevertheless close enough approximations.
3. As a computer scientist, I did a lot of programming finding approximate solutions to NP-hard problems. There are even computer science problems we can prove have no mathematical solution. Knuth famously ended one of his papers with "I don't know whether this algorithm works or not; I haven't implemented it, I've only proved it correct." 🙂 Even in computer science, we can trust our models only insofar as they bear out in practice.
Now for a few more probing questions:
1. Suppose we lived in a universe which was a physicist's nightmare, but an engineer's heaven 🙂 I.e. there really were no laws of nature. Nothing really held true at all times and places. Nevertheless, engineers could gradually accumulate a grab bag of techniques, some resembling mathematics, some resembling poetry, some which resembled worship of a god or gods, etc, which eventually lead them to possess a civilization not unlike our own. What would your counterpart say in such a universe? Would he be able to advert to the anthropic principle as well to justify how his engineers were able to do their thing?
2. Imagine a universe which was created by a benevolent deity, ordered such that there were mathematical laws, and populated by creatures who could do math and find out these laws. What would your counterpart say to justify the existence of that universe? Could you even prove that *this* universe is not such a universe?
Now for some brief observations of my own as to why math is so effective.
1. Let's start with what we mean by a law of nature. Best answer I could find is something like a) it hold at all times in all places, under all circumstances, i.e. it is utterly exceptionalness, and b) any two observers will be able to agree that those laws hold.
2. Unpacking the agreement among agents, at the very least, if two agents are to agree on something, that something has to be *communicatable* from one agent to the other. Which means that it has to be expressible in a language which both agents can understand and agree on the meaning of the messages expressed in that language--in particular, they need to be able to agree that they can refer to the same object or situation.
3. Unpacking communicable: what are the conditions under which a language is communicatcble: the best answer to this question was given IMHO by Turing, in his original paper where he introduces the Turing machine. For space reasons I can't put his full argument here (I do recommend the paper to you if you haven't read it! It's marvelous). Turing had to formalize a notion of communicatability in order to argue that his machines were indeed could do any possible computation. Namely, the symbols on the tape, and the symbols used to represent the states of the machine, must be communicatable because, really, when a machine goes from state to state, it is communicating with itself as to where it is in the computation.
Bad description--pleae read Turing's paper if that isn't clear.
4. So lets telescope back: For something to be a law, any two observers must agree that it holds everywhere excepitionlessly. For them to agree, they have to communicate. For them to communicate, they must share a language which is computable. And that's where the math comes in: computation is just math in action, as it were.
5. So that's why math is all over the place: anything we can communicate with each other, anything we can agree on--or even specify with enough exactness to be able to disagree about, must be expressed in a computable language.
Thanks again for the very enjoyable essay.
I hypothesise that, actually, everything can be modelled mathematically but at levels of precision and complexity that are beyond human comprehension, maybe.
Higher kolmogorov complexity is difficult to model but I doubt that it’s a physical property as much as a human limitation. Thoughts?
Also, I’m skeptical that a law of nature requires two observers who can communicate. I think it’s very often a property that tends to naturally follow but I’m unsure if it’s a requirement. I posit the existence of several laws of nature that we cannot observe, and thus have no observers. Unless I’m misunderstanding any of the commenter’s arguments.
This is a fascinating post. I am a huge believer in a deep biological origin of mathematics. Specifically, the emergent complexity of biological systems is one of many ways in which our mathematical system of logic seems to build itself.
Life started out in small havens of predictable sources of chemical complexity (primordial deep sea vents). From there, life began to colonize the rest of the world through a process of biological evolution and ecosystem engineering — e.g. photosynthesizers stabilizing our atmosphere over time, forests creating their own ideal growing conditions.
This process is pretty analogous to the semantic colonization of our world by rationalist societies. Math started out in controlled, predictable corners (like Euclidean geometry) and was built and expanded to cover any practicable area. You are right that math proved surprisingly able to predict many unrelated aspects of our universe, just as life proved capable of colonization many surprising niches.
One thing you are missing though is that math, like organic life, creates many of its own areas of application. For example, our financial system emerged and dominated in its current form because that is the form that allows clear and simple analysis by mathematics. However, not every aspect of our universe is equal in this regard. Quantum mechanics is starkly concise in mathematics, and biological life has no real effect on that (quantum Darwinism is an interesting theory related to that though). On the flip side, day to day weather events are still entirely unpredictable and mathematically ugly. The process of biological and financial evolution is mainly the expansion of semantics into these frontiers, and the logical corrections needed to maintain systemic consistency.
In short, complex life and mathematical reason both originate in small havens of predictability and reason. They then spread into available areas of logical consistency through evolution, or create new pockets through semantic colonization.
I rly like how comprehensive this is! Will defo be giving this a re-read or few.
The bit about anthropic thinking seems pretty accurate and likely satisfies several other questions but I’d be wary of how quickly it could slide into circular reasoning territory.
Say, if you’re particularly interested in the question of how we came to evolve minds that can do mathematics, you might like Dennett’s book From Bacteria to Bach and Back. It attempts to answer this very question and it’s got a disproportionately high S/N ratio.
Also I’m intrigued by any attempts to formalise this question (or other questions within this area) further. If you come across anything like that I’d be interested.
In any case, I don’t know if answering the question of “why is mathematics so unreasonably effective” is possible, really, or if the question makes sense to begin with. Maybe the very question is faulty. I don’t know. It gets really messy, really quickly.
And because of that, I know writing this essay wasn’t easy, so honestly mad respect for pulling it off so well. Thanks.
Thanks for the book recommendation! I’ll definitely consider it! So much to read, so little time!
You posit that our universe necessarily possesses a "gradient of mathematical tractability" and this explains the effectiveness of mathematics. But it's not obvious to me that a gradient of mathematical tractability is more likely to continue far beyond the patterns that the mathematical abilities evolved to handle than it is to stop at that level. As a kind of example: if we set up artificial universes with a small number of simple rules, then we can often apply very deep mathematics to analysing them, so we could get this gradient from "merely" from the combination of a few elementary tractable patterns, and it's not obvious that tractability needs to generalise beyond the elementary patterns.
I do find the anthropic argument compelling, though, and it suggests to me that it must be the case that it's actually hard to construct universes that feature tractable patterns at all without it having more tractable patterns than you intended.
Thanks! As a partial answer to both you and a subpoint of Shlomo’s, to be clear my argument doesn’t presuppose a form of “mathematical realism” (probably not the right technical jargon) where simple mathematics correctly describe literally everything in the universe. As alluded to in the article, we should be less surprised if things very far from normal human range (like very big or very small things) are less well-captured by simple mathematics. And my argument doesn’t give us additional reason to be surprised if we see a specific subspace in the universe that strangely doesn’t obey the normal mathematical laws, except for all the usual reasons (consistent with other theories) you might find that surprising.
“You posit that our universe necessarily possesses a "gradient of mathematical tractability" I searched for “necessarily” and I don’t think I used it in this way in my post. Can you point to where I said it? (Not being difficult, I edited the article in several rounds so it’s very likely that a quantifier skipped past me). I did mean for my argument to be probabilisitic.
True that your argument doesnt assume that simple mathematics governs everything...but science does assume that...sort of...like we obseve mathematical laws by looking at the solar system and then by default expect them to apply outside the solor system as well. So my point is we need a general theory of why we expect that and my answer is that it is just a law of epistemology that simple things are more likely to exist than complicated ones.
I see Linch’s argument as trying to explain why that epistemological law is helpful. Consider a world where we assume it a priori but it’s also completely unhelpful. The assumption doesn’t explain the helpfulness.
You didn’t say necessarily, I misspoke. Though you say you don’t really know how to construct priors, so we’re both speaking a bit loosely. In any case my argument about gradients was probabilistic.
One possible solution to the issue I pose: maybe the surprising part is the tractability of micro-scale stuff. Macro scale “has” (with high probability) to have some tractable features, and maybe via “complex universe from simple rules” type principles this almost always ends up being many tractable features, but micro scale stuff which is not directly observed doesn’t seem to have the same anthropic constraints.
One possibility is that an ordered macroscale requires an ordered microscale unconditionally. Another is that “macro scale tractability” means that macro scale stuff is predictable via micro scale processes (i.e. brains), so the micro scale has to be ordered enough to do the prediction.
I think that scale separation between the predictor and the predicted could be needed for “intelligent life”, because if the machine to predict the phenomenon is the same size and speed as the phenomenon it’s not clear that there’s any way to use the prediction in order to grow. So at least one of (smaller, faster) seems needed.
(actually I don’t think any of this explains gravity)
I disagree with this post.
In your 3 alternatives you have:
1. The Weierstrass Universe.
I have no idea what this universe would look like so I can't confidently say that minds can't evolve in it to seek patterns. Like I also don't even now what this universe is…or how minds would look.
Is it really true that patterns don't "exist at every scale and abstraction level" in such a universe? Also, maybe other properties of the Weierstrass function would remove other constraints we have to developing minds. Like sure, in our universe we can develop minds evolutionarily because as you say simple pattern recognition helps you get to advanced pattern recognition. But also in our universe we are constrained by the fact that our lives are finite and short as are our sense capacities. As we have no idea what beings in Weierstrass-Universe even look like I can't confidently say that that's true about them.
2. Chaotic Dynamics
To some extent this is our universe. So I'm not sure what you mean. Unless you are using "dominated by" to mean that it is dominated by Chaotic Dynamics so much so that there are no observable patterns…in which case, … sure … it's tautological that if there are no noticeable patterns then there are no noticeable patterns
3. not governed by any mathematical rules at all.
This is incoherent. Like, if there are no rules to the universe than such a universe might have brains that are capable of doing math. There are no rules against such brains popping up. You might say "well they could pop up but they wouldn't be very likely to" but that means you are asserting a pattern to such a universe… namely the pattern of consistently not having brains pop up. And this is why the notion is incoherent… any universe where stuff happens…you can count the frequency of that stuff happening and boom! you have math. The intuitive-notion of a "non-mathamatical" universe could possibly be made more precise into something that is coherent but until that's done I don't know what this means.
But more importantly, this argument misses a bigger point…
You are assuming mathematical reasoning does work in our universe. I agree with this point, but it's important to consider why we think this is indeed true.
For instance, suppose it was the case that the general laws of gravity exist everywhere in the universe, except for one tiny rock which has a slightly smaller gravitational constant than everywhere else. The slightly smaller constant for this rock is not noticeably different from the main gravitational constant so we wouldn't notice the difference.
(maybe if the rock were the size of Jupiter we could tell the difference, but since it's the size of a pebble and its gravitational force is so small anyway, we can't).
So our experimental results couldn't tell us it's not the case. And it's not logically impossible for that to be the case either. Yet we don't think that's the case and we assume that our universe does in fact obey mathematical laws.
The reason we do this is because we a priori would consider a universe where all rocks to obey the same laws of gravity more likely than a universe with one exception. We express this by saying "the law of gravity has been tested so many times…what are the chances this one rock is different?" Indeed. What are the chances? Low. Because we apriori don't expect the universe to work that way.
We a priori expect the universe to be simple (hence Occum's razor). Simple things follow mathematical rules (by definition) so there is an apriori expectation that our universe would work this way.
No Anthropic reasoning required.
This is an excellent post which I really enjoyed reading. I'm currently doing an undergrad in mathematical physics and hoping to start a PhD soon so it's very refreshing to see original takes on Wigner's problem. A few points though:
1) I agree that Kolmogorov complexity is a good starting point, but it seems to me like the defining feature of mathematical simplicity is very often beauty, or, as you mentioned, symmetry. So you could make the case that complexity could be, instead, measured by the order of a group, or more specifically its representation, given that the group (on the scales you're considering) could be infinite. In general I feel like making a link between aesthetic property and simplicity could give the specific measure you're looking for, although this inevitably leads you down the cascade of defining aesthetics. But the main supporters of my argument would be Dirac and Einstein, since they both said that it was beauty which guided them to simplicity.
2) What about the possibility that a mathematically 'complicated' world (e.g. the Weierstrass world) evolves a brain which adapts to this? You mention that the gradient would be flat, but I don't think this is the case - it doesn't seem impossible to me that a universe governed by things which seem like random discontinuities in this world would not be able to give rise to minds which interpret the patterns differently, and begin seeing deep patterns, albeit still tailored to a more intricate system. But for them the baseline of simplicity would be the upper bound of our current perception; i.e. the baseline of what's considered 'simple' in the Weierstrass world is just much higher. In that case the evolution gradient would be the same if not steeper than in our world, for the same reason (I'm assuming laws of physics still exist). The gradient would still exist, even though this is not a mathematically 'simple' universe. What seems like turbulence to us would be intuitively easy to the inhabitants of that universe.
3) Point already mentioned in the comments below and once considered by Taleb - it very much doesn't seem to me like evolution selected for mathematical thinking at all. This is why every single human being, including mathematicians, find the subject very difficult. We evolved to recognise patterns, of course; but my claim is that human minds are at the lowest minimum bound to even recognise that mathematical patterns or truth exists. I think this is more of a byproduct of a more relevant 'module' in our minds (recognising patterns) than a special ability selected for by evolution. The natural thing to compare with here is language; sure, not everyone can write like Shakespeare, but every single human can speak fluently before the age of six. The same is not true for math; our brains are not adapted to be fluent with it, and this takes up a lot of compute. If this was selected for, then we'd have the same fluency in math as we do with natural language; and the level of abstraction in something like calculus is quite far above any abstraction you need in the daily world. The gradient exists, definitely, but evolution-wise, you quite quickly begin hitting diminishing returns once the abstraction level is for something like mathematics. Evolution did not strongly select for mathematical ability, and I think humans are just barely smart enough to do any kind of math (I mean this in the sense that even our greatest discoveries are, in contrast to the population of a hypothetical mathematically-selected universe in which everyone is completely fluent in mathematics, quite trivial).
Thank you for writing this, I'd love to see more of your work!
I like this post and the idea. Though as with the response you pinned, I don’t actually think mathematics is as good as we think. We approximate all the time. A lot of math is unsolvable. Pythagoras thought that a complete musical tuning system could be created with perfect math. It still hasn’t happened. We instead just use approximations.
It appears to me that what we are really trying to do is find better ways of understanding the universe with language. Mathematics is a better language than our culturally dependent spoken languages when dealing with objective “non human” problems/ideas, (anything cultural and anthropological, ie, “human”, should be left to spoken languages) but the main thing is that mathematics is still just a language. This is why it seems both made up and yet very accurate, the same way our other languages are.
The power is not in math itself, it’s in the language. I think at the root of these questions is a question of linguistics, and perhaps the universe itself is actually fundamentally linguistic. We just need to find better and better languages to use.