If we come up with a language that can be used to describe what’s true - that’s just how languages work.
the so-called "surprising effectiveness" is not that formal languages can describe reality. as you point out, that's a property of any descriptive language. no, the surprise which must be explained is that many of our specific sentences in those languages later turned out to be physically relevant.
i don't necessarily agree with the surprising effectiveness crowd, but you misrepresent their point.
Can you come up with an example of a mathematical statement that describes reality while not being physically relevant?
sorry, i think we misunderstand each other.
i guess my comment was unclear. i'm just saying that "mathematics is a language, so of course it can be used to describe reality" is arguing against something the surprising-effectiveness crowd does not claim. they do not say "it is surprising that mathematical language can be used to describe reality", but rather that "it is surprising that specific mathematical objects of interest to mathematicians often end up useful in physics or other empirical sciences."[1]
i feel the original essay -- in this section -- elides the difference between "mathematical language" and "mathematical object", and thereby argues against a strawman.
note that i'm making a very narrow point, here. for example, i think the next section of the essay is strong: mathematics has catalogued a LOT of true statements, and it's not necessarily surprising that some of them can be used in models of reality.
to your direct question: no, of course not? any description of reality is physically relevant.
- ^
the distinction between mathematical language and object is fuzzy. mathematics is happy to treat languages themselves as objects of study. nonetheless, while we're speaking philosophically, i think we can allow it.
1
1i'm just saying that "mathematics is a language, so of course it can be used to describe reality" is arguing against something the surprising-effectiveness crowd does not claim. they do not say "it is surprising that mathematical language can be used to describe reality", but rather that "it is surprising that specific mathematical objects of interest to mathematicians often end up useful in physics or other empirical sciences."[1]
Do I understand correctly that you claim that the surprising part is that mathematical objects are "discovered" first for their own sake and only later are revealed to be physically relevant? Because if this is the case, I address this here:
We can still be somewhat surprised if the language has necessary words to describe certain parts of reality before they were encountered. But this is an improbability of a much smaller degree, reduced by the fact that creating new words in this language is an interesting and rewarded activity, which people tend to do for its own sake, as exactly is the case with mathematics.
I've met people just generally confused why math can describe reality at all. And math being a language is a part of the answer here. I think Lorenzo Elijah, whose post I'm answering to, is confused in a similar manner. He acknowledges that math is a language but then thinks this is problematic and uses the middle-earth-planaet analogy which would make sense only if math was a story. So I think it's totally appropriate to point that math is a language and not a story and that this resolves the huge part of the apparent improbability, before addressing the smaller improbability related to the order in which mathematical objects are invented/discovered.
i feel the original essay -- in this section -- elides the difference between "mathematical language" and "mathematical object", and thereby argues against a strawman.
Mathematical objects are parts of the language. Like words pr other semantic constructions. I think it's pretty straightforward.
Therefore, there is nothing weird about mathematical efficiency. Nor any incompatibility between math and empiricism or math and physicalism. We do not need the natural laws of the world to be “written in the language of mathematics”. It’s enough, that we can speak math ourselves.
Indeed. I think the reason people get caught on questions like "what is math?" (and "what is reality/morality/consciousness/etc.?") largely comes from what I might call "metaphysical confusion". It takes the form of feeling like there's some meta question to be answered because they struggle to keep the whole picture in their head as they reason, and so it seems like concepts must bottom out somewhere other than experience because when they look at experience they struggle to see how it's enough.
But experience is enough, and there's no reason to do metaphysics, in that if a question is a question of metaphysics, we can generally reason equally well with and without the metaphysical model, except insofar as a metaphysical theory might provide useful intuitions to power counterfactual thinking to extend non-metaphysical theory, or in that we need some implicit metaphysics to say anything at all.
I doubt the point of posts like this. There are countless philosophy of mathematics papers out there which have discussed these topics at a much higher level of sophistication with many references to previous literature. Someone who is unfamiliar with this literature is unlikely to add any original ideas at this point.
Of course, people, one way or the other, are very unlikely to dig up and read this dry prior literature, because that takes a lot of work. (Even if LLMs can help with the digging-up part.) So we have either a) uninformed but fun controversial blog posts to read and discuss, or b) nothing. Which is the better option? I'm not sure.
Frankly, I don't trust academic philosophy. The density of good ideas is too low. If you think otherwise, then why don't you write about what we're missing?
Someone who is unfamiliar with this literature is unlikely to add any original ideas at this point.
...was Wigner familiar with the literature, when wrote "The Surprising Effectiveness of Mathematics"? Sure, Wigner was an unusual genius, but I feel like a competent mathematician with a habit of writing philosophical blog posts could've also come up with it.
Frankly, I don't trust academic philosophy. The density of good ideas is too low.
Even if you believe the average quality in academic philosophy is lower than in LessWrong philosophy, that doesn't mean that the best philosophy of mathematics papers, which build on each other, aren't ahead of this post, which doesn't build on much.
If you think otherwise, then why don't you write about what we're missing?
As I said already, because this would require a substantial time investment.
...was Wigner familiar with the literature, when wrote "The Surprising Effectiveness of Mathematics"?
No, but it would have been a lot better if he was. I'm not even sure how novel his idea was relative to the prior literature.
Sure, Wigner was an unusual genius, but I feel like a competent mathematician with a habit of writing philosophical blog posts could've also come up with it.
I don't think that this is true, because we don't usually see novel philosophical insights from people who are unfamiliar with the prior literature. People who just wing it are unlikely to compete with people who stand on the shoulders of giants.
Lorenzo Elijah, PhD shares his fascination with math and echoes a common idea among philosophers that the “surprising efficiency of math” is a problem for empiricism and physicalism
The phrase "surprising efficiency of math” comes from a paper by Eugene Wigner, a physicist. It's not like the philosophers are all in one side, and scientists on another. About 40% of philosophers are Platonists.
On this fictionalist view, mathematics is just a language like any other
There is a reasonable objection to fictionalism, that mathematical truth is nowhere near as arbitrary as fiction. To address it, Fictionalism is usually combined with formalism. They answer different but complementfary questions. Fictionalism answers the ontological question, what mathematical objects are: they are fictions and have no extra mental existence. Formalism answers the epistemological question, what mathematical truth is: it's derivability from axioms ... there is no more to truth than proof. It's important for anti realists to emphasise that they need to resist the objection that it's an anything-goes theory. So long as the standard axioms are used, the standard set of truths is obtained.
the surprise which must be explained is that many of our specific sentences in those languages later turned out to be physically relevant.
How many?
“it is surprising that specific mathematical objects of interest to mathematicians often end up useful in physics or other empirical sciences.”[1
How often?
The oddity of this well worn debate is that the realists make a 1 ) quantitative claim about 2) maths, which they 3) don't quantify.
The ability of.mathematical to come up with physically avoid maths using pencil and paper methods is nowhere near enough for physicists to dispense with particle colliders and other expensive tests. The actual effectiveness is nowhere near the greatest imaginable effectiveness.
In fact, it can easily be seen that the amount of maths that is physically valid is no more than infinitesimal. For instance , the dimensionality of space is an integer, and whichever integer it is, 3 or 4 or 10 or 11, there is a countable infinity of integers which isn't it.
If you look into the history of non Euclidean spaces, the mathematicians involved did not have any intuition that they were dealing with a better physical theory of space, that insight came from the physics side...promoted by empirical results.
ETA: Then, in 1829–1830 the Russian mathematician Nikolai Ivanovich Lobachevsky and in 1832 the Hungarian mathematician János Bolyai separately and independently published treatises on hyperbolic geometry. Consequently, hyperbolic geometry is called Lobachevskian or Bolyai-Lobachevskian geometry, as both mathematicians, independent of each other, are the basic authors of non-Euclidean geometry. Gauss mentioned to Bolyai's father, when shown the younger Bolyai's work, that he had developed such a geometry several years before,[13] though he did not publish. While Lobachevsky created a non-Euclidean geometry by negating the parallel postulate, Bolyai worked out a geometry where both the Euclidean and the hyperbolic geometry are possible depending on a parameter k. Bolyai ends his work by mentioning that it is not possible to decide through mathematical reasoning alone if the geometry of the physical universe is Euclidean or non-Euclidean; this is a task for the physical sciences.
Of course, most people have never heard of 45,971 dimensional spaces, because people are mostly taught maths that has an application. Which gives us a way of dissolving the problems: maths only seems unreasonably effective because we are selectively familiar with the effective parts.
However, empiricism doesn’t explain the relative simplicity of the mathematical axioms required to describe reality
They are relatively simple because they describe reality very inefficiently. An axiom system where most mathematically valid axioms are also physically true would be much more complex.
I would guess that you distinguish between platonic reality, physical reality, perhaps simulated or thought reality, and maybe others. However I would consider all of these to be real, and it is clearly impossible for something to be real in the physical or other sense while not existing in the platonic sense.
Which tells me that physical existence is platonic existence plus something else, but not what the something else is. A guess would be the possibility of causal interaction. It's a,standard argument against Platonism that Platonic entities can't interact with physical ones, and therefore, given physicalism about the mind, can't play a role in the thought processes of mathematicians.
Was Wigner familiar with the literature, when wrote “The Surprising Effectiveness of Mathematics”?
Probably not, but I don't count it as good for the reasons explained above.
Frankly, I don’t trust academic philosophy.
Are you supposed to trust it?
The density of good ideas is too low
How are you judging "good"?
Indeed. I think the reason people get caught on questions like “what is math?” (and “what is reality/morality/consciousness/etc.?”) largely comes from what I might call “metaphysical confusion”. It takes the form of feeling like there’s some meta question to be answered because they struggle to keep the whole picture in their head as they reason
Well, realism is an answer , anti realism is an answer , and empiricism is an answer.
it seems like concepts must bottom out somewhere other than experience
You can't bottom out all the concepts in experience , [in the strong sense that every concept refers to a possible experience]because you cant experience infinities or imaginary numbers. That is not an argument for Platonism, because a concept can just be a concept. In terms of the Fregean sense/reference distinction -- one of the useful things you can learn from philosophy -- there doesn't have to be a reference , physical or Platonic.
Confused by this weird multi-reply comment, but I'll address just the part to me:
You can't bottom out all the concepts in experience , because you cant experience infinities or imaginary numbers. That is not an argument for Platonism, because a concept can just be a concept. In terms of the Fregean sense/reference distinction -- one of the useful things you can learn from philosophy -- there doesn't have to be a reference , physical or Platonic.
Both infinities and imaginary numbers are artifacts of the model. That is, they are things we know about only because the map says they are there. This is both historically true (both things were discovered because the theory had a place where they should be) and true on the model itself (e.g. any experience of boundlessness I have is necessary a bounded experience that can only partly capture the infinite, supposing the infinite is even something we can meaningfully talk about as existing to experience).
And yes, there doesn't have to be a direct reference. You can just make up concepts. But where did you make up that concept from? As I argue here, it comes ultimately from experience, because all concepts are grounded in experience via ostensive concept construction, and then we can build concepts on concepts, but those concepts not directly tied to experience are indirectly tied through their relationship with other concepts that are.
As far as I know it's an open question of how LLMs learn language. It's clearly different from humans because they don't have the same kind of learning-from-sense-experience process that we do to ground the meaning of words defined in terms of other words. It's possible they don't learn words the same way we do and the sense in which an LLM gives a word meaning is different from how a human does it, and maybe that happens in such a way as to be meaningful even if humans and LLMs can communicate and each experiences the other as producing words that they interpret as grounded in the way they ground meaning.
As for mathematical concepts made out of relationships to each other, the story from the human perspective is the same: grounded in experience with words that back up those other words. When we try to do mathematics where everything floats free, it's possible, but those symbols seem to come to take on meaning only because they get grounded by how they are used, which, arguably, is what humans do, so maybe that's what LLMs do, too, only they don't have the point-at-a-thing-to-know-it operation, so their use of words have meaning grounded in use, but not use to describe sensory experience.
If they are being applied to pure mathematics they are being used to do mathematics. Math, in an important sense, doesn't exist when it's not being done.
Math exists in the map, not the territory. Math is a map to make sense what we experience (I phrase it this way to avoid making excess metaphysical commitments about the nature of what's experienced). It's an abstraction of symbols, and it's useful to the extent it accurately models and predict what we experience. Modeling is an active task. When there's no mind, there's no modeling happen, and hence no math. There may well still be things happening that could be modeled by math and you could argue that the abstractions of math are latent, but they don't meaningfully exist, as far as we know, when we're not mathing.
this implies that mathematics itself is not real, which has not been demonstrated at all.
Taboo the word "real". What exactly do you mean here?
it is unconditionally true that, for example, the inscribed angle theorem follows from the axioms of Euclidean geometry
What is meant by "follows" is not unconditional. Implication is itself conditional on logical axioms.
However, empiricism doesn't explain the relative simplicity of the mathematical axioms required to describe reality.
Our brains evolved in the reality and developped the kind of reasoning that is useful for it. No more explanation is required.
The word real is rather important and I don't think the conversation can proceed without it unless it is replaced with something else. I would suggest 'existent' as such a replacement.
You are failing at the rationalist taboo game. Read this, please, and try again.
The word itself is just a label. The meaning is important. Try explaining the gears behind it, instead of comming up with a synonym.
I disagree that this is the kind of simplicity meant when pointing out the unreasonable effectiveness of math. If it is then I believe this line of thought has a glaring hole. Simple systems do not imply simple dynamics in general. Emergence exists. An elementary cellular automata can generate fractals from 8 binary rules, and a simple set of axioms is not held back by this simplicity when it comes to what that those axioms generate.
So the simplicity of the reality describing math generating axioms is interesting, but not surprising.
But lets keep going, and look at the anthropic principle. Assume we're 18th century Deist gods who get to decide how to build a new universe and we do this by deciding a set of axioms and watching the universe unfold. We have two choices, a simple set of axioms or a complex set of axioms. Which is more likely to generate the human race?
I'd argue the simple set of axioms is more likely to generate intelligence. The more moving parts you have the more likely something is going to break. A pendulum is orderly, a double pendulum exhibits chaos, and the
But then again this is a bit circular isn't it? Using dynamical systems (math) to argue the dynamics of mathematical formalism and all.
Sorry, I was making an argument from analogy for what I now realize was for no apparent reason. I saw you mention the anthropic principle and got overexcited.
I agree that before the development of cheap widespread compute such a conclusion on the likely simplicity of the generating rules of the universe would be in the reverse (again with the Deism).
I agree, but I think there's another way to look at it.
I think the answer to the question, "Does complexity require complexity to come into existence?" is directly related to the nature of god, and the logical relationship between the two questions does more for explaining Deism's existence in history.
If complexity requires complexity, then some form of complexity must have spawned the complexity of the world, and we can call this complex form 'god'.
If complexity does not require complexity and rather can form from simple systems, then god is no longer implied. (This assumes we do not wish to worship a 'simple' system, denoting it some low-dimensional phase-space 'god')
What I'm getting at is that the existence of Deism was a product of the technology of the time, but in multiple ways simultaneously. The drawing of the equivalence between the mind and computation is one such way technology of the time produced Deism. Another way is how the understanding of complexity forming from simplicity requires simulation, and at the time only analog simulation was available. There was no reproducible or scalable way to test abstract systems for emergence.
What you said is the push, the cause of why (some) people of the 18th century were Deists.
What I'm pointing out is the pull, a reason why Deism as it was, no longer exists.
Fun topic :)
In Does the Universe Speak a Language We Just Made Up? Lorenzo Elijah, PhD shares his fascination with math and echoes a common idea among philosophers that the “surprising efficiency of math” is a problem for empiricism and physicalism:
Personally, I’m a big fan of empirical observations and physicalism does appear to me as the most probable ontology. I’ve written several posts critiquing metaphysical reasoning for trying to guess nature of reality from the armchair. And yet, I don’t see anything weird in the fact that math is the “backbone all the hard sciences”.
This may have something to do with the conflict of interest. My university diploma claims that I’m a mathematician, after all. Alternatively, maybe I have a better model of math than Dr Elijah. Confusion is the property of the map. If reality seems weird to you, then it’s you who has a problem, not reality. I’ll let my readers to deside which one of these factors is more relevant here.
Reality and Fiction
In his post Dr Elijah argues against mathematical fictionalism. He describes it like this:
I would not call myself a fictionalist, however, I think the view described here is basically correct, capturing core truths about the nature of math. Indeed, mathematics is a very precise language that allows us to talk about specific things instead of something else by logically pinpointing them. And then we can preserve truth relations and reach conclusions that follow from the premises.
Yet, Lorenzo Elijah thinks this view is problematic as it denies that mathematical objects “really exist”.
Before we engage with his argument, I’d like to share one tip for disentangling standard philosophical confusions. A lot of philosophical categories are a grab-all-bags of multiple meanings linked together for semantic reasons. And “real” is one of them. So, whenever philosophers talk about realness of something - pay close attention to what exactly is meant.
To Lorenzo Elijah’s credit, he tries to preemptively address this by explicitly stating that we are not talking about “plus sign floating in space”. This is commendable, but not enough. We haven’t tabooed the word “real”, haven’t got to the substance of the meaning that is being explored here.
What does it mean that a referent for symbol “4” really exists? What even is this referent?
So, it seems that 3. is our crux of disagreement. Now let’s get to the argument that Lorenzo Elijah presents to resolve it.
The Argument
Here is Elijah’s argument against fictionalism:
I agree that if mathematicians indeed were inventing/discovering math from their armchairs without any contact with reality and then it just so happened to perfectly coincide with the way reality is, that would be extremely surprising. Which is a huge hint that it’s not what is actually happening.
Remember, your strength as a rationalist is being confused by fiction more than by reality. If something is extremely unlikely to happen, then most likely it didn’t, and something is wrong with the theory that it did. In our case, with the argument above. So, what’s wrong with this argument?
Several things.
Math Is a Language, Not a Story.
First of all, let’s notice the core problem with the Middle-Earth-planet analogy. Math is a language that can be used to describe reality. Not a story about reality. This difference is quite crucial.
If we come up with some story and it just so happens to be true, it’s an incredible coincidence. If we come up with a language that can be used to describe what’s true - that’s just how languages work. Wouldn’t be a huge surprise if objects on the other planet can be described via the language of Tolkien’s elves, would it? Even less so with math which is much more general by design.
We can still be somewhat surprised if the language has necessary words to describe certain parts of reality before they were encountered. But this is an improbability of a much smaller degree, reduced by the fact that creating new words in this language is an interesting and rewarded activity, which people tend to do for its own sake, as exactly is the case with mathematics.
Math Is Not a Single Thing
When people are baffled by how well math describes the fundamental nature of reality, they tend to forget that math is a compilation of many different axiomatic systems. That it consists of multitude of “crazy weird objects”. Another failure of the Middle-Earth-analogy is that Lord of the Rings is the work of Tolkien, while the structure of quantum mechanics is described by yet another mathematical object, no more special than any other.
Imagine if Tolkien wrote thousands upon thousands of books, essentially brute forcing through all possible coherent settings. And then some planet with life happened to resemble one of them. That would be much less of a coincidence, wouldn’t it?
The same principle applies here. It would’ve been super impressive if mathematicians could a priori deduce which mathematical object in particular describes the fundamental nature of reality - to the point that we would indeed have to rethink the importance of empiricism as well as the established laws of thermodynamics.
On the other hand, when you try to explore all kind of premises and their conclusions, it’s no surprise that one non-specific of them happened to fit reality. We would still need to do the actual work and look through all of them, comparing them to the reality to figure out which one it is.
Realism Doesn’t Help
Last but not the least, is that, even if fictionalism performed poorly in accounting for this aparent coincidence, realism does no better.
Let’s suppose, for the sake of argument, that there exists a separate platonic realm about which mathematicians are reasoning. And then it just so happens that reasoning about this realm are applicable to our physical universe. Why would it be any less of a coincidence?
Previously we were surprised by the correspondence between our armchair reasoning and physical reality. Now we are, likewise, surprised by the correspondence between physical reality and platonic realm. The fact that our reasoning got an external referent doesn’t make this correspondence less improbable.
In fact, we would now also need to account for coincidence between platonic realm and our mathematical reasoning. How comes our brains have access to this separate realm, if they evolved in the physical reality? We can invent some just-so dualist explanation, but it will come with extra complexity penalty.
Imagine if someone tried to explain the similarity between Middle Earth and some discovered planet A by the fact that Tolkien was in contact with aliens from some completely different planet B. Not only this would not reduce the improbability of the initial coincidence, but it’s also an additional extraordinary claim that one has to justify.
What would help, is if there was some connection between planets A and B. If platonic realm was connected both to the mathematicians reasoning and to the physical world. But this connection is also an additional assumption which we’d need empirical evidence to back up. If a theory postulates something like this, it has to make some testable predictions about the nature of the connection, that have to come true to outweigh the complexity penalty.
And if we are ready to consider that there is a connection between our mathematical reasoning and physical universe through the platonic realm… why involve the platonic realm at all? Why not consider that there is a direct connection between mathematical reasoning and physical universe? That’s a strictly simpler theory, isn’t it?
Math As a Generalization of Observations
Wait… am I claiming that math is empirical? But isn’t it clearly absurd? Everyone knows that mathematical truths are certain while empiricism produces only probabilistic estimates!
I’ve briefly talked about this misconception in Give Skepticism a Try:
But let’s focus on this a little bit more. Consider these empirically testable hypotheses:
If there is one apple on the table and you put another apple on the table, there are now two apples on the table.
If there is one apple on the table and John puts another apple on the table, then there are two apples on the table.
If there is one orange on the couch and Mary puts another orange on the couch, then there are now two oranges on the couch.
If there is one apple on the table and an orange on the couch, and no other fruits on either of them, then, between a table and a couch, there are two fruits.
We can write a lot of such statements. For every type of object, for every type of actor, for every type of counting mechanism. Or we can generalize our observations and say:
For every object and process that work exactly like addition of natural numbers, one of such objects and another one of such objects submitted as an input of such process result in two of such objects.
Or even simpler:
1+1=2This is a tautology, of course. True by definition. True only for things that it’s true for; false otherwise. This is where the absolute certainty comes from.
Valid theoretical reasoning conserves truth. It allows us to outsource improbability from one part of the theory to another, while keeping total the same. With math we simply outsource all the improbability to the question of whether a particular real-world scenario fits our mathematical model.
This truth preservation property can be very powerful. We can deduce consequences of some assumptions and immediately apply them to any part of the real world about which we managed to become quite confident that it satisfies these assumptions. The latter is something that still require us to go outside and look and isn’t something we can be totally certain about.
And I understand why it may feel magical. But this is a magic rooted in empiricism about our physical reality. Only empiricism can reduce the improbability of a particular part of reality fitting specific mathematical axioms. Moreover, our ability to reason about properties of the universe and generalize patterns is the result of evolving inside this universe that has such properties in the first place.
Therefore, there is nothing weird about mathematical efficiency. Nor any incompatibility between math and empiricism or math and physicalism. We do not need the natural laws of the world to be “written in the language of mathematics”. It’s enough, that we can speak math ourselves.