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.
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.
@Ape in the coat
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.
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.
@DirectedEvolution
How many?
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.
@Horosphere
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.
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.
@XelaP
Probably not, but I don’t count it as good for the reasons explained above.
Are you supposed to trust it?
How are you judging “good”?
@Gordon Seidoh Worley
Well, realism is an answer , anti realism is an answer , and empiricism is an answer.
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:
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.
Comment withdrawn.
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.
Comment withdrawn.
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.
Comment withdrawn.
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.
Comment withdrawn