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Surely this implies these LLM generated proofs require the LLMs to have mathematical intuition . . . and honestly I don't think many people believe that, and ri

Surely this implies these LLM generated proofs require the LLMs to have mathematical intuition . . . and honestly I don't think many people believe that, and rightly so, certainly not in the way Poincaré was on about. Maybe it's been done, but I'd like to see an LLM recreate Euclid from questioning without having seen it during training.

watwut 7d ago
It does not imply that. He is talking about how people do math. Intuition is what you use when deciding what to try and how to think about things. Proof is the rigorous outcome. LLM running probabilistic loop is different kind of process.
- fidotron 7d ago
  The parent comment literally said "You cannot do proof without intuition". Therefore, according to that logic, an entity producing proofs must have intuition. Edit to add: the parent commenter has now confirmed my interpretation of their statement.
  - bunderbunder 7d ago
    Your unstated major premise here is that their intent was to make a universal statement about how proofs work and not just talking to humans about how they teach humans. That premise seems unlikely to be correct.
    - fidotron 7d ago
      Why? The entire subject of conversation is triggered by things which are not humans producing proofs. If it's possible for a machine to produce a proof without intuition then clearly a human could also do it too. (And in fact I'd argue I've seen many people like that, simply very good at pattern matching over memorised items).
      - bunderbunder 7d ago
        Because regardless of the point TFA is making, that interpretation makes less sense for the specific comment. It doesn’t fit with the immediate context, which was a response to a thoughtful comment about how humans do math. And it requires assuming a math professor doesn’t understand a very basic and obvious thing about their area of expertise. That doesn’t really read as good faith engagement in the discussion. At best, it reads as being so AI pilled that you can’t even fathom that others might want to have a little side discussion about something other than AI.
        fidotron 7d ago
        You realize the commenter has now confirmed my interpretation was right? What is up with this whole sub thread of obvious hole digging?
        deleted 5d ago
        [deleted]
      - watwut 7d ago
        That math prof was talking about his stance in discussion between mathematicians long before AI. Plus, I studied math, I am from that environment. His description matches how math is done by people. People who are good at pattern matching and memorize are, frankly, shit mathematicians. They are find in fun culture around math, but rarely in actual math. They cant really do it as science.
bananaflag 7d ago
> Surely this implies these LLM generated proofs require the LLMs to have mathematical intuition . . . and honestly I don't think many people believe that Yes, I believe that, it's part of what I was implying (I believe the LLM weights have some internal representation of math in the same way brains do that allow them to produce proofs)
- fidotron 7d ago
  Appreciate the clarification, even if I disagree! I think we differ on what "mathematical intuition" is then. I've seen people that do well in undergrad math degrees simply by massively memorising things and learning how to join them up to some level of degrees-of-separation, but seemingly completely fail to understand, for ezample, why even calculus is how it is. Because they are able to regurgitate the results and "produce proofs" this is never questioned. The Euclid example also shows my bias towards spatial intuition of mathematical concepts (which is deeply unfashionable) but also exposes exactly where at least current LLMs break down; they do the symbol based pattern matching version, but they cannot leap outside of that, at least today.
  - bananaflag 7d ago
    > I've seen people that do well in undergrad math degrees simply by massively memorising things and learning how to join them up to some level of degrees-of-separation, but seemingly completely fail to understand, for ezample, why even calculus is how it is. Because they are able to regurgitate the results and "produce proofs" this is never questioned. If you want to catch them, surely you can find proofs they aren't able to produce.
    - fidotron 7d ago
      That's easy: basically all the spatial ones. I used to be a game dev, and one of the interview questions someone came up with consisted of working out the surface area of a variant of Menger sponge to some given level of depth. The bifurcation for people that could do this vs those that couldn't was incredible, and did not follow obvious trends for academic achievement. (The same interview also included the gem "How wide is a pointer?" which also catches a frightening number of people).
lupire 7d ago
LLMs have LLM intuition, not human intuition. (See the movie Her.) LLM cannot reinvent Euclid from scratch, but a larger system including LLM might.
keeda 7d ago
My hunch (or intuition, hah!) is that intuition is an instinctive mental shortcut required to navigate large problem spaces that can’t entirely fit into our heads. Maybe LLMs do not need intuition because they can scale their “cognitive capacity” with hardware and brute force their way through these problem spaces.
- fidotron 7d ago
  > Maybe LLMs do not need intuition because they can scale their “cognitive capacity” with hardware and brute force their way through these problem spaces. My view is that is certainly true of smaller LLMs but becomes less true as they scale up. To quote the parent bananaflag in a sub-comment: > I believe the LLM weights have some internal representation of math in the same way brains do that allow them to produce proofs I think as the sort of spare space adjacent to pure language processing in LLMs grows the probability of the sort of reasoning bananaflag is getting at (or spatial reasoning, or anything else) emerging in that space grows enormously. One of the questions for AI development over the coming months or years is going to be if deliberately cultivating the architecture of those sub models for specific reasoning types beats any emergent reasoning mechanisms or not.
  - keeda 7d ago
    Hmm we may be talking of two different interpretations of intuition here. I agree that LLM weights contain representations of abstract concepts, as a lot of prior research has shown. This surely includes Mathematical concepts. But to me that is analogous to what human brains do, and a bit different from intuition. I think of intuition as “heuristics”, typically developed through experience, that may link seemingly unrelated concepts via vague, hard-to-define associations, but which let us make mental leaps (or shortcuts) while reasoning. (Maybe analogous to System 1 / 2 thinking.) On the other hand, LLMs can do both: build “intuition” from patterns in data AND brute force a huge amount of potentially unrelated concepts. This gets fuzzier when we realize that even these “concepts” themselves are gleaned from patterns in data! But my point is we necessarily have to take shortcuts to scale, whereas machines can scale with hardware. This is of course a layman theory! But it could explain why these models are progressing so fast.
adastra22 7d ago
That has been done, like back in the 1960’s.
famouswaffles 7d ago
>Surely this implies these LLM generated proofs require the LLMs to have mathematical intuition Well yeah they do, obviously.
jltsiren 7d ago
I think it's actually the opposite. Intuition is the only thing LLMs can do, which is why they are prone to hallucinating when they can't validate their intuition against reality. There is an idea that human intuition, expertise, and critical thinking are largely pattern recognition. When you encounter a situation, your brain gives you a plausible starting point, based on what it has experienced before. You then continue with explicit reasoning, which is slow and inefficient, and try to validate your ideas. The more relevant the patterns you have learned are to the situation, the more likely you reach a useful conclusion. LLMs are largely the same, except that they cannot learn from experience in normal usage. And except that they experience the world only through symbolic data, while the human brain has access to plenty of sensory data.
- pegasus 5d ago
  Kind of. First of all, reasoning LLMs can also do (a form of) reasoning. Second, yes you could say LLMs form a sort of intuition, but its domain is the space of human-produced text. It only translates to real world intuition to the degree that those intuitions make their way into the corpus the model has been trained on. You could say that, when it comes to anything other than textual prediction, their intuition is secondary, a reflection of a reflection, so it will always lag behind that of humans.

watwut 7d ago

It does not imply that. He is talking about how people do math. Intuition is what you use when deciding what to try and how to think about things. Proof is the rigorous outcome. LLM running probabilistic loop is different kind of process.

fidotron 7d ago
The parent comment literally said "You cannot do proof without intuition". Therefore, according to that logic, an entity producing proofs must have intuition. Edit to add: the parent commenter has now confirmed my interpretation of their statement.
- bunderbunder 7d ago
  Your unstated major premise here is that their intent was to make a universal statement about how proofs work and not just talking to humans about how they teach humans. That premise seems unlikely to be correct.
  - fidotron 7d ago
    Why? The entire subject of conversation is triggered by things which are not humans producing proofs. If it's possible for a machine to produce a proof without intuition then clearly a human could also do it too. (And in fact I'd argue I've seen many people like that, simply very good at pattern matching over memorised items).
    - bunderbunder 7d ago
      Because regardless of the point TFA is making, that interpretation makes less sense for the specific comment. It doesn’t fit with the immediate context, which was a response to a thoughtful comment about how humans do math. And it requires assuming a math professor doesn’t understand a very basic and obvious thing about their area of expertise. That doesn’t really read as good faith engagement in the discussion. At best, it reads as being so AI pilled that you can’t even fathom that others might want to have a little side discussion about something other than AI.
      - fidotron 7d ago
        You realize the commenter has now confirmed my interpretation was right? What is up with this whole sub thread of obvious hole digging?
        deleted 5d ago
        [deleted]
    - watwut 7d ago
      That math prof was talking about his stance in discussion between mathematicians long before AI. Plus, I studied math, I am from that environment. His description matches how math is done by people. People who are good at pattern matching and memorize are, frankly, shit mathematicians. They are find in fun culture around math, but rarely in actual math. They cant really do it as science.

bananaflag 7d ago

> Surely this implies these LLM generated proofs require the LLMs to have mathematical intuition . . . and honestly I don't think many people believe that Yes, I believe that, it's part of what I was implying (I believe the LLM weights have some internal representation of math in the same way brains do that allow them to produce proofs)

fidotron 7d ago
Appreciate the clarification, even if I disagree! I think we differ on what "mathematical intuition" is then. I've seen people that do well in undergrad math degrees simply by massively memorising things and learning how to join them up to some level of degrees-of-separation, but seemingly completely fail to understand, for ezample, why even calculus is how it is. Because they are able to regurgitate the results and "produce proofs" this is never questioned. The Euclid example also shows my bias towards spatial intuition of mathematical concepts (which is deeply unfashionable) but also exposes exactly where at least current LLMs break down; they do the symbol based pattern matching version, but they cannot leap outside of that, at least today.
- bananaflag 7d ago
  > I've seen people that do well in undergrad math degrees simply by massively memorising things and learning how to join them up to some level of degrees-of-separation, but seemingly completely fail to understand, for ezample, why even calculus is how it is. Because they are able to regurgitate the results and "produce proofs" this is never questioned. If you want to catch them, surely you can find proofs they aren't able to produce.
  - fidotron 7d ago
    That's easy: basically all the spatial ones. I used to be a game dev, and one of the interview questions someone came up with consisted of working out the surface area of a variant of Menger sponge to some given level of depth. The bifurcation for people that could do this vs those that couldn't was incredible, and did not follow obvious trends for academic achievement. (The same interview also included the gem "How wide is a pointer?" which also catches a frightening number of people).

lupire 7d ago

LLMs have LLM intuition, not human intuition. (See the movie Her.) LLM cannot reinvent Euclid from scratch, but a larger system including LLM might.

keeda 7d ago

My hunch (or intuition, hah!) is that intuition is an instinctive mental shortcut required to navigate large problem spaces that can’t entirely fit into our heads. Maybe LLMs do not need intuition because they can scale their “cognitive capacity” with hardware and brute force their way through these problem spaces.

fidotron 7d ago
> Maybe LLMs do not need intuition because they can scale their “cognitive capacity” with hardware and brute force their way through these problem spaces. My view is that is certainly true of smaller LLMs but becomes less true as they scale up. To quote the parent bananaflag in a sub-comment: > I believe the LLM weights have some internal representation of math in the same way brains do that allow them to produce proofs I think as the sort of spare space adjacent to pure language processing in LLMs grows the probability of the sort of reasoning bananaflag is getting at (or spatial reasoning, or anything else) emerging in that space grows enormously. One of the questions for AI development over the coming months or years is going to be if deliberately cultivating the architecture of those sub models for specific reasoning types beats any emergent reasoning mechanisms or not.
- keeda 7d ago
  Hmm we may be talking of two different interpretations of intuition here. I agree that LLM weights contain representations of abstract concepts, as a lot of prior research has shown. This surely includes Mathematical concepts. But to me that is analogous to what human brains do, and a bit different from intuition. I think of intuition as “heuristics”, typically developed through experience, that may link seemingly unrelated concepts via vague, hard-to-define associations, but which let us make mental leaps (or shortcuts) while reasoning. (Maybe analogous to System 1 / 2 thinking.) On the other hand, LLMs can do both: build “intuition” from patterns in data AND brute force a huge amount of potentially unrelated concepts. This gets fuzzier when we realize that even these “concepts” themselves are gleaned from patterns in data! But my point is we necessarily have to take shortcuts to scale, whereas machines can scale with hardware. This is of course a layman theory! But it could explain why these models are progressing so fast.

adastra22 7d ago

That has been done, like back in the 1960’s.

famouswaffles 7d ago

>Surely this implies these LLM generated proofs require the LLMs to have mathematical intuition Well yeah they do, obviously.

jltsiren 7d ago

I think it's actually the opposite. Intuition is the only thing LLMs can do, which is why they are prone to hallucinating when they can't validate their intuition against reality. There is an idea that human intuition, expertise, and critical thinking are largely pattern recognition. When you encounter a situation, your brain gives you a plausible starting point, based on what it has experienced before. You then continue with explicit reasoning, which is slow and inefficient, and try to validate your ideas. The more relevant the patterns you have learned are to the situation, the more likely you reach a useful conclusion. LLMs are largely the same, except that they cannot learn from experience in normal usage. And except that they experience the world only through symbolic data, while the human brain has access to plenty of sensory data.

pegasus 5d ago
Kind of. First of all, reasoning LLMs can also do (a form of) reasoning. Second, yes you could say LLMs form a sort of intuition, but its domain is the space of human-produced text. It only translates to real world intuition to the degree that those intuitions make their way into the corpus the model has been trained on. You could say that, when it comes to anything other than textual prediction, their intuition is secondary, a reflection of a reflection, so it will always lag behind that of humans.