Comment by fidotron
21 hours 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, 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.
> 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)
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.
This feels very related to the issues re: the presence or absence of world models in LLMs. Insofar as they have world models (or "intuitions"), these would seem to have to be primarily verbal-linguistic (or symbolic, when using math). LLM world models are not likely (currently) very spatial, in contrast to e.g. V-JEPA-2 models, which likely do have some basic spatial models (and perhaps "intuitions").
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> 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.
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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.
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.
> 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.
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.
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>Surely this implies these LLM generated proofs require the LLMs to have mathematical intuition
Well yeah they do, obviously.
That has been done, like back in the 1960’s.
LLMs have LLM intuition, not human intuition. (See the movie Her.)
LLM cannot reinvent Euclid from scratch, but a larger system including LLM might.
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.
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.
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.
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