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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 d

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).
D-Machine 7d ago
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").
- fidotron 7d ago
  Yes, I think the augmentation of LLMs with (hopefully eventually higher dimensional) world models will prove very interesting for all this.
bluecheese452 5d ago
I don’t understand this. You seem to look down on people who put in effort and instead celebrate those who have natural talent. Like the people the people are somehow cheating by studying. Seems backwards.
- pegasus 5d ago
  Intuition also takes effort, just of a different type and quality. It's the difference from mindlessly applying memorization and simple inference/recombinations (basically delegating to our own internal LLM) versus deeply contemplating the meaning of certain mathematical objects, looking for intuitive analogues etc. I would say the second takes more effort and seriousness. The first is more like brute-forcing a problem.

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).

D-Machine 7d ago

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").

fidotron 7d ago
Yes, I think the augmentation of LLMs with (hopefully eventually higher dimensional) world models will prove very interesting for all this.

bluecheese452 5d ago

I don’t understand this. You seem to look down on people who put in effort and instead celebrate those who have natural talent. Like the people the people are somehow cheating by studying. Seems backwards.

pegasus 5d ago
Intuition also takes effort, just of a different type and quality. It's the difference from mindlessly applying memorization and simple inference/recombinations (basically delegating to our own internal LLM) versus deeply contemplating the meaning of certain mathematical objects, looking for intuitive analogues etc. I would say the second takes more effort and seriousness. The first is more like brute-forcing a problem.