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Comment by warkdarrior | Hacker News Reader

If you know all this, can you explain how these models produce advanced mathematical proofs? (as recently done by OpenAI, for example)

I tried to generate the next word to the best of my ability, starting with a mathematical problem, but I did not create a valid proof. How do these LLMs work when they create math proofs to problems not yet solved?

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How did you generate the next word? Did you first read pretty much every written work ever published, including blog posts, forum posts, books, etc? Learn how to imagine everything as a point in a gigantic abstract space where similar meanings cluster together? How did you manage training with gradient descent? And then did you do a lifetime of matrix multiplication for each token you predicted?

It turns out being orders of magnitude faster at searching with the aid of a strong verifier is a great way to generate proofs

I agree these are central components, but to avoid oversimplification and the mistaken belief that modern LLMs do a lot of search during inference: If it was so simple, the traditional computer algebra systems would have reached similar breakthroughs when deployed at large supercomputer centers. This didnt happen because the search space is huge. You definitely also need a fancy learning algorithm. Although these ingredients would suffice (depending on what the learning algorithm is), you probably also want to learn in the absense of a strong verifier at every step, to allow building a fuzzy/erratic sense of the search space that can lead to planning/intuition and allow distant jumps in a targetted direction.

The search space is far too large for a mere order of magnitude to make any difference at all.

You don’t have the computational ability to process as many calculations as a datacenter. You can hardly transpose a 5×5 matrix in your mind, so you won’t be able to do what datacenters do.

This is like saying we don‘t know how a car works because a car can beat the best human athletes in 100 meter dash.

Yes, it turns out that matrix math over a feature space of math works pretty well because unlike poetry or real world work, maths are internally coherent and entirely theoretical.

Funny how you said "it turns out" when the whole point is that we don't understand LLMs - we just empirically see what they are good at.

Claiming that you understand LLMs is similar to saying that you understand how our biology work because you understand evolution. No - you understand the mechanism behind evolution, but not the complexity it produces.

stop it. This is such reductionist bullshit. By your infinite reductionism definition no one knows how anything works

This not understanding "understanding".

> maths are internally coherent and entirely theoretical

Nope. This kind of wish-washy thinking is not what we mean by understanding.

https://iep.utm.edu/math-inc/

stop with reductionist absurdum.

You can see how an LLM works here https://bbycroft.net/llm they are not magic.

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