Ten advances in mathematics and theoretical computer science

LLMs have made math proofs more computable, in the sense that a computer can both generate potential solutions and check the validity of its solutions on its own, with a reasonable chance of converging on something correct. I assume this was already doable to some extent, but it seems like it’s now exponentially easier. That still doesn’t mean that all math is automatically solved.

This is somewhat similar to things like molecular dynamics or protein folding or finite element simulations, etc. Some problems that were previously intractable via computation became tractable. Others - the vast majority of other problems - remain unsolvable by these computational techniques, because the scale of compute required is beyond imagination. These are simple things like simulating the dynamics of a cubic millimeter of water molecules for 1 second. Unfathomably beyond current capabilities (and LLMs aren’t going to change that).

I think LLMs are great, I use them every day and I think they have a ton of value. But if these things were as revolutionary as people promote/fear them to be, you should immediately point them at the highest value math problems and see progress. Like the Millenium Prize problems. Haven’t seen a solution to those.

So there are limits - but we’re about to learn a lot about the new normal of what constitutes a layup math proof vs the truly difficult.

I don't know if I agree with that but it doesn't seem like an irrational claim and does seem credible to me.

eg. "we spent another 2k on searching for pre-existing proof but found only the weaker result xyz by abc in 1972" would be in the spirit of academics quoting prior work.

- travel and restaurant recommendations. my last few outings have been entirely LLM-advised and they turned out excellent. LLMs seem to have ingested every single Google review, photo, and menu of every business on Earth and can answer very nuanced questions like "is the garlic chicken at garnished with coriander?"

- fitness, nutrition, accounting, therapy, medical, legal, immigration advice (sure it's not a real professional but you know what, it's pretty fucking close, and any capability gap is made up by having perfect two-way communication which you don't get when talking with a human)

- coding (work, side projects, personal tools, documentation & pricing questions, "review this code", etc).

- I start reading most articles with the prompt "Summarize this article: ". I just started a non-fiction book by pasting into Claude: "There are 12 chapters in the book . Can you give me a 2 sentence synopsis of each chapter?". It reduces the "activation energy" hump and screens if it's worth reading at all.

- I use the LLM in my Tesla for on-the-fly advice for parking and other things. You can simply ask "what's the best Boba place around here?" and it will give you a decent recommendation. You can also follow up with "does this place have ample parking?".

- I use the LLM in YouTube to summarize videos and ask specific questions and/or get timestamps to the parts I care about.

If your critical thinking skills are strong then LLM is a literal superpower.