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What sort of maths are LLMs good at?

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Re: What sort of maths are LLMs good at?

#141

Post by annzabelle »

najmbajwa123 wrote: post #46

From my experience, you have to be good at math to trust an LLM to do the math.

That's been an interesting thing for me with Claude Code and working with non-technical coworkers. Often they ask me to write something that could be a simple script and takes maybe 3 prompts to get it right, and I find myself wondering why they can't just do it themselves. But then I have them asking me what a .md file is and how to open it, and they're unable to follow readme instructions to use commandline programs, so now I have to throw a "make a gui" onto every prompt.

Re: What sort of maths are LLMs good at?

#142

Post by empath75 »

gowld wrote: post #139
empath75 wrote: post #70

Earlier quoted context omitted.

IME, LLMs are primarily good at grinding through cases, which is why you see them pushing upper and lower bounds and finding counter examples. I spent a few weeks working on a number theory proof with Claude off and on and it spent hours and hours and hours grinding through one shape of polynomial after another, reporting "progress", and it's true, it proved what I was trying to prove for more and more classes of pol…

How much API usage cost (or what subscription did it fit in)?

oh, it was probably thousands of dollars in "cost" but it was on a personal max plan.

Re: What sort of maths are LLMs good at?

#143

Post by redox99 »

CamperBob2 wrote: post #132
zahlman wrote: post #119

Earlier quoted context omitted.

> Sampling is what AI is good at. You might think so, but I tried asking ChatGPT to solve one of the puzzles from https://en.wikipedia.org/wiki/Countdown_(game_show) (which a Python script can brute-force on my 12-year-old hardware in half a second) and it made an elementary arithmetic error that's decidedly not human-like.

"ChatGPT" could mean anything from Instant to Pro. What was the prompt, and which model(s) did you try?

Almost surely the free model on the web ui. So a dumb model, no harness, and probably a bad prompt.

Re: What sort of maths are LLMs good at?

#144

Post by black_knight »

TuringTest wrote: post #134
wyager wrote: post #124

Earlier quoted context omitted.

> But ask them to enumerate all the intermediate steps required to create a formal direct proof, and it will loose attention and forget important details as they go out of their input window size. It's interesting how people will comment on LLM capabilities despite clearly not having engaged with frontier models in any meaningful way in a long time Having models write Lean proofs of mathematical claims is standard op…

Yeah but the LLM can only handle proofs that hold inside its context window. Proofs for novel theories requiring thousands of pages with dozen millions of steps will need support from external tools to organize the full structure of the formal document; it cannot be done by the LLM inference process alone, which was my point. It would be like asking a mathematician to proof theorems without pen and paper; external to…

Not sure how true this restrictions is once you have the agents hammering on at a big code base of formalised proofs.

Proofs stretching thousands of pages are split into lemmas, grouped into sub theories.

What I haven’t seen agents do yet is to develop new ideas for entire such theories. I have usually seen them bite into some existing idea and grinding out related results. But I am less sure than ever that they won’t!

Re: What sort of maths are LLMs good at?

#145

Post by willmarch »

YeGoblynQueenne wrote: post #115
ozgung wrote: post #85

Earlier quoted context omitted.

This is not "brute-force" though. It's an iterative search algorithm. You learn things at each iteration. You also don't search blindly. You use "something" (heuristics, experience, intuition) to come up with "ideas" at each iteration. You don't try 650 random programs. You try 650 different ideas each learning from the results of previous trials. Yes this is the "Universal Problem Solving Algorithm". It's actually t…

>> This is not "brute-force" though. It's an iterative search algorithm. You learn things at each iteration. You also don't search blindly. You use "something" (heuristics, experience, intuition) to come up with "ideas" at each iteration. You don't try 650 random programs. You try 650 different ideas each learning from the results of previous trials. But, learn what? All those ideas where wrong. How does an LLM "lear…

Ask Edison about the light bulb

Re: What sort of maths are LLMs good at?

#146

Post by antonvs »

n4r9 wrote: post #2

A thoughtful and measured post, as usual from Gowers. The final note is neat and worth pasting out here in full: > A good sign that LLMs have reached human level for a much wider class of problems will be if they start proving theorems using methods that, like much of the very best human mathematics, are new and surprising but that with hindsight come to seem beautiful and natural. They should also be methods that ar…

> with hindsight come to seem beautiful and natural.

…because they’ve been internalized. I wonder if the author has ever questioned where his notions of “beauty” and “natural” come from.

Re: What sort of maths are LLMs good at?

#147

Post by antonvs »

root-parent wrote: post #60
vhantz wrote: post #45

Earlier quoted context omitted.

Stop wasting your time and use actual code for most of what you give an LLM to do. Make them write the code even. Anything that can be verified mechanically should be code. Only use LLMs to fill in the gaps where things are fuzzy. Don't fall for the idea that those harnesses are general purpose, make your own fit to your task with the guards and verification steps you need. Make the LLM create the harness even. There…

The irony about your comment is that, this is probably the most likely opinion and most consensual around many technological practitioners. But the mathematicians here in this thread, are having a hard time with these clearly dumb models, doing so well in proving theorems in their domains :-)

Writing code and proving theories are flip sides of the same coin. See e.g. Phil Wadler’s “Proofs are Programs”: https://philarchive.org/rec/WADPAP

The statement should probably really be “programs are proofs” since it’s difficult to make it a true bidirectional isomorphism, which is why the underlying principle is properly called the Curry-Howard(-Lambek) correspondence: https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon...

The point is that the fact that LLMs are good at writing code directly implies that they should also be good at certain classes of mathematical proof.

Re: What sort of maths are LLMs good at?

#148

Post by inigyou »

antonvs wrote: post #147
root-parent wrote: post #60

Earlier quoted context omitted.

The irony about your comment is that, this is probably the most likely opinion and most consensual around many technological practitioners. But the mathematicians here in this thread, are having a hard time with these clearly dumb models, doing so well in proving theorems in their domains :-)

Writing code and proving theories are flip sides of the same coin. See e.g. Phil Wadler’s “Proofs are Programs”: https://philarchive.org/rec/WADPAP The statement should probably really be “programs are proofs” since it’s difficult to make it a true bidirectional isomorphism, which is why the underlying principle is properly called the Curry-Howard(-Lambek) correspondence: https://en.wikipedia.org/wiki/Curry%E2%80%93H…

Not in any useful way, though. In the Curry-Howard correspondence, being able to write a function of type "int -> string" proves that there is at least one valid string or there are no valid ints. The code of the function is a proof of this statement. For example you might prove directly that there is a string, by writing return "hello"; or you might write return bool2string(int2bool(myInt)); thus chaining two implications to arrive at a shortcut implication.

Re: What sort of maths are LLMs good at?

#149

Post by snaking0776 »

gowld wrote: post #130
Jensson wrote: post #121

Earlier quoted context omitted.

> Would you trust a bridge built by a single human? But an AI doesn't replace a single human, it replaces every human in the project, so it needs to match a team of humans not a single human.

That's trivially solvable by deploying multiple AIs on one project, so that can't be the true obstacle.

I’m not so sure this is trivial and strongly depends on the biases of the agents that the group is composed of. It seems like some species of animals (humans, ants, wasps, etc) seem to select for intelligence in group settings more than individual intelligence. One of my favorite facts about human evolution is that in the process of becoming homo sapiens our brains actually shrunk relative to our ancestors possibly due to our reliance on social ties and specialization allowing for each individual unit to be “dumber” while the collective becomes more intelligent than any individual ever was.

It’s not apparent to me that current LLM’s have the right biases for a trivial unit of “more LLM” = smarter. It seems apparent that LLM’s are under many metrics more “intelligent” than the average person but they still don’t rival our corporations or collectives in terms of intelligence and I don’t see them being plug in replacements for humans and maintaining the benefits of the collective structures yet. Not impossible but definitely not trivial in my mind. We and our cultures evolved to work together and the modern world is the result of the emergent structure that resulted.

As a plug my favorite field relating to this is called stigmergy and basically describes how individual units like ants build intelligent collectives that are far greater than the sum of their parts.

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