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Can AI do maths yet? Thoughts from a mathematician

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Re: Can AI do maths yet? Thoughts from a mathematician

#161

Post by ndriscoll »

zmgsabst wrote: post #151
vouaobrasil wrote: post #146

Earlier quoted context omitted.

I was just thinking about this. I already posted a comment here, but I will say that as a mathematician (PhD in number theory), that for me, AI signficantly takes away the beauty of doing mathematics within a realm in which AI is used. The best part of math (again, just for me) is that it was a journey that was done by hand with only the human intellect that computers didn't understand. The beauty of the subject was…

Presumably people who get into math going forward will feel differently. For myself, chasing lemmas was always boring — and there’s little interest in doing the busywork of fleshing out a theory. For me, LLMs are a great way to do the fun parts (conceptual architecture) without the boring parts. And I expect we’ll such much the same change as with physics: computers increase the complexity of the objects we study, wh…

Even current people will feel differently. I don't bemoan the fact that Lean/Mathlib has `simp` and `linarith` to automate trivial computations. A "copilot for Lean" that can turn "by induction, X" or "evidently Y" into a formal proof sounds great.

The the trick is teaching the thing how high powered of theorems to use or how to factor out details or not depending on the user's level of understanding. We'll have to find a pedagogical balance (e.g. you don't give `linarith` to someone practicing basic proofs), but I'm sure it will be a great tool to aid human understanding.

A tool to help translate natural language to formal propositions/types also sounds great, and could help more people to use more formal methods, which could make for more robust software.

Re: Can AI do maths yet? Thoughts from a mathematician

#162

Post by ComplexSystems »

nyrikki wrote: post #153
busyant wrote: post #138

As someone who has a 18 yo son who wants to study math, this has me (and him) ... worried ... about becoming obsolete? But I'm wondering what other people think of this analogy. I used to be a bench scientist (molecular genetics). There were world class researchers who were more creative than I was. I even had a Nobel Laureate once tell me that my research was simply "dotting 'i's and crossing 't's". Nevertheless, I…

What LLMs can do is limited, they are superior to wet-wear in some tasks like finding and matching patterns in higher dimensional space, they are still fundamentally limited to a tiny class of problems outside of that pattern finding and matching. LLMs will be tools for some math needs and even if we ever get quantum computers will be limited in what they can do. LLMs, without pattern matching, can only do up to abou…

"LLMs, without pattern matching, can only do up to about integer division, and while they can calculate parity, they can't use it in their calculations." - what do you mean by this? Counting the number of 1's in a bitstring and determining if it's even or odd?

Re: Can AI do maths yet? Thoughts from a mathematician

#163

Post by joe_the_user »

jokoon wrote: post #155

I wish scientists who do psychology and cognition of actual brains could approach those AI things and talk about it, and maybe make suggestions. I really really wish AI would make some breakthrough and be really useful, but I am so skeptical and negative about it.

Unfortunately, the scientists who study actually brains have all sort of interesting models but ultimately very little clue how these actual brains work at the level of problem solving. I mean, there's all sort of "this area is associated with that kind of process" and "here's evidence this area does this algorithm" stuff but it's all at the level you imagine steam engine engineers trying to understand a warp drive.

The "open worm project" was an effort years ago to get computer scientists involved in trying to understand what "software" a very small actual brain could run. I believe progress here has been very slow and that an idea of ignorance that much larger brains involve.

https://en.wikipedia.org/wiki/OpenWorm

Re: Can AI do maths yet? Thoughts from a mathematician

#164

Post by dinosaurdynasty »

pixl97 wrote: post #139
lupire wrote: post #131

Earlier quoted context omitted.

That's not useful by itself, because "anything cam model anything else" doesn't put any upper bound on emulation cost, which for one small task could be larger than the total energy available in the entire Universem

I mean, that is why they mention super-Turning processes like quantum based computing.

Quantum computing actually isn't super-Turing, it "just" computes some things faster. (Strictly speaking it's somewhere between a standard Turing machine and a nondeterministic Turing machine in speed, and the first can emulate the second.)

Re: Can AI do maths yet? Thoughts from a mathematician

#165

Post by dinosaurdynasty »

lazide wrote: post #22
LittleTimothy wrote: post #21

It's fascinating that this has run into the exact same problem as the Quantum research. Ie, in the quantum research to demonstrate any valuable forward progress you must compute something that is impossible to do with a traditional computer. If you can't do it with a traditional computer, it suddenly becomes difficult to verify correctness (ie, you can't just check it was matching the traditional computer's answer. I…

If constrained by existing human knowledge to come up with an answer, won’t it fundamentally be unable to push human knowledge forward?

There are likely lots of connections that could be made that no individual has made because no individual has all of existing human knowledge at their immediate disposal.

Re: Can AI do maths yet? Thoughts from a mathematician

#166

Post by exprofmaddy »

red75prime wrote: post #148
exprofmaddy wrote: post #144

Earlier quoted context omitted.

The universal approximation theorem is set in a precise mathematical context; I encourage you to limit its applicability to that context despite the marketing label "universal" (which it isn't). Consider your concession about empiricism. There's no empirical way to prove (i.e. there's no experiment that can demonstrate beyond doubt) that all brain or other organic processes are deterministic and can be represented co…

Function is the most general way of describing relations. Non-deterministic processes can be represented as functions with a probability distribution codomain. Physics seems to require only continuous functions. Sorry, but there's not much evidence that can support human exceptionalism.

Some differential equations that model physics admit singularities and multiple solutions. Therefore, functions are not the most general way of describing relations. Functions are a subset of relations.

Although "non-deterministic" and "stochastic" are often used interchangeably, they are not equivalent. Probability is applied analysis whose objects are distributions. Analysis is a form of deductive, i.e. mechanical, reasoning. Therefore, it's more accurate (philosophically) to identify mathematical probability with determinism. Probability is a model for our experience. That doesn't mean our experience is truly probabilistic.

Humans aren't exceptional. Math modeling and reasoning are human activities.

Re: Can AI do maths yet? Thoughts from a mathematician

#167

Post by alphan0n »

cruffle_duffle wrote: post #160
alphan0n wrote: post #64

As far as ChatGPT goes, you may as well be asking: Can AI use a calculator? The answer is yes, it can utilize a stateful python environment and solve complex mathematical equations with ease.

It still has to know what to code in that environment. And based on my years of math as a wee little undergrad, the actual arithmetic was the least interesting part. LLM’s are horrible at basic arithmetic, but they can use python for the calculator. But python wont help them write the correct equations or even solve for the right thing (wolfram alpha can do a bit of that though)

You’ll have to show me what you mean.

I’ve yet to encounter an equation that 4o couldn’t answer in 1-2 prompts unless it timed out. Even then it can provide the solution in a Jupyter notebook that can be run locally.

Re: Can AI do maths yet? Thoughts from a mathematician

#168

Post by aithrowawaycomm »

aithrowawaycomm wrote: post #9

I am fairly optimistic about LLMs as a human math -> theorem-prover translator, and as a fan of Idris I am glad that the AI community is investing in Lean. As the author shows, the answer to "Can AI be useful for automated mathematical work?" is clearly "yes." But I am confident the answer to the question in the headline is "no, not for several decades." It's not just the underwhelming benchmark results discussed in…

Just a comment: the example o1 got wrong was actually underspecified: https://anokas.substack.com/p/o3-and-arc-agi-the-unsolved-ta...

Which is actually a problem I have with ARC (and IQ tests more generally): it is computationally cheaper to go from ARC transformation rule -> ARC problem than it is the other way around. But this means it’s pretty easy to generate ARC problems with non-unique solutions.

Re: Can AI do maths yet? Thoughts from a mathematician

#169

Post by HDThoreaun »

lordnacho wrote: post #93
jampekka wrote: post #19

I just spent a few days trying to figure out some linear algebra with the help of ChatGPT. It's very useful for finding conceptual information from literature (which for a not-professional-mathematician at least can be really hard to find and decipher). But in the actual math it constantly makes very silly errors. E.g. indexing a vector beyond its dimension, trying to do matrix decomposition for scalars and insisting…

The only thing I've consistently had issues with while using AI is graphs. If I ask it to put some simple function, it produces a really weird image that has nothing to do with the graph I want. It will be a weird swirl of lines and words, and it never corrects itself no matter what I say to it. Has anyone had any luck with this? It seems like the only thing that it just can't do.

The agentic reasoning models should be able to fix this if they have the ability to run code instead of giving each task to itself. "I need to make a graph" "LLMs have difficulty graphing novel functions" "Call python instead" is a line of reasoning I would expect after seeing what O1 has come up with on other problems.

Giving AI the ability to execute code is the safety peoples nightmare though, wonder if we'll hear anything from them as this is surely coming

Re: Can AI do maths yet? Thoughts from a mathematician

#170

Post by bongodongobob »

jokoon wrote: post #155

I wish scientists who do psychology and cognition of actual brains could approach those AI things and talk about it, and maybe make suggestions. I really really wish AI would make some breakthrough and be really useful, but I am so skeptical and negative about it.

If you can't find useful things for LLMs or AI at this point, you must just lack imagination.
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