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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

#81

Post by ludwik »

bambax wrote: post #50

> As an academic mathematician who spent their entire life collaborating openly on research problems and sharing my ideas with other people, it frustrates me [that] I am not even to give you a coherent description of some basic facts about this dataset, for example, its size. However there is a good reason for the secrecy. Language models train on large databases of knowledge, so you moment you make a database of mat…

> But if all models were truly open, then we could simply verify what they had been trained on

How do you verify what a particular open model was trained on if you haven’t trained it yourself? Typically, for open models, you only get the architecture and the trained weights. How can you reliably verify what the model was trained on from this?

Even if they provide the training set (which is not typically the case), you still have to take their word for it—that’s not really "verification."

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

#82

Post by jampekka »

fn-mote wrote: post #66
dartos wrote: post #58

Earlier quoted context omitted.

What’s worse about it? It never tells you the wrong thing, at the very least.

Its understanding of problems was very bad last time I used it. Meaning it was difficult to communicate what you wanted it to do. Usually I try to write in the Mathematica language, but even that is not foolproof. Hopefully they have incorporated more modern LLM since then, but it hasn’t been that long.

Wolfram Alpha's "smartness" is often Clippy level enraging. E.g. it makes assumptions of symbols based on their names (e.g. a is assumed to be a constant, derivatives are taken w.r.t. x). Even with Mathematica syntax it tends to make such assumptions and refuses to lift them even when explicitly directed. Quite often one has to change the variable symbols used to try to make Alpha to do what's meant.

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

#84

Post by bondarchuk »

newpavlov wrote: post #38
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…

>in the quantum research to demonstrate any valuable forward progress you must compute something that is impossible to do with a traditional computer This is factually wrong. The most interesting problems motivating the quantum computing research are hard to solve, but easy to verify on classical computers. The factorization problem is the most classical example. The problem is that existing quantum computers are not…

No, it is factually right, at least if Scott Aaronson is to be believed:

> Having said that, the biggest caveat to the “10^25 years” result is one to which I fear Google drew insufficient attention. Namely, for the exact same reason why (as far as anyone knows) this quantum computation would take ~10^25 years for a classical computer to simulate, it would also take ~10^25 years for a classical computer to directly verify the quantum computer’s results!! (For example, by computing the “Linear Cross-Entropy” score of the outputs.) For this reason, all validation of Google’s new supremacy experiment is indirect, based on extrapolations from smaller circuits, ones for which a classical computer can feasibly check the results. To be clear, I personally see no reason to doubt those extrapolations. But for anyone who wonders why I’ve been obsessing for years about the need to design efficiently verifiable near-term quantum supremacy experiments: well, this is why! We’re now deeply into the unverifiable regime that I warned about.

https://scottaaronson.blog/?p=8525

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

#85

Post by red75prime »

voidhorse wrote: post #63

Eventually we may produce a collection of problems exhaustive enough that these tools can solve almost any problem that isn't novel in practice, but I doubt that they will ever become general problem solvers capable of what we consider to be reasoning in humans. Historically, the claim that neural nets were actual models of the human brain and human thinking was always epistemically dubious. It still is. Even as the…

> there is no reason to believe a connection between the mechanical model and what happens in organisms has been established

The universal approximation theorem. And that's basically it. The rest is empirical.

No matter which physical processes happen inside the human brain, a sufficiently large neural network can approximate them. Barring unknowns like super-Turing computational processes in the brain.

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

#86

Post by ivansavz »

Yesterday, I saw a thought provoking talk about the future of of "math jobs" assuming automated theory proving becomes more prevalent in the future.

[ (Re)imagining mathematics in a world of reasoning machines by Akshay Venkatesh]

https://www.youtube.com/watch?v=vYCT7cw0ycw [54min]

Abstract: In the coming decades, developments in automated reasoning will likely transform the way that research mathematics is conceptualized and carried out. I will discuss some ways we might think about this. The talk will not be about current or potential abilities of computers to do mathematics—rather I will look at topics such as the history of automation and mathematics, and related philosophical questions.

See discussion at https://news.ycombinator.com/item?id=42465907

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

#87

Post by swalsh »

Every profession seems to have a pessimistic view of AI as soon as it starts to make progress in their domain. Denial, Anger, Bargaining, Depression, and Acceptance. Artists seem to be in the depression state, many programmers are still in the denial phase. Pretty solid denial here from a mathematician. o3 was a proof of concept, like every other domain AI enters, it's going to keep getting better.

Society is CLEARLY not ready for what AI's impact is going to be. We've been through change before, but never at this scale and speed. I think Musk/Vivek's DOGE thing is important, our governent has gotten quite large and bureaucratic. But the clock has started on AI, and this is a social structural issue we've gotta figure out. Putting it off means we probably become subjects to a default set of rulers if not the shoggoth itself.

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

#88

Post by asadotzler »

ludwik wrote: post #81
bambax wrote: post #50

> As an academic mathematician who spent their entire life collaborating openly on research problems and sharing my ideas with other people, it frustrates me [that] I am not even to give you a coherent description of some basic facts about this dataset, for example, its size. However there is a good reason for the secrecy. Language models train on large databases of knowledge, so you moment you make a database of mat…

> But if all models were truly open, then we could simply verify what they had been trained on How do you verify what a particular open model was trained on if you haven’t trained it yourself? Typically, for open models, you only get the architecture and the trained weights. How can you reliably verify what the model was trained on from this? Even if they provide the training set (which is not typically the case), yo…

The OP said "truly open" not "open model" or any of the other BS out there. If you are truly open you share the training corpora as well or at least a comprehensive description of what it is and where to get it.

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

#89

Post by tananan »

voidhorse wrote: post #63

Eventually we may produce a collection of problems exhaustive enough that these tools can solve almost any problem that isn't novel in practice, but I doubt that they will ever become general problem solvers capable of what we consider to be reasoning in humans. Historically, the claim that neural nets were actual models of the human brain and human thinking was always epistemically dubious. It still is. Even as the…

> Unless you believe in determinism and an overseeing god

Or perhaps, determinism and mechanistic materialism - which in STEM-adjacent circles has a relatively prevalent adherence.

Worldviews which strip a human being of agency in the sense you invoke crop up quite a lot today in such spaces. If you start of adopting a view like this, you have a deflationary sword which can cut down most any notion that's not mechanistic in terms of mechanistic parts. "Meaning? Well that's just an emergent phenomenon of the influence of such and such causal factors in the unrolling of a deterministic physical system."

Similar for reasoning, etc.

Now obviously large swathes of people don't really subscribe to this - but it is prevalent and ties in well with utopian progress stories. If something is amenable to mechanistic dissection, possibly it's amenable to mechanistic control. And that's what our education is really good at teaching us. So such stories end up having intoxicating "hype" effects and drive fundraising, and so we get where we are.

For one, I wish people were just excited about making computers do things they couldn't do before, without needing to dress it up as something more than it is. "This model can prove a set of theorems in this format with such and such limits and efficiency"

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

#90

Post by WhyOhWhyQ »

Bootvis wrote: post #6
casenmgreen wrote: post #4

I may be wrong, but I think it a silly question. AI is basically auto-complete. It can do math to the extent you can find a solution via auto-complete based on an existing corpus of text.

You're underestimating the emergent behaviour of these LLM's. See for example what Terrence Tao thinks about o1: https://mathstodon.xyz/@tao/113132502735585408

I'm always just so pleased that the most famous mathematician alive today is also an extremely kind human being. That has often not been the case.
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