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The cultural divide between mathematics and AI

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Re: The cultural divide between mathematics and AI

#81

Post by zipy124 »

nybsjytm wrote: post #60
EigenLord wrote: post #50

Is it really a culture divide or is it an economic incentives divide? Many AI researchers are mathematicians. Any theoretical AI research paper will typically be filled with eye-wateringly dense math. AI dissolves into math the closer you inspect it. It's math all the way down. What differs are the incentives. Math rewards openness because there's no real concept of a "competitive edge", you're incentivized to freely…

> Many AI researchers are mathematicians. Any theoretical AI research paper will typically be filled with eye-wateringly dense math. AI dissolves into math the closer you inspect it. It's math all the way down. There is a major caveat here. Most 'serious math' in AI papers is wrong and/or irrelevant! It's even the case for famous papers. Each lemma in Kingma and Ba's ADAM optimization paper is wrong, the geometry in…

I'd be interested to read about the gibberish in UMAP, I know the paper "An improvement of the convergence proof of the ADAM-Optimizer" for the lemma problem in the original ADAM but hadn't heard of the second one. Do you have any further info on it?

Re: The cultural divide between mathematics and AI

#82

Post by tylerneylon »

I agree with the overt message of the post — AI-first folks tend to think about getting things working, whereas math-first people enjoy deeply understood theory. But I also think there's something missing.

In math, there's an urban legend that the first Greek who proved sqrt(2) is irrational (sometimes credited to Hippasus of Metapontum) was thrown overboard to drown at sea for his discovery. This is almost certainly false, but it does capture the spirit of a mission in pure math. The unspoken dream is this:

~ "Every beautiful question will one day have a beautiful answer."

At the same time, ever since the pure and abstract nature of Euclid's Elements, mathematics has gradually become a more diverse culture. We've accepted more and more kinds of "numbers:" negative, irrational, transcendental, complex, surreal, hyperreal, and beyond those into group theory and category theory. Math was once focused on measurement of shapes or distances, and went beyond that into things like graph theory and probabilities and algorithms.

In each of these evolutions, people are implicitly asking the question:

"What is math?"

Imagine the work of introducing the sqrt() symbol into ancient mathematics. It's strange because you're defining a symbol as answering a previously hard question (what x has x^2=something?). The same might be said of integration as the opposite of a derivative, or of sine defined in terms of geometric questions. Over and over again, new methods become part of the canon by proving to be both useful, and in having properties beyond their definition.

AI may one day fall into this broader scope of math (or may already be there, depending on your view). If an LLM can give you a verified but unreadable proof of a conjecture, it's still true. If it can give you a crazy counterexample, it's still false. I'm not saying math should change, but that there's already a nature of change and diversity within what math is, and that AI seems likely to feel like a branch of this in the future; or a close cousin the way computer science already is.

Re: The cultural divide between mathematics and AI

#83

Post by tylerneylon »

tylerneylon wrote: post #82

I agree with the overt message of the post — AI-first folks tend to think about getting things working, whereas math-first people enjoy deeply understood theory. But I also think there's something missing. In math, there's an urban legend that the first Greek who proved sqrt(2) is irrational (sometimes credited to Hippasus of Metapontum) was thrown overboard to drown at sea for his discovery. This is almost certainly…

PS After I wrote my comment, I realized: of course, AI could one day get better at the things that make it not-perfect in pure math today:

* AI could get better at thinking intuitively about math concepts. * AI could get better at looking for solutions people can understand. * AI could get better at teaching people about ideas that at first seem abstruse. * AI could get better at understanding its own thought, so that progress is not only a result, but also a method for future progress.

Re: The cultural divide between mathematics and AI

#84

Post by troymc »

jasonhong wrote: post #13
nicf wrote: post #8

I'm a former research mathematician who worked for a little while in AI research, and this article matched up very well with my own experience with this particular cultural divide. Since I've spent a lot more time in the math world than the AI world, it's very natural for me to see this divide from the mathematicians' perspective, and I definitely agree that a lot of the people I've talked to on the other side of thi…

Interestingly, the main article mentions Bill Thurston's paper "On Proof and Progress in Mathematics" ( https://www.math.toronto.edu/mccann/199/thurston.pdf ), but doesn't mention a quote from that paper that captures the essence of what you wrote: > "The rapid advance of computers has helped dramatize this point, because computers and people are very different. For instance, when Appel and Haken completed a proof of…

Another example akin to the proof of the 4-color map theorem was the proof of the Kepler conjecture [1], i.e. "Grocers stack their oranges in the densest-possible way."

We "know" it's true, but only because a machine ground mechanically through lots of tedious cases. I'm sure most mathematicians would appreciate a simpler and more elegant proof.

[1] https://en.wikipedia.org/wiki/Kepler_conjecture

Re: The cultural divide between mathematics and AI

#85

Post by lmpdev »

I did a fair bit of applied mathematics at uni

What I think Mathematicians should remind themselves is a lot of prestigious mathematicians, the likes of Cantor or Erdos, often only employed a handful of “tricks”/heuristics for their proofs over their career. They repeatedly and successfully applied these strategies into unsolved problems

I argue would not take a tremendous jump in performance for an AI to begin their own journey similar in kind to the greats, the only thing standing in their way (as with all contemporary mathematicians) is the extreme specialisation required to reach the boundary of unsolved problems

AI need not be Euler to be an important tool and figure within mathematics

Re: The cultural divide between mathematics and AI

#86

Post by senderista »

mcguire wrote: post #61
nicf wrote: post #8

I'm a former research mathematician who worked for a little while in AI research, and this article matched up very well with my own experience with this particular cultural divide. Since I've spent a lot more time in the math world than the AI world, it's very natural for me to see this divide from the mathematicians' perspective, and I definitely agree that a lot of the people I've talked to on the other side of thi…

Many years ago I heard a mathematician speaking about some open problem and he said, "Sure, it's possible that there is a simple solution to the problem using basic techniques that everyone has just missed so far. And if you find that solution, mathematics will pat you on the head and tell you to run off and play. "Mathematics advances by solving problems using new techniques because those techniques open up new area…

Really? I've always had the impression that "elementary" proofs of hard problems are highly valued.

Re: The cultural divide between mathematics and AI

#87

Post by joe_the_user »

lmpdev wrote: post #85

I did a fair bit of applied mathematics at uni What I think Mathematicians should remind themselves is a lot of prestigious mathematicians, the likes of Cantor or Erdos, often only employed a handful of “tricks”/heuristics for their proofs over their career. They repeatedly and successfully applied these strategies into unsolved problems I argue would not take a tremendous jump in performance for an AI to begin their…

What I think Mathematicians should remind themselves is a lot of prestigious mathematicians, the likes of Cantor or Erdos, often only employed a handful of “tricks”/heuristics for their proofs over their career.

I know this claim is often made but it seems obvious that in this discussion, trick means something far wider and more subtle than any set computer program. In a lot of ways, "he just uses a few tricks" is akin to the way a mathematician will say "and the rest of the proof is elementary" (when it's still quite long and hard for anyone not versed in a given specialty). I mean, before category theory was formalized, the proofs that now are possible with it might classified as "all done with this trick" but grasping said trick was far from elementary matter.

I argue would not take a tremendous jump in performance for an AI to begin their own journey similar in kind to the greats, the only thing standing in their way (as with all contemporary mathematicians) is the extreme specialisation required to reach the boundary of unsolved problems.

Not that LLMs can't do some impressive things but your narrative seems to anthropomorphize them in a less than useful way.

Re: The cultural divide between mathematics and AI

#88

Post by rssoconnor »

nicf wrote: post #17
jasonhong wrote: post #13

Earlier quoted context omitted.

Interestingly, the main article mentions Bill Thurston's paper "On Proof and Progress in Mathematics" ( https://www.math.toronto.edu/mccann/199/thurston.pdf ), but doesn't mention a quote from that paper that captures the essence of what you wrote: > "The rapid advance of computers has helped dramatize this point, because computers and people are very different. For instance, when Appel and Haken completed a proof of…

The Four Color Theorem is a great example! I think this story is often misrepresented as one where mathematicians didn't believe the computer-aided proof. Thurston gets the story right: I think basically everyone in the field took it as resolving the truth of the Four Color Theorem --- although I don't think this was really in serious doubt --- but in an incredibly unsatisfying way. They wanted to know what underlyin…

Is the proof of the Four Colour Theorem really that unsatisfying?

The Four Colour Theorem is true because there exists a finite set of unavoidable yet reducible configurations. QED.

To verify this computational fact one uses a (very) glorified pocket calculator.

Re: The cultural divide between mathematics and AI

#89

Post by lairv »

> A revealing anecdote shared at one panel highlighted the cultural divide: when AI systems reproduced known mathematical results, mathematicians were excited, while AI researchers were disappointed

This seems very caricatural, one thing I've often heard in the AI community is that it'd be interesting to train models with an old data cutoff date (say 1900) and see whether the model is able to reinvent modern science

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