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Leiden Declaration on Artificial Intelligence and Mathematics

leidendeclaration.ai

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Re: Leiden Declaration on Artificial Intelligence and Mathematics

#61

Post by josh-sematic »

bcjdjsndon wrote: post #54
Spacecosmonaut wrote: post #51

Accelerationists may argue that the eroding of proper attribution and proof verification by humans is a meaningless short term struggle of a dying field. Mathematics seems to be entering an era where human + machine maximizes performance, much like chess in the 1990s. However, imagine a future where even talented mathematicians are nothing but noise in the machine (as is the case in chess now). A future where AI gene…

> However, imagine a future where even talented mathematicians are nothing but noise in the machine (as is the case in chess now). Isn't chess more popular than ever? Ai dominating the game didnt seen to matter

Chess is a game where we play and observe for the sheer joy of mastery. Mathematics has some of that “done for its own sake” aspect but not as cleanly. There’s always the chance that some applied mathematics might help something in science or engineering. Whether we care more about process vs outcome is now more important than ever before.

Re: Leiden Declaration on Artificial Intelligence and Mathematics

#62

Post by wslh »

Spacecosmonaut wrote: post #51

Accelerationists may argue that the eroding of proper attribution and proof verification by humans is a meaningless short term struggle of a dying field. Mathematics seems to be entering an era where human + machine maximizes performance, much like chess in the 1990s. However, imagine a future where even talented mathematicians are nothing but noise in the machine (as is the case in chess now). A future where AI gene…

> imagine a future where even talented mathematicians are nothing but noise

That would be AGI. My conjecture is that LLMs alone are not enough for that future. They are incredible, but AGI needs other breakthroughs.

In that sense, I think math is very different from chess or Go. Chess and Go are complete-information games with fixed rules and a fixed board. Math is open-ended.

Re: Leiden Declaration on Artificial Intelligence and Mathematics

#63

Post by FergusArgyll »

Spacecosmonaut wrote: post #56
bcjdjsndon wrote: post #54

Earlier quoted context omitted.

> However, imagine a future where even talented mathematicians are nothing but noise in the machine (as is the case in chess now). Isn't chess more popular than ever? Ai dominating the game didnt seen to matter

Sure, but if we continue that analogy it does mean that there will be no human contributions to frontier mathematics.

(This doesn't disprove your point because it seems like it should be trivial to make but interestingly)

I don't think there's any computer system which autonomously come up with new directions in openings. As far as I know; a GM looks at stockfish's evaluation of the top x moves and analyzes one that hasn't been played a lot etc.

Re: Leiden Declaration on Artificial Intelligence and Mathematics

#64

Post by gobdovan »

Spacecosmonaut wrote: post #51

Accelerationists may argue that the eroding of proper attribution and proof verification by humans is a meaningless short term struggle of a dying field. Mathematics seems to be entering an era where human + machine maximizes performance, much like chess in the 1990s. However, imagine a future where even talented mathematicians are nothing but noise in the machine (as is the case in chess now). A future where AI gene…

Chess has set rules and is a closed world with a set objective. In maths, you make up the rules. For every clearly defined problem that everybody cares about and is not yet proven, there's someone who first who recognised the importance of that given problem and managed to define it clearly enough for it to be recognised as such.

There's also the separate, less glamorous issue that people don't want to talk about, which is proof reliability. [0] If you have systems to help you formalise the problems and leave an algorithm or AI or whatever solve it in a verifiable way, that's a win for both the mathematicians and the rest of the world.

The deeper question is whether AI can replace the human role in deciding what mathematics should be done and what concepts matter. If that's automated, then yeah, we're screwed.

[0] https://lamport.azurewebsites.net/tla/proof-statistics.html

Re: Leiden Declaration on Artificial Intelligence and Mathematics

#65

Post by adrian_b »

applfanboysbgon wrote: post #15
Supermancho wrote: post #13

Earlier quoted context omitted.

I'm not sure it's constructive to explain our differences, point by point. eg > 2. is nothing about "following established traditions" > undermine the traditional system of attribution Literally does. Suffice to say, I find your interpretations to be surprising and disconnected and it has not changed my views.

The actual thesis of point 2 is about plagiarism, and the thesis would remain the same if the sentence you quoted were removed completely. Your portrayal of it moves the out-of-context snippet to the forefont of the argument and makes it sound like an issue of "tradition for tradition's sake" or something similarly indefensible, but you refuse to engage with the real argument being made, hence why I suspect you are a…

I do not agree that the thesis of point 2 is about plagiarism.

Plagiarism is a minor concern in research papers. Plagiarism means that the paper has 0% new content and everything in it is extracted from older works and combined.

However, even very valuable research papers may have only 1 new idea or only a few new ideas, which can be applied only together with a much greater amount of old knowledge, which might be mentioned explicitly in the paper, with proper attribution, enabling the reader to search the original sources, or the necessary old knowledge may be implicit, the writers assuming it to be known by the readers.

It is important that the writers should not imply that some old idea is new, by omitting to attribute it, but much more important is that when they mention some ideas as being old, they must also provide sufficient information about the sources of those old ideas, so that the readers will be able to access them.

Re: Leiden Declaration on Artificial Intelligence and Mathematics

#67

Post by magicalist »

Mattasher wrote: post #44

Interesting document. Especially interesting compared to the Bourbaki movement from a century ago, which was much more focused on universality and correctness, and much less focused on process and attribution (in fact, demanded anonymity).

> Especially interesting compared to the Bourbaki movement from a century ago, which was much more focused on universality and correctness, and much less focused on process and attribution (in fact, demanded anonymity).

The Bourbaki group was (is) about documenting existing mathematics rigorously, not doing original research. This document is specifically about the future of mathematical research.

Re: Leiden Declaration on Artificial Intelligence and Mathematics

#68

Post by joe_the_user »

pfdietz wrote: post #11
curt15 wrote: post #3

It's worth remembering Thurston's essay on mathoverflow ( https://mathoverflow.net/questions/43690/whats-a-mathematici... ): "The product of mathematics is clarity and understanding. Not theorems, by themselves. Is there, for example any real reason that even such famous results as Fermat's Last Theorem, or the Poincaré conjecture, really matter? Their real importance is not in their specific statements, but their ro…

That's the product of math from the point of view of mathematicians. But is it the point of view of those funding math? I suggest if one looks at the history of funding for mathematics and science, the product of these efforts is not understanding, but rather power. Funding went way up after WW2 when the war demonstrated that power flows from them. Math not only contributed to the scientific weapons of the way, but w…

The debate on the funds of scholarship and education has been between democratic vision of "an informed citizenry" versus the technocratic vision of education for jobs and technical progress. Both of those might be "power" (Foucault said "everything is power" and if so, it tells us nothing). Thus I don't think your particular framing is useful.

For many who pursue mathematics, it is a refuge from ordinary life (or normal "power"). Within mathematics, there has historically been a tension between the "Babylonian" model of pure calculation and the "Greek" model of advancement through understanding. If current AI models subsume mathematics entirely, the Babylonian model will have "won" and the possibility of an informed citizenry will be in doubt.

Re: Leiden Declaration on Artificial Intelligence and Mathematics

#69

Post by applfanboysbgon »

Lerc wrote: post #49
applfanboysbgon wrote: post #10

Earlier quoted context omitted.

This appears to be a very bad faith post that intentionally misrepresents what is being said. 1. pertains to the quantity of output adding stress to review processes; LLMs can feasibly produce a million plausible but incorrect 'proofs' in the time that a human can produce one. We already see this effect in software development, with bug bounty programs shutting down and open-source software rejecting AI contributions…

> 2. is nothing about "following established traditions" but rather the general concept of crediting people for their prior work, unless you think that "not plagiarising" is a trifling established tradition. But that is the nature of establishment, when something is a sufficiently firmly established tradition, people see it as a truism. Crediting people is a social convention. Plagiarism is a social construct. It can…

I did say in another reply already that if this was the person I replied to's position, they should stand on it and argue for it. I don't believe it's some kind of sacred cow that is beyond discussing. My problem with what they were saying is that they weren't solidly taking a stance and trying to make such an argument, but rather taking cheap shots by quoting things in misleading ways. Tackle the argument head on if this is what you believe, don't try to mislead people who might not've (fully) read the article by misstating the article's position in an attempt to make it look weaker than it actually is. If your argument depends on misrepresenting your opponent's position rather than standing on your own convictions, well, it doesn't speak much for your convictions.

Re: Leiden Declaration on Artificial Intelligence and Mathematics

#70

Post by derpism999 »

scarmig wrote: post #48
briandw wrote: post #28

Most of the arguments here feel like gate keeping and resistance to change. I didn't see any arguments that were directly about advancing the state of knowledge of math. “Current automated techniques can produce plausible but unreliable (or even incorrect) arguments which are difficult to distinguish from correct mathematical proofs.” That seems like a problem for mathematics with or without AI. Isn’t this a problem…

> Isn’t this a problem with human proofs as well? Human proofs are themselves a kind of a proof of work. They certainly write flawed proofs, but you can expect a human author of a paper to have put in more effort--substantially more--than the human reader needs to verify it. Arguably, this asymmetry disappears for generated proofs. Automated theorem provers help a bit here, but they don't eliminate the human verifica…

That doesn’t mean it isn’t a problem for human proofs as well. Mathematicians use affiliations and c.v.’s as their primary means of gatekeeping for review, not the effort that went into a proof or its superficial appearances. Especially in hard and big problems which have had any number of flawed proofs.

Even Grigori Perelman took years to have his proof of Poincaré conjecture accepted, and he had a Ph.D. and a Berkeley fellowship.

Expecting a human author to put substantially more effort into a proof than needed to verify it is oversimplifying. It is more a matter of credentialing and collegiality in mathematics whereby someone’s reputation and work-product demonstrates that a purported proof put forward for review is likely to be true or at least a valuable or interesting contribution even if imperfect.

AI makes this a much bigger challenge.

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