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…
It is, but it is somewhat worse for machine-generated proofs, especially when the proof is very long and was done by brute force (eg the 4 colour map theorem[1] is the famous example), or depends on a lot of niche results in disparate areas (which LLMs are wont to sometimes do). Even when the proof is produced by the llm in a formal system like Lean4 it may not be “honest”[2] and it can be hard to tell if the proof i…
Leiden Declaration on Artificial Intelligence and Mathematics
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Re: Leiden Declaration on Artificial Intelligence and Mathematics
#42Most 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…
You aren’t really engaging with the substance or heart of the post, and your reading feels a bit knee-jerky and bad-faith to me.
Re: Leiden Declaration on Artificial Intelligence and Mathematics
#43Most 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…
Can't all proofs be eventually broken down into their fundamental pieces and then it's clear as day if it's right or wrong? Seems like a proof would be the best place to determine if an AI is right or not because the output is either right or wrong, there's no subjectivity and the, now common, excuse "well a human would have done the same" won't hold up.
Generally, yes, but once broken down you end up with a large number of items that individually each is obviously true, so you know the combined statement Is true, but you don't find out if it is saying what you think it is saying.
In combining the parts you have the correct answer to a question, but is it that question you want to know?
Consider a proof that in the future all people will be happy.
You can methodically show this to be true but at the same time inadvertently include a proof that the number of people in the future will be zero.
It doesn't make the claim wrong, it stays undoubtedly true. It's just not what you assume it means.
Re: Leiden Declaration on Artificial Intelligence and Mathematics
#44Interesting 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).
Re: Leiden Declaration on Artificial Intelligence and Mathematics
#45Most 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…
There is some of that but I wouldn't call it gatekeeping. Universities lately promote citations and publications so there's a sense that results are all that matters. Results matter, yes, but there's a human side too where we're kind of asking about human creativity and ability. To me an appropriate analogy is in climbing Mt. Everest. Proving something, or even writing a thesis, is like climbing Mt. Everest. A lot of the value is actually in the effort you put into it. You could take a helicopter ride up to the top and then climb a few steps and claim "You climbed to the peak of Everest". That's like using AI. But if you asked them about what it was like, how they prepared, etc. their answer would not be helpful. So I think there is a lot of value in the journey itself and outsourcing all this to AI would destroy the human part of it.
Re: Leiden Declaration on Artificial Intelligence and Mathematics
#46So they are recommendations. At global level.
Re: Leiden Declaration on Artificial Intelligence and Mathematics
#471. 'incentives' hmm
"Now, here, you see, it takes all the running you can do, to keep in the same place."
Re: Leiden Declaration on Artificial Intelligence and Mathematics
#48Most 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 verification cost.
Re: Leiden Declaration on Artificial Intelligence and Mathematics
#49The potential threats section reads like panic, rather than a critique of AI. I can see where #2 has some legs, if I thought tradition was sacrosanct. 1. AI proofs might be incorrect and difficult to demonstrate why. This implies they are not like human proofs in these qualities. 2. AI proofs are difficult to attribute correctly, because they don't follow established traditions. Nothing to do with the math, but ok. 3…
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 be useful, in many areas of science, to reference to support your arguments. This is less important in proofs, because a proof is a proof, but references aid in understanding.
These are all reasons to reference and attribute that benefit the writer, and could be done voluntarily. The notion of a duty to reference or attribute has no impact on the validity of the claims being made. It is a collective decision to proportion prestige.
Turning the duty to do so into an unquestioned truism means it has to be done regardless of whether it accurately represents any property of merit.
There are many instances where prestige delivered grossly mismatches what an impartial observer would consider a fair balance of effort and ability.
We should at least recognise that this is so because we have chosen to let it be this way.
Re: Leiden Declaration on Artificial Intelligence and Mathematics
#50In a word, they're cooked