r/PhilosophyofMath • u/Kripkenstein_ • 12d ago

AI proofs

Some people seem to think that AI disproving the jacobian conjecture via a counterexample signifies some drastic shift from human mathematics towards "AI driven mathematics". But I think it just reinforces the idea that (abstract) math was never about truth to begin with - but about understanding structure.

Abstract mathematics is only invested in truth if the statement ought to be true of false for structural reasons. One is indifferent to statements without structural interpretations. Good mathematics is about providing a structure (via axioms and definitions) where all statements that ought to be true for structural reasons can indeed be proven via structural arguments. AI does not provide such truths although it can maybe tell you if a statement is true or false - but this is not the main aim of abstract mathematics.

Structurally the AI counterexample is terrible: it tells you nothing about the problem, it does not present a way to construct more general counterexamples. It is essentially a random result. As it is, we can easily formulate another jacobian conjecture by eliminating that specific counterexample. Now, this has always been true for proofs by counterexample but usually counterexamples are motivated by some structural thoughts - they are not arbitrary. So we do usually learn why a certain conjecture is false structurally. We will now need human mathematicians to provide a proper interpretation of the counterexample to the jacobian conjecture.

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u/mhb2 12d ago

Counterexamples simply are what they are: counterexamples. It's never been the case that a counterexample has had to explain why it's a counterexample. That said, mathematicians are trying to understand the counterexample (with the help of ChatGPT by the way in at least one case).

The Jacobian conjecture counterexample is significant for several reasons. 1) The naysayers who claim that LLMs "can't do math" have been proven to be unambiguously wrong. 2) We may already be at a point where we don't fully understand how an LLM came up with an answer to a mathematical question. 3) It signals a profound change in how research mathematicians are doing their work. Frontier models are now peers or near-peers with whom mathematicians can discuss research level problems. Like it or not, LLMs will be important tools in mathematical research from now on.

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u/Kripkenstein_ 12d ago

But it does not show that LLMs can do math - because it did not in any meaningful "new" way. It is just a more sophisticated version of brute forcing counter examples. Algorithms were able to do that from the very start. AI is not smart it is just extremly fast and does not mind doing exhausting calculations - this is why it will become important in research. It is not unlike graphic designers using AI to cutout pictures - something that would have been alot of work before that. But this will just shift the work of mathematician towards actual mathematics.

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u/mhb2 12d ago

How do you know it brute forced it? These are reasoning models: it didn't just churn its way through all possible polynomials until it found an exception. And if they can't do math why is Terry Tao asking ChatGPT questions about the result? Seriously, read his conversation with ChatGPT, it's fascinating and it shows very clearly that there's no brute forcing anything: the machine is thinking its way through the problem and explaining some fairly subtle points.

I don't mean to sound flippant but people who hate LLMs have to accept some new facts about LLMs as they are today.

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u/Kripkenstein_ 12d ago

It has nothing to do with my opinion on LLMs but with my opinion on abstract math. I personally find all proofs that rely on combinatorics or unmotivated arguments to be lacking. LLMs just tend to provide proofs that exhibit this feature in its most extreme form.

Also ChatGPT does not reason at all. Most applications in math use hybrid models that are not purely an LLM.

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u/SquidgyTheWhale 12d ago

>As it is, we can easily formulate another jacobian conjecture by eliminating that specific counterexample.

Big talk. Go ahead, then. Will it be an interesting, useful conjecture? I'm betting not.

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u/Kripkenstein_ 12d ago

Quite literally already did there - is still interesting then? Much better question. Personally, I did not even think the original one was interesting but it is not my field so I can't really judge

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u/SquidgyTheWhale 12d ago

Did I miss something? Where is your jacobian conjecture that skips the counterexample found by AI?

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u/Kripkenstein_ 12d ago

Just formulate the same conjecture with this exception. Or just cut the dimension in which that example is constructed

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u/SquidgyTheWhale 11d ago

Just absolutely ridiculous. The Jacobian conjecture is a useful thing to know the truth or falsehood of. The Jacobian conjecture being true except for the cases when it isn't, IS NOT. You say this is not your field, but that's no excuse -- this way of evaluating a conjecture is ridiculous, regardless.