After Math
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After Math
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Re: After Math
#2I feel like things have changed dramatically overnight. The field of mathematics seems to be moving at an extraordinary pace, especially following the recent developments around the Navier-Stokes problem.
25 Field Medalist and 5000+ mathematicians from leading institutions around the world endorsed an open letter expressing concerns about the impact of AI on mathematics:
More than 1,900+ mathematicians have also shown concern over the Caltech Mathathon:
https://docs.google.com/document/d/1IL0b2oG2KvvSnxn_DuXsNxuH...
James Maynard, a Fields Medalist, has also publicly expressed concerns about the implications of AI for mathematics:
Re: After Math
#3The first assumption [1. AI really did solve a problem in mathematics] is wrong because to really solve a mathematical problem, providing a mere answer (even if formally certified) is not sufficient.
This subjective attitude turns mathematics into nothing more than number-poetry.
That would reduce mathematics to something very pathetic.
Focus instead on attribution. Yes, OpenAI took the last tiny step in the process of solving this problem (= proving it). But it cannot attribute credit to all the mathematicians whose chat logs from the past few months were fed into its training data. Unlike a human, it can't even remember where it learned things from! For many theorems, I can still recall which exposition was the one that "sank in" for me (often not the first one!) a decade after grad school.
In my mind, this makes current LLMs unfit to deserve any credit at all -- they cannot give credit to others, so they and their owners deserve no credit themselves. OpenAI's LLM took the last tiny step, but not any of the important ones.
Re: After Math
#4The first assumption [1. AI really did solve a problem in mathematics] is wrong because to really solve a mathematical problem, providing a mere answer (even if formally certified) is not sufficient. This subjective attitude turns mathematics into nothing more than number-poetry. That would reduce mathematics to something very pathetic. Focus instead on attribution. Yes, OpenAI took the last tiny step in the process…
I don't think that's particularly true. The whole point of the Millennium Prize problems as the article states was not on absolute difficulty of the problems but on the high chance of a proof producing fruitful results leading to new concepts and theories. Any pursuit of capital T truth will of course aim for better and more clarifying abstractions and is not a subjective turn by any means.
Re: After Math
#5The first assumption [1. AI really did solve a problem in mathematics] is wrong because to really solve a mathematical problem, providing a mere answer (even if formally certified) is not sufficient. This subjective attitude turns mathematics into nothing more than number-poetry. That would reduce mathematics to something very pathetic. Focus instead on attribution. Yes, OpenAI took the last tiny step in the process…
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Re: After Math
#6The first assumption [1. AI really did solve a problem in mathematics] is wrong because to really solve a mathematical problem, providing a mere answer (even if formally certified) is not sufficient. This subjective attitude turns mathematics into nothing more than number-poetry. That would reduce mathematics to something very pathetic. Focus instead on attribution. Yes, OpenAI took the last tiny step in the process…
I don't think that's particularly true. The whole point of the Millennium Prize problems as the article states was not on absolute difficulty of the problems but on the high chance of a proof producing fruitful results leading to new concepts and theories. Any pursuit of capital T truth will of course aim for better and more clarifying abstractions and is not a subjective turn by any means.
What you describe is Tao's rebuttal of #2. I fully agree with him there.
It's his rejection of #1 that makes me sad.
Re: After Math
#7Wouldn't AI make the field of mathematics more ambitious? In software development it feels that way: there are often tasks I can take on that would have been too risky in 2025, because it was unclear if they were worth it. Now you generate a prototype and can make much better judgement calls what is possible and what is worth pursuing.
Re: After Math
#8The first assumption [1. AI really did solve a problem in mathematics] is wrong because to really solve a mathematical problem, providing a mere answer (even if formally certified) is not sufficient. This subjective attitude turns mathematics into nothing more than number-poetry. That would reduce mathematics to something very pathetic. Focus instead on attribution. Yes, OpenAI took the last tiny step in the process…
Focus instead on attribution. Yes, OpenAI took the last tiny step in the process of solving this problem (= proving it). But it cannot attribute credit to all the mathematicians whose chat logs from the past few months were fed into its training data. Unlike a human, it can't even remember where it learned things from!
In my mind, this makes it unfit to deserve any credit at all -- it cannot give credit to others, so it deserves no credit itself. It took the last tiny step, but certainly not any of the important ones.
Yes I fear a lot of doom and gloom around AI is unearned and only really serves to prop up the valuation of AI companies. It's still very much unclear how much work OpenAI actually did versus just copying the nearly complete homework of someone 5 minutes earlier.
Re: After Math
#9The first assumption [1. AI really did solve a problem in mathematics] is wrong because to really solve a mathematical problem, providing a mere answer (even if formally certified) is not sufficient. This subjective attitude turns mathematics into nothing more than number-poetry. That would reduce mathematics to something very pathetic. Focus instead on attribution. Yes, OpenAI took the last tiny step in the process…
Focus instead on attribution. Yes, OpenAI took the last tiny step in the process of solving this problem (= proving it). But it cannot attribute credit to all the mathematicians whose chat logs from the past few months were fed into its training data. Unlike a human, it can't even remember where it learned things from! In my mind, this makes it unfit to deserve any credit at all -- it cannot give credit to others, so…
[deleted]
Re: After Math
#10Earlier quoted context omitted.
I don't think that's particularly true. The whole point of the Millennium Prize problems as the article states was not on absolute difficulty of the problems but on the high chance of a proof producing fruitful results leading to new concepts and theories. Any pursuit of capital T truth will of course aim for better and more clarifying abstractions and is not a subjective turn by any means.
What you describe is Tao's rebuttal of #2. I fully agree with him there. It's his rejection of #1 that makes me sad.
If you mean > What is missing is an intelligible proof that human mathematicians can understand and use to advance the aims of mathematics.
Then that is not necessarily subjective either if an AI can produce an actually intelligible proof. The problem is that as mathematicians with PDE expertise have mentioned on Twitter the actual solution seems to devolve into an unreadable mess focusing on irrelevant details after a more readable first few pages in the proof. If it wasn't a Lean compiled proof and presented as a human artifact, it would be hard to assess if the deviser of the solution had any actual understanding of the solution.