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Notes -
Too vast and beautiful to be left to the mathematicians
Three recent results in the intersection of mathematics and AI provide and insight into universality of man's flaws: Jacobian Conjecture, Non-Sofic Group, Proportion of non-Trivial zeroes of the Riemann Zeta Function on the Critical Line (zeta zeroes result, for short).
As has been widely publicized, a counterexample to JC was tweeted by an employee of Anthropic, who is otherwise also a mathematician. It requires only taking partial derivatives of polynomials, taking a determinant of a 3 by 3 matrix with polynomial entries, and evaluating a 3 variable polynomial at 2 points. AP Highschool level stuff. Thus those seeking to spread FUD about the correctness would face a vastly more numerous group capable of checking. So a different strategy was devised: redefine the conjecture as being primarily about the 2 variable/dimensional case, and the 3 or more case a niche generalization. JC’s Wikipedia article shows that moving the goal posts was successful. Checking the version of the article of this conjecture that was live when Anthropic posted the announcement, shows this.
The concept of a sofic group is a vastly nicher subject1, familiar to a far smaller set of people. Any result about them thus can't be checked by anyone who knows only high school math. Accordingly, the construction (PDF of the 10 proof batch, page 78) by OpenAI of a non-sofic group requires an actual paper, not merely a tweet. OpenAI went above and beyond the standards required of human mathematicians and even published lean machine checkable formalizations.
Mathematics instills a rigour found in no other field of human inquiry. Thus if one wants to discredit a result, one can't reach for the weapons academics in other fields have in their disposal. A mathematician has to be confident that an error in reasoning or in consequences exists, before he dares to question the result itself. So Gary Marcus found himself in the company of only a crank when he denied the validity of a proof, mathematicians directionally aligned with him, took a different route: argument from ignorance: "Won't read it." and nitpicking the introduction: "The introduction section says the conjecture was that all groups are sofic, but actually nobody believed that. This is the most important part of the paper.".
Zeta zeroes result is the most obviously merely incremental of the three. Notability relies on the borrowed light of the Riemann Hypothesis, despite even if it would have proven the limit is one, not merely greater than or equal to 3/2-1/tan(1/sqrt(2))/(sqrt(2)) (which is about .67250070…), Riemann Hypothesis would remain unproven. Here again, only an expert can fully understand the proof written for humans, but also again, the AI developer released a lean formalization. So what discrediting tactic now? Appeal to irrelevent tradition: "The paper lists as an author an entity which is not a homo sapiens. Didn't read further.".
This is where a split in the anti-AI argumentation emerges: the subfaction coming from art considers the person who writes "pretty blond girl in frilly pink dress" in a image generation model to be unworthy of being credited with the resulting image, demands that AI be given praise, or for them blame, for it. But as the linked comment says, and the declaration linked further down in the linked comments to the same reply ("affirming the humanity of authorship"), the human (Jarred Sumner) who writes "keep going" , "believe in yourself", "push it to 2/3" somehow did more than 60 agents scouring literature, forming hypothesis, checking them. (An Anthropic employee on hackernews clarified further the role of humans in this result. Beyond the annotated extract from logs of what the model did, and what the human wrote to it).
On this question of what was the human contribution, and what AI's, and what should be done given a certain allocation between these two, discussion moves into waters that are less rigorous than underlying results. AI labs usually publicize results where they say the human involvement was minimal. This is logical for them as the addressable market of field experts is smaller than that of dilettantes. The prompter for the zeta zeroes result is a programmer by work, his prompts are unlike those of Terence Tao, interested highschooler could have wrote them. On the other hand, results which working mathematicians unaffiliated with AI labs achieve using non-zero AI are more difficult to evaluate. It requires admission against interest for them to say the crucial construction came from an AI, and not their brilliant mind. Some are honest2 but without full transcripts of conversations with AI, we only have the human’s word, and thus only his subjective and non-rigorous evaluation and apportionment.
(1)”Jacobian Conjecture”’s English Wikipedia page was in 2025 viewed 16777 times, “Sofic Group”’s 2659. For comparison: “Always: Sunset on Third Street”’s 16743, “Sonoko Kawai”’s 2421.
(2)This paper's first listed author gave his thoughts on it ("The paper was written under time pressure, due to the current AI environment, and because we already had discussed some of the ideas on an online public forum."), and also on the 9th results ("This one hurts, as I've spent a lot of time thinking about it,[...] From the positive side, I guess this means that my coauthors and I have retroactively solved (half of) an Erdös problem."), which is about Multicolor Ramsey numbers, in the OpenAI's 10 proofs batch (page 229), which included the non-sofic group.
Edit: Incorporated ToaKraka's improvements. To add balance, to portray more accurately, it should be noted that the author of the paper considered crucial for ChatGPT's construction of a non-sofic group, Gabor Kun, and the author of the detailed mathoverflow answer linked above, explaining what was crucial, Andreas Thom, already published a paper building on ChatGPT's work. So some mathematicians still place interests of mathematics first.
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