Quotations from the "REMARKS ON THE DISPROOF OF THE UNIT DISTANCE CONJECTURE" paper that might help non-mathematicians to appreciate its relevance to other scientific fields.
Especially relevant for biomedical problems are the noted ability of AI to find connections among scientific facts that are part of different specialized fields, and the persistence in exploring the implications of these connections.
Noga Alon
" AI was able to do here what lots of excellent human researchers tried and failed to do. Like other mathematicians who had the opportunity to experiment, even if only briefly in my case, with ChatGPT Pro 5.5, my impression has been that AI tools are capable of changing research
in mathematics in a dramatic way. The new spectacular solution of the Erdős unit distance problem convinces me that it is hard to overestimate the full potential impact of this change."
Thomas Bloom
" On examining the construction, it becomes more clear how people had missed this before – it requires the confluence of several different unlikely events: that a good mathematician is
(1) spending significant time in thinking about the unit distance conjecture in the first place;
(2) seriously trying to disprove it, despite the oft-repeated belief of Erdős that it is true;
(3) believes that there is mileage in generalising the original construction to other number fields, and so is willing to expend significant time in exploring such constructions; and
(4) sufficiently familiar with the relevant parts of class field theory to recognise that the appropriately phrased question about infinite towers of number fields with appropriate parameters can be solved using existing theory.
The AI met all of these criteria, and its success here echoes previous achievements: it often produces the most surprising results by persevering down paths that a human may have dismissed as not worth their time to explore, combining superhuman levels of patience with familiarity with a vast array of technical machinery."
" One aspect of this proof should not be overlooked: while the original proof produced by AI was completely valid, it was significantly improved by the human researchers at OpenAI and the many other mathematicians involved in the present paper. The human still plays a vital role in discussing, digesting, and improving this proof, and exploring its consequences.
The frontiers of knowledge are very spiky, and no doubt the coming months and years will see similar successes in many other areas of mathematics, where long-standing open problems are resolved by an AI revealing unexpected connections and pushing the existing technical machinery to its limit. AI is helping us to more fully explore the cathedral of mathematics we have built over the centuries; what other unseen wonders are waiting in the wings?"
WT Gowers
" Perhaps what we are seeing at the moment is not that AI is about to overtake human mathematicians, but rather that there are certain styles of problem where it has a distinct advantage. It has an encyclopaedic knowledge of mathematics, and it does not have to worry nearly as much as we do about time management, so it is good at finding surprising connections, and it can afford to try quite hard to prove statements that seem unlikely to be true – provided, in both cases, that the complexity of the proofs it finds is not too high."
" My current bet is that progress in AI mathematics is not about to reach a plateau, and that we will soon see AI solutions to many problems that we will find hard to explain away as easier than expected with hindsight. "
" In any case, there is no doubt that the solution to the unit-distance problem is a milestone in AI mathematics: if a human had written the paper and submitted it to the Annals of Mathematics and I had been asked for a quick opinion, I would have recommended acceptance without any hesitation.
No previous AI-generated proof has come close to that. Furthermore, even if it is correct that AI cannot yet find a proof that needs a long hint sequence, such proofs are very difficult to find for humans as well, so in the unlikely event that progress in AI mathematics does suddenly stall, we have still probably entered an era where it will become very difficult for humans to compete with AI at solving mathematical problems."
Daniel Litt
"There are a few examples of relatively well-known open problems resolved via a fairly short, clever argument: famously, the finite field Kakeya conjecture, proven by Dvir; the sensitivity conjecture, proven by Huang; and a few others. Arguably, this solution to the unit distance problem has the same flavor. [...]
One possible explanation for such low-hanging fruit: those working on the problem have anchored onto a non-optimal approach or belief about the truth (for example, in this case, that Erdős’s conjecture on the topic was true). Another: the solution requires ideas from areas with which most of those working on the problem are unfamiliar. These explanations, if correct, should cause us some discomfort. They suggest that incentives towards specialization and siloing, though understandable, have cost us some high-quality science."
" Finally, it is illuminating to contrast the most productive current approach to doing mathematics by AI to the way humans mathematicians work. At any given time a human will, driven by their personal curiosity, choose a small number of questions and try to understand them deeply.
By contrast, the best autonomous AI mathematics has been produced by trawling through entire problem lists and solving some portion of the listed problems. This is a vast expansion of the attention aimed at mathematical problems, and perhaps will serve to better focus future human attention and curiosity."
Arul Shankar
" The model’s CoT (Chain-of-Thought) is deeply interesting. It is noteworthy that a significant majority of the thoughts are trying to construct a counterexample to the widely believed upper bound, rather than trying
to prove it. This argues that the model has some combination of good intuition, willingness to try approaches considered long-shot by the community, and a predisposition to attempt constructions.
The CoT (Chain-of-Thought) showed the model trying out a vast array of ideas from a wide range of mathematics for the required construction. The model went through ideas pretty quickly, but when it reached the crucial idea (in the paragraph starting with "Suppose optimistically that..."), it honed in on the proof quite methodically.
In my opinion this paper demonstrates that current AI models go beyond just helpers to human mathematicians – they are capable of having original ingenious ideas, and then carrying them out to fruition.
Jacob Tsimerman
" This may indicate one way that AI systems have an edge: it’s not just that they can try all known methods, but they can play for longer and in more treacherous waters than mathematicians without getting overwhelmed. Of course this is not yet robustly true, but this may be a foreshadowing event."
Melanie Matchett Wood
" We can all be reminded by this development of how frequently interesting and powerful things happen mathematically when one applies ideas from one field to another, and think about how AI can help us find more cross-field applications.
This result does not show us all the times AI has claimed to have a proof of something and been wrong. Without that context (which many of us have just from personal experience), it is also easy to draw incorrect conclusions about the current state of AI and research mathematics. In many cases, it will be easier for AI to convince humans it has a proof than to come up with a correct
mathematical argument, and I believe that we as mathematicians are not sufficiently prepared for this.
One other concern that directly arises in this development is that there is a history of closely related ideas in the literature, some of which are mentioned above, but which are not appropriately referenced in Chat GPT’s paper. If a human came up with this argument and didn’t cite such previous work, we would assume that they were unfamiliar with the previous work and came up
with the ideas independently, since our professional norms require us to cite previous work whose ideas influenced our work. On the other hand, Chat GPT is in some sense “familiar” with all the previous work. In the future we can expect humans to write many papers that include ideas suggested by AI. Mathematicians need to think about what best practices and proper citation is in these kind of situations, and come to a common understanding as a community. Properly contextualized, we can see from this and other developments that AI will play an increasingly important role in research mathematics. As a mathematics community, we urgently need to plan for how we can keep our work rigorous and correct, properly acknowledge the influence of previous ideas, and preserve a high level of human understanding of mathematics as we move forward in our use of AI as part of process of research mathematics."