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On one particular Friday afternoon, he stated a conjecture that he hoped was true, and invited us to try to help him prove or disprove it. It was the sort of thing that he really wanted to be true; he liked things smooth and beautiful. I, on the other hand, hoped it was false as I like the weird and exceptional in mathematics. It was also the case that I had absolutely no command of the sort of machinery that one would use to prove such a thing, but I could certainly look for a counterexample.
I learned on Monday that he had spent the entire weekend trying and failing to prove it. I, on the other hand, had put all my energy into finding a counterexample and had one within an hour.
My single (quite small) contribution to mathematical research was a counterexample because it was all I could do. The story does illustrate that it can be helpful to have people with different tools, hopes, and motivations working on a problem, though. I was not, and will never be, even a shadow of that great mathematiciam I studied under, but on that occasion, I had reason to look in a different direction than he did.
Hm, as a mathematician, my experience feels opposite. A proof would be an adaptation of a proof I know, some tweaking it here and there. A counterexample would require some deep understanding of the structure of the objects involved, which frequently is beyond my comprehension.
But probably this is because I think of quite abstract objects which are harder to grasp. For numbers or polynomials, this would be the other way round.
He spent an entire weekend before having the wisdom to pause, and let someone else contribute their time to finding a counter.
Interestingly, Yitang Zhang of the twin-prime-conjecture fame spent 7 years working on the Jacobian conjecture under the advisor Tzuong-Tsieng Moh at Purdue. A key step in his thesis used a corollary of Moh's. It turned out that the corollary was incorrect. As a result, Moh refused to write any recommendation letter for Zhang, and Zhang couldn't find any teaching or research job and ended up spending years working at a Subway[1].
Imagine Zhag had ChatGPT in 1986 when he started working on the Jacobian Conjecture.
[1] Of course now this has become an inspiring story. That said, the story definitely invokes complex emotions. The best way to describe it is probably this Chinese poem, which I have no idea how to translate: 庾信平生最萧瑟，暮年诗赋动江关
Sometimes, of course, the proof is just wrong. That is the dangerous case, which will cause either major corrections or failure.
Naomi Wolf 'death recorded' on BBC[1], skip to 5:51. After this the book was pulped and she had some sort of psychotic break during COVID and allied with ultra-right and COVID denialist loonies.
[1] https://www.bbc.com/news/av/world-us-canada-48639663
> After this the book was pulped and she had some sort of psychotic break during COVID and allied with ultra-right and COVID denialist loonies.
That's quite an extreme shift considering she had previously been a leading figure in third-wave feminism and an OWS activist.
That makes sense. She wouldn't want to be burned by people doing fact checking again. Write books for the crowd that will never do the legwork to discover that you're full of shit.
I'm tired of tales like this in academics though. That's not a criticism of you for telling the tale, I'm just so tired of this kind of thing in academics in general. So, so, so much politics and public reputation management. Zhang should have never had to suffer like that.
As my own research has drifted more into math, I've been surprised at how many assertions in the literature turn out to be false. Not just false, but propagated into the applied literature extensively, and even when you point out the problems a lot of defensiveness and denial about it along the lines of Zhang's story.
I agree about wondering what would have happened if LLMs had been around in 1986. My guess is the outcome would have been the same for the same reasons?
My experience with LLMs in proofs is they can be very helpful, but also very wrong. It's like having another person with another set of hunches about what path to go down.
Very true. Unfortunately, when there are people, there will be politics. I remember when reading Yau's autobiography, I kept marvel how much calculation, or "politics" if you will, that Yau mentioned or implied in the book.
> My guess is the outcome would have been the same for the same reasons?
At least Zhang didn't have to spend 7 years working on the Jacobian conjecture. He said in an interview that he always wanted to work on number theory. Moh asked him to work on Jacobian, and he obliged.
Yu Xin’s was a life of utter desolation; in old age, his poems and rhapsodies stirred the riverlands.
No life ran more bleak and desolate than Yu Xin's —
yet in his twilight years, his verses stirred the rivers and the passes.
and for now as long humans are going to judge of what counts as an elegant or illuminating proof, there's going to be work for human mathematicians
[Dis]proof by counterexample is the most straightforward way to show a statement to be false. What better way is there to disprove a general statement like 'all x are y' than finding an 'x' that isn't 'y'?
Unless you want to spend time "proving" that 2 * 2 = 1.
But mathematics is not a collection of facts. Mathematics is the study of abstraction. And what do you learn from a single data point? What can you abstract from that?
That's why just being a counterexample isn't really interesting. There has to be more than "counterexample" for there to be something to abstract. Was it generated from an analysis of the problem? Can the counterexample be generalized to explore the problem further? Is the counterexample a surprise in a way that suggests something is missing from current understanding?
Being a counterexample doesn't mean that something isn't interesting to a mathematician. But it's also not the interesting part.
P.S. It helps that in CS lots of theorems are about either inductive or coinductive definitions.
1) I recommend the wonderful 1976 book Proofs and Refutations by Imre Lakatos.
2) There is a considerable list of books dedicated to counterexamples, e.g. in topology, probability, analysis, etc.
[1] https://en.wikipedia.org/wiki/Proofs_and_Refutations
[2] https://www.amazon.com/s?k=counterexamples
Yeah, given how much it accelerates grad students to produce meaningful output more quickly, why wouldn’t you make an investment of $2400/student/year. Seems like pennies overall.
[1] https://en.wikipedia.org/wiki/John_Henry_(folklore)
[2] https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK
Maybe in the future the work of Mathematicians will be like the ones of SWEs with AI, check thousands of lines of AI generated proof and find the subtle errors
Of course Lean proofs are rarely a good way to understand proofs, but hopefully they can be used to generate more human understandable arguments.
https://en.wikipedia.org/wiki/Compressed_sensing
Occasionally large improvements in our models of the universe have been associated with the development of mathematical tools that allow those models to be expressed and/or tested.
If we do it probably won’t be for a long while. We’re barely using math from a couple hundred years ago for most applied usage.
It's not easy to find a counterexample to the Jacobian conjecture, by any means – by which I mean to say that naïve brute-force search will take too long – but the scope of existing searches listed on Wikipedia[0] suggest that many tricks are already known, and that people just hadn't looked, systematically, for a counterexample in three variables before. Wikipedia writes:
> Tzuong-Tsieng Moh checked the conjecture for polynomials of degree at most 100 in two variables.[17][18]
where reference 17 is from 1983, and reference 18 is a preprint with no given date. Knowing very little about this problem, my impulse is to side with the unnamed faculty member cited in the article:
> [who] said to me that the fact that the counterexample was so easy to find just indicated that humans had not spent enough time thinking about the problem,
For context, the auto-generated counterexample is in three variables, has degree 7, and was discovered in 2026.
One thing that he mentions that already quite surprising is that AI was able to autoformalize the Golod-Shaferevich theorem and proof.
https://aaronlou.com/jacobian_counterexample_prompt.pdf
Obviously it is not random, but it's very generic. No mention of search space or how to reduce it.
We're a good team, and that's ok.
Maybe the prompt contained a part like this: "the search space is too big to just try all the combinations of 3 variable polynomials, so be clever about it". Or maybe this part was omitted from the prompt, because modern LLMs are smart enough to figure this out without us having to mention it.
We really need to stop thinking about LLM training data as the knowledgebase they build from and instead consider it more the skillset they start with.