By now most of us have heard about the “mathpocalypse” giant drop of AI-generated unreadable slop-papers by open-AI. Many with attached lean formalization, which prove correctness of the result beyond the shadow of a doubt, and suggest more or or less that the natural language proofs are correct as well.
Among that, a lot of very important results! Unique games conjecture theorem, incredible progress on the Riemann hypothesis, faster fast fourier transform (and integer multiplication) in O(n\,\mathrm{log}^{1-\varepsilon}n), faster fast matrix multiplication in O(n^{2.25}), and much more. Much bigger than the already incredible Navier-Stokes moment.
Every maths-adjacent community needs to grapple with this. We should have a thread for talking about that. And not just about the technical aspects: Emotional outbursts and mutual support explicitly invited.
No, it appears to be impractial. But it’s very WTF, and opens the surprising hypothetical possibility that future progress may find something practical.
Same appears to apply for the matrix multiplication. (heck, even good old Strassen is barely practical!)
In the short term, it seems like a bigger problem for pure math than applied math.
In pure math, almost the entire value really comes from the process, the understanding and the new abstractions that are generated in the course of grappling with a problem. The actual answers to these grand challenges may have little or no practical utility, but the byproducts of the process are often useful — especially the generations of creative mathematicians who are trained as a result, capable of taking on many sorts of problems with a wider range of applications. But the career incentive structure in pure math requires “breakthrough results” in order to decide whom to hire, and now that incentive structure seems in danger of being irrevocably broken. In its wake, a whole generation of students may be deterred from pursuing a career in pure mathematics, its rich culture of deep intellectual challenges replaced by a “radioactive wasteland” of incomprehensible proofs of unexplained origins (“mathslop”) that disincentivize further study.
Whereas, in applied mathematics, the mathematical results are usually a means to an end, so in the short term we may be able to solve many practical problems more quickly, just as software engineers can now produce useful software more quickly in many cases.
But in the long run, we may all be poorer for the loss of trained mathematicians and mathematical culture. How will you learn to do mathematics when all homework problems, and even many research-level questions (once they are carefully chosen and clearly defined, which takes experience), have solutions that are a click away? Just as it is not clear to me how the next generation of software engineers will be trained. How many people will want to devote years of their life to learning these things? We are already seeing severe strain on systems of higher education. So who will be able to guide and evaluate the results of these tools in 20 years?
Not to mention the resulting concentration of power and information in the hands of a wealthy few.
Like poetry, mathematics is a creative pursuit. If the creative process is automated, people may feel their vocation is devalued.
School curricula, including math, often compound the problem by neglecting the history and human struggle behind the ideas, so students meet finished results without the people who made them.
AI may deepen this, with theorems that arrive with no human story at all.
I came across a video that was very much click-bait, and the opening line was “Millenials may be the smartest generation”, including all future generations.
While I didn’t watch the video, the basic premise was that every generation of students was taught based on the knowledge built from the combined work of all previous ones – standing on the shoulders of giants. And to get good jobs, a person had to possess that previous knowledge and have the ability to go further.
But now Gen Z and Gen Alpha can turn to LLMs for answers. And if job security in today’s market means devoting your energy to creating the best LLM-prompts, then there is automatically less incentive and less time someone can devote to learning the subject matter itself.
I doubt that the “human intelligence has peaked” will come to fruition, but it is at least something worth contemplating.
Two unrelated casual questions: 1. Since LLMs are incredibly good at literature search, why can’t OpenAI spend a few percent of their math budget on generating proper citations? 2. Should grant agencies like the NSF generously support pure mathematicians in purchasing LLM token credits or local AI inference hardware, at a level comparable to supercomputer costs of computational sciences like CFD and lattice QCD?
They could, but if their main goal is just showing off that their LLMs are capable of solving these open problems, what is their incentive to do this once they produce a valid Lean proof? If they wanted to produce good written papers, they would need to do more than just citations though (actually rewrite it by mathematicians, do a backward pass to ensure the natural language proof matches what the Lean proof does, etc).
Not really an answer, but at my university mathematicians already get a LLM subscription paid for at least.
I don’t think it’s possible to redesign the field of mathematics to reward understanding and exposition. Understanding is inherently private, how could we possibly measure that? Should we started applying proctored test to mathematics professors? And exposition is something that LLMs are not good at right now, but what if in a couple of years they can write perfectly clear, insightful and didactical exposition of the proofs they find? What are mathematicians supposed to do then?
And even if it were possible, who would want to do that? At least the mathematicians I know are motivated by solving problems. I’m a physicist, not a mathematician, but I often need to develop some mathematics as part of my work. And I absolutely love proving new theorems. Learning other people’s theorems, not so much.
I think that mathematics has just died, and there’s nothing to do but mourn.
Steve, I think you may have the impacts reversed. The AI providers do stupid math tricks just for publicity, and that will very soon gain diminishing returns, if it doesn’t already. There’s no money in it, so they will stop. Meanwhile, mathematicians have always decided for themselves what questions are worth asking and what answers are worth thinking about. And they have incredible new options; with time, the good ones will learn to set their sights higher.
Now, there are other severe pressures on academia and the wider economy that could still devastate mathematics, but it’s not clear that the field will be special in that regard. It’s hard for me to accept that solving hard problems in a week rather than multiple lifetimes will be a net negative.
On the other hand, applied mathematicians justify their (our) work in no small part by its usefulness, which has instantly been deeply discounted: domain experts have far easier access to mathematical and computational expertise than they did last year, and it will be even easier and more comprehensive next year. But the applied mathematician has no meaningfully better access to, say, a chemistry lab. If applied domains are no longer publishing about challenges they can’t tackle themselves, what is the source for new problems?
Current AI capabilities are already impressive next to our biologically capped human brains, and AI systems keep scaling. It seems fair to say that we cannot reliably anticipate what kind of science machines will soon produce, nor be sure that we will be able to follow it.
The capabilities of the best LLMs is still built on human knowledge. Let’s take the worst case scenario: LLMs become so good that they run human mathematicians and theoretical physicists completely out to the point where college students stop studying those fields. At the point, can LLMs continue to push science and math further with zero additional human expertise?
I say no. All the problems LLMs are solving currently were developed by humans, all the techniques LLMs use were developed by humans. Can an LLM decide on its own what problems in math/physics are worth pursuing? Can an LLM create, test, and prove completely new and novel mathmetical techniques for proofs? The only reason OpenAI is able to even attempt these things is because the LLM models don’t have to start from scratch.
If we could somehow limit an LLM model to only data before 1997 and asked it to study black holes, would that model be able to come up with the AdS/CFT correspondence without a single human expert guiding it? Will it be possible in a year? 10? 50? ever?
In the interest of empiricism, maybe we can be take this literally and turn it into an actual benchmark. Example prompt:
You are a visionary theoretical physicist. Propose a new correspondence between two string theories or quantum field theories, and present strong arguments supporting the correspondence.
Unfortunately, I don’t have enough money to run the benchmark. I hope OpenAI will dump $10M on a prompt like this to see if something useful comes out.
That seems very difficult to answer no? You would need a clear measurable definition of what expertise even is and whether it would transfer to a machine. That’s a philosophical debate.
Is that true? Aren’t anthropic and openAI using their own tools to develop AI?
The question of “what is intelligence” or “what does expertise mean” truly is a philosophical debate, but I still think my overall question
Can LLM models advance scientific knowledge without a human telling it what to study?
is absolutely answerable in a non-philosophical way for current models. Predicting the future obviously though is more of a debate.
But take my question from above, could an LLM model come up with an acceptable idea for a PhD thesis with no human interaction except what its already been trained on? That seems answerable today if OpenAI or Anthropic wanted to use resources to answer it.
I was strictly speaking of open problems in theoretical mathematics and physics when I wrote that statement, however, I think it holds for many other fields as well. I will say I don’t know what it being done to train models these days, but assuming OpenAI is training using LLM agents to further its own research, I would still ask the question, are the LLM agents simply implementing ideas that (“expert”) humans prompt them with or are they coming up with truly novel algorithms/theoroms/etc without any human guidance?
I don’t think this is easy or necessary to answer.
Subjectively, people have observed LLM-based software doing apparently novel things. Even if we argue that it’s just remixing prior human accomplishments, that itself is a widely acknowledged source of novelty. With more objective verification, which is present in mathematics as opposed to say music, this sort of novelty is a more reliable basis for growth. It even gets around model collapse, though there are claims it still occurs in some ways.
The more important question I always suggest people ask is whether we want LLMs to do something for us. It’s much easier to justify automating hard labor that results in chronic illnesses, but no offense intended, cushy desk jobs with very indirect-at-best and dubious-at-worst economic gains? What are we doing here?
