"Element Pro" is not Open Source. Nor does it even come from the EU. From the outside it's incomprehensible anyone in EU bureaucracy would want something like this, given that there are good OS alternatives.
There are good reasons not to delay publishing at all:
> They should release all their results immediately. (Imagine working on one of the problems they already solved.)
This is the most popular answer to a question regarding AI advisory group and immediate access on a popular website for professional mathematicians: https://mathoverflow.net/a/515442/473286
The whole debate regarding the behaviour of OpenAI is a red herring. Mathematics need to redefine their profession and how they work (like us software developers too). There are very good reasons to believe mathematics has an important role to play. If they could just stop talking about OpenAI and get back to work - they are very much needed, in particular now!
> If they could just stop talking about OpenAI and get back to work - they are very much needed, in particular now
This kind of phrasing sounds particularly empty. We are not in WWII researching the nuclear bomb. What are they so urgently needed for to drop everything and work on understanding openai's proof on partition principle and axiom of choice?
Where did I say they should "drop everything"? I hope to read more about how they envision their future, instead of all this regretting and whaling how one company (that I very much dislike too) published a large amount of proofs. They will get more of them, very soon - if they like it or not -, and I wish that would be the primary subject of the discussion. And yes, I'd also hope they engage with the published proofs. The more raw these proofs are, the better. If AI companies start selecting mathematicians to write nice expositions of their proofs, this is doomed to become a very elitist science.
Says who? The professional mathematician writing the article disagreed. Why should people start dancing the tune that openai wants to play for their own reasons and interests? And I do not see how taking the time and effort to write a proper exposition makes it "a very elitist science" when this exact effort and time is needed to actually get other experts understand and build on a result. Unless you equate spending time and effort learning math as "elitism", which is the ai-shilling moto some time now with everything time and effort related. I cannot see how spending time and effort to understand a field and then spend time and effort to make a proper exposition so that other people can also understand it as "elitist" vs throw everything out there "in raw form".
This was my opinion. But anyone arguing to publish "results immediately" is likely to imply something like it. I guess in chemistry we have the situation you envision - for different reasons: Laboratories holding back their data, until their scientists have published their papers or developed their products. There is a real danger AI companies will do something similar too.
Who do you expect they will select for the exposition?
On which basis do you want a (likely US based) AI company to decide who is to untangle a proof that their latest internal model has just spit out?
Do you expect this to fall to an aspiring, but still unknown mathematician at - say - the mathematics department of Nairobi university?! This is what I meant with my rather unclear "elite" reference: The first publication will always show the name of a mathematician already known to the field, more likely than not to come from the same country as the company ("Our message ... is, you’re a great American company, but you’ve got to hire great American workers"). Are you not worried at all? Don't you think it would be good, if anyone in mathematics had a chance to write that first paper on a new proof?
There are parts of mathematics where the result is the important part. That's not what we're seeing here. Knowing whether the partition principle implies the axiom of choice doesn't meaningfully shape downstream knowledge and decisions. For these more foundational problems, clever proof techniques and the exposition around them are literally the point. Without that, neither humans nor AI can take this slop and derive anything useful.
And if AI can do that then great. I don't care about being elitist or not, and I'm fine with AI taking over math. That's not what it's done though, at least not yet.
> Mathematics need to redefine their profession and how they work (like us software developers too). There are very good reasons to believe mathematics has an important role to play. If they could just stop talking about OpenAI and get back to work - they are very much needed, in particular now!
The root issue is OpenAI et al.'s thoughtlessness in their engagement with a field.
OpenAI has resources.
That they fail to allocate enough of those to cleaning up pre-print papers (that seem to be a corporate PR priority for them to release) so they can be consumed and engaged with by the field they're targeting is... acting like a jackass?
It's the same "Meta / Alphabet can't vs won't hire more human reviewers" problem.
OpenAI could, at an immaterial salary level to them, pay a ton of PhD students and mathematicians just to clean up their proofs and papers.
Not doing so is a leadership and financial choice.
I'd prefer AI companies don't decide who's "cleaning up pre-print papers". This should remain the job of mathematicians at universities, which I am happy to pay with my taxes. Ideally there was something like a Bermuda Principles declaration for mathematics (https://en.wikipedia.org/wiki/Bermuda_Principles). This gave mathematicians even at poor universities and beyond the chance to participate in mathematical progress.
What do you think is gained, if AI companies manage "cleaning up"? Tax money?
No one is "responsible". If the paper is read, depends on the interest of the individual scientist. Many mathematicians interested in a theorem proven/disproven in a new AI paper will be curious, even if it is utterly cumbersome to extract the relevant line of thought. But don't you agree that the job can only be done by a mathematician anyway - be she/he paid by the company or by a university?!
What you are arguing for is somewhat like the "proprietary period" in astronomy (e.g. see https://www.scientificamerican.com/article/nasas-plan-to-mak... ). But there is no mathematician who asked for the run of AI, i.e. there is no mathematician, who can be regarded as the owner of the result, even if ownership was temporary. Complex proofs might take years to explain in detail (think of the ternary Goldbach problem, https://en.wikipedia.org/wiki/Goldbach%27s_weak_conjecture ). It's just unfair, if AI companies held back with their data until a proof has been put into a nicely readable article, and at the same time mathematicians elsewhere are spending all their time trying to solve it.
I'm not. To the extent my taxes are used for maths research (which is minute as a proportion of them) I want them to be used for the development of human mathematical understanding, culture and education, not trawling through a mountain of slop mechanically generated by a Silicon Valley startup that's about to IPO for trillions in the hope there might be some nuggets of insight hiding in there. If the latter is an activity with some value the startup in question can pay for it.
Is it not more democratic, if AI companies announce new theorems and proofs, and the discipline of mathematics furthers "our understanding" - instead of that the wealthiest universities in existence (all of them in the US) use their advantageous access to AI to produce them, along with their exclusive furthering of our understanding?
In a world, where expensive AI models running thousands of agents produce the most remarkable results, how should the discipline of mathematics look like, when money and access to compute plays such a pivotal role?
With the Navier-Stokes solutions being out in the wild, any mathematician at an impoverished university can make a stab at it and help our understanding. Assume the solutions being held back by those who produced them at Harvard & Co. (until they have their nice articles ready), what would the role of mathematicians be at less affluent universities?
I do not assume AI companies will be ready to provide equal access to their AIs to worldwide mathematics.
I’m hardly an expert, quite the opposite, but I imagine coordinating all those groups across multiple countries that don’t have a unified bureaucratic system would undo any disadvantages of just working on them independently and open sourcing at all. In theory they could all start borrowing from each other eventually.
It's not just a structure problem. The needs are different. And monoculture is always bad, so developing a plurality of solutions is a good idea.
I'm not saying every small administration of a 200-souls village needs to do this, but on the scale of a nation who has a GAFAM budget in the hundreds of millions of euros (probably, haven't made the calculation), it's not crazy to invest a few million on a side project like this to reduce dependency.
Thanks for pointing me to this video - it's been interesting to follow the discussion! (I personally don't see that math has lost its purpose at all in the past months. I mean, where would we be, if we were thrown at these AI based mathematical proofs and had no mathematicians and specialists?! Much of this discussion is about a disciplin readjusting its way of work and tasks.)
I would like to ask a question to you as a math professor: I think we all agree we do not know what the discipline will look like in ten years. But doesn't the rapid surge in mathematical proofs and methods imply that - at least for the coming years - there will be more, not less, work for mathematics?
Consider the "Jacobian conjecture counterexample": the work doesn't simply end once Terence Tao explains the computer-generated proof to a wider specialist audience.
1. I assume that the counterexample will give rise to a host of new questions, each of which will in turn need to be resolved. In the long run, the process of formulating questions might also be automated by AI - but likely not within the next few years to such an extent the growth of knowledge results in a decline in relevant questions.
2. Mathematicians will have a great deal to do in terms of meaningfully formalizing results within Mathlib - and hopefully Isabelle/HOL and other systems as well. From what I have read, the way current AI formalizes theorems makes them unsuitable for these libraries. I envision this as an undertaking not unlike the development of the Linux kernel. Throughout this formalization process, there should always be a human who has actually grasped the reasoning to ensure the AI hasn't simply exploited a flaw of the system.
3. Physics, chemistry, and many other sciences are currently benefiting from AI to a lesser extent. I anticipate significant changes at the interface between mathematics and other sciences as the body of mathematical knowledge expands dramatically. I cannot imagine this resulting in anything other than an increased workload, at least for the next few years.
Isn't it likely that mathematicians' workloads will initially rise rather than fall, provided they are willing to accept a shift in the nature of their tasks?
In practice, mathematicians' workloads have been a function of their work ethic, motivation, and competing demands on their time. There's no big-picture question of "how much math there is to do now"; the amount of remaining math to discover has long been presumed to be, for all intents and purposes, infinite. (Similar questions are relevant on a much smaller scale -- for example in case of someone who has specialized in a narrow specialty which goes dead.)
Your (1) is most certainly true.
As for your (2), most mathematicians I know have at most a passing interest in formalization, Mathlib, and Lean. My understanding, which is admittedly quite superficial, is that AI is actually getting quite good at translating human-readable mathematics. I could be mistaken about this, but even if there is a lot of human work to do, it sounds like a lot of anal-retentive oversight of work you didn't do yourself -- the sort of task that academics love to complain about!
Perhaps human interest in Lean will grow, but I don't anticipate it occupying the attention of more than a small slice of the community.
Your (3) is an interesting question. I work on the theoretical rather than applied side, but what you describe might very well be true for applied mathematicians.
Where I see models having a huge impact is in simulation code development.
One blocker for years now has been the adoption of GPUs. LLMs can fairly successfully and very quickly port to GPU and suggest/implement useful optimisations. Once it's verified, a code can go from anywhere between 2x to 1000x faster (mainly because CPU codes are so poorly optimised). Some science can reach much greater problem sizes, while some can run the same problems in hours rather than months and both can be revolutionary. Even more than that, LLMs seem to be finding fundamental performance bugs in both open and closed source core libraries so there's a bit of a whole-ecosystem uplift.
Can't comment on the more theoretical, less computational applied maths impacts!
I clicked on the dot for Khartoum, the only one in Sudan. It shows the Nile in Egypt, related to a movie that plays (only) in Egypt. A bit disappointing.
> You've seen them?
> Yes sir.
> You realize there's a government directive stating that there is no such thing as a flying saucer?
> Yes sir.
> Do you stand by your statement that you've seen flying saucers?
> Well, uh yes sir.
Plan 9, https://upload.wikimedia.org/wikipedia/commons/e/e8/Plan_9_f...