contubernio

I'm a professional mathematician. Today I proved what for me is a very solid theorem. It's something I had thought about for a few years. With a few weeks of serious use of AI I've found a proof that I am currently trying to write up, but which appears correct. The change in the workflow is enormous, but so is what one can do if one has clear what to do and how to do it.

As a math professor, I care much more about the key idea, heuristics, and motivation than the proof. With the others in place the proof is clear, something an AI or a student can do.

They should just make them read an old chess book like Lasker that gives low level heuristics

This is the essence of why disciplinary, authoritarian, stick based teaching of humans generally fails. It teaches succeed at any cost.

One sees this in math research. The model reports it has proved X. In fact it has given an erroneous numerical check of Y in a few atypical cases.

What makes math approachable is that the context is so well delimited (semantically) that one can guide the model with adequate correction.

Tha Jacobian conjecture is (was?) notorious for publication of false proofs. There have been many, not all by those who shouldn't have known better.

Perhaps all these fields medalists are just worried that AI mitigates their competitive advantages over a mediocre professional mathematician like me. Their main advantages are a kind of penetrating clear thinking that not all of us have and an ability to work obsessively without tiring or losing focus while retaining with a clear head a great number of facts - AI removed these advantages - as Tao has written it shifts the focus to asking good questions and using available tools to address them - and the traditional elite may not be better than I am at that part - they were just better at the solving the problems part ...

Tao is quite the opposite of what you describe. He's had an ongoing dialogue intellectually with the challenges created by ai for mathematicians for some time (years) now and his viewpoint is quite balanced. He is not alarmist but rather correctly addresses the real problems ai advances create for mathematicians traditional way of working.

"I suspect that with six good mathematicians spending their whole careers primarily focused on it, and working together closely, Navier-Stokes might well have fallen already"

There were more than six doing that and it's essentially why it was ripe for AI to finish it off. But the finishing off was quicker than anyone expected

The idea that mathematicians were not involved in actively directing the and structuring the search for solutions is absurd to any professional mathematician who has tried to prove things using these models.

What you say is true but ... This is qualitatively different than calculators or computers.

I'm a professional mathematician and all the better mathematicians I know are in crisis mode. Most of us hadn't taken this sufficiently seriously and don't know how to use these models effectively but we play with them and immediately see that the entire way we've worked all our professional lives has to change. We worry less about ourselves than about the younger folks. I've got good ideas ai still doesn't know about ... Younger folks may not get the chance.

Math problems are highly structured, very precisely defined, and already heavily studied and not very complicated compared to problems in engineering or finance. There's a lot of quality material on which to train and it's easy to tell quality apart from crap. The search spaces are a priori much smaller than in other areas and the people using the tools to study them are themselves good mathematicians.

Success in such problems does not automatically extrapolate to other contexts.

People are underestimating the costs in terms of money and energy.

The third law of thermodynamics is an essential barrier in all engineering.

In math and theoretical physics the arxiv is the default place to publish and where one goes to find the best and most current work. Papers take a long time to review (years) particularly hard and good papers and publication in journals mainly serves the needs of bean counting dean's and quality control morons in the administration. Actual scientific communication is via the arxiv. Self selection mitigates most ills you describe someone who regularly posts crap on the arxiv will do themselves damage professionally and disappear.

Bubeck is familiar with how publishing works in mathematics.

What is specifically alleged is that a particular approach to the problem - itself not easily discoverable - was copied. This is what is meant in the text "I should say here why I interpreted their statement the way I did, the in- terpretation I will discuss below. The route to the Clay problem through a smooth force, options c and d in Fefferman’s statement of the problem, is the route Luis and Diego opened and the one Levent and I had quietly chosen to attack. Almost nobody else I know of was working on it. It is not the direction one arrives at in a few days by giving a model the problem statement. When I heard “forced,” it was a bright red flag."

For those who know nothing about the context - the Diego mentioned was a student of Fefferman and Luis was a student of Diego's - these people have all worked hard on these problems for a long time and are genuine experts. The mathematicians at OpenAI are strong mathematicians, but not expert on these particular problems. The particular approach is claimed to be the key to the whole thing.

The allegation is not different in spirit to alleging that a particular group of astronomical researchers "discovered" a new planet because they had access to the logs of another group that had already pointed its telescope at the planet.

This post is not intended to assess the correctness of the allegation.

A country like Spain has a formal system of grants and awards for athletes designated as "elite" and supports them roughly at the same pay as it supports junior university professors.

Nobody is in panic, but we are using these systems and seeing what are their capabilities and it's clear that their use has a big impact on the practice of mathematical research, probably far more impact than it has in other areas. Part of mathematics involves classifying structures - think of describing combinatorial objects - and this is a sort of game with which a well directed AI tool can be very effective. Part of mathematics involves being able to bring to bear on a problem a diversity of techniques, and AI is also very helpful in this regard.

The point is that people who spend their time classifying nilpotent Lie groups that admit structure X are out of work if they don't change their perspective. Maybe such problems were never really that interesting, although it was useful to have a group of people acting as human computers to work them out - but well used AI yields for such problems more complete and more reliable results - and so allows researchers to spend their time on other more interesting things. The problem for your run of the mill professional mathematician is that more interesting things are harder ...

Calling the arxiv a vanity press shows you aren't a researcher. In math and physics all the best stuff is on the arxiv and the general level is well above the level of most journals. Journals mainly serve as accreditation and many are basically mediocre - the review process resta more value than it adds overall.

It is capable of applying know heuristics and general principles in places where they haven't been applied and in this sense very much capable of generating conjectures in much the same way a person does. It's ability to employ a diversity of techniques coupled with it's computational power differentiate it from a human researcher. It still needs guidance to work well, but I've already changed my daily work flow as a research mathematician to incorporate use of AI.

As a mathematician not expert kn these things, yes it scans as reasonable, and yes it has me and most of my colleagues reconsidering what we do for a living.

Sounds just like standard behavior for gaming incencyives in large institutions.

AI is clearly of great utility for defense/military/security. That alone guarantees continued support financially.

As a middle-aged mathematician, AI is provoking a serious crisis among the better mathematicians I know. Many are only half jokingly talking about retiring. We use AI a lot in research already and it's obvious that whoever doesn't will fall behind fast. At the very least it gives access to the full library in a few minutes, serving as a kind of knowledge oracle generally more useful than the expert who ought to be in the office next door but isn't. And we see what can happen when it is effectively coupled with lean to automate theorem proving and discovery.

We aren't dumb enough to see this as the end of the profesión - Esther it's clearly a shift in how we will work - but we like doing computations and playing around with examples and how one does that just changed a lot. The other problem is we know we don't have the energy of youth to learn to use AI as effectively as the kids, although we are wiser and have better judgment and do know some things.

Mathematicians who are not taking seriously how to adapt to AI are deluding themselves.

Institutions like Duke benefit enormously from having people like this around. They can claim they have lots of highly cited and well published researchers, a bit of controversy is free publicity, etc. They make money off him and so they defend him. It helps support the myth that they are good educational institutions. US private universities are grift mills.

Part of running a profitable business is doing the right thing. Following socially obligated rules is a cost just like buying drywall.

The profits at all cost mentality is a criminal mentality. Maybe it gets away with not being formally criminal because laws or enforcement are weak (as is the case in the USA) but that doesn't justify the mentality.

As a dual American-European (citizenship in both, lived in both) I celebrate the EU data privacy laws and lament the absolute lack of protection of data or privacy that prevails in the US. Because of US based websites the genealogical history of my extended family (to many degrees), my past US residence history, etc. are easily available online, while in the EU it is very difficult to obtain any information online whatsoever about me or my immediate family (based in the EU), where we live, or even what we do for a living.

Most of the websites that are bad (operationally) in the GDPR sense are based in the US or represent US based entities. It is primarily US based entities that engage in bad faith fake compliance.

All this reflects the complete deterioration of basic business ethics in the US that has been led by the piracy culture that dominates in the tech sector, where the mentality is to bend or break every rule as much as possible, suffer the fines as business cost, and so on.

ODB understood this dynamic years ago - "Wu Tang is for the children"

Instrumental education is less effective at producing innovation and utility than is basic education.

Forgetting this is (and has been) destructive to universities.

People on the street, so to speak, don't know better even in advanced nations with strong educational systems. Look how much rejection of vaccination there is in spite of "knowing better". In many African nations there is very reasonable suspicion of European authorities - as there is a long history of abuse and degradation that has moreover been weaponized politically kn the post colonial era. The solution is trustworthy and pedagogical people on the ground doing a lot of hard work and that requires lots of institutional support.