An example of how mathematicians and AI can complement one another in probing the frontiers of knowledge: Number theorist Youness Lamzouri has constructed a new, conceptually simpler proof of Claude's Theorem that 2/3 of the zeros of Riemann's zeta function lie on the critical line and are simple.
> Levent Alpöge and Ralph Furman, two of Anthropic’s own mathematicians, examined Claude’s work to understand the new results and how they related to the prior work mentioned above.
Anthropic seems to be challenging the traditional way math gets published. As far as I understand, these results did not get submitted to journals, and did not get Arxiv preprints; they are released only as self-hosted pdfs, and we don't even know the names of their authors.
The canonical reference for the counterexample to the Jacobian conjecture is a tweet with no puntuations nor capitals.
The current consensus in mathematical publishing is that LLMs are tools, so they don't get listed among the authors. But nothing would have prevented them from posting these preprints on Arxiv, with the human prompters as authors and the LLM's contribution acknowledged in the text.
One of the 2 names on here says they aren't being listed as an author on the paper because their contributions don't meet the standards of authorship. People can call LLMs 'tools' or whatever they want but if you had essentially nothing to do with the breakthrough, you're not an author.
My wife did her mathematics PhD under an analogous structure in Germany where it’s fairly common. She was supervised by a professor at the University of Munich and a lab at Bosch, wrote and defended a thesis as usual. Employed in industry immediately after graduating. A great experience.
During my academic career I worked at a top German research university and advised multiple of these projects. On the surface it’s a good deal for everyone: the professor gets increased publication and graduate counts without having to pay for it, the company gets a top student working on their research problem very cheaply, and the student gets a PhD, an employment contract, and often a visa.
The reality was unfortunately often not so great. What would happened is that the student would get 3-4 year contract with 50% allocation to the PhD research in addition to normal employment duties. That is not enough to graduate if the professor has high standards.
As a result, the student would transition to full time employment after the 3-4 years, try to work on research in the evenings, and give up after some time. Or they would graduate with a very low quality degree (I saw students graduate with a 0 or 1 publications and a very thin dissertation) that has practically no value to anyone beyond providing the student with a Dr. title.
In the end I was actively advising students against it, and to just go into industry with a masters degree or find a fully academy funded PhD position if they are serious about research.
I guess it depends. While doing my PhD at a top German research university, our chair advised multiple of these projects, too (e.g., BMW, Siemens, Audi).
Yes - the downsides you mention are all true. But similar downsides apply to most PhD students working directly at the university - either you have some teaching load and administrative duties, or you work in an externally funded project and have to write project reports and do a lot of non-research stuff, too.
As I mentioned elsewhere in the thread, if you want to have an academic career, doing a PhD in industry is not the best choice. But if you want to work in R&D or as a group leader in industry, these PhD positions might be a good stepping stone.
While I understand the upside, the reality of why it’s being pursued in the United States is that this is intended to drive academia into an advanced version of a trade school.
Do the math: The number of professorial positions is not growing (or not very quickly). Under steady-state, each professor only needs to train one single tenure-track PhD in their entire career. (Or maybe 10 professors need to train 11-12 students, to account for various forms of attrition.)
If you want more PhD's than that, you have to figure out a place for them to work; industry is a good "pressure relief valve".
This is true but it doesn't necessarily mean that academia ought to shift its focus to professional training. Also keep in mind that this has ~always been the case. Sometimes you train with a specific task in mind. Other times you do something open ended and unrelated "for the experience" with the idea that it will benefit you in various ways in the future. Neither of those approaches is always wrong or always right. Sometimes you need to focus but there are other times narrowing your view would materially diminish the benefits of the experience.
I do think it would be good for all levels of academia to be more explicit about the goals of any given program. I think these things are often blurred together to the detriment of both student and staff.
The goal of PhD programs has never been to have every student become a professor. Also, strong STEM PhD students generally don't need much help finding jobs after they graduate. Claiming that PhD students "aren't prepared" for industry careers is simply saying that students lack industry experience (obviously, being students!).
The problem here -- and why universities even do research at all -- is that industry's support of basic science is not very stable. At least when I was in school, industry research generally was more applied and engineering-focused.
i think major federal funding agencies should track department-level stats on "how many phd students did you admit?" versus "how many tenure track positions did you hire?" and use that to implement mechanisms that encourage a healthier balance (e.g., changing funding lines on grants, adjusting their overhead cuts, boosting trainee grants at certain places).
this would hopefully stop departments from over-admitting and encourage less adjunctification.
That sounds like an excellent thing based on that last sentence. Anecdotally, I hear people talk about trade school as being much more practical compared to the majority of traditional 4 year degrees. So I am interpreting a "move towards trade school" as an increase in practicality and therefore utility to the student.
I just wanted to echo your comment. Most people, in my experience, pursue an undergrad program with the intention of being employable. This is my personal experience, but also a learned opinion from being a teaching assistant for some years. Places like waterloo stress co-op, it is built in to the program. IMO the average person seeking an education are served under a model that places them in or near industry related work at some point. While I understand the original post is relating to PHDs, I think most programs (or rather, participants in programs) would love to have this sort of integration despite it being a hard problem.
Interestingly enough, Kitchener, adjacent to Waterloo has a large population of German-descent. I wonder if the OP's spouse's experience in Germany has any relation.
There is no reason to assume it is “more practical” or that it will increase utility for the student. Especially at the graduate level, the deeper you are able to explore a subject and the more you can connect that subject to a broader range of topics the more utility you get from the degree. Less time to finish a degree means you are less able to do that. I would think this will decrease the likelihood of new ideas being developed which will quickly become a problem for society in general. Also no one gets a PhD because it is practical, you get it because you want to explore a topic and push the boundaries of human knowledge.
My impression is that many of the indian engineering schools are some form of trade school and it shows. A lot of people are very good at churning out things, but they vastly prefer to use well defined tools for their trade and do a lot of work impassionately. Taking the creative path or even reflecting on failures isn’t natural.
On the other side of things, you have PhDs coming out of the US with too many ideas and the inability to deal with basics such as sprint planning and CI/CD.
Not sure if this will help, but I would love it for the Colleges of Sciences peoples instead of a postdoc.
The problem is that this makes it friction-free for the student ... and the actual trade? Your wage. It takes away risk, which everybody likes, until the costs of that start accumulating. It's an excellent way to give the student a very strong reason to avoid applying anywhere else (and make applying elsewhere far less successful), and so effectively to pay them (much) less.
Get a PhD with years of experience, who knows your company, for the price of a secretary.
Practicality and utility to the student must be the reason I started getting junior software engineers that are computer illiterates out of whatever Java IDE they were taught in.
"Learn what we believe to be the textbook-correct, fundamental, foundational theories for a topic; learn about the frontiers of new research in the topic; find a niche on that frontier where there's something new to try; try something new; write down the results (no matter how boring); share with your friends; rinse, lather, and repeat" is a perfectly respectable trade.
No reason why the good plumbers, electricians, or truck drivers of the world ought to look down on academics-- no matter how stuffy or highfalutin or soft-handed they might seem.
Every once in a while, academics come up with something useful. (And in fact, I think that that might just be the whole point).
There is a non-insignificant segment of the population that would have been better suited and employed as painters or HVAC technicians than for their chosen careers. Some software developers (and 80% of the city of Portland) would also fall into this category. The world has a surplus of PhDs opening cupcake shops.
i think there are a lot of college-educated folks that would love to switch into a trade, but there are cultural barriers to starting such a career. a not-insignificant number of tradespeople have been taught to see college as the enemy and college-educated folks as soft, coddled people with poor work ethic. if you showed up at a job site after getting a four-year liberal arts degree, you'd be met with lots of mocking and "oh, you thnk you're better than me?" defensive posturing.
It already is. Most PhDs in, let's say CS but probably true for a lot of STEM, just get a job at a tech company. I know a guy who did a PhD in meteorology and now he just programs at Google. The promise of contributing to science and being an academic probably doesn't work out for most. There's actually a big cost to this too, this isn't just an alternative path. It's also costly to non-PhDs who suddenly find themselves competing for jobs against them in careers that really don't need them. At least if we're going to continue printing PhDs it would be great to make them shorter and less costly.
Germans are big on trade schools. Despite being one of the richest western european countries, Germany (and Switzerland) have along the lowest college graduation rates, because they prioritize trade schools.
I know the German system quite well and know people who did their PhD in industry, people who did their PhD in an externally funded research project, and people who pursued a more self-directed PhD while working as a research and teaching assistant.
I don't think that there is a general 'PhD inflation' in Germany (though there are some disciplines with this problem). It is well understood by most PhD students that an academic career is the exception, not the rule. Most of them choose to do a PhD because they like the academic environment and want to learn more. Most PhDs go on to work in industry research labs, science-adjacent roles (e.g., museums) or as group leaders in tech companies. There is sufficient need for PhDs in most fields.
Industry-embedded PhD students are required to meet the same criteria as other PhD students. They also publish their research at the same conferences - but it is often more on the applied side. One could argue that such research has "little scientific value" - but so does most research.
The most important outcome of a PhD is not the list of publications but a person who deeply understands a domain, knows how to critically analyze a problem, and finds good solutions. Doing a PhD 'in industry' also allows you to do this. And it gives you a foot in the door at that company.
FWIW, many PhD students I knew, e.g. at BMW, complained a little bit about the side-projects they were expected to do, or the bureaucracy at such large companies. And, because you don't have to do any teaching and rarely supervise undergrads in industry labs, you are less qualified for an academic career than PhD students who work at the university.
"The most important outcome of a PhD is not the list of publications but a person who deeply understands a domain, knows how to critically analyze a problem, and finds good solutions. Doing a PhD 'in industry' also allows you to do this. And it gives you a foot in the door at that company."
Interesting though. Not sure you can have an extensive publication list without a deep domain understanding. But an extensive publication list normally assures that you have it.
My professor always said, I don't want to see you PhD thesis, just show me your publication list. Actually, while we were obligated to write a thesis, in his opinion it should be sufficient to just submit your publications.
In all OECD countries, but in particular in Germany, there seems to be an enormous overproduction of PhDs – in case one sees a PhD as the starting point of an academic career or as an important asset on the job market.
I am aware of that system, doing a PhD in a company as an external PhD student. The "research" is extremely applied. That is good for the company, possibly even a better career choice, but has limited scientific value. You are also unlikely to switch back to academia and become a professor.
Most people mistake "feeling" for "thinking". Just because you don't like my post, does not mean it is not true.
No, I'd say ambiguous means "context dependent" and the context is unclear.
Pronouns like you/me/he/she/they/them are context dependent in everyday English writing but they're only ambiguous when the context is unclear, otherwise most people have no trouble dealing with them at all!
tl;dr: Being a homomorphism from a multiplicative structure into an additive structure isn't enough to grant it the logarithm title.
Although logarithms are certainly ubiquitous in mathematics, I don't think that the mappings that the article's author identifies as logarithms are appropriately viewed as such.
I can't endorse viewing dimension as a logarithm. It appears superficially logarithm-like because we typically (and somewhat unfortunately) write the direct sum of n copies of a vector space V as V^n rather than nV. Writing nV, we simply get the dimension identity dim(nV) = n dim(V). Writing nV instead of V^n also conveniently frees up V^n for the tensor product of n copies of V, with corresponding dimension identity dim(V^n) = dim(V)^n. So I don't think there's any "multiplicative-to-additive" business going on here at all.
Also, I don't think it's advisable to view the p-adic valuation ord_p as a logarithm, even though it's a homomorphisms from the multiplicative group of the rational or p-adic field into the additive group of the rational field. In fact, in many number theoretic contexts, the ratio log_p/ord_p is of particular interest.
I think a good rule of thumb for viewing a mapping as some kind of logarithm is that it has to have some relation with the Taylor expansion of log(1 + x) around x=0. Being a homomorphism from a multiplicative structure into an additive structure isn't enough to get the logarithm title.
I think the OpenAI model that resolved the Unit Distance Problem would be capable of solving a significant proportion of mathematics PhD thesis problems.
> Now if I know anything about math for the sake of math, and academics, these are the same people that lament the idea of intelligent people going to the finance sector or any other trade they just happen not to respect as much
IME a vastly more common sentiment among mathematicians regarding mathematical talent leaving the nest to apply their skills in other fields is that those other fields are lucky to get them!
To me, the most interesting feature of the OpenAI solution of the Unit Distance (Erdös) Problem is that the solution - using deep algebraic number theory as a source of extremal combinatorial/geometric constructions - is much more interesting than the problem’s elementary statement might lead one to expect.
Writing off Erdös’s problems as random, useless, or meaningless dismisses his mathematical intuition, second-to-none, and strikes me as somewhat uncharitable.
Finally, I agree that AI threatens mathematical training by rendering an entire class of acolyte-level research problems solvable by prompt. But the Unit Distance Problem is not of this class.
> much more interesting than the problem’s elementary statement might lead one to expect
This is reinforced by the immediate (human) use of the idea to resolve in the negative another significant problem, the sum-product conjecture on reals.
I don't think Erdos problems are useless myself, I put "useless" in quotes to emphasize that they are the sort of research that doesn't have an immediate application, and so their automated resolution should be weighed against the sociological cost.
I am not a mathematician and did not read the unit distance solution too carefully, but my impression was that it used a variation of a known technique to solve the problem. And that makes perfect sense to me, there are a lot of techniques and lot of less relevant problems, I am not surprised that one can solve some of them with known techniques that just nobody has tried [hard enough] before. I am much more sceptical when it come to the important unsolved problems where every known technique has probably been tried several times over. In those instances it will probably take a true leap in understanding to solve them and I am sceptical that large language models are well suited for that because of the way they work.
We're very fortunate to have had some very eminent mathematicians backfill the OpenAI proof with history, context, and a literature review [1]. Ideas behind the proof seem to have been "in the air". Indeed, looked at certain point of view, the OpenAI construction can be viewed as a high-dimensional generalization of a known low-dimensional one. In this vein see the remarks of Gowers, Sawin and Tsimerman in [1]. Are LLMs capable of "true leap[s] in understanding"? I have absolutely no idea. But LLMs keep surprising me.
Not sure about these books as a self-study curriculum — their unifying theme seems to be that they require a reasonable level of mathematical maturity going in. But, they absolutely comprise an excellent “greatest hits” list of math books in the most influential subdisciplines. You’re guaranteed to learn a tonne if you study any one of these books.
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