Just about anyone who has played a sport has heard that they have to target where the ball is moving to, not where the ball is right now. In this regard I am consistently disappointed with the commentary that mathematicians have been producing lately.
As a student, I am actively making decisions which will shape my career for the next forty or so years. At a minimum a discussion like this should acknowledge the possibility that the current rate of AI progress continues apace. I understand the desire to be encouraging, but the the best preparation for students involves the consideration of possibilities that current mathematicians negligently paper over.
Of course it will get better at exposition than humans. And it will prompt itself in due time.
You could being playing best jai alai player always knowing where the ball goes.
Whoops, people stopped caring about your sport!
> Just about anyone who has played a sport has heard that they have to target where the ball is moving to, not where the ball is right now. In this regard I am consistently disappointed with the commentary that mathematicians have been producing lately.
Reading this it feels like you haven't played a sport.
Do you care to share then? Because anyone I know with any sense right now is aware that "planning for the next forty or so years" is genuinely impossible at this point, especially in mathematics, and the right approach is to remain on your toes, ride the waves, and diversify as much as possible. Heck, even planning for the next five years seems absurd. This is a time of immense uncertainty, and to think otherwise is foolish.
A few years ago I was in a pure math degree, but I switched and graduated with a combined degree in math and computer science. Once I finish my masters degree I intend to work on industry applications of formal methods, which I anticipate will be a high-growth area for the next few years. The idea is to take advantage of the growing ability to generate proofs/code and work on keeping the messy human side of the equation up to speed. So I plan on addressing the social and operational aspects which come with delegating more work to AI. I'll work on integrating these gains into the industry, on building trust in the work that AI produces, and I'll work on the interface that lets us ensure that this trust is deserved.
But nothing is certain. I am trying to avoid overspecialization and remain flexible, just as you advise. If this path fails I might become a paramedic or physiotherapist.
The trajectory of a ball can be predicted. The trajectory of the job market for mathematicians is far too uncertain at this point to make reasonable predictions. This is not a matter of intelligence.
No one succeeds in making accurate predictions of the future over a 40 year time scale. Look back at the 20th century, and see the range of range of events, and the kind of technological change, that could occur in a span that long.
Go into things expecting you might have to change careers.
> No one succeeds in making accurate predictions of the future over a 40 year time scale.
The sports analogy holds; the future location of the ball may be uncertain, but that is the student's target, and not overcommitting to the present location of the ball. The commenter is expressing disappointment that the commentary seems to be reacting to the present too much.
Science was quite stable in terms of how it works over the last 100 years.
Sure you wouldn’t predict what you would work on exactly. And there were better tools, better communication and so on sure, but no disruptive innovations or seismic shifts really.
Freeman Dyson said no scientific (or, I think, technological) project should last more than about five years, because the context under which it was conceived changes too much in that time.
It's not a good analogy because the ball is constrained inside a court with fixed rules about how it can travel. No one can predict accurately how AI is going to change things. Look at all the predictions so far. We have the whole spectrum from "the end of humanity" to "utopia".
We might not be able to precisely predict where the ball will move, but I think GP is implicitly saying "don't assume the ball doesn't move", which IMHO is kind of baseline, but seems to be over the head for some people that are now complaining.
Some of those complaining today are the ones who keep saying "LLMs are just stochastic parrots predicting the next word", "how can AI do math if they can't even count the number of 'r's in strawberry?", etc.
It doesn't take a genius to realize AI will start getting better and possibly become a "threat" even before all this mathocalypse stuff happened.
> In this regard I am consistently disappointed with the commentary that mathematicians have been producing lately.
For all the arrogance the hard sciences have, in the end they are just as human as their humanities counterparts. They just thought they were above becoming worthless because science was elevated above the softer sciences.
> At a minimum a discussion like this should acknowledge the possibility that the current rate of AI progress continues apace.
But the post does do this: "[...] even if their capacity becomes far superior, there will always be a need for mathematicians at all levels to guide research in paths that make sense for humans to walk (not run)."
> Of course it will get better at exposition than humans.
I'm not so sure about this, because I don't know how one would train towards better exposition as it's not easy to check for good or bad exposition at scale.
As the article points out, though, all of this fundamentally depends on current AI approaches "hitting a wall", that is where there are some inherent limitations in LLM-based tech that leaves humans some areas where we are still superior.
It's definitely (again, to the article's credit as it points out) up in the air whether LLMs have inherent limitations. But I can't fathom how someone can say "I don’t expect a jobs apocalypse, not even in math." Because if AI does end up surpassing humans, what exactly will there be left for humans to do? And even if they don't surpass humans, there will still be a jobs apocalypse. Probably the majority of people today work in jobs where AI can surpass their performance.
Simple case in point: years ago I used TurboTax to do my taxes, but I eventually needed to hire a CPA because I had some complicated situations that TurboTax couldn't handle, and I also needed some tailored advice. I ended up finding a great CPA. Now though, AI agents can literally do 100% of the job I hired my CPA to do, including asking me the right questions and offering advice. I'm sure there may still be tasks for a CPA in the corporate world, but for personal taxes, I literally can't imagine a CPA providing value over what an AI agent can provide. And to emphasize, I would have probably considered that a ludicrous statement a year ago with all the mistakes LLMs made. But so many of those mistakes have been fixed, and you can get better-than-human performance by having agents check each other's work. And AI will only get better when tax season rolls around next year.
You must be thinking of physics and other sciences. Mathematics has very little to do with the real world and can be better described as a game, similar to chess, but much more complex. It's no wonder it now went the way of chess, go and similar games
> Mathematics has very little to do with the real world
But it's a construction based on very basic properties of the real world, at least on those properties we accept and perceive as very general and we use them as building blocks.
> Mathematics has very little to do with the real world a
I don't necessarily think that that is true, as almost every social science has been turned into maths one way or another, beginning with Economics. Heck, even mundane stuff like sports (think football/soccer) has now been absorbed by data, which you can see as applied maths.
Yes, I do know that advanced maths doesn't currently explain Arteta's system at Arsenal but we're generally talking about that same semantic area of interest, i.e. if the interest on maths were to subside as a result of AI taking that interest away from us then we will have no more data/stats-focused Artetas in the future (and possible no AI-like thingie also).
Almost all sciences (at least, parts of them) have been also expressed as code, mathematics is no different. It's a tool used by sciences, but it is not itself a science and it is not necessarily rooted in anything "real". Not better or worse, just different.
I’m saying that this wouldn’t have happened had our society not become “mathematics-ised” at some point (I’m talking about said society’s technical and scientific elites, of course). I’m also saying that outsourcing our “mathematics-mind” to AI (which seems the current process we’re now part of) will not allow for similar “mathematics-ing” to take place into the future. I’ll come with an even stronger claim and say that we risk losing the “mathematics-ing” we now associate with most of our sciences.
So learn math to teach math? You see the recursion here right? What’s the end value for society? Why would anyone fund this? How is this different from a hobby?
In a future where learning math has about the same economic value as learning to play chess or the flute, you can see why many mathematicians aren’t very excited about their chosen career paths.
A lot of math is useful to society. I'm not sure the more advanced research is very useful but you could say that for a lot of advanced research in the non science subjects.
There's going to be a massive reshaping of how mathematics is done. If you do not like where math is going (using computers to explore fully new ideas), and you have not committed to this path, it seems totally correct to opt out of it. There probably isn't going to be a technical field that isn't reshaped by this in the immediate future though, so there is unlikely to be anything that slots in cleanly as a replacement. Maybe the philosophy department.
This post in the blog comments is quite compelling. Some observations that publishing of research is drying up because results can be easily retrieved at any time. Over the long-term, I wonder whether this will result in accumulation of knowledge grinding to a halt:
> With the latest ChatGPT models, these problems are more equivalent to homework questions: the answer is `in the back of the book.’ I am not discovering new solutions. Instead, I am working on problems whose answer exists and is simply waiting to be retrieved by a user of the model. In fact, I mentioned a problem that I was interested in working on to my advisor and he informed me that he and a collaborator had completely resolved it using ChatGPT – they have no plans to write up the result, so it will sit there until another `researcher’ pulls the proof slot machine.
There is another good point that the most tenured researchers have a sense of what problems are most worth exploring and therefore are more likely to feel excitement than younger researchers:
> I have also heard the contention that math research has `gotten more exciting,’ mainly from established researchers. They have decades of open problems that they care deeply about and want to see resolved. I have no such problems.
It reminds me of The Hitchhiker’s Guide to the Galaxy. The mice asked for the answer to everything in the universe, and it produced the answer:
“42.”
But no one could understand what the answer even meant. So they designed a computer to build the question itself again, and that was Earth. Then the story begins with Earth being destroyed because of a cosmic highway problem (I won’t write more since that would be a spoiler). In the opening background of this work, I found it interesting that after calculating for 7.5 million years, they didn’t even know what they had originally been asking. The story now feels similar to that story from back then.
Isn't there analogy here to what's happening in software engineering? People keep saying things like "software engineering is so much more than programming". Couldn't one say that "mathematics is so much more than writing proofs"?
At least for now someone still has to decide what to prove and why. Like why are you trying to prove that thing to begin with? Presumably it's a step along some journey, right? Maybe the journey is where you need to start deriving your satisfaction from, then.
I’m frankly a bit disappointed that Tao published the blog post I linked to above.
Telling OpenAI they shouldn’t test frontier math on their internal models is just plain nuts and illogical. It’s surprising that the advanced math community can lack so much logic.
There are lots of similarities currently within software engineering and mathematics and a lot of analogies which apply to Software engineering apply to mathematics and vice versa.
> At least for now someone still has to decide what to prove and why. Like why are you trying to prove that thing to begin with? Presumably it's a step along some journey, right? Maybe the journey is where you need to start deriving your satisfaction from, then.
Beautifully said. I had been thinking about the same thing and the analogy b/w CS and mathematics and these were some that I had found:
1.) to prove/disprove from the proofs that OAI created, you needed an mathematician to do so and OAI had to withdraw three mathematical proofs.[0]
But it was only because an expert within the field could verify if it was true or not, I feel as if software engineering is the same as well. We are/can be paid to prove/disprove if a software is working as intended or not.
2.) for someone to be that said mathematician who disproved it, he had to learn the basics of mathematics and multiple branches of it to then perhaps specialize in one thing that he most strongly resonated with and within all this learning, there was some struggle definitely involved. They had to learn algebra etc.,
this analogy can also extend to how we teach children algebra/calculations and other things even though we have had calculators for a long time, yet, we teach children how to do calculations because it is still valuable enough and either teaching maths can help them perhaps in future make a mathematician or it can help them be less reliant on simply calculators and more confident on on the spot calculations and help them within this skill.
As knowing calculation has become the norm rather than exception, even though we have calculators. In fact knowing how to do calculation by hand can perhaps better help you write a problem to calculator. Knowing the technical aspects of CS can help you express a problem to AI with much more depth and effectiveness as well.
One thing with programming is that LLMs are actually pretty bad at it, and use of them for that purpose is driven entirely by hype. I'm not remotely qualified to judge the mathematical realm, but mathematicians seem to think that LLMs can do the math well. So there's a significant difference there. In programming, we will see a major pull back from LLMs as the abysmal quality of vibe software becomes too prominent to ignore, but the same might not happen in math if they can actually do the job.
I know most people here are already well into their careers so they might not even notice at this point, but the CS job market isn't just 'going through a tough spot' for new grads. Entry Level Hirings have literally collapsed. Roughly 65-75% of entry level hiring dissapeared since 2019 at major segments of the tech industry, and it's much worse than the overall industry average. Software Engineers are still getting hired, but the proportion that should be juniors is catastrophically less.
The difference between math and all the other professions AI has been obsoleting is that math is actually not a useful pursuit. The tired old line about math someday discovering something that will be useful in an entirely unrelated field is mostly baloney, and the problems that mathematicians spend their years broadly have no real world applications at all. A modern day mathematician is much closer to a monk than a productive worker.
I'm not a mathematician. I was a college math major, got a PhD in physics, and still enjoy math.
It's not like "should I get a PhD" is a new question. It's not unheard of for unpredictable events to drastically affect the career prospects of PhDs in my field. During my lifetime:
1. Mandatory retirement of professors was ruled illegal. While good for civil rights, it created a 10+ year gap in faculty retirements.
2. End of the cold war.
3. Transition of college teaching from tenured professors to gig workers, aka "adjuncts."
The one constant during this time was the perpetual optimism of the faculty for the employment prospects of PhDs. "There will always be a need for physicists." My dad, also a PhD, confirmed that this goes back as early as the 1950s.
I would add one question to the student's letter: What are the ethics of AI and its owners?
Indeed, I left academia immediately after finishing my degree. Now in my 60s, I'm at the "senior staff" level of a F500 company that makes measurement equipment, so I've done OK even if others have done better.
One thing that helped me a lot is that both of my parents were industrial scientists, so the idea of life beyond academia wasn't foreign to me.
Not even a compiler, but a search engine. That's what LLMs are: an NLP search engine. Maybe even slightly worse than that since the only option is the "I'm feeling lucky" button.
why invoke move 37 as evidence against AI creativity? Move 37 was made by an AI. Is it so insane to think that the same selfplay training regimen that AlphaGo underwent to make that move couldn't be applied to LLMs trying to solve math problems?
> However, a proof of the Riemann hypothesis, say, may need new ideas that are strictly outside of the convex hull of current mathematical ideas
[...]
> After I finished writing this blog post, and had already sent it to Terry, OpenAI released a huge treasure trove of results in mathematics [including] the resolution of the so-called quasi Riemann Hypothesis
We ought to all be careful about underestimating the speed and magnitude of the change that is coming.
> if you are a student who is passionate to learn what is new and what is left to do, then a PhD is definitely the right path for you
This is an awful lot of confidence to put behind career advice in a wildly changing world. Markets are real and tradeoffs bite. We're not in gay communist space utopia yet.
I am not sure what you are trying to imply, but quasi-Riemann hypothesis is in fact not Riemann hypothesis and resolution of the former, while truly fucking amazing (it wouldn't be an exaggeration to say it is a discovery of century), gives no hint how to resolve the later.
> On the other hand, none of the proofs so far seem to contain “alien ideas,” a move-37, or completely novel arguments or new concepts that were not present in the literature in some form or another.
I have the same optimism as Prof Tao. That said, I can understand why so many mathematicians have been so upset or stressed out. It turned out much of the mathematical work is about clever combination of existing methods - this already requires enormous amount human ingenuity and years of dedicated learning. Unfortunately, or maybe fortunately, AI can be very good at knowledge transfer and finding combination of existing ideas to solve seemingly impossible problems. Even though mathematicians are extremely smart and capable, only a small number of them are capable of truly inventing "alien ideas", discovering new ground-breaking mathematical structures, or coming up with new problem-solving techniques. That is, AI can eat many mathematicians' cake. That said, I'm still hopeful. Mathematicians still understand mathematics deeply. If someone can prompt AI to solve an important problem, that person is more likely a good mathematician than an average joe like me. So, I think mathematicians do have a bright future: leverage AI, and make more and bigger math discoveries. It's still the same north star: we must know, and we shall know. It's just that with AI, we will know sooner and more.
It's quite amusing that this mistake gets made on every single one of these guest posts when the first words on the page are "This is a guest post by Álvaro Lozano-Robledo" and the first paragraph starts with "I would like to give Terry my heartfelt thanks for giving me the opportunity to contribute a post to his blog." It's pretty clear that everyone is skimming these at best. Maybe he should put the disclaimer in a bigger font or something.
It's interesting, what would be Tao's views on the creativity question. Like, I don't find the empirical observation of LLMs now, the notion these proofs are mostly "just" sophisticated grinding, to be a strong indicator of what could/couldn't happen.
It's pretty much the same story with software development. There are two camps and the bigger camp got hit hard. Nowadays I feel entitled to use the "software engineer" title. Essentially, all the decisions-making and nuanced trade-offs which AI is not capable of making; that's engineering. AI can only do programming and code analysis... At least so far.
That said, in the past couple of month, AI has gained a lot of ground on the 'project management' front in terms of planning capability for a lot of simple to medium-complexity projects.
I was recently working on a project where I had to come up with a novel solution to a problem the company was facing. The problem has been solved before, but in suboptimal ways that wouldn’t work for this specific company. I came up with a pretty interesting solution, one that combines different statistical methods, and was a little more complicated, but gave better results.
I asked Claude what it thought about the technique. It hated it. It loathed the idea that I was not sticking with traditional methods that were pointed to in papers and pre-existing open source packages. Even after talking through the idea and how it was better and showing it the results, it was weirdly hesitant to admit it was a valid method, simply because it seemed nervous that the idea was novel.
I don’t have a real moral to the story, I just thought I would add my anecdote here.
Interesting! My experience is that Claude has a tendency to always suggest the true and tried, but likes off-center ideas when I present them. Fable especially has a "yes, and!" attitude where it understands and plusses the idea.
I had a similar experience with not being able to convince Claude of a different approach to a problem. Then I shared the idea with Codex and it liked it. As another commenter said, you can have the same thing happen with humans.
I don’t think anything is going to change for actual mathematicians. The only jobs they were getting were in universities and schools as teachers and professors anyways and you’re always going to need those.
You tell them to cancel their Astra subscriptions and get to work. Hundreds of mathematicians, including Peter Scholze, have already agreed to not use AI at all. There’s absolutely no reason to help train an AI that will just make your own life more miserable.
That's going to be a very short term solution, no?
You can't really swear off looking at problems solved by AI, or keep moving away from fields whenever AI starts contributing, and have a viable career.
Today's frontier models, are everybody's local models in the near future.
If this were a change that had expected stalls or reprieves it would be different.
Doing mathematics looks to be the next chess, er, I mean protein folding challenge. The latter's computation demands have dropped significantly.
You have many fields that will be protected from ai, or even enabled by it - if I could bet, sociology will be a hot new field that will by definition still need humans always, but will gain both speed and significance.
There are an infinite number of problems that require a unique proof (you can prove this btw). Yes maybe you need to work harder to find a more obscure problem now but why give up the fight already? I believe that students should have the right to intellectually stimulating education. If you really need to use AI, go for something that isn’t trying to put you out of business or destroy your right to a decent education, like a local open weights LLM.
A friend of a friend had a grant project written up for the next 10 years, with habilitation on the horizon.
His end goal was one of the things released in the recent OpenAI publications. It took a model 3.5 hours to do.
Things are wild now on math and theoretical faculties now.
And for theoretical fields labs don’t need human data - synthetic one works well if not better.
The best thing you can do if you’re a scientist, imho - wrapup whatever grant you have now asap with ai, use ai to get more grants if possible, and spend the rest od the year learning what new science you can do with the new tools and participate in creating the new paradigm in your field.
That’s when you’re an established scientist. If you’re new in the field then this is the most exciting time you could wish for - an opportunity to make a name for yourself.
and that generation was educated without AI. going to be interesting how future generations of mathematicians will fare who have been using AI from day 1. I suspect they'll be simply much worse at math.
That generation will say "How on earth people could do this without AI", just like we have said "How on earth people could do this without the internet".
Without agents - sure, 100x slower. But without LLMs? Some jobs I do would be borderline impossible.
I gave an LLM recently a job of analysing past sessions to estimate how much time it would take a human to do the jobs requested.
Ingesting 600k word document and drawing conclusions/patterns from it is something I routinely give to Terra. It would take a human 33h non stop to read it. But finding patterns with good precision - personally I wouldn’t be able to.
Imagine you want to understand how well a certain strategy works for finding antiderivatives. You can get meaningful insight into that question by constructing synthetic examples - complicated functions which you intentionally designed to have or not have an antiderivative - and studying how well your strategy works when applied to those functions. This won't necessarily be worse and could easily be better than testing random functions you found which just so happen to be discussed in existing literature.
In areas where you can easily recognise valid data (math, programming, gaming, to some extent also visual and mechanical), the synthetic data can be way broader. Using manmade data as input speeds up early stages of the process but is not really necessary. For later stages you run out of manmade data, and it’s quite easy to generate new one.
With writing/art it’s different because there you need and always will need human taste to judge. To some extent it can be overcome - you train models to be judges on what output is good in taste, but it can only go so far.
In principle you could have a training system that just gets hello world as the only program and develop all the algorithms and structures that we have now and more.
In art, if you gave it a single cave drawing, it would either stick to cave drawings or - more likely - develop its own art that would be good according to internal taste, but non enjoyable to humans.
I think there is a bug misunderstanding if this in the art community. Most people there seem to assume AI needs human data to be trained. To create something that people like yes, to create something that works in math/programming/science/engineering - not really.
> The best thing you can do if you’re a scientist, imho - wrapup whatever grant you have now asap with ai, use ai to get more grants if possible, and spend the rest od the year learning what new science you can do with the new tools and participate in creating the new paradigm in your field.
> That’s when you’re an established scientist. If you’re new in the field then this is the most exciting time you could wish for - an opportunity to make a name for yourself.
s/scientist/mathematician/g
There are plenty of fields of research where AI is not nearly as destructive as this, currently. Most experimental science is pretty safe for a while.
You got this right, these are incredibly exciting times. Take your friend of a friend's as an example. AI's are trained with existing information, so obviously whatever he expected to discover was already out there. It took the machine 3.5 hours to find and sort out something that would take a human 10 years. Now your F of a F can dedicate the next 10 years to do what the machine cannot do: find meaning and insight.
> AI's are trained with existing information, so obviously whatever he expected to discover was already out there.
I've wondered about that myself. Does the fact that LLM's were able to solve these problems so easily indicate that most of the solution already existed, scattered in the training data, and the LLM was simply able to put together the pieces?
I think it's an interesting conjecture but nothing more without proof.
I think we should apply AI into human, so we could be better and then start doing more advanced stuff required for the Universe explorations. Current AI/LLMs can be considered as first step to that. Take and look about our history. We could never achieve so many if we stopped doing great things because of <any reason>.
Why do we ever use PCs, smartphones, Internet, electricity, medicine and so on? Why do we live? For what?
I don't get this kind of thinking. I was talking to my 13-year-old last night about web tech and I had Gemini generate a quick explainer on how CGI-based web servers work because I thought it could explain it a little bit more compactly than I could.
Similarly, in a computer science curriculum, you could use LLMs to visualize various cache invalidation strategies, sorting algorithms, etc. As I've said here before, when I was at university, I would have Tuesday and Thursday classes that were 80 minutes long. I attended a research university, so the professors were not big on teaching what things were for. They would just sort of dive in and start talking while they were writing on the board. This led to a situation where we'd get 10 or 15 minutes into an 80-minute lecture, and I would have lost the thread, missed some step in how they'd proceeded with the proof, etc. Having an LLM there to ask for the missing piece so that I could keep up with the lecture would have been invaluable to my education.
All the LLM hate feels a lot like "screen time", where, for some reason, it took decades for people to realize it was what you were doing on the screen that mattered, not the fact that you were looking at a screen. Similarly, it's what we do with the LLMs that matters, not the fact that we're using one.
To be fair I see two separate ways CS students use LLMs, one is to just complete assignments in the same way you could by asking someone else to do it, the other way is by getting an LLM to explain certain concepts in another way, or create more examples of them to aid in understanding quicker/in more depth. I actually think in the 2nd case they can be incredibly useful and can assist in creating a deeper understanding, but in the 1st case they obviously create fundamental gaps in understanding.
In a way it's on the student to make sure whatever they're doing, they actually understand it, or come exam season the gaps in their knowledge sans Claude will become quite obvious.
way back in the dark ages when I was in college I used the hell out of back files. we all did. Over the years the teachers had added more and more homework because the students using backfiles were answering more and more homework correctly. The teacher swore that home work should take about 3 hours a week. the smartest guy I knew did it once without backfiles, because the fraternties, the holders of the backfiles, were all at an off campus event. it took 17 hours. he just gave up and used the backfiles.
I don't think homework load has adjusted in some classes. it was hard to do the homework and learn honestly, it was just copying and finding previous answers. it actually detracted from the learning experience.
it would be nice to have something that could explain it to me, like a tutor. My mom paid for me to have a private tutor to get me through a compressed calc 3 summer class. Best I scored in a math class other than diffy-q.
so I think it's, like everything, on the student. but fuck the homework load at some places, it's not a good solution.
Students are there to study to get the permission to use the LLM. Maybe they can use it for querying information but not to do their job (which is thinking).
Yeah, sometimes the comments that are easiest to vote for don't actually have serious things to say. In this case, it's in tension with the article we're replying to.
> Undoubtedly, LLMs are already an incredibly powerful tool. If used correctly, and if we set up sensible academic conduct expectations around the use of LLMs, these tools can accelerate progress in our discipline unlike in any previous era. I fully expect that we, the community, will adapt and adjust to this new period, and we will harness these tools to achieve truly great things that just a few months ago seemed far out of reach. And I fully expect that human mathematicians will be front and center in these wonderful achievements to come. I will add reasons that support my optimism below.
One natural step of AI usage, the way I see it, is that even mediocre researchers can use AI as a harness to become prolific researchers.
So if you're part of a pure "human only" researchers that publish 1/10th of what the rest are doing, how are you going to survive?
The more prolific researchers will eat your grants for lunch.
This is exactly the same thing a lot of software devs are going through now. AI models have lifted up EVERYONES ability to produce code, so there's simply no premium anymore for those that will only code by hand. And the people paying their salary are asking why they aren't more productive, when even business analyst Joe is pumping out new products left and right.
People will pay good money for artisanal, hand-crafted clothing, or watches, or cars, or whatever, but I doubt that the market for artisanal mathematics is now or will ever be very large.
It's just stubbornness, in the end, that is manifesting as a kind of gatekeeping Luddism. I see the same thing in my Mastodon feed from folks in the software engineering world, particularly people who have a great love for writing code and hold it as a core part of their identities. I understand being irritated at a machine having been trained on (potentially, inter alia) your work with no compensation to you, and now it is perhaps better at (some of) what you do than you are; or maybe companies are now making a lot of money off of something that benefited from your contributions; but I find it extremely improbable that we will be able to refuse our way out of further progress now that so much money has been invested and so much momentum has been gathered.
I don't think I feel any discomfort about a machine being better at anything that I can do than I am at it. Maybe it's because I'm a mediocre person. I personally think that if there's something that I'm better at than a machine, then we need to build a better machine. Imagine what I'd be able to do if I used that machine!
There are many things you are better at than a machine. Giving love for example. I’d bet your dog or a child would rather spend 5 seconds with you than 5 hours with an LLM.
So others will write their PhD thesis faster, will publish more, you will not publish so you will perish. The rules of the game in this business are brutal. One can count on inventing something entirely new that AI will not be able to figure out, but this is a really high bet.
It’s more like avoiding looking at the answer key while doing homework to be able to pass an exam and to understand the material better, especially for the grad students who haven’t gotten their PhDs yet.
Just reading the letter from the student is depressing, and then you think of all the students who are not in a first world nation and how much worse their chances are.
God, the absolute decimation that is going to come for middle income countries.
Hell, right now the people quitting frontier labs are rediscovering trust and safety isues, like language parity problems, except all the work being done is in the first world and exported back to the rest.
That not all humans are born equal. Equal inquisitive, equally providing of original thought and inspiration. The new, the step back and above is where AI still has weak spots and needs help.
This is a great moment to be a hassadeur, a madmen, somebody who wants to jump of the cliff with just a rope on his feet.
If you are that- great times- otherwise- good luck.
158 comments
[ 0.27 ms ] story [ 23.4 ms ] threadAs a student, I am actively making decisions which will shape my career for the next forty or so years. At a minimum a discussion like this should acknowledge the possibility that the current rate of AI progress continues apace. I understand the desire to be encouraging, but the the best preparation for students involves the consideration of possibilities that current mathematicians negligently paper over.
Of course it will get better at exposition than humans. And it will prompt itself in due time.
Whoops, people stopped caring about your sport!
> Just about anyone who has played a sport has heard that they have to target where the ball is moving to, not where the ball is right now. In this regard I am consistently disappointed with the commentary that mathematicians have been producing lately.
Reading this it feels like you haven't played a sport.
https://news.ycombinator.com/newsguidelines.html
Sincerely,
An "old" person in their 30s.
But nothing is certain. I am trying to avoid overspecialization and remain flexible, just as you advise. If this path fails I might become a paramedic or physiotherapist.
Go into things expecting you might have to change careers.
The sports analogy holds; the future location of the ball may be uncertain, but that is the student's target, and not overcommitting to the present location of the ball. The commenter is expressing disappointment that the commentary seems to be reacting to the present too much.
I think the problem is the reverse. We no longer can make good predictions for four years ahead, certainly not eight.
And yet, people have to make choices - so they need advice and some kind of hope of a guess.
> you might have to change careers.
I hope this is some dust dry humor!
Sure you wouldn’t predict what you would work on exactly. And there were better tools, better communication and so on sure, but no disruptive innovations or seismic shifts really.
Some of those complaining today are the ones who keep saying "LLMs are just stochastic parrots predicting the next word", "how can AI do math if they can't even count the number of 'r's in strawberry?", etc.
It doesn't take a genius to realize AI will start getting better and possibly become a "threat" even before all this mathocalypse stuff happened.
For all the arrogance the hard sciences have, in the end they are just as human as their humanities counterparts. They just thought they were above becoming worthless because science was elevated above the softer sciences.
But the ball is flying out of the court and into the abyss.
But the post does do this: "[...] even if their capacity becomes far superior, there will always be a need for mathematicians at all levels to guide research in paths that make sense for humans to walk (not run)."
> Of course it will get better at exposition than humans.
I'm not so sure about this, because I don't know how one would train towards better exposition as it's not easy to check for good or bad exposition at scale.
I expect it will remain true into the future (10+ years). I have moderately high confidence (75% or higher) in this prediction.
I use frontier AI models in my work all the time. I think they accelerate my work by helping me understand faster and prompt better.
The models are most useful and most productivity-enhancing in the hands of experts and in the area of their expertise.
I don’t expect a jobs apocalypse, not even in math.
It's definitely (again, to the article's credit as it points out) up in the air whether LLMs have inherent limitations. But I can't fathom how someone can say "I don’t expect a jobs apocalypse, not even in math." Because if AI does end up surpassing humans, what exactly will there be left for humans to do? And even if they don't surpass humans, there will still be a jobs apocalypse. Probably the majority of people today work in jobs where AI can surpass their performance.
Simple case in point: years ago I used TurboTax to do my taxes, but I eventually needed to hire a CPA because I had some complicated situations that TurboTax couldn't handle, and I also needed some tailored advice. I ended up finding a great CPA. Now though, AI agents can literally do 100% of the job I hired my CPA to do, including asking me the right questions and offering advice. I'm sure there may still be tasks for a CPA in the corporate world, but for personal taxes, I literally can't imagine a CPA providing value over what an AI agent can provide. And to emphasize, I would have probably considered that a ludicrous statement a year ago with all the mistakes LLMs made. But so many of those mistakes have been fixed, and you can get better-than-human performance by having agents check each other's work. And AI will only get better when tax season rolls around next year.
But it's a construction based on very basic properties of the real world, at least on those properties we accept and perceive as very general and we use them as building blocks.
I don't necessarily think that that is true, as almost every social science has been turned into maths one way or another, beginning with Economics. Heck, even mundane stuff like sports (think football/soccer) has now been absorbed by data, which you can see as applied maths.
Yes, I do know that advanced maths doesn't currently explain Arteta's system at Arsenal but we're generally talking about that same semantic area of interest, i.e. if the interest on maths were to subside as a result of AI taking that interest away from us then we will have no more data/stats-focused Artetas in the future (and possible no AI-like thingie also).
I’m saying that this wouldn’t have happened had our society not become “mathematics-ised” at some point (I’m talking about said society’s technical and scientific elites, of course). I’m also saying that outsourcing our “mathematics-mind” to AI (which seems the current process we’re now part of) will not allow for similar “mathematics-ing” to take place into the future. I’ll come with an even stronger claim and say that we risk losing the “mathematics-ing” we now associate with most of our sciences.
In a future where learning math has about the same economic value as learning to play chess or the flute, you can see why many mathematicians aren’t very excited about their chosen career paths.
> With the latest ChatGPT models, these problems are more equivalent to homework questions: the answer is `in the back of the book.’ I am not discovering new solutions. Instead, I am working on problems whose answer exists and is simply waiting to be retrieved by a user of the model. In fact, I mentioned a problem that I was interested in working on to my advisor and he informed me that he and a collaborator had completely resolved it using ChatGPT – they have no plans to write up the result, so it will sit there until another `researcher’ pulls the proof slot machine.
There is another good point that the most tenured researchers have a sense of what problems are most worth exploring and therefore are more likely to feel excitement than younger researchers:
> I have also heard the contention that math research has `gotten more exciting,’ mainly from established researchers. They have decades of open problems that they care deeply about and want to see resolved. I have no such problems.
“42.”
But no one could understand what the answer even meant. So they designed a computer to build the question itself again, and that was Earth. Then the story begins with Earth being destroyed because of a cosmic highway problem (I won’t write more since that would be a spoiler). In the opening background of this work, I found it interesting that after calculating for 7.5 million years, they didn’t even know what they had originally been asking. The story now feels similar to that story from back then.
At least for now someone still has to decide what to prove and why. Like why are you trying to prove that thing to begin with? Presumably it's a step along some journey, right? Maybe the journey is where you need to start deriving your satisfaction from, then.
I’m frankly a bit disappointed that Tao published the blog post I linked to above.
Telling OpenAI they shouldn’t test frontier math on their internal models is just plain nuts and illogical. It’s surprising that the advanced math community can lack so much logic.
> At least for now someone still has to decide what to prove and why. Like why are you trying to prove that thing to begin with? Presumably it's a step along some journey, right? Maybe the journey is where you need to start deriving your satisfaction from, then.
Beautifully said. I had been thinking about the same thing and the analogy b/w CS and mathematics and these were some that I had found:
1.) to prove/disprove from the proofs that OAI created, you needed an mathematician to do so and OAI had to withdraw three mathematical proofs.[0]
But it was only because an expert within the field could verify if it was true or not, I feel as if software engineering is the same as well. We are/can be paid to prove/disprove if a software is working as intended or not.
2.) for someone to be that said mathematician who disproved it, he had to learn the basics of mathematics and multiple branches of it to then perhaps specialize in one thing that he most strongly resonated with and within all this learning, there was some struggle definitely involved. They had to learn algebra etc.,
this analogy can also extend to how we teach children algebra/calculations and other things even though we have had calculators for a long time, yet, we teach children how to do calculations because it is still valuable enough and either teaching maths can help them perhaps in future make a mathematician or it can help them be less reliant on simply calculators and more confident on on the spot calculations and help them within this skill.
As knowing calculation has become the norm rather than exception, even though we have calculators. In fact knowing how to do calculation by hand can perhaps better help you write a problem to calculator. Knowing the technical aspects of CS can help you express a problem to AI with much more depth and effectiveness as well.
[0]: https://news.ycombinator.com/item?id=50002650
You haven't tried the latest gen of frontier models, I take it? These things aren't just hype.
https://digitaleconomy.stanford.edu/news/canariesaug26/?trk=...
1. There was no overhiring in 2019.
2. It's much worse at the entry level than the overall industry.
3. We're years out of COVID. It's not relevant anymore. Not sure when people will stop blaming it for any and everything.
It's not like "should I get a PhD" is a new question. It's not unheard of for unpredictable events to drastically affect the career prospects of PhDs in my field. During my lifetime:
1. Mandatory retirement of professors was ruled illegal. While good for civil rights, it created a 10+ year gap in faculty retirements.
2. End of the cold war.
3. Transition of college teaching from tenured professors to gig workers, aka "adjuncts."
The one constant during this time was the perpetual optimism of the faculty for the employment prospects of PhDs. "There will always be a need for physicists." My dad, also a PhD, confirmed that this goes back as early as the 1950s.
I would add one question to the student's letter: What are the ethics of AI and its owners?
This is a huge, and likely permanent, change.
https://philip.greenspun.com/careers/women-in-science
Basically, it's arguing that science (really academia) is generally an awful career path for most Americans.
One thing that helped me a lot is that both of my parents were industrial scientists, so the idea of life beyond academia wasn't foreign to me.
What actual industries where you could get a phd in, died?
If I was to ever suggest one, it would be Philosphers.
Yet they have found ways to get tenure and/or other jobs for as long as the field exists.
[...]
> After I finished writing this blog post, and had already sent it to Terry, OpenAI released a huge treasure trove of results in mathematics [including] the resolution of the so-called quasi Riemann Hypothesis
We ought to all be careful about underestimating the speed and magnitude of the change that is coming.
> if you are a student who is passionate to learn what is new and what is left to do, then a PhD is definitely the right path for you
This is an awful lot of confidence to put behind career advice in a wildly changing world. Markets are real and tradeoffs bite. We're not in gay communist space utopia yet.
I have the same optimism as Prof Tao. That said, I can understand why so many mathematicians have been so upset or stressed out. It turned out much of the mathematical work is about clever combination of existing methods - this already requires enormous amount human ingenuity and years of dedicated learning. Unfortunately, or maybe fortunately, AI can be very good at knowledge transfer and finding combination of existing ideas to solve seemingly impossible problems. Even though mathematicians are extremely smart and capable, only a small number of them are capable of truly inventing "alien ideas", discovering new ground-breaking mathematical structures, or coming up with new problem-solving techniques. That is, AI can eat many mathematicians' cake. That said, I'm still hopeful. Mathematicians still understand mathematics deeply. If someone can prompt AI to solve an important problem, that person is more likely a good mathematician than an average joe like me. So, I think mathematicians do have a bright future: leverage AI, and make more and bigger math discoveries. It's still the same north star: we must know, and we shall know. It's just that with AI, we will know sooner and more.
This is not written by Tao, but by Álvaro Lozano-Robledo.
That said, in the past couple of month, AI has gained a lot of ground on the 'project management' front in terms of planning capability for a lot of simple to medium-complexity projects.
I asked Claude what it thought about the technique. It hated it. It loathed the idea that I was not sticking with traditional methods that were pointed to in papers and pre-existing open source packages. Even after talking through the idea and how it was better and showing it the results, it was weirdly hesitant to admit it was a valid method, simply because it seemed nervous that the idea was novel.
I don’t have a real moral to the story, I just thought I would add my anecdote here.
https://www.ahmath.org/members
You can't really swear off looking at problems solved by AI, or keep moving away from fields whenever AI starts contributing, and have a viable career.
Today's frontier models, are everybody's local models in the near future.
If this were a change that had expected stalls or reprieves it would be different.
Doing mathematics looks to be the next chess, er, I mean protein folding challenge. The latter's computation demands have dropped significantly.
When things really start going downhill, if we are still in the current mess, it will grease the the wheels, and the brakes.
His end goal was one of the things released in the recent OpenAI publications. It took a model 3.5 hours to do.
Things are wild now on math and theoretical faculties now.
And for theoretical fields labs don’t need human data - synthetic one works well if not better.
The best thing you can do if you’re a scientist, imho - wrapup whatever grant you have now asap with ai, use ai to get more grants if possible, and spend the rest od the year learning what new science you can do with the new tools and participate in creating the new paradigm in your field.
That’s when you’re an established scientist. If you’re new in the field then this is the most exciting time you could wish for - an opportunity to make a name for yourself.
I gave an LLM recently a job of analysing past sessions to estimate how much time it would take a human to do the jobs requested.
Ingesting 600k word document and drawing conclusions/patterns from it is something I routinely give to Terra. It would take a human 33h non stop to read it. But finding patterns with good precision - personally I wouldn’t be able to.
How could that be true? I am a mathematical rube, but surely data that works "better" than real data should be extremely suspect?
With writing/art it’s different because there you need and always will need human taste to judge. To some extent it can be overcome - you train models to be judges on what output is good in taste, but it can only go so far.
In principle you could have a training system that just gets hello world as the only program and develop all the algorithms and structures that we have now and more. In art, if you gave it a single cave drawing, it would either stick to cave drawings or - more likely - develop its own art that would be good according to internal taste, but non enjoyable to humans.
I think there is a bug misunderstanding if this in the art community. Most people there seem to assume AI needs human data to be trained. To create something that people like yes, to create something that works in math/programming/science/engineering - not really.
> That’s when you’re an established scientist. If you’re new in the field then this is the most exciting time you could wish for - an opportunity to make a name for yourself.
s/scientist/mathematician/g
There are plenty of fields of research where AI is not nearly as destructive as this, currently. Most experimental science is pretty safe for a while.
I've wondered about that myself. Does the fact that LLM's were able to solve these problems so easily indicate that most of the solution already existed, scattered in the training data, and the LLM was simply able to put together the pieces?
I think it's an interesting conjecture but nothing more without proof.
Why do we ever use PCs, smartphones, Internet, electricity, medicine and so on? Why do we live? For what?
Sometimes HN is crazy and out of touch.
Maybe there should be a class on how how to use LLMs too.
But leaning computer science using LLMs all the time will be like learning to ride a bike using training wheels, and never taking them off.
I expect people that read things here to be literate. i choose optimism.
Or like learning to add fractions by looking up the answer key instead of actually struggling to figure it out.
Similarly, in a computer science curriculum, you could use LLMs to visualize various cache invalidation strategies, sorting algorithms, etc. As I've said here before, when I was at university, I would have Tuesday and Thursday classes that were 80 minutes long. I attended a research university, so the professors were not big on teaching what things were for. They would just sort of dive in and start talking while they were writing on the board. This led to a situation where we'd get 10 or 15 minutes into an 80-minute lecture, and I would have lost the thread, missed some step in how they'd proceeded with the proof, etc. Having an LLM there to ask for the missing piece so that I could keep up with the lecture would have been invaluable to my education.
All the LLM hate feels a lot like "screen time", where, for some reason, it took decades for people to realize it was what you were doing on the screen that mattered, not the fact that you were looking at a screen. Similarly, it's what we do with the LLMs that matters, not the fact that we're using one.
In a way it's on the student to make sure whatever they're doing, they actually understand it, or come exam season the gaps in their knowledge sans Claude will become quite obvious.
I don't think homework load has adjusted in some classes. it was hard to do the homework and learn honestly, it was just copying and finding previous answers. it actually detracted from the learning experience.
it would be nice to have something that could explain it to me, like a tutor. My mom paid for me to have a private tutor to get me through a compressed calc 3 summer class. Best I scored in a math class other than diffy-q.
so I think it's, like everything, on the student. but fuck the homework load at some places, it's not a good solution.
Maybe "people who need to be able to make decisions about rocket use should understand them deeply not just ride on them"
- Can get you somewhere very quickly.
- Requires technical knowledge and ability to use correctly.
- If you're not careful, you will get to the wrong place, and will probably blow up by the time you arrive (if not before).
P.S. Your analogy is dumb. (Just arguing at your level here.)
However, to say they "should never an LLM" entirely is going to make them unemployable.
Being the greatest [anything] alive probably puts you at an extreme risk of being "out of touch". Is LeBron James out of touch?
> Undoubtedly, LLMs are already an incredibly powerful tool. If used correctly, and if we set up sensible academic conduct expectations around the use of LLMs, these tools can accelerate progress in our discipline unlike in any previous era. I fully expect that we, the community, will adapt and adjust to this new period, and we will harness these tools to achieve truly great things that just a few months ago seemed far out of reach. And I fully expect that human mathematicians will be front and center in these wonderful achievements to come. I will add reasons that support my optimism below.
This seems like a reasonable take.
One natural step of AI usage, the way I see it, is that even mediocre researchers can use AI as a harness to become prolific researchers.
So if you're part of a pure "human only" researchers that publish 1/10th of what the rest are doing, how are you going to survive?
The more prolific researchers will eat your grants for lunch.
This is exactly the same thing a lot of software devs are going through now. AI models have lifted up EVERYONES ability to produce code, so there's simply no premium anymore for those that will only code by hand. And the people paying their salary are asking why they aren't more productive, when even business analyst Joe is pumping out new products left and right.
It's just stubbornness, in the end, that is manifesting as a kind of gatekeeping Luddism. I see the same thing in my Mastodon feed from folks in the software engineering world, particularly people who have a great love for writing code and hold it as a core part of their identities. I understand being irritated at a machine having been trained on (potentially, inter alia) your work with no compensation to you, and now it is perhaps better at (some of) what you do than you are; or maybe companies are now making a lot of money off of something that benefited from your contributions; but I find it extremely improbable that we will be able to refuse our way out of further progress now that so much money has been invested and so much momentum has been gathered.
I don't think I feel any discomfort about a machine being better at anything that I can do than I am at it. Maybe it's because I'm a mediocre person. I personally think that if there's something that I'm better at than a machine, then we need to build a better machine. Imagine what I'd be able to do if I used that machine!
You should also tell them “better luck next time” as they have to create and find their own luck unfortunately.
This game was meant to be unfair because the ones who have “won” want to keep it unfair for others, and cannot stand losing due to complete greed.
The truth is they (students) have to find a way to outsmart the incumbents. The best advice is to find your own strategy and listen to no-one.
God, the absolute decimation that is going to come for middle income countries.
Hell, right now the people quitting frontier labs are rediscovering trust and safety isues, like language parity problems, except all the work being done is in the first world and exported back to the rest.
My own suspicion/guess is that AI can break out of the bounds of the convex hull in math, and that’s going to become apparent soon.
This is a great moment to be a hassadeur, a madmen, somebody who wants to jump of the cliff with just a rope on his feet.
If you are that- great times- otherwise- good luck.