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 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.
If AI continues to improve at the current rate for 40 years then there is unlikely to be such a thing as a career.
Manual labor and trades is likely to be one of the last things to go, so the whole "dropout of school and go into trades" crowd may have been more correct than ever, though their original reasoning was not.
> Manual labor and trades is likely to be one of the last things to go
Yes, this is likely true. Plumber & Electrician on existing construction will likely take a long time (relatively speaking) to automate. And even if we end up being able to do it, it could be that humans will be the cheapest option when it comes to a lot of manual jobs especially given that there will be a large supply of unemployed humans in many scenarios.
The challenge is not will robots fix toilets but will the economics support it if we end up with most of the world unemployed and without careers. Yes of course we can hand wave basic income and government intervention but reading the room (e.g. last 70+ years of US labor and capital trends) it’s not the most likely outcome, even if it is the most game theory optimal for all.
Realistically the outcome of highly capable advanced AI is an economic dark age more than anything else.
I read The Mountain in the Sea by Ray Nayler a couple years back. In it AIs were running ocean fishing fleets. They figured out that robots were too expensive and prone to malfunction at sea, so they would have the robots kidnap people on shore, drug them and then have them wake up on the ships too far out in the ocean to be able to escape and thenm tell them they could get back home if they work.
They most certainly will, for at least two reasons. First, jobs aren't just about raw productivity, they're also about responsibility / liability diffusion. A CEO managing an army of agents is possibly on the hook if anything goes wrong - for example, if one of the agents does tax fraud or hacks a competitor. A CEO paying an army of employees, even if the employees are there just there to supervise agents, usually has little to worry about.
The second reason is that we often prefer humans to do it even if a machine can do it faster / cheaper. Sometimes, it's a status thing. For example, some people will buy Ikea furniture, some will have it custom made by a local craftsman. In fact, it's sort of the hallmark of the upper middle class that you can spend more for that human touch. Cupcakes from an artisanal bakery, private banker, etc.
Now, I don't think there's enough people who want to pay extra for human-made software. From the current trends in the industry, my impression is that we don't value our craft, so it would be surprising if others did. For mathematicians, I don't know; it's a bit painful to watch.
Some careers that will undoubtedly exist are the 'human manipulation' trades: politicians, salespeople, con artists, religious leaders, basically anyone who convinces people to feed them.
What specific direct advice would you give then if you could talk to someone who is currently studying math?
Do you know where “the ball” is going? Do these math students? Do the professors? It’s easy to say a vague nothing platitude like “target where the ball is going to” but such empty vague platitudes do not help anyone and just waste time.
I was recently trying to work out what advice I should be giving my 15 year old son, I asked various AI for their recommendations given "possible" AI time frame and it seemed pretty much study whatever you want but always add " and business". Investing in things that AI will just make better. However, in the case of AI improves robotics to the point it's better than humans... who knows. But all that may not pan out as reality. For me, AI is still making a lot of the same mistakes it was making last year, it's very marginally improved in some aspects (not quite sure how to categorize what its failing at, but its like an over eager worker who really doesn't understand what they are working on but super keen to do stuff). It's gotten a lot better at doing more from single prompts. But it feels like, for LLMs at least, they are bumping into limits. But it can be hard to see because at the same time they are actually doing more. More and more I feel like what I'm doing is getting more ambitious in what I'm asking and then a big cycle of correcting.
Then what? math, the kind LLMs are making "obsolete" is a pursuit unlike any other. What other field would tickle the kind of mind attracted by math the same way? I doubt they went into it for the money.
Literally what else? Either you believe the future is AI + Humans, in which case most professions are open to you or you don’t in which case almost none are.
Ironically, this very attitude could lead to AI creating one of the biggest slowdowns in progress in history.
It's this one, but different careers will diminish at different rates. Mathematics was already not financially well rewarded, and current AI is basically better than everyone in the field.
> this very attitude could lead to AI creating one of the biggest slowdowns in progress in history
I think there are enough young people with pre-AI experience that we'll probably develop superintelligence before they retire, so I don't really think a near-term lack of fresh talent will result in any significant slowdown.
Chess has always been "just a game". Sure, it teaches valuable life lessons, and it is fun to defeat your opponents through superior training and strategy, but it is "just a game".
Mathematics was historically "just a hobby". Much theoretical work is still "just a hobby". But, for the past few hundred years, humans that better understood the theory could find very profitable applications.
That is no longer true. Computers have solved mathematics, like they did chess two decades ago. A human will never find a better application than a computer, just like they will never find a better chess move. Yet people still play chess, and people still make money tutoring chess.
That is the future of humans in mathematics. Mathematics will become mathematics competitions. It will be just another intellectual sport. Sure, a sport that teaches valuable life lessons, but still "just a game". If you are going into mathematics today, your future career is as a competition mathematics coach.
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.
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.
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.
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?
That may be true, but mathematicians still need to eat, and for the they'll need jobs. 'Will there be more or less demand for mathematicians in the future?' and 'What will the work entail?' I think are valid concerns
the vast majority of people who pursued a maths degree, including myself here, didn't expect to compete with Terrence Tao or solve outstanding research problems.
We studied maths because it is interesting, because it teaches you how to think, you often pair it up with something that's more employable like economics or software development, or you go for a teaching career. Unless you're one of the few people who are heavily invested in cutting edge research I don't even see how new research tools change the profession.
An undergraduate maths education isn't going to change because you have people with computers churn out 400 page proofs. It's like being worried about 30 move Stockfish opening theory if you're a club level chess player.
Isn't there an 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.
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.
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.
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.
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.
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.
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.
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 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.
Manual labor and trades is likely to be one of the last things to go, so the whole "dropout of school and go into trades" crowd may have been more correct than ever, though their original reasoning was not.
Yes, this is likely true. Plumber & Electrician on existing construction will likely take a long time (relatively speaking) to automate. And even if we end up being able to do it, it could be that humans will be the cheapest option when it comes to a lot of manual jobs especially given that there will be a large supply of unemployed humans in many scenarios.
Why are meta glasses such a big deal? Because they need training data for manual labor.
Realistically the outcome of highly capable advanced AI is an economic dark age more than anything else.
A lot of people put a lot of their self-worth into their work and that’s a good thing that won’t change.
The second reason is that we often prefer humans to do it even if a machine can do it faster / cheaper. Sometimes, it's a status thing. For example, some people will buy Ikea furniture, some will have it custom made by a local craftsman. In fact, it's sort of the hallmark of the upper middle class that you can spend more for that human touch. Cupcakes from an artisanal bakery, private banker, etc.
Now, I don't think there's enough people who want to pay extra for human-made software. From the current trends in the industry, my impression is that we don't value our craft, so it would be surprising if others did. For mathematicians, I don't know; it's a bit painful to watch.
The honest answer is via politics not education, but this debate is often had in "apolitical" circles that try their best to ignore this.
Sincerely,
An "old" person in their 30s.
Do you know where “the ball” is going? Do these math students? Do the professors? It’s easy to say a vague nothing platitude like “target where the ball is going to” but such empty vague platitudes do not help anyone and just waste time.
Human-crafted religious items and artwork may be more appealing than machine-crafted replicas.
I assumed the student ultimately wants a career that is financially viable.
> Then what?
Literally anything else?
Ironically, this very attitude could lead to AI creating one of the biggest slowdowns in progress in history.
It's this one, but different careers will diminish at different rates. Mathematics was already not financially well rewarded, and current AI is basically better than everyone in the field.
> this very attitude could lead to AI creating one of the biggest slowdowns in progress in history
I think there are enough young people with pre-AI experience that we'll probably develop superintelligence before they retire, so I don't really think a near-term lack of fresh talent will result in any significant slowdown.
Mathematics was historically "just a hobby". Much theoretical work is still "just a hobby". But, for the past few hundred years, humans that better understood the theory could find very profitable applications.
That is no longer true. Computers have solved mathematics, like they did chess two decades ago. A human will never find a better application than a computer, just like they will never find a better chess move. Yet people still play chess, and people still make money tutoring chess.
That is the future of humans in mathematics. Mathematics will become mathematics competitions. It will be just another intellectual sport. Sure, a sport that teaches valuable life lessons, but still "just a game". If you are going into mathematics today, your future career is as a competition mathematics coach.
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.
> 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.
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 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.
We studied maths because it is interesting, because it teaches you how to think, you often pair it up with something that's more employable like economics or software development, or you go for a teaching career. Unless you're one of the few people who are heavily invested in cutting edge research I don't even see how new research tools change the profession.
An undergraduate maths education isn't going to change because you have people with computers churn out 400 page proofs. It's like being worried about 30 move Stockfish opening theory if you're a club level chess player.
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.
> 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
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.
[...]
> 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.
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.
“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.