What should we tell our students?

> 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.

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.

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.

Nice piece. I think the fundamental message that there is still demand for mathematicians and mathematics done by humans is fundamentally correct.

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.

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.

Treat the model as a compiler, not an oracle. You still must know how to specify the problem correctly, or you will only generate garbage faster.

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?

Learning mathematics is about understanding the world, not proving things.

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.

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.

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.

"Solve the Riemann hypothesis, make no mistakes"

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.

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.

> This is a guest post by Álvaro Lozano-Robledo.

just work hard as much as they can