[ashleyn]

They might be low-hanging fruit but two things immediately come to mind:

* As more of the small stuff is just proven for free, the more they can be used as a basis for other proofs. If you know something is true or false for certain, that can be a significant tailwind for the much harder, much more important problems. Fermat's last theorem looks deceptively simple and invited many failed amateur attempts at solving it, but Wiles' proof drew on a diversity of seemingly-distant subfields within mathematics that were better understood.

* What are aspiring math Phd's supposed to do, now that the bar is much higher these days? The net effect of this appears to be that we'll see far fewer, but far more elite math Phd's, potentially discouraging many young people from the field.

[ndriscoll]

The bar for math PhDs already ruled out like 99.9% of the population, so I don't see it having much effect on discouraging people. The gap between even a bright student who takes AP calculus or whatever and someone studying e.g. spectral sequences is already incomprehensibly large. Like you literally could not even convey to a smart young person how far away they are from the boundary of today's understanding. I don't think I even have a reasonable sense with a math bachelor's!

People who get into math do it because they can't not do it.

[ian-g]

I think it'll become increasingly important to have folks thinking about how to explain new math in a way that makes sense to your math bachelors students. And to your bright AP Calc students.

If we can point ChatGPT at these problems and get eventual answers, that's awesome, but it'll still be important to figure out how to tell people why this matters in ways they understand.

[throwaway0123_5]

> The net effect of this appears to be that we'll see far fewer, but far more elite math Phd's, potentially discouraging many young people from the field.

It seems plausible that the value of education will go down for the vast majority of fields and as a result less people will be getting degrees of all types.

Not a good outcome I think for humanity to be less educated, even if people are provided for when they can't get jobs... things like mathematical and scientific literacy, as well as history knowledge (which even STEM majors often receive via undergraduate degree breadth requirements), etc. I would expect strongly result in more informed and harder to deceive citizens.

[CWuestefeld]

A large proportion of degrees awarded today are not useful for any practical application.

But having a degree of some kind still serves as a signaling mechanism, demonstrating lots of things including the ability to "play the game". to follow instructions, and so forth.

[throwaway0123_5]

I agree somewhat, but for the most part the signal is that you can succeed in some kind of knowledge work. If AI replaces ~all knowledge work, that signal isn't actually going to be valuable I think. While I don't find it certain that AI will do that, it seems increasingly plausible with every model release. In that world, I don't think very many people are pursuing degrees.

[zarzavat]

If I were 18 I'd be sceptical about the value of degrees compared to the value of trades.

Yes degrees are a prerequisite for white collar jobs but if there is competition for ever fewer white collar jobs that's a life gamble perhaps not worth taking unless you are supremely confident in your abilities in a field.

If we're being honest, the value of a degree has been in decline for a long time. It was only being held up by STEM and even that is on shaky ground now. The days when an e.g. English degree would open doors are long over.

[cedws]

I never understood why being able to “play the game” is considered a positive signal. Getting a degree is the safe path, it doesn’t require any risk taking, and not much initiative. It’s about having the money (or going into debt for most people) and sitting your ass in a chair until you’re told you’re worthy. That idea is repulsive to me.

[hgoel]

I don't believe this is raising the bar for minting fresh PhDs.

We need to maintain perspective here, a PhD is essentially work done by a researcher at the least experienced, least skilled point of their career. Their primary goal is to demonstrate that they are capable of contributing to research.

They are not competing against AI to publish a counterexample to a known conjecture.

[chermi]

Maybe masters will become more popular. I have no problem with bars being raised on PhDs, but it's crazy history that math may be the first one to have it raised (or brought back to old levels).

[StableAlkyne]

IMO, I don't think this will raise the bar for PhDs.

In every field of science, PhD student research is largely incremental. Very rarely is a thesis groundbreaking. The point of it is all is to function as an apprenticeship for that PhD student to become a scientist.

Sometimes a particularly gifted or lucky one hits an important result, but that's rare and isn't the purpose.

[chermi]

I agree, but there's degrees (no pun intended!). Many PhDs are just an advisor telling a student exactly what to do. The training is still valuable, but it's not really what a PhD is "supposed" to mean. The example I give because of familiarity is many PhDs in chemistry/physics are just clicking run on simulations + running predefined analysis with results already predicted by the PI. Or in chemistry specifically OChem wet lab, it's often basically a minimum wage factory job.

[julianeon]

Is the bar really higher? Those math PhD's can use GPT too; they benefit equally from AI assistance.

[fn-mote]

This approach totally changes the game, in ways we are still discussing.

The current consensus is that domain experts get the most out of using AI on problems. How will domain expertise develop when AI is doing the work?

I’m not saying that there won’t be another approach that builds on the strengths of AI, but we have to look for that and develop it.

There’s a lot to talk about.

[hgoel]

A decent chunk of the work a student does during a PhD is stuff that their advisor would have done in much less time, but are intentionally delegating to the student to allow them to learn.

So, I don't think AI changes the way expertise is earned. It might change the cost tradeoffs depending on where AI costs eventually settle.

[altairprime]

As a potential elite math Ph.D, though sadly I can't afford to complete studies in that direction yet, both before and after A.I. I have been interested in the field for synthesis of different fields of mathematics into novel outcomes. A.I. is, as with protein and antenna folding, excellent at constraint solving and optimization — but human creativity continues to excel at creating new frontiers that are not found within the 'local' maxima explored by algorithms. Consider that we have brute-force calculation tests that can be applied to an entire floating-point 2^32 space and I guarantee a lot of A.I. effort invested in finding more efficient storage methods, but there's still room for human invention of floating-point storage simply by a person making a leap between Field One and Field Two that hasn't been proven yet:

> We would like to stress that the key idea behind ALP is to design for vectorized execution; it led us to analyze and uncover unexploited opportunities from a vector perspective in a variety of datasets.

— doi.org/10.1145/3626717, via ALP: Adaptive lossless floating-point compression (6 days ago, 4 comments) https://news.ycombinator.com/item?id=49051355

Perhaps future mathematicians use A.I. to prove it, but that's still a human invention — and even if we brute force the entire space of mathematics, it's functionally useless without an ontology to help humans narrow an A.I. 'problem-solving' response from 'all possible solution spaces' to 'the interesting solution spaces', and even if that ontology is A.I. created, a human is still going to be evaluating 'interesting' through their own cognitive experiences, biases, and dissonances in order to identify novel connections for other humans to build with. Just because we can `echo 0..2^64-1 >> file.txt` doesn't mean that we're deriving value from it, even if it's indexed by number of digits or nearest power of two or whatever. OEIS exists, and cannot be easily replaced by an A.I., because it's not just a list of numbers indexed by sequence (which no doubt computerized proofs plus generative algorithms will eventually automate the generation of), it's a list of interesting to humans sequences, and that's something A.I. can't be substituted for.

> Director Cary Fukunaga mentioned a complex narrative structure in his 2018 big pharma miniseries Maniac being nixed because of the audience loss predicted by the data.

— via Bland, for fans of everything (11 months ago, 7 comments) https://news.ycombinator.com/item?id=45049412

[jauntywundrkind]

Notably we have had very few conjectures proven true, so it doesn't feel like there is that widening base of established map to draw yet more proofs upon.