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.

> 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.

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.

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).

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

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.

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

Real talk.

AI solving these makes me feel like mathematicians put far more importance on their work than was actually there. Many solutions seems to be tautological games, and games of logic where conjecture puzzles that few work on or care can be solved by AI which doesn’t care what it works on.

It seems to always be some form of this:

Mathematician: “Propose conjecture a and conjecture b can’t be true simultaneously”

AI: “they can”

Everyone: “ok…”

I know this might be unfair or out of ignorance but it genuinely is how this field feels today. Games of games with self importance added in.

Edit: the point I should have made is, should we be using AI to figure out what proofs MATTER now vs games of proofs?

"AI solved another conjecture" is exciting for a lot of people who have otherwise no interest in math. So the fact that it goes viral may have very little connection to the mathematical importance of the result.

My university had a great Physics department (Nobel laureate level), but every year you’d have a bunch of junior year (year 3) physics students desperately trying to switch into engineering as they realized that making a go of a pure physics degree was going to be more difficult. It was a consistently repeating pattern.

Pick any field and you'll find countless PhDs eager to tell anyone who will listen all the ways it ruined their lives.

Counterpoint: my entire physics phd cohort is doing great now and it didn’t ruin our lives.

I think better advice is “do you want to work in this specific lab for 5 years?”, not “do you really want to do physics”? Talk to the lab members, learn what they do.

At least in europe you can do a 3 year Eng Phys BSc and if it doesn't pan out you can do a master in EE or MEng.

That's funny because so many of us in my PhD philosophy program would say: I should've been a physicist, philosophy ruined my life. I had a physics prof who told me it was foolish to go into philosophy when I could've been a physicist and I thought I knew better. Oh well, I probably would've been a software engineer in the end no matter what.

I know multiple Physics PhDs who are living rich, happy lives.

Though, mostly by abandoning the field and moving to Anthropic.

i would make a similar argument for entrepreneurship and most things: do it only if you can’t bear the thought and reality of doing something else.

The term “currently” is the sticking point here. As far as I know, there is still no one who can provide evidence as to why it should remain that way. And the "by luck with RL" is soon not by luck anymore. The more stupid and not so stupid ideas are discussed with the LLMs, the chances are growing that it is not a question about luck but by growing probability.

Frankly, such paradigm shifts are almost impossible for humans as well. If a mathematician proposes a truly radical paradigm shift, they're either a once-in-a-decade genius or a crackpot.

Barely better than brute force (I can't believe it's not brute force!™) aside, presuming we get super intelligence it will all be depressing when it comes to intellectual pursuits like this.

Why is it “just a matter of time”? Why do we assume and say this?

The amount of times humanity has said this and time itself was not enough of an ingredient to achieve some anticipated outcome are legion. But we filter those out and go back to making more predictions based on the current linear derivative we’re observing.

I'm not sure that elegance will be so easy to train for, the same way that writing skill has plateaued (or arguably declined) since earlier models. "Have you solved the problem" is verifiable, but questions of taste are harder to pin down.

I've actually been involved in annotation projects doing RLHF to train LLMs to do exactly that. It's not a matter of time, it's already happening - it's just seemingly lower priority than "profitable" projects like post-training LLMs to replace white collar workers.

I've driven back roads in Ireland. Super messy and chaotic. I was still able to use a map to get to my destination.

formal verifiability e.g. vi Lean

Eh, some conjectures have counterexamples.

This one might be one of those. ;-)

That very much does not describe what happened. Someone found the bug and used it to disprove the Collatz conjecture as a demonstration of the bug. Nobody ever claimed it as an LLM proof.

I’m pretty sure “counterexample” is the wrong word here.

"They" don't "care" about anything. It is a stateless computational run across thousands of semiconductors. There is no objective this software has other than the computational function completing. To care would mean the model would have a level of discernment that goes along with sentience.

> Minimizing tokens out, or generally "token efficiency," is part of the objective function for these systems.

First time I heard that, and I doubt it. Don’t customers pay for output tokens? If so, why would a company specifically spend time training their LLM to generate fewer?

You clearly haven't used Claude to generate code or documentation.

Even if you use AI tools?