I propose the following reconceptualization of the goal of a mathematics PhD: to become a world expert on some interesting, deep topic, and to be able to convey that interest and understanding to others. Part of operationalizing this might be a thesis, but the degree would be awarded primarily on the basis of a rigorous defense, in which the student explains the topic to their examiners until they are satisfied.

I think this is a refreshingly forward looking idea and I agree with it 100%, especially the the "rigorous defense" part. That is a good measure of how well the topic has been researched and understood by the researcher. This is where the humans can be "in the loop".

How different would this look from current PhDs? I think students would still meet with an advisor, who might suggest a topic. That topic could be explored with AI assistance, or not, but the student would be responsible for understanding it; it might be much more open-ended and larger than the typical PhD is currently.

Interesting point about "more open-ended" and "...larger than the typical PhD". I think the author has a point. Earlier, the bottleneck was the candidate's/researcher's understanding and knowledge. Now with AI tools, it is so much easier to zero in to relevant knowledge, get your questions answered quickly which might lead to understanding more quickly.

For e.g., before the advent of public libraries and printing press, the knowledge was inaccessible and guarded. So that was the bottleneck.

Then books became ubiquitous and the bottleneck to knowledge and understanding was people's motivation AND knowledge of WHAT books and topics to research.

Then came the internet and free PDFs of books and research articles. Now, the bottleneck was still people's motivation and a mild version of what books and topics to research. I say "mild" because one can lookup articles and newsletters, and book reviews and come up with a list of reading.

Now comes AI and it looks like the only bottleneck is people's motivation.

I believe there was also a silent, yet potent, bottleneck all along which is also removed by AI: personal tutor/coach/teacher/professor etc. Let's say if I am reading a textbook on manifolds or some research paper and I have a question about a specific theorem or even a mathematical operator being used. Before AI my only way to get my questions answered was to read more books (PDFs or print), or ask on math exchange or math overflow and wait for someone to answer, or to ask a professor. This could take up to a week.

Now all of that has been cut down to 1 hour or less with an interactive chatting session.

!!!!!

So....the only bottleneck is people's motivation! QED

Exciting time!

Why is "Doctor of Philosophy" the correct certificate of "becoming expert in a topic"?

That's a radical departure from "PhD" being a certificate that someone is qualified to produce new research.

What you describe is more like a Masters Degree.

I think you have misunderstood.

PhD has nothing to do with expertness.

If you have a PhD, you have completed some kind of research training.

That's all there is. Says nothing about knowledge or whether or not you're a genius.

You cannot conclude anything else, and nobody claims that you can.

If someone has a PhD, they have some training in doing research.

Another bottleneck is money required to pay corporations who own the technology to "do mathematics". In that future, there will never be another Ramajunan.

That's a good point. Low level (for a lack of better term) mathematics will become extinct/saturated.

Perhaps a parallel to this is what's happening with youth sports in the US. It is becoming increasingly inaccessible.

Yeah I think it was in Terence Tao's recent post where he pointed out that the "low-hanging fruits of mathematics" are being harvested by the AI companies, when traditionally those problems were available for aspiring mathematicians to prove themselves in the community. Maybe there's a parallel in the software world too.

a refreshingly forward looking idea

What?

"a rigorous defense, in which the student explains the topic to their examiners until they are satisfied."

is exactly how PhDs were awarded for hundreds of years.

Even my BSc in Applied Physics (1977) had a viva voce that was a substantial fraction of the final exam.

The more I see these posts about mathematics institutions reforms and challenges from AI advancements the more it looks like they may need to go through a death. Or to put it another way they may need to start again from first principles.

If math is truly about spreading intuition and understanding then our institutions have dropped the ball decades ago and have not been able to grab hold of it since (if they ever had it to begin with)

Excellent optimistic post in a sea of negativity, and with actual suggestions, too. After reading, my mental image is this: think of Olympiads in Ancient Greece.

* A weightlifter was only awarded a laureate if he were able to lift a heavy stone (have no idea what they were lifting, for illustrative purposes only :-)

* Along comes Archimedes who invents what we would call an exoskeleton. Now any regular guy can lift twice as much as last year’s athlete.

* What to do? You can cancel the Olympiads, but they are actually useful as training, motivation, etc So now you have to give the prize on other factors, eg how well he can lift, has he opened a gym in the city, etc

BTW, physics and bio are not exempt, so those researchers better read and try to stay ahead.

I don´t think so. For programming agents can run code, check compiler output, etc. For mathematics, it is almost the same once you factor in the usage of lean.

For the reality, you can´t close the loop that fast, or with that precision. You will have to slow down by several orders of magnitude.

Doesn’t have to. There are petabytes of experimental physics data that can be fed to AI to extract additional insights. The only holdup is this is slightly harder to than math. With bio, you’re right, generally designing and conducting an experiment goes hand in hand and theoretical biologist is not a common label.

I am not a mathematician, but I can’t see how we are going to address the problem which we already see in coding:

Impossibility to independently validate all AI results

And in math it goes even worse. In coding code reviews are typically still the form of action you do within days. In math, historically, the lifecycle of proof is months if not years. Take as an example Millennium problems. They require at least two years of validity after publishing. Two years! In modern times with amount of output AI can produce, it feels like infinity.

We are inches close if not at the moment already when humans can’t reliable validate proofs and mathematics produced by AI. Then next research will be based on this AI-written-no-human-in-the-loop results. And we will end up in just few years in a world where novel and frontier problems will be articulated by AI and proven by AI based on AI results and humans will be incapable of understating the mere nature of the solution.

I don't think that's actually the real problem. Along with the progress in answering mathematical questions, recent progress on AI-powered autoformalisation has been astonishing. All the recent AI discoveries have been accompanied by Lean proofs.

And, yes: that doesn't absolutely guarantee correctness. The Lean kernel has had soundness bugs, and may have some still. But it's pretty strong evidence of correctness nevertheless.

The concern among mathematicians is not mainly that they doubt the correctness of any of these discoveries, but that human understanding may be devalued.

I am not that worried, but rather just observing. Humanity is about to enter the phase when we will be using things based on ideas no human ever properly understands. This thought … disturbing, somehow?

It is perfectly valid counterpoint to say that we already do it. We everyday use myriad of things, tools, and software we have 0 clue how it operates. But for us as humans it was reassuring that we know that at least there are a few other alive humans who know it, who create it and who can explain it.

With AI soon that comforting zone will be gone.

Humanity is about to enter the phase when we will be using things based on ideas no human ever properly understands. This thought … disturbing, somehow?

This is just normal though. We were building sophisticated bronze and steel tools long before any complex understanding of metallurgy or chemistry. Medicine is still the wild west.

I actually think it's the opposite: Lean proofs and autoformalization make it very easy to announce proofs alongside proofs of the correctness of those proofs (Lean certificates). It's not an absolutely fool-proof combination (the Lean kernel could still contain bugs), but it does immediately attach a very substantial degree of credibility to the result.

And that I think is essential to why some of the world's leading mathematicians are taking this so hard. In a world where we "merely" have AI systems capable of superhuman informal reasoning, verification, correctness, and acceptance could still only be conferred or anointed by human mathematicians. But a world that combines superhuman informal reasoning with superhuman autoformalization is a fundamental shakeup in the institutional order.

The recent proof of Fermats Last Theorem is interesting: it is (iirc) 13 million lines of lean code. And type-checking takes 5 hours or so on a pretty beefy machine. I cannot independently verify the proof, and I have to take Anthropics word for it that it actually type-checks.

That seems like a red herring. Have you independently verified the human generated proof of FLT? Surely someone else will try to verify Anthropic's formalization on different hardware. Plus, it seems likely that FLT formalizations will improve / get shorter over time, requiring less compute. And computers (and type-checkers) will continue to get faster over time as well. So maybe in 5 years you could own a computer fast enough to verify a/the proof in say a week, instead of 5 hours.

I think so. This machine might produce answers we value, but it would not, in itself, produce human understanding of those answers.

It's nice that the author is optimistic, but won't the AI be best placed to dumb down its increasingly complex proofs into a language us lowly humans can understand? To keep thinking until it can refactor complex proofs into ones from 'the book'?

AI will never be able to dumb down a proof to a level simple enough for someone to understand who has never studied math and put a lot of effort into it. Some concepts just need time and effort to absorb no matter how relatively simply they are phrased.

As someone who has a degree in math, I still can't help but think mathematicians are getting a little bit of a comeuppance. In a lot of areas of mathematics there had been little effort to make the work understandable and leaves numerous folks who could benefit from the knowledge on the outside looking in. Now AI comes along and do the same to mathematicians. Makes me chuckle a little bit.

Lowering the bar for the masses isn't always a good thing

Examples, preferably academic?

Half of modern academia, for one. Universities have turned into degree mills with the expectation that >50% of the population requires a college degree, regardless of whether they actually have any interest or need for one beyond doing it because it's a prerequisite for a good career, independent of whether said career actually uses the knowledge in any way.

Not that I actually agree that math was at the right level of gatekeeping. It definitely feels intentionally opaque beyond reason, and I think it's why LLMs are able to cut through the obfuscation and solve problems that maybe wouldn't actually have been considered quite so hard if mathematicians did a better job of making their work accessible.

<pedant>28% of adult Americans</pedant>

66% in Japan. Who said anything about America?

We have seen the thing in programming. I'm not a gatekeeper, I love seeing more people to code. On the other hand, along the way, we lost the joy of the journey and only fixated at the destination.

Result is more software at lower quality. The reason is statistics. When you increase the population, you increase the population of every kind of programmer, and people who want the result are favored in most competitive sectors because corporations want something somewhat working yesterday.

...and here we are.

Now programmers talking about code quality is stoned en-masse. If it's somewhat working then it's good. Efficiency, resiliency, maintainability and sustainability is an afterthought. Some of my friends who loved debating programming language theory now don't even care about the code. They don't write it, just vibe, and they don't plan to come back to "older, caveman style of development".

Some universities are also adding fuel to the fire: They "prepare students for the job", not teaching the science, but the parts that corporations need for the job only.

Though, hardware was cheap and people were expensive, and now code is cheap and hardware is expensive now. We'll see.

this comparison makes little sense.

we lost the joy of the journey and only fixated at the destination.

that's because you can make money with code and so it went the same way as everything else that's capitalized/commodified.

here a better comparison would be art which is generally still pursued on its own merits/pleasures.

The point I was trying to make was not "enjoying the process for the sake of it", but paying attention and spending effort on the journey created better software at the end.

When you look at older software, most of it was higher quality than the things we have today. A web site contained more information in a more readable way, more features in a smaller footprint. Same for native applications.

Now we slap what we found online together and calling it done. Everything is sparse, takes ages to load, centuries to submit and everything is so disconnected and async that some simple features are straight out impossible.

This is what fixating on the destination brought us. I dare you to download something you purchased 3 months ago via a 4096 character S3 link they have sent you, and double dare you to ask customer support for a new link. I'll bet that with a 80% chance they have no way to verify your serial number, even.

Have people completely forgotten how to read? I'm saying your analogy between software and math is completely, irreconcilably flawed. I don't need to read the rest of your screed because this is a thread about math not software.

I see.

Somebody said that lowering the bar is not good. Somebody else asked for examples, and I provided an example. So, if you want to be pedantic, that doesn't track well, because what I answered was not about mathematics.

If you want something about mathematics, computation is mathematics, as software is. So, my example tracks the same way in mathematics.

Finding solutions without understanding its parts or the path is equally detrimental to mathematics as it is detrimental to software.

Maybe you need to read a bit slower and think along the way. Using AI too much blunts critical thinking skills in some, as I read.

What was the end of the proof thing you mathematicians use, was it "Q.E.D."?

Q.E.D.

here a better comparison would be art which is generally still pursued on its own merits/pleasures.

What about the crafts? Carpentry, blacksmithing, stonesmithing, etc. are all commodified yet people still pursue these for pleasure.

this comparison makes little sense.

Stop gatekeeping what people find personally fulfilling.

Carpentry, blacksmithing, stonesmithing

they're literaly not - do you know what the word artisanal means?

Stop gatekeeping what people find personally fulfilling

wut? i can't even parse this in relation to what i wrote (which is that people who enjoy math will continue to be able to).

It is not lowering the bar to find better ways to demystify and explain things. In fact, I would say those who can explain it well understand it the best. Richard Feynman would be my best example.

Do you have any examples of places you felt like there was a lot of gatekeeping? Perhaps having studied mathematics I am a bit blind to the issue here and would like to learn more.

I agree making simple things sound complicated to appear more impressive is bad but there are limits. Even with Feynman he could only go so far, e.g. his interview about why questions and magnetism.

I am not thinking of intentional gatekeeping, but more of the kind where mathematicians build themselves their own island of concepts and notation with no thought of building a passageway for others to engage and make use of the theory. I would find modern algebraic geometry highly useful as someone who works in computer graphics and computer vision, but much of the theory is akin to learning a new language and I don't get the sense that those who publish their work in this field care if I enter their world.

Kind of makes me think of Arnold's, "On Teaching Mathematics":

https://www.maths.tcd.ie/pub/Maths/Courseware/ProblemSolving...

  > Mathematics is a part of physics

This feels backwards. I frequently joke "physics is the subset of mathematics that reflects the observable world". Math can be as abstract as it wants, but physics has a constraint. It must model an observable world (related, this us part of why people say String Theory is math and not physics)

What about the next 3,900 words after the first sentence? (yes, it is that V.I. Arnold https://en.wikipedia.org/wiki/Vladimir_Arnold )

I think you're a little naive if you think this is because of gate-keeping by mathematicians rather than the essential complexity of mathematics. Mathematicians individually and as a group would love nothing more than a world which is capable of understanding their work more deeply.

In the exact same way that LLMs allow anyone to vibe code an app but do not replace real understanding of system design due to its essential complexity, non-mathematicians will quickly learn that asking an LLM to pump out advanced mathematical statements to you, even if they are correct (and even if you could verify them) does not constitute understanding, and that the human brain is the bottleneck either way.

It is only if the LLM is super-human at simplification and explaining that a difference will be noted. This would be excellent for mathematics but its not a foregone conclusion (and the argument of most mathematicians, such as Terence Tao, is that this distillation process is one of the key parts of doing mathematics, and that LLMs so far seem to be going in the opposite direction. I suspect its probably user error and leveraging the tools better will produce different outcomes, but mathematicians are only just starting the journey that software developers have been going through, so patience is needed).

Folks keep using the word "gate keeping" -- I never used that term nor implied intentional gate keeping. It's simply not caring if those outside the club understand -- there is no one guarding the door its just folks don't care if anyone finds it.

Gatekeeping is a human universal. Even those who preach maximum inclusion do exclude plenty, of course there is always explanation why that doesn't count. If you try to run any community you quickly learn the importance of gatekeeping. Eternal September etc. See any site or fandom that gets mainstream etc. I don't think anyone has any duty to actively pull in as many as possible new people. Zen masters used to chase away students or make them sit on their doorstep for days and send them away anyway. Now, the west is essentially Christian and so the missionary impulse is real, but it doesn't work without the other stuff in the package.

Again with the gatekeeping. An AI generating a convoluted proof for some conjecture is not gatekeeping. It just does not care if you understand. It only cares enough that it can be verified -- usually using some other opaque proof tool like Lean.

resulting in the production of an abundance of PDFs. The contents of some of those PDFs may even have important applications.

I hope, from the depths of my soul, that the static typeset report format for transmitting knowledge and understanding will finally die and be laid to rest.

What would you prefer instead of "static typeset report"?

The author argues for evaluating Ph.D. candidates based more on the oral thesis defense than on the actual thesis.

By essentially the same reasoning, I’ve been arguing for prioritizing in-person design/code reviews over code-only async PR comments.

The important thing is to verify that the human has a coherent design in mind and can demonstrate that it got implemented, regardless of who or what was at the keyboard. “I dunno, I guess Claude thought this was a good idea” is not a coherent design.

I can see how this trend is going to go.

Manager: “Why isn’t feature X available?”

Person B: “key pieces are delayed due to the developer not understanding all of the LLM doesn’t and implementation.”

Manager: “does it work? What are the risks?”

Person B: “well yes it works for now but we’re accumulating tech debt due to a lack of understanding and potential flaws that haven’t been thought out yet”

Manager: “they want feature X, ship it, we can deal with it later, I don’t care if it’s not coherent as long as it works.”

How many decades at this point has these been a push for functionality over everything at all costs? And you have a mechanical snow plow now. Most businesses don’t care about later risk or any future planning beyond the quarter horizon, they’re not concerned about how it will effect their performance in 3 quarters or lead to instability or issues, those are future problems for a future person and we’re here for money now.

The simulated dialog you have provided is very much representative of what many of us have heard first-hand. However, I disagree with:

  Most businesses don’t care about later risk or any future 
  planning beyond the quarter horizon, they’re not concerned 
  about how it will effect their performance in 3 quarters or 
  lead to instability or issues, those are future problems 
  for a future person and we’re here for money now.

Businesses care about "later risk" and what it implies.

Individuals within an organization do not unless it will specifically affect their bonuses/promotions.

It is also true developers can have a myopic view of what should be valued and that can be incongruent with the business.

You can make a perfectly rational business choice to take on tech debt to get a feature sooner.

An oral defense is ultimately a meeting, with the same weaknesses as other types of meetings. In particular, if new information comes up in a meeting, you cannot reasonably expect to get useful responses from the other participants. First impressions maybe, and plausible-sounding bullshit from those prone to generating that. If you want anything more, you need to provide the information in advance or schedule another meeting later.

Oral defenses in academia are largely rituals. If a student fails their defense, it's almost always the supervisor's fault. The supervisor is supposed to be the primary quality control. With their regular meetings with the student over the years, they should be able to tell whether the student has achieved sufficient understanding and contributed enough to graduate. If the supervisor thinks the student is ready to defend, the reasonable expectation is that the student will pass and graduate. The defense is mostly there to let the other examiners validate the supervisor's judgment.

The tricky part is that live review is much more expensive than async review. But maybe that just means we were getting away with a cheap proxy because it happened to work reasonably well.

I admit I just don't understand this attitude.

People had the same complaints that the code produced by early coding models was messy, lazy, poorly commented, had terrible architecture and so forth. The central complaint was that it was just too difficult for humans to review. The answer is just to improve the models and move on.

Similarly now we're getting AI doing math. The proofs compile but are a mess. So just make the models better at writing clean proofs and explaining what they're doing to humans. That's the end of it.

Rather than just go on and on about how it's the end of the world if we don't do this, why don't we just do it?

There's a fundamental difference between the goal of code and math. The goal of code is to produce software that does something useful. As long as the code does what it's supposed to do, arguably, it's good to ship. (As you imply, we want the code to be good enough to also be reasonably certain there are not too many bugs, that it is maintainable and can be extended etc. This is what early models failed at but now seems broadly fine.)

But for math: what is the point of a proof if no one will read it and no one uses its result? To quote the article, "AI systems will [...] result in the production of an abundance of PDFs. The contents of some of those PDFs may even have important applications." But if there's no one reading the PDFs, what's the point - no matter how good your AI model.

The point of math is understanding. So mathematicians should feel free to use AI as much as you want, but in the end, they should've gained some understanding on what happened.

I keep seeing people repeat that the goal of math is "understanding".

I do think that is one goal of math but I don't think it's the only one.

I think an additional goal is simply "truth", which can be found without understanding as we've seen with these human-incomprehensible proofs.

Yet another is practical applications. While there's less of these in pure mathematics than in most domains, they do still exist.

The primary goal of all basic sciences is human understanding. "Truth" is no more a goal for mathematicians than the physical laws are a goal to physicists; they simply exist in nature. The goal is rather to develop useful language and conceptual frameworks for reasoning and communicating. That understanding underpins all practical applications.

Similarly now we're getting AI doing math. The proofs compile but are a mess. So just make the models better at writing clean proofs and explaining what they're doing to humans. That's the end of it.

The assumption here is that the true/false of the theorem is the important outcome. While it is certainly part of it, a big part of maths is the understanding you gain from a proof. Many of the best proofs elegantly explain some aspect of the maths which was previously unclear and expand our understanding of the world.

To use a programming related example, imagine that an LLM spits out a solution to the travelling salesman problem which works in O(n) time. On the one hand that's very convenient for whatever problem you happen to be trying to solve at the time...but there's also an answer to P=NP in there! The former means your delivery drivers app works a bit faster on their busy days, the latter fundamentally shifts how humanity thinks about certain problems.

Going back to the maths, there have been theorems that were proved (by people) where the proof is broadly seen as 'unsatisfactory' in that it doesn't really expand our understanding. I assume some of these LLM proofs are a bit like that: we now know that the thing is true, but we really want to know why it's true, and how that changes our understanding.

> models better at writing clean proofs and explaining what they're doing to humans. That's the end of it.

The assumption here is that the true/false of the theorem is the important outcome

You're replying to a comment about "clean proofs" and explaining to humans. You talk about true/false anyway. Re-read please.

Are you always an ass? He had a great comment, and you respond with this low effort hostile bullshit.

We already interview faculty hires; we must now do the same for graduate admissions

They hire PhD students without hearing them give a talk and then doing interviews? In Germany, the applicant gives a talk (30-40 min) to the research group they want to join, usually presenting their master's thesis, engage in discussion, often share lunch with the group, then do 1 on 1s with individual members of the group and a longer one with the PI. Obviously this can vary within Germany too, but I couldn't imagine hiring someone without something like this.

1. US a lot of international PhD applicants, so traveling before even being accepted is difficult, and 2. lots of people don't do a masters.

when I applied to PhD programs (not in math) it was basically CV + personal statement + recommendation letters + short chats with interested faculty :shrug: Maybe it was because my CV was "strong" but the chats were more see if interests were aligned, rather than actually interviewing me.

Remote talks are also a common alternative in such cases. Zoom etc.

I also know that another major difference is that American universities tend to hire without a professors involvement, into a generic "program", then the PhD student seeks an advisor after being accepted, so applicants have to woo some unconnected committee pursuing various goals misaligned from the PIs instead of convincing the PI. In much of Europe a professor basically "owns" a chair and really is boss and decides hiring pretty much alone.

Anyways, if they don't have a masters' yet, they could present their bachelor thesis. But I already think it's a bad model to combine the masters (courses) and the phd (research) into this American hybrid that's the direct-bachelor-to-PhD jump, but that is somewhat unrelated.

I agree the combined masters + phd is weird tbh. I think almost everyone treats it like "just focus on research and spend as little time on course work as possible." I would prefer it if courses were more flexible.

I'm not a big fan of the US application setup. IDK how it is in Europe, but in the US, it feels like there's a lot of not-very-meritocratic "secret" stuff you need to know to up your chances.

Unlike Europe, in the US, most PhD applicants do not have a masters. 10 years ago, most had not even conducted substantial senior projects (e.g a semester or year). Though increasingly senior projects / undergraduate "theses" have become more common.

In my opinion, that's not a good time to really know you want to pursue academic research. A masters gives you a little taste at least and allows to show aptitude. Doing a student research assistant job, working on a thesis, maybe even a paper submission, diving deep into specialized courses. Then you can know if it's for you as well as the PI can know if it's for you. Straight from a bachelor and them interleaving the phd with coursework is a bad model I think, but of course I'm from Europe so it's what I'm used to.

In the UK I had a panel interview with 4 people, in person. But for international students who often brought their own funding sponsored by their government that didn’t happen.

"A computer or monkey could easily start at the axioms of ZFC and iteratively apply deduction rules....simply conjecture all mathematical propositions in alphabetical order...The prospect of automating mathematics by enumerating all conjectures, and all proofs of ZFC, is probably not so disturbing to you."

I thought we were going to get at least some brief comment on Godel here?

Gödel effectively says ZFC must be incomplete, otherwise it would not be sound, but does that stop you from listing all mathematical propositions it can generate in some well-defined order?

That's right. You could list all those propositions and search for proofs of them. In fact this is very similar to Hilbert's very program to which Godel's First Incompleteness Thm was a response (https://en.wikipedia.org/wiki/Hilbert%27s_program).

Godel showed that there are true statements that cannot be proven, and also that among the unprovable statements from within the system is the consistency of the system itself.

As I understand, mathematicians are still trying to figure out how much this matters. One of the best examples of its mattering is may be the Continuum Hypothesis: CH is consistent with ZFC, and ~CH (not CH) is also consistent with ZFC. In other words, you have enough flexibility in constructing your ZFC world such that in some ZFC-consistent worlds CH is true, and in others CH is false.

Litt's statement is not wrong; it's just that what he wrote sounds so much like Hilbert's program, that I'm surprised we didn't get some even minor comment on what kinds of truths we could reach if we embarked on such an effort.

People always bring up the Continuum Hypothesis as though it has something to do with Gödel's incompleteness theorems, but it doesn't really. The Continuum Hypothesis being neither proven nor disproven by some particular axioms is a similar phenomenon as that the group axioms neither prove nor disprove commutativity, the ordered field axioms neither prove nor disprove the existence of a square root of 2, etc. There's no particular reason to expect any particular formal system to be complete, sans some demonstration that is.

Gödelian incompleteness is the specific kind established by Gödel's proof, where theory T can't prove Con(T) without being inconsistent. But the inability of ZFC to consistently decide the Continuum Hypothesis is established in a completely different manner, with the Continuum Hypothesis not consistently decided by ZFC + Con(ZFC) either, or any such thing.

I’m not a mathematician, but if I were I would imagine the main attraction would be the act of thinking hard and solving problems. Maths felt like the ultimate profession for someone who loved puzzle solving. If you delegate the thinking, any percent of it to an LLM, I don’t think it’s maths anymore. Might as well join any other job and earn better money now. A completely ludicrous proposition, that maths will no longer involve mathematical problem solving.

My worry is that the frontier of math is too far away for most humans to reach. AI is only going to make that worse. If today it takes twenty years of math study to reach the frontier (in a narrow field), what's it going to be like when it takes forty years? Or four-hundred years? Will the fields just get narrower and narrower to accommodate finite human intelligence?