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.

* 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.

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.

PS. The validation problem, being one.-

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?

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)

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

With AI, it might be the case that we don't know, we won't know, but the machine does.

The central question, namely whether humans should be in the loop, will be repeated again and again in the years to come for all industries, starting with mathematics.

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 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.

I'm heartened to see this kind of thoughtful, reasoned discussion around long-needed reforms in the educational community, with AI as the catalyst. We certainly have interesting times ahead.

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'?

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!

One of the things that struck me was that most people have an area of their workflow that they would be happy to hand off to an AI, so that they can focus on the thing they love. However, that area is different for each person. One person's grind-work is another person's love-work.

This is why it's so hard to build consensus around where the "red lines" are in this space.

1. It will take much longer to understand the output of the machine that it takes to prompt and create it. 2. The only? best? one? way to /verify/ that you /in fact/ understand the output of the machine is to explain it to someone else.

So there will be a machine generating koans which need to be meditated upon and discussed with human social back-pressure validating understanding. I think this could be much more cooperative and at a minimum this will be a way different math social construct.

In code, any developer whose differentiaton was clean code and knowledge of different technologies is now average.

In math, any mathematitian whose differentiation was to manipulate formal systems and know tricks of different domains will be average.

Fortunately, humans do more than syntax and search.

The bad news for developers is that if you know what the output of your program should be (which happens most of the time), almost all of the job is syntax and search to build the code that reproduces the output.

The good news for mathematitians is that for the majority of problems you never know the output, or you just know the output is either "True" or "False". There are some cases where you need something else, for example "a solution that blows up in finite time". For those cases AI will outperform you easily (see new Navier-Stokes solution)

So if as a mathematitian you were doing more than syntax and search, then keep doing that and use AI just for what its best.

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.

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

I also have a degree, but mostly taught myself the important undergrad-level concepts as a kid. I just went to the library and borrowed any of the hundreds of books written to clearly communicate maths to beginners or downloaded any of the free ebooks / lecture notes.

Name a single field of endeavor that is more open in 2026. Software certainly isn't one of them - the best stuff has always been gatekept.

But also, with things rapidly changing, perhaps in ten years chat programs will not only do superhuman but make their proofs marvelously accessible and provide incredible tutoring sufficient to bring any curious up to a super high level quickly. Then what can you say and what can you complain of.

But I think "do everything machines" are necessarily inevitable but the situation does make it uncertain where the limits are.

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).

Why should we expect everyone to be both a great researcher and communicator? The obvious result is that it is as effective as engineering managers. Sure, there's some amazing ones, but most aren't. Though that also doesn't mean no expertise in the field (i.e. non-engineering manager) is any better. It is just that managing/communicating is a different and orthogonal skill.

What needs to happen is we need to make it okay for people to specialize in more things. More nuance to this rather than trying to throw everyone into nice easy to manage buckets. Those buckets are just unrealistic abstractions filled with hope, denial, and laziness. Reality is surprisingly complex. Math can do a really good job helping you understand that, but it's a sufficient condition, not a necessary one

Pedagogy is the primary purpose of educational institutions. That includes universities. However, when research is placed first, you are often left with mediocre teachers, because "who cares?" They were hired to do research and lend the university that kind of prestige; teaching is an afterthought squeezed into the spaces that remain.

And frankly, we haven't the faintest clue today what education is even for. That doesn't bode well.

Add to this the other troubles facing the university, like declining attendance, skyrocketing costs, and AI cheating and we're in for a very interesting transition indeed.

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.

They can write great code, they can gain understanding of something for themselves, but they still kinda suck at explaining it to people. As the article says, if you want understanding, it seems like there is no replacement for getting in the weeds yourself. A model can help, but it's not going to magically download knowledge into your brain.

You can understand this as learning a mapping between the model's internal "world" (i.e., 'meaning,' which is hopefully coherent and consistent -- but definitely not always! see, e.g., https://arxiv.org/html/2505.11581v1) and language (i.e. 'form') that reflects that world.

For this to work, you need both coherent / consistent internal model worlds, and also good mappings onto human language. Supervision by mathematicians has provided the signal for both internal coherence (though this can also come from interacting with a proof oracle) and for good explanations. If models exceed human capacities, you could imagine that aligning their explanations potentially becomes harder (though not necessarily). Also, humans naturally have to do the same thing: as researchers we must find analogies to make our work legible to collaborators or laypeople. Often in doing this, we further clarify our own understanding!

More deeply I think the "end of the world" vibe arises not only from the practical need to have models that explain, but also Litt's (and many other fields' researchers) grappling with being relegating to not mattering.

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.

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.

If we use AI well here we could actually understand math much better than we do now.

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.

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 we often delude ourselves as to how well we understand problems and their solutions. Some instructors even make you feel that you understand better than you do by pointing to a few approximations or simple solution spaces that obscure the larger complexity. Just looking in wonder at the many categories of three-body solutions (currently on hnews) is enough to remind me of this.

In my country, that's exactly how it is.

Yes, you need to have a thesis to defend, but ultimately it all comes down to the (oral and live) defense/disputation.

Who pays the salary of the mathematicians and the cost of accessing the AI?

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.