I’m reminded of that famous debate between Poincaré and Hilbert at the International Congress of Mathematicians in Paris in 1900. It was then that everyone decided to follow Hilbert’s path, and proof came to be valued more than intuition. I think modern math at school and at applied university kind of lost this intuitive part. I try to teach my students that mathematics is, first and foremost, a very precise language of communication. It’s sometimes amusing to ask those who don’t like math to do without it entirely, just to see how much harder it becomes to describe the things around them. Second thing I tell them, formulas are the essence of mechanisms in their purest form. And in this form, they’re much easier to grasp and mentally manipulate. It always amused me, after taking a mechanics course, to imagine that for any formula, you could visualize a mechanism or process that implements it. And third thing, I suppose, the ability to verify one’s own statements as proof. Although, of course, mathematicians would probably tear me apart here for my heresy:sorry, I’m not a mathematician, but an engineer. You can make mistakes by using incorrect assumptions, but at some point, analysis itself will show you that you were mistaken. There’s a wonderful book, How to Prove It by Daniel Velleman, which provides an introduction to proof for the uninitiated like me. I really enjoyed it.

The math field is confronting something that coders have been dealing with for a few years now, only far more violently. Today's moat for software seems to be that AI can automate tasks but not a full job (yet). But for a large proportion of mathematicians, doing these tasks really was _the_ job. It's the bit they wanted to do, and if they completed a sufficiently difficult set of tasks, they got tenure. Now this model is failing, they frantically need to pivot the role of humans to save their profession from funding cuts.

I remember when "writing code was never the point" became a mantra here. There was truth in it, but removing the coding has certainly taken away a lot of the texture of the work and enjoyment of the craft. Many of us feel this loss as we tech-lead teams of agents as our source of income. I am not optimistic the mathematics pivot is going to work, but I'm certain that most will be depressed with the outcome even if they succeed.

We are all staring at the same existential dread, just seeing it unfold slower. We're being told that utopia is to be obsolete, and that is a jarring idea to contend with.

Part of the controversy here is that now the skill advantage that some Field Medalist had is much narrower. The fact that fields medals have an age limit implies that it favors brain power over understanding. And that was the guiding light award of the community. So i find it "funny" (and natural) when they are offended by AI. That is the main "crisis" of mathematics.

In my opinion there has never been a better time to be a mathematitian, and there has never been a better time to be a software builder.

But there has never been a worst time to have the need to prove your economic value as a mathematitian or software developer alone. Because "understanding" is not something you can prove in one afternoon, its something that you prove with a life.

I have been losing interest in this proof-oriented approach into extremely abstract concepts (what seems to be the core of academic mathematics today). Obviously, proofs are very attractive because they are the closest thing we have to a universal truth (at least under the assumed axioms). Having mechanisms to reliably show a proof, and computational methods to handle complicted proofs is great.

But.. the navier stokes proof was the last straw for me. People spent over a hundred years arguing whether a continuum approximation of a particle system would behave oddly. In the mean time, other folks went ahead and completely revolutionized the world of computational fluid dynamics (with multi-billion $$$ impact on society) by just doing better numerics (Kahn-style numerical analysis).

Making my complaint more general: I find modern math is exploring areas that are interesting to mathematicians, but increasingly irrelevant to society. And certainly not moving us towards "human understanding". The biologists are the ones working on that, the math folks should try working with them on neuro stuff to understand how human brains can do math at all, given their architecture.

I'm a professional mathematician. Today I proved what for me is a very solid theorem. It's something I had thought about for a few years. With a few weeks of serious use of AI I've found a proof that I am currently trying to write up, but which appears correct. The change in the workflow is enormous, but so is what one can do if one has clear what to do and how to do it.

I find myself less worried about it than at first. I think what we'll see are that some things are low-hanging fruit and can be solved just by tireless search. Maybe half the millennium and other such high-visibility problems will fall this way.

Others, I think, will be beyond both human and AI. And so what then? Mathematicians just throw in the towel and say it's not worth trying? Of course not. We will continue that pursuit, and as we do, new ideas will arise and new problems will need to be solved. It's math. There is no end.

It's easy to look at the current landscape and see AI ticking off solutions to problems and imagine that soon there will be nothing left. Machines replaced the need for much manual labor, but they also established a basis for an economy that provides the opportunity for more labor. This is the situation with math now. It will take some getting used to. There will be little-to-none pencil-to-paper working out of problems anymore, but there will always be work to do, things to solve, curiosities to unravel. And it will still be professional mathematicians who are the ones most capable of directing that effort. Because, if nothing else, they're the ones whose curiosity is piqued by the problems. Which, let's face it, has been 99% of the motivation for graduate-level math in the first place.

There's the the old question: is math invented or discovered? I think it's both: the problems are invented, and the solutions are discovered. In the age of AI, the discovery part will be greatly affected, but the invention part will remain firmly in the human domain.

The value in academics is teaching and research. The value in research is discovery. Proof was a useful function for humans to do towards discovery until recently. Understanding is a useful property insofar as it helps you teach and it is a prerequisite for generating hypothesis. Humans will always be driving discovery, the tooling and focus of work may just be a little different. Attachment to one particular modality of discovery is an aesthetic choice, not a moral one.

While I understand and emphatically with Tao's concern I'm afraid it's missing the forest from the trees. Unless you can make a claim that AI will never be able to perform intellectually at the same level as any human at a much lower cost, there's an outstanding utility problem that remains unaddressed. Sure enough, the AI may not have taste or goals, or many human traits, but that's irrelevant to the much thornier (and much broader than mathematics or even academia) question related to who's getting paid how much and for what.

This goes in a necessary direction, from my personal take away of Gower's recent post on the subject.

Mathematics is suffering from Goodhart's Law:

"When a measure becomes a target, it ceases to be a good measure."

One thing that concerns me from all this is "understanding" is very important to human progress. The fact it took 400 years to crack Fermat's theorem resulted in a lot of "Side Quests". These side quests helped grow other fields (for instance elliptical cryptography). Im concerned with AI that we will loose these side quests.

> It was a short film called Outside In, perhaps the earliest example of a viral video about substantive math, visualizing the key idea of Thurston’s own construction for sphere eversion.

This is really interesting and available here: https://www.youtube.com/watch?v=IbGNZQvobkc

> we might imagine what it could look like to have an analog of the Millennium Prize Problems for open exposition problems

The core idea seems to me that we should shift the standards for professional evaluation from generating proofs to generating explanations. Makes sense that such a proposal would come from the 3B1B guy, and I actually agree with it, irrespective of AI. But what eludes me is how that could be a defensive mechanism against AI automating humans out of mathematics. AI is likely no less good at producing natural language explanations as it is at generating rigorous proofs. It's telling that even Terrence Tao turned to AI to understand AI-generated results [0]. It seems that the essay doesn't address that issue at all.

[0] https://news.ycombinator.com/item?id=49010345

It's time to stop solving logic problems and start solving the more difficult philosophical problems, like the hard problem of consciousness.

How about we stop moralizing technology so much and start focusing on how we want to spend our time in the real world which now contains it

Let's see how long (if it ever happens) it takes for models to generate motivated explanations (possibly done via the Manim library or something like it) along with their Lean proofs. Grant Sanderson is right that this is kind of subjective but so is Art and I'm very enthusiastic about AI generated Art.

Hmmh. I like motivated explanations, but, as acknowledged in the text, this is a subjective thing to measure. What is a great motivated explanation for Tao, might be hard to grasp for me. So I guess judging how well an explanation motivates something depends on two things: 1) My way of thinking, and 2) what I already know and how well I recall it in this context.

There is a third thing: how well does the motivation chime with or go against my current belief system? You would think this is not much of an issue in mathematics, but it can be, and I had my fair share of frustrations because of it.

Anyway, all of the above points to one thing: the best motivated explanation will be generated by an AI, knowing the subject and you in a deep way that no other human will, and being able to interact with you during the explanation.

Last summer Grog was still celebrated and admired for bravely piercing animals with a spear and bringing home the meat. But now Goong made this newfangled arrow and bow thing and any cowardly fool can now shoot animals from a distance. Grog devalued. Grog sad.

If this suggestion were to come to pass, I wonder how new math PhDs would think about choosing between a 'normal' R1 faculty job vs. the "teaching route" (teaching professorships, lectureships, community college professorships, or SLAC professorships).

It's been my understanding that traditionally the ones who care about "motivated explanations" in this sense go for the latter, but if the research community has now decided they care about teaching and understanding, it might "even the playing field" and make the jobs more similar.

If I have to take the risk of simplifying,

1. We humans have managed to take huge amount of information and compress it using a loss function containing some bias we have about the information.

2. We now ask ourselves to decompress the same information with some additional cross-entropy. As a side effect of this process we sometimes spurt out information that may or may not have any meaning since the compression was lossy.

3. Now, we ask ourselves to present this some-what newly decompressed information with brevity in order to understand what we've learned from it.

Knowing that this process is happening on a larger scale, this resurfaces the argument if meaning can be reduced to computation only.

Although some might favor this argument but we are at the risk of anthropomorphizing this process.

The idea presented in the post itself is perspicuous (in Grant Sanderson own words) as he always does.

The last days we are served these high goals about understanding, "digestion" and so on.

But if you look at the practice of present mathematics, in the last 20 years it is all about publishing solutions to problems.

There are famous problems to be solved, there is a hierachy of conjectures to be solved. A quick search here on HN gives pearls like "Theory building papers are dime a dozen and don't get published in high tier journals unless they solve a problem".

And all of a sudden it turns out that problem solving can be automatized.

So then what will problem solvers do? Well, from now on they will "digest" problems solved by AI.

In a way or another they will find a way to stay on top.

That's the goal, at least, but mathematics as a living practice does not have much to do with these games of power.

Taught proofs too, and plenty of students fake intuition with pattern matching.

The guest posts are from a self selecting group of course, but so far all we have is "inevitability", "adaptation", "exiting times" and, most importantly:

"We want SAIR or the EU shell out $10 billion for a gated AI for privileged academics!"

The last point is particularly troublesome, since the same people were gushing about "democratization by AI" before the N-S proof.

So the subset of mathematicians that is vocal on the internet wants their AI toys, only paid for by the state like in the best academic tradition.

None of these people cares about other professions or wants to slow down the industrialization of academia.

It would be interesting to see what would happen if we had two competing mathematical institutes, a sort of First/Second Foundations:

1) Rejection of AI for anything but trivial applications while still using computers at their full capacity. Researchers would ensure full human understanding of proofs and methods. This Institute believes on Math as a process of discovery, Mathematicians as explorers/poets/storytellers and not proof machines.

2) Unrestricted, all-embracing use of the latest AI, including potentially research in creating even better AIs as part of the program. These researchers would be okay with not understanding proofs if verified to be correct. This group is focused on rapid problem resolution and believes Mathematicians are theorem creators and provers.

After X years (100?), which one would advance Mathematics and humanity the most (we'd need to define "advance")?

Software is logic applied to intersubjective truth. It's not physical truth which is the subject of the hard scientific fields such as physics and chemistry, as well as biology for the most part.

So no, software is much less than science.

I'm not sure that this new approach will be AI-resistant. Why would people not use AI to help in creating the "motivated explanations". Maybe they can't be one shotted today, but AI also makes this easier.

Assume in 2 years we have a heap of these motivated explanations, all as high quality as Grant's videos and the best books. But who will read them? There is limited interest in this genre. Grant reaches a large fraction of this audience but most people really don't want to think about math either way, no matter how good the explanation is.

Indeed, there is now "edutainment slop" online and AI can use 3blue1brown's manim library to copy his style and AI can use blender and video generation to mimic 3d animations of other explainer channels. Today it's still slop, but it may not be for too long. And then people will have to reframe their job until it's "doing X while also farting and burping every now and then", and then a machine will be better at that too eventually.

Also, this new style of doing math will appeal to a different set of people. Many mathematicians aren't super social, they just like to explore a problem on their own. Think Grigori Perelman. They will still face the problem and their temperament may not make it easy to switch to being a communicator.

I enjoy learning math from LLM proofs with the help of LLMs https://github.com/htzh/flt_for_human . It is amazing how well models do when they are well grounded by formalized proof traces (even if created by other models).

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Give teachers and professors 1% of all future earnings of every student

Well, so Tao now invites literal industry boosters to lure mathematicians into a pro-AI stance. This is the guest poster:

https://www.3blue1brown.com/talent

The only concrete step any mathematician on the internet, including on the other AI concern site https://proofsandprompts.com/ , is demanding funding for an academic frontier AI.

Strange that the Poincare conjecture was solved by a hermit without all this AI bullshit. Maybe reject AI, ignore all AI proofs and retreat from the internet.

I fully agree that motivated explanation is more important than proof. This doesn't resolve mathematicians' feelings of existential dread, however. Machines will get better than human mathematicians at motivated explanation in another year.

There will be no more glory in mathematics, but at least the joy of understanding will remain, and it will come without deciphering the tortured proofs that machines output today. Each bit of understanding will come with much less struggle, but this just means we can get more understanding for a given amount of struggle.

Even if we can proof/disproof any statement in Math (not possible due to halting problem), Human still need to decide which statement to be called "theorem".

The theorem thing is invented by human to help other people better understand Math structure in a easier way.

Math academia 2025

> Sorry, only epic problem solvers allowed here

Math academia 2026

> We were more than just problem solvers

I think people are overblowing this though. Wake me up when GPT-whatever writes gcc from scratch, then by the Curry-Howard I'd be impressed

I can’t help but feel a little schadenfreude. STEM folks may soon find themselves masters of skills as esoteric as translating Ancient Greek poetry or analyzing 18th century novels. The ability to construct complex mathematical proofs will become a party trick, rather like the ability to mentally multiply 10 digit numbers. The arguments that STEM snobs dismissed in favor of the study of the humanities will be the very same arguments that they now turn to. We will hear about how math and science make you a better rounded person, have inherent as well as instrumental value, etc. etc.

Interesting headline.

It's interesting because, as far as I'm aware, the vast majority of people already believe that math is more than proof. A slightly smaller but still very large majority don't even include proofs in their mental concept of what math involves.

Jacob Tsimerman claims [1] we might have superhuman expositors by April, so then what?

[1] https://youtu.be/H7_d_sgui6o?t=4436 (timestamped url)

It's too late for this. Much of my work in math has been classified as trivial or best described as not math at all. When I was working on chatbots and described deep learning algorithms to enhance them, it was deemed as a pseudoscience. Mathematicians sound very disingenuous with their backtracking.

I'm afraid that much of mathematicians work is too trivial to be taken seriously and they should just find something completely different to do.

Logic is the foundational weapon operating on sentences.

The act of stitching together, a series of sentences as true is what logic is.

If you make the stitching as airtight as possible, congratulations, you are in the realm of math.

If you are stitching together reasonably similiar to how the masses do, congratulations you have common sense.

If you stitch together completely random sentences, you are in the realm of nonsense and you may be classified as a retard.

The weapon is the same. The discipline differs and hence the effort to produce the chain.

So I am not at all worried about LLMs producing math proofs.

Godel with his incompleteness theorem helps one sleep easy. Rest assured no LLM can fly above Godel Incompleteness theorem.

There will always be statements that are true. So yes, it is time to celebrate.

It starts to sound like medieval science - "understanding" instead of proofs. And like a medieval army loosing a battle in the open field tries to retreat back into the fortress, people, facing the prospects of machine doing intelligent tasks better than humans, start to retreat into areas like intuition which supposedly aren't reachable by the machine. Some go even further starting to talk about religion. It is very Hegelian that the crown jewel achievement of our civilization starts to drive people away from the foundational principles of that civilization.

Mr. Tao is an excellent politician. Lots of awards and texts, yet no major problem solved.

It seems now that NS is solved he is mobilizing the community to convince taxpayers continue to pay even though AI may do a better job in his work.

Also, his opinion of AI has continually changed in the past years, after the capabilities were demonstrated.

Its a reasonable view to take that "human math" [ math residing in human minds ] is the only math that counts.

Math that only resides in the weights of models, or arcane forms such as a long lean proof or even an unread textbook .. is not the math that we should be striving for.

Likewise all other technology [ and culture ].

LLMs and AI / AGI / ASI could lead to a new renaissance of math discussion and expansion of human math and science. Or the opposite, where we outsource all our thinking to the AI, and no new generation of artisans is trained by doing hard problems, and in a generation we have killed off human math.

Likewise all of the fields of human intellect. We need to make sure we protect future generations of doctors, biologists, software developers, architects, engineers, librarians, musicians, artists ...

A moratorium on AI development might be the only way to achieve this preservation of human culture.