The Dark Night of Mathematics

Mathematicians should realize: instead of spending decades on individual theorems, you can now create whole new subfields of mathematics. Of course, you won't be doing the creation directly, in the way the mathematicians of old did; that's just not the name of the game anymore. But you will still be directing the research and involved in the creation.

Someone said:

> No one is stopping you from doing what you enjoy.

But that's wrong. LLMs are creating an environment in which some things are no longer enjoyable.

For example there is less joy to be had in learning a programming language in 2026 than there was in 2016, because there is less utility in it now.

As for squeezing in the time to do mathematics if the paying jobs disappear, there is a good chance we will have a world of plenty very shortly. I have a belief that the Austrian economists are right that there will always be plenty for humans to do and get rewarded for, but I also think that the costs of living will be so low that someone who wants to do mathematics will have plenty of time to do so. I hope this person holds the faith long enough and tries to see how the AI's can help accelerate one's own understanding. I am no fan of AIs generating a bunch of papers that no one reads or cares about, but I could say the same about the immense number of papers made by humans as well. If anything, the AIs make all of that immense knowledge more valuable as it is now knowable and accessible through these tools. There is also the fact that the mathematics job market has been oversaturated for at least 20 years. What our new tools allow is for people to be able to do mathematics outside of having access to a research library and top mathematics department.

This is really beautiful. And no matter what happens, two things have to be true: some form of lively philosophic and religious richness will keep being a core part of our history and will touch every person in some way; and it will change enormously over time (maybe with math continuing to be a small part, maybe not).

There is a part of math that is like that. There's also another part (as Vladimir Arnold provocatively said):

> All mathematics is divided into three parts: cryptography (paid for by CIA, KGB and the like) hydrodynamics (supported by manufacturers of atomic submarines) celestial mechanics (financed by military and other institutions dealing with missiles, such as NASA).

> Cryptography has generated number theory, algebraic geometry over finite fields, algebra, combinatorics and computers.

> Hydrodynamics procreated complex analysis, partial differential equations, Lie groups and algebra theory, cohomology theory and scientific computing.

> Celestial mechanics is the origin of dynamical systems, linear algebra, topology, variational calculus and symplectic geometry.

Those parts can just be delegated to AI the same way they used to be delegated to mathematicians. But Arnold continued:

> The existence of mysterious relations between all these different domains is the most striking and delightful feature of mathematics (having no rational explanation).

When a new technology comes along, it diminishes some aspect of the human experience - because whoever had a talent for doing things the older, more difficult way becomes less important.

In that regard, AI is the extreme, because it threatens to make so many of our intellectual talents useless.

I am not a mathematician, but this seems like an entirely non-problematic answer to me. Mathematics is ultimately the discovery of relations and their consequences, so of course large language models were eventually going to catch up. But precisely what they do lack is ability to appreciate these relations.

The author says mathematics is a spiritual pursuit.[1] I have asked LLMs about theological topics before and they have also been able to produce perfectly cogent answers (even heavy questions like “how did Aquinas view Pseudo-Dionysius’ negation-laden description of God”). I am not the least bit shaken by this, because it is still on me to evaluate and understand, and appreciate these answers.[2] The author seems to think LLM’s pattern recognition makes redundant human understanding and appreciation, which is a logical leap that doesn’t make sense to me. Maybe the author is conflating utility and purpose? I feel bad, and hope the author takes care of themself, but I really think the author is taking a massive leap here. It is not that deep.

[1] I begrudgingly agree, but with the caveat that tending to a vegetable garden in your backyard is also a spiritual pursuit. Mathematics isn’t some special discipline that elevates you beyond other people, as much as it is useful and interesting.

[2] I of course wouldn’t consult an LLM if I actually wanted to educate myself or reason about these topics. Not only do I want to reason through them myself, but when it comes to issues in philosophy/theology/heavy stuff, you need to take in to account the perspective and experiences of the human writing them. LLMs muddle everyone’s perspective together.

LLMs just democratized that process.

> Something fundamental to the experience of mathematics is being taken.

That is the near term. At least "near term" like we talked about it five years ago. And it won't stop.

AI isn't a tame long term transition - in scale it is bigger than any transition since the first individual cells. In speed it is happening faster than web adoption. No human adoption bottleneck. They are taking the reins of our tools, and don't need human adaption to improve.

Ineffables, and our intellectual primacy are going away, right now, even as we think about it.

I am not making light of it. But not surprised, because how could AI not redefined everything.

But maybe not everything:

20 years from now, 50, 100, there will still be ineffable experiences at the frontiers, by different beings. I believe esoteric curiosity and the intrinsic rewards of discovery will continue. Our propensities to find idiosyncratic interests and pursuits, seek answers and adventures, exist in our psychology because the low median return is decisively outmatched the extremely hight mean returns of unanticipated progress. The benefits of many pursuits will be even greater for them.

It seems unlikely to me, that future beings will become less interesting.

I relate to the author. Anyone who isn't feeling butterflies or stones in their stomach, isn't really processing the moment.

I'm conflicted in feeling there's now a lot more to do with AI, but also if it ain't me doing most of it, where's the fun in that? Days feel meaningless, and don't know who will pay for what I want to do, after all these years of heavy training.

There are a lot of reasons you might make a conjecture. Perhaps you are motivated by a search for structure: "We suspect that these objects behave like this, therefore we naturally imagine that...". But you might make a conjecture out of empirical evidence: "We checked 1 million examples, time to make a conjecture." While empiricism has its place in mathematics, it might not be the best motivation for a conjecture. As Bishop said, "Do not ask whether a statement is true until you know what it means." If you pose a conjecture based on empirical evidence, perhaps you should be thinking more about why you would imagine your statement be true in the first place.

I know some will think that this rings of goalpost moving, but when I hear about all these counterexamples spelling the end of mathematics as we know it, I just wonder how strong those conjectures really were in the first place. Again, I haven't actually dug into the specifics of any of the results so admittedly I could be totally off base, but I do know that not all conjectures are created equal.

It was a struggle with a lot of dead ends but it is just software engineering. It’s not a world away from getting a Rust program to type check.

The main problem I had with it is that LLMs will happily grind away case checking in Lean until the end of time and it’s up to you to see patterns and find dead ends. For example it wasn’t until I suggested to try translating the problem to a different characteristic that Mythos one shotted the proof (and found a counterexample for a related question I was working on).

My main problem now is _what to do with it_. I am not an academic, don’t know any academics and it’s a minor problem that I picked because I thought it was tractable and turned out to not be in the literature and fairly complicated, and the only reason I spent as much time on it as I did is that I thought I was an hour away from cracking it for about 10 of those days.

(In case anybody is curious about the proof, it’s that you can’t compose a single two variable polynomial over the integers with itself and any number of integer constants via substitution to generate all polynomials, but you can with x^2 - y and 1/2 if you allow rational numbers)

> ... It revealed that the process of prompting novel proofs will be as auraless as ordering doordash. Watch as magic and mystery evaporate. Watch as the sun sets on our heroic age. Is there not something evil in the act of blocking all future generations of mathematicians from the experience of discovery? Forget about accuracy or even attribution. Something fundamental to the experience of mathematics is being taken."

These passages resonated with me, as someone who has been enchanted by writing software for almost 60 years. It crystalizes something that has been nagging at me for many years: I like writing software. Reviewing, testing, spec-ing, designing, etc. are all important, but they are all incidental to the actual creation of software. They are all necessary for me to do if I'm going to write software, but they are peripheral. I didn't latch on to computer programming because I got into flow state reviewing code, or spec-ing it.

And this is happening in one profession after another. For example, fighter pilots. I suspect that a fighter pilot feels about flying jet fighters the same way that I feel about programming. And he or she will soon be exactly as useless: Doing things related to flying, from the sidelines, but not doing the thing him or herself.

AI is stealing all the fun parts.

There's a Chinese idiom: 生不逢时 "to be born at the wrong time." The story was about a man who, when young, lived in a peaceful kingdom (they needed older men then). Then, when he was old, he lived in a kingdom at war (they needed young men then). You can't choose when you are born or what environment you are born into.

I also wonder what it would be like to be this age in, say, the year 2226 and to think "hey, I could have been a great mathematician if I had been born 200+ years earlier!" Or to think, "hey, if I had been born 500 years earlier I would have been a great explorer."

Do what you can with the time you are in. It will go by just as fast.

As a mathematician who publishes theorems, I think this is mistaken. New theorems in abstract and pure fields of mathematics never had the economic value of what mathematicians are paid. Even creative works of art can be displayed but mathematical papers are unread by anybody except a tiny number of people who wouldn't want to pay all that much for them if they had to.

The economic value of mathematicians is to teach mathematics. Proving theorems is how we build the credentials we need to be trusted to be allowed to teach it. Teaching young and less young people how to navigate through a mathematical problem is how we make ourselves useful.

LLMs coming up with counter examples to obscure conjectures (albeit old and important to a small number of people) is only a threat in that it removes this ritual of belonging.

Many people are excellent teachers (and very competent mathematicians) without publishing new theorems, and many theorem producers are atrocious teachers.

Perhaps the pecking order may change. Or maybe something else will happen. Maybe society will decide to replace all its teachers with AI, or that mathematics is no longer worth learning. I don't know. But what I know is that nobody saw any of the papers I wrote or the theorems I proved and thought "that's money well spent".

For a mortal like myself, I've never discovered any new mathematics. All I can do is appreciate what I'm taught. But I can still appreciate it. I still watch videos of people solving high school/uni level questions. Why can't you appreciate this new counterexample, just because someone used an LLM to find it?

It's also worth looking at other times technology has changed our world. We invented various engines, so there's not a whole lot of economic value left in being a big strong guy anymore, but plenty of people still exercise because keeping in shape makes them happy.

If you want to make the case that the journey of creating the proof is part of doing mathematics, then you should also document that path and make it reproducible, as science should always be.

There were some examples of mathematicians publishing their AI proofs like that in the last days, but by far not all.

We already have desk lamps and heating fans, and many other things which make daily life convenient. But if you hand me a piece of paper (or better, a notebook), I'll fill it just enumerating the amount of diseases I want gone from the world. At least in theory, we know the laws of physics and how things are put together, and we know many of the impossibles of today are simply due to extreme scarcity and cost of qualified intellectual labor.

Maybe what we need is a shift from extreme pride on a specific set of technical skills, to priding ourselves in understanding enough of the world to solve some of its problems.

Maybe he should just touch grass.

Would a mathematician please unpack this for me? A theorem is the opposite of ineffable. A conjecture may be ineffable but its pursuit reifies it. I am missing something.

Both groups love math, but the later group is in a real bind because they have confounded their career with their hobbies/passions.

For years I've wanted to get back into self-studying mathematics, not to make serious contributions but just to appreciate its beauty; but the recent observation that I'll never be able to answer a pure mathematics question that a clanker could not has been off-putting, to say the least.

Unsaid here is that the companies involved have stolen so much (both literally in their plagiarism, and in their breaking of people's spirit), and given back so little. You still need to pay them cold, hard cash for them to help you prove theorems, and meanwhile, mathematicians working at these companies have themselves done little to contextualize and interpret results. That insight has mostly come from outsiders.

You need to be thinking about an entirely different form of life for all of humanity, happening in the very near future. Focusing on one specific field that may be earlier in the obsolescence chain is a distraction.

Merit based society based on intelligence or work of nearly any kind is about to cease existence.

For some reason it feels to me that this should make mathematicians feels better. :)

Another case of Deep Blue.

If you are interested in mathematics because it can model things precisely, and you want precise answers about these models, and you want to know how it is all connected (Langlands anyone?), AI is fantastic news. There is plenty of new and interesting and beautiful and elegant mathematics to be had this way, as well.

This is not a time to be scared or frightened. This is a time to be excited as fuck.

There will always be open questions. Now, there will be actually many more of them, because many more people will be asking questions.

[1] http://abstractionlogic.com

Some mathematicians (like SWEs) view AI as a boost to their ability to explore and understand possibilities - it is a great tool. Others see it as taking the soul out of what they do (loss of flow or craft). Most worry about losing their jobs.

I think for both mathematicians and SWEs, there is real loss for how we worked in the past, but I don't think job loss will be quite as severe as feared (and may actually increase). Because we still need, more than ever, people who understand the output and, more importantly, can ask the right questions.

The joy and pride of lifting heavy chunks of mathematical infrastructure into place are being made redundant by industrial machinery that any amateur can rent or build for themselves. But it seems to me there's plenty of room to discover new mathematical vistas. The future of mathematical discovery is not cracking hard open problems that everyone in the math community agrees would be an impressive lift, but by bigging into things that nobody else thinks are interesting or important.

Suppose any theorem you set out to prove had already been proved in 100 wonderful ways

No matter how brilliant you are, no matter what intellectual heights you scale, you'll never be Pythagoras or Euclid or any of many famous mathematicians whose insights purchased immortality. Why even live?

To be frank, I occasionally feel this way because I am still absolutely knocked out by very simple things like plane geometry, powers, irrational numbers, exponentiation and logarithms etc. I smile and nod politely about reports of contemporary breakthroughs linking this obscure subfields with another - partly because I haven't put in the years of study to know a great deal about advanced and frontier topics, partly because I'm not smart enough to fully appreciate them, but mostly because they're often about the surprising obverse of some feature in a corner of a utility corridor in the dusty cellar of an annex in the grounds of the Grand Mathematical Temple. Nobody will be able to experience lighting a candle and illuminating the great structures of the main hall for the first time, just like no chemist can ever hope to wake up in the morning and discover a new element and most physicists have abandoned the idea that they will ever be able to do more than tinker around the periphery of the discipline in the hope of extending the precision of measurements by another decimal place.

But the amazement and perplexity about the unreasonable coherence of mathematics (and its equally unreasonable effectiveness in the natural sciences) are what make the field compelling in the first place. The capacity for curiosity and obsession are what yield big discoveries, more so fascination with extending a well-defined knowledge boundary out a little farther. Put another way, pointing out the existence of a problem can be more significant than solving it.

Some notions:

- finding counter-examples is one of the most exploratory uses of calculation

- there is no evidence, not even in the hands of the inimitable Terence Tao, that Approximate Intelligence (AI) is insightful rather than inferential*

- people who enjoy Mathematics, even amateurs, do so for the beauty

- AI is only good in the hands of those who can provide corrective context

- AI unchecked is a remarkably inefficient noise engine

- Mathematics will provide joy to the curious 'til the heat death of the Universe

- AI is contributing to the heat death of planet Earth

- anthropomorphism, of which I am otherwise not a fan, may yet be a saving grace as without proprioception we may not morph

- we may need to redefine the Turing Tarpit of Perlis

If any of this nonsense really troubles you return again to thermodynamics and revel in the joy of being the local universe's pinnacle of meaty evolution. And go outside and smell the flowers.

* Caveats abound

lol who is paying for this? Mathematicians are generally paid to do other things and get to do this kind of work as a side effect.

Although I love these new tools this is pretty much exactly how I feel about hand written code. I miss that addiction viscerally and spirituality. I don't want to go back but I also wish I could. I wish I could go back to typing basic at a glowing orange Apple IIe CRT, but I can't.

I hope that this doesn't mean that you need write proofs while streaming Twitch. ;)

100% yes. They brag about it!

Regarding math, the new tools are humanity's achievements, they weren't handed down to us from the sky. These tools are spiritual achievements. Just as much as calculators and computers in general. Just as much as the invention of writing and notation. You could also mourn the days when we'd calculate on our fingers only and would orally memorize certain calculation heuristics of geometry handed down the generations.

Regarding jobs, people aren't being paid for having a good time or for having fun and spiritual feelings. When you get paid, someone gives you money. The person or institution giving you the money has to have a reason for doing this. It is very basic logic, but academics don't seem to get this. If your work is pleasing for someone on an artistic or spiritual level, you can find a rich patron, the same way learned scholars did back in the day.

It's not any different from cashiers who get replaced by self-checkout or bank tellers by ATMs. You're not special. It's a job.

And I say this as someone who really appreciates the feeling of gaining insight when cracking a math puzzle or grokking how the definitions fit together and why something is the way it is, why a theorem works, I like mathematical elegance etc. But I don't think that is diminished in the least by being able to consult a smart AI about it.

I bet the same argument was used when the calculator was invented.

Besides, doing pure math as a vocation was and still is an incredibly niche and privileged profession that 1-O(e^-n) of the population had access to anyways. If anything we will need more mathematicians, more engineers and more physicists to keep up with the increased throughput. Maybe journals will finally start EMPLOYING (gasp!) some referees instead of relying on free labor from broke graduate students even!

First, this is about not just a job or career, but someone's identity. And we must have compassion that they are losing something that is fundamental to who they are. To them, it doesn't just feel like they're losing it, it is being taken by these companies that have so often acted in ways we despise

And that is a tragedy! And there are so many tragedies like this that will happen regularly as the technology advances

But my second reaction is one of this great shared experience. I studied math. Many of my friends are mathematicians. And it is now possible for anybody to access mathematical insights or think deeply about strange conjectures and theorems that were previously incomprehensible to anyone who hadn't at least studied mathematics in college

I don't know if 3b1b's Grant Sanderson considers himself a mathematician or science communicator, but I think of him as both. And while I expect he is someone with the mental capacity to find and prove new things in the world, I am grateful for the time he spends instead understanding things at a fundamental level and explaining and celebrating those concepts with his audience.

Not every mathematician can be or wants to be _that_ kind of mathematician. But for the moment, it is enough for me that higher mathematics is more accessible than ever.

And with some trepidation, I predict that the "traditional" job of a mathematician will change (of course it will). And it will change in big obvious ways and also subtle little ones. How will it change, though?

Importantly, math is actually going to be one of the fields where the fundamental things we thought we knew are shaken, because in the next five years we are going to start to see connections between things that were previously considered completely separate.

And so one way the job will change is that anyone who discusses math regularly will need to learn new things.

To me, this is exciting. It's almost like finding a bunch of new dinosaur fossils that fundamentally reshape our understanding. There's going to be a lot of work to do!

I stopped reading there.

Seriously how are we supposed to evaluate whether these were counterexamples to conjectures that real mathematicians had ever put any amount of effort into or not, if the author won't even link to them?

There are plenty of junk conjectures out there that even the conjucturee never spent time on.

Anyway, your "spiritual journey" doesn't matter. We'll automate mathematics because we can, because it's useful. Don't like it? Well, should have not chosen a capitalist economic system that rewards scientific progress so much.

What it has done in software engineering is kill all human open source spirit. There is hardly any new software out there, people do not talk about interesting things but just how AI "generates value" or similar nonsense. AI has stolen at least three potentially productive years.

The Leiden declaration is fine, but if people who make OpenAI ads like Tao sign it what do we make of it? Professors who are truly concerned should ban AI in universities, talk about IP theft to politicians and so on.

Found a new organization "Mathematicians against AI".