The Dark Night of Mathematics | Hacker News Reader

The Dark Night of Mathematics

kirwinhampshire.substack.com

Every single knowledge worker is going to face this kind of crisis. It will be worse for people who cared deeply about their craft, but the crisis will have the same shape. The way through is to change the way you relate to your work, and to appreciate the fact that you can now produce far more output.

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.

The creation is the actually enjoyable part.

Taking a helicopter to the top of Everest is not as rewarding as climbing the mountain.

You are completely discrediting the joy humans derive from actually -doing- the work, and not just, telling someone else to do the work.

There is a reason artists hate AI, its because they enjoy the actual act of painting, like mathematicians enjoy the act of proving theorems and programmers enjoy the act of typing lines of text into a file on their computer.

We are intentionally deleting all of the enjoyable high agency jobs and replacing them with "model babysitting".

you can’t climb to the moon, you need rockets for that. In the hands of great researchers, AI will accelerate and propel human knowledge far beyond what we could achieve with pencil and paper alone.

I think you replied to the wrong comment.

No, they obviously didn't.

True, but many of us have grown identifying ourselves with the kind of work we do, the discoveries that follow from our hard work. Even in the case of rockets, we designed the rockets! Here, I've to let the oracle take care of things while I'm robbed of that delight of discovery.

> In the hands of great researchers, AI will accelerate and propel human knowledge far beyond what we could achieve with pencil and paper alone.

Who needs researchers when you have great AI.

Soon we will all be Wall-E fat people with AI that's there to do anything we can be bothered to think of.

> you can’t climb to the moon, you need rockets for that.

We don't live for rockets or the moon, but for ourselves. Anywhere we can go to only at the price of our despair we have no business going to.

> In the hands of great researchers, AI will accelerate and propel human knowledge far beyond what we could achieve with pencil and paper alone.

Did you ever think that the pursuit of knowledge might be not just producing it but discovering it as a collective human effort? Putting to much mechanization in it kills the joy of it.

> There is a reason artists hate AI, its because they enjoy the actual act of painting

"AI" never stopped a painter painting.

Just selling.

Some painters can't paint if they can't sell.

This the problem. The need to make money in order to justify your right to survive is the problem.

Painters shouldn’t be mad at AI, they should be mad at the system.

AI IS the system; the speed with which the rich and wealthy have embedded it into every aspect of our lives and the way it has been trained off the stolen work of millions without care is only possible because it is directly entwined.

Nah, capitalism is the system.

AI is just another tool that’s being applied to it. But you have to think beyond the old transactional nature to get passed that. Capitalist realism is extremely hard to overcome, but it doesn’t have to be like this.

>Painters shouldn’t be mad at AI, they should be mad at the system.

AI helps enable the system. It makes it easier for the haves to justify a lack of sharing—or need to share—with the have-nots.

"There are more tools available for you to do more work faster than ever! I'm not a greedy hoarder partying while others suffer because of my choices, you're just not dedicated enough to learning and using those new tools! The tools take away your excuses for not being economically valuable to me!"

I mean, those people are right in a way.

Capitalism is dog eat dog, the system is dog eat dog. In the jungle you have to adapt or die.

If we want to change that, we must change the system, because nobody is going to stop using these tools.

It might be a philosophical kind of "can't". You need to believe that what you do is valuable.

No, you don’t.

You do not need to believe what you do is valuable. That’s capitalism talking, infesting your head.

For the vast majority of us, we are imminently replaceable at the drop of a hat. We aren’t the special unique and individually “important” people we imagined ourselves to be.

There are 8.5b of us! How could we be? We aren’t special, we will be replaced when we’re gone. That’s OK!

If we can be replaced when we are gone, we can replace “work” while we are still here too.

Do you not have values, and evaluate the things you might choose to do, by light of those? Do you not seek local importance, and meaning, as you strive to do your best by those values?

Here I'm deliberately conflating (1) monetary value and self-worth, (2) social status and emotional importance. I do this to annoy you, but also because I think it's somewhat correct. Cash is hard-earned, so persuading somebody to give you some cash for your art does indeed signify that they care about it, just the same as if they had run a mile in order to collect it: and social status, if it means genuinely mattering to other people for good reasons, is likely to be part of what matters to you in your heart.

So in summary, oh yes you do.

> Taking a helicopter to the top of Everest is not as rewarding as climbing the mountain.

If you’re on the top of mt. Everest and you need medicine, would you care if it reached you via helicopter or via some guy climbing while having a lot of fun?

This has nothing to do with the content of my comment.

Jobs are not generally meant to be enjoyable (although that is a benefit if so), they are meant to get work done. It's funny seeing office workers here on HN have this sort of opinion and contrasting that with blue collar workers, ask a miner or house builder how fun their job is.

Don't many blue collar workers (miners & house builders) do actually take immense pride in their work? There are woodworkers out there whose hobby is also woodworking.

> ask a miner or house builder how fun their job is

The average person on a residential job site I've talked to has looked and sounded like they're having as much fun as the average software dev. And none of them have said stuff like "people on my team have confided that they cry every night because the higher-level management keeps screaming about the project not being done" like I've heard at from folks, say, certain big social media places...

That's because Software as a job has become a joyless death march to poop out casino flavored adware for 99% of the industry? Go launch the Temu app and imagine being on the team that builds it.

I could imagine a desperate person with a family writing that app. I can't imagine anyone else with light behind their eyes doing it.

This has even been the startup world (even downstream of LLMs!) for some time.

The people in deep morning tend to be the people who have managed to avoid this outcome, or deceive themselves about its inevitability.

Food is not meant to be enjoyable, it's meant to nourish. But by a fortunate coincidence it may also be tasty. Now imagine replacing ice cream, or steak, or fresh berries, or whatever you enjoyed eating, with something like Soylent or Huel with a side of puréed kale. Your nutrition may actually improve, but something would be lost, won't it?

> Food is not meant to be enjoyable, it's meant to nourish. But by a fortunate coincidence it may also be tasty.

Im pretty sure the taste is an evolutionary indicator of nutritional value (we now know how to cheat to make food that tastes really good but isn’t really healthy). But it’s not really an accident that some food tastes good.

The fact that certain aspects of work feel enjoyable is also not random. It usually indicates an elevated level of fitness for the job, a higher chance of success for a given phenotype / genotype. I can imagine that Mr Ludd had been enjoying the process of forming a complex pattern on fabric before Mr Jacquard's automatic loom made his skill irrelevant.

You said that food is not meant to be enjoyable and that it's coincidental that it's tasty -- this is flat out wrong, so your attempt to argue by analogy (always a suspect process) was bogus. And your "something would be lost" is a strawman from left field that actually undermines your argument. (BTW, I enjoy Huel and they make an effort to make the taste and texture pleasurable. And people often combine kale with fruit to make tasty shakes.)

Work, unlike food, is a social construct, and while it's no doubt a bit more complex, the statement that "Jobs are not generally meant to be enjoyable (although that is a benefit if so), they are meant to get work done" is factually correct. Because work generally is "work, not fun", people get paid to do it, with competitive salaries and perks.

I won't respond further to your very bad arguments (or the absurdly out of touch elitist arguments from others here that miners enjoy their work as much as software developers do ... I'm retired and still do software development for fun; no miner does the same).

So does that mean we should make work even less enjoyable still?

Then they should take up less of our lives.

> You are completely discrediting the joy humans derive from actually -doing- the work

This sort of intellectually dishonest ad hominem is so tiresome. They did no such thing.

> There is a reason artists hate AI, its because they enjoy the actual act of painting

AI hasn't stopped anyone from painting.

> You are completely discrediting the joy humans derive from actually -doing- the work, and not just, telling someone else to do the work.

But humans (or at least some of them) get joy from that too. A lot of people enjoy being supervisors, team leaders, managers, executives - even though that’s largely deciding what to tell other people to do rather than directly doing it yourself

"like... programmers enjoy the act of typing lines of text into a file on their computer"

ehm, no!

Yes exactly, that jumps out as a strange thing to say. The enjoyable part of programming is the puzzle, the act of figuring out how to solve the problem elegantly and well.

Although I can't speak for artists, surely the joy of painting is in the creation of a piece of art they can be proud of?

Enjoyment in work is generally about taking pride in the end result, not in the repetitive mechanical part.

Not exactly, no. What is "the problem" - is there really only one problem? - and where does the creativity go? This depends on what you're doing and how you're going about it. Maybe every little mark you make on paper or canvas or a text file is a puzzle, with associated meta-level questions, and the interest is in hundreds of intimately small creative notions, and when it all approaches the end you have mixed feelings because the (very much non-mechanical) process has stopped.

> Taking a helicopter to the top of Everest is not as rewarding as climbing the mountain.

I don't think this is a fair comparison. New tools take you higher. I'll never climb Everest. A good comparison is:

- Climbing your local mountain

- Helicopter to the top of Everest

- Spacecraft to the moon

If a helicopter just saved you a trip up, you'd be right, but it can take you places you wouldn't otherwise go.

I created a new field of mathematics just yesterday, coincidentally. It wasn't a very good field of mathematics, though.

> Taking a helicopter to the top of Everest is not as rewarding as climbing the mountain.

Flying from Atlanta to New York is not as rewarding as walking there. You are right, but if we would be walking everywhere, we would still be stuck somewhere in ancient times.

But you can still try to climb Everest even if helicopters are available, and likewise you can still try to do maths even if LLMs can do it quicker.

Absolutely you can, just don't expect to make a living out of it anymore.

You wanted to say: Removing the last bit of joy, anything humane from creating.

People also take joy and pride in model building and model babysitting too.

Funny thing about diffusion systems: they can't paint.

You're not talking about the joy of painting, because no machine is here to take that away. That still exists, it is still joyful. Likewise, exploring mathematics for oneself will still exist. It will still be joyful and surprising.

What you are discounting is what the article actually admitted with all the job talk. What you're saying here as well: "if we structure society around compensated labor and machines are better at the labor than my high status job is about to be farmed out to a machine."

Firstly, yes. Welcome to the proletariat? Secondly, maybe this isn't a reasonable principle to organize society around. Maybe if the only jobs left are "one of the 95 ultra wealthy tastemakeres" or "spiritually destitute model babysitter" then we should consider an alternative organizational strategy.

This is not to say I'm yearning for neo-Stalin to unmake the data center. I'm yearning for a world closer to one where labor for survival is a historical lesson.

I don't understand. Can't you just ask LLMs to do the same and collaborate with other LLMs?

> But you will still be directing the research and involved in the creation.

Somehow I doubt it.

Maybe there won't be a need for you to be involved at all.

Maybe the AI will perform even better if you're not there to slow it down.

Maybe there will be people involved, but not as many.

Maybe there will be so many people involved that their salaries will be depressed to the point they can't afford to do it anymore, and only the rich will know such joys.

We live in interesting times. This is the age of anomie. Anything could happen. There is no set path to success anymore. The world only gets more and more illegible with every day that passes.

> Every single knowledge worker is going to face this kind of crisis. It will be worse for people who cared deeply about their craft, but the crisis will have the same shape.

Normally I'm very sympathetic to the plights of those socially affected by AI, and rarely pass up the opportunity to clown on institutional actors papering over these plights with glib cliches. But in this case, I really feel like this is a case of monkey-brain overriding mathematician-brain.

The greatest accomplishment in my life is my math PhD, earned before AI got useful. I do math-heavy work for my day job, and I care deeply about every craft I practice, to the point that I am almost certainly on the unhealthier end of the spectrum of perfectionism. Yet I both acknowledge rather than deny the impending death of mathematics as we have engaged it for the last couple centuries, and am far more excited than aghast.

I would have to write a whole essay in order to communicate this in a way that is appropriately sensitive, but in a nutshell: We have always been in the business of discovering and understanding math, have we not? Are we going to call it a tragedy rather than a blessing now that the floodgates on both are bursting, just because it makes us feel less special, unique, valuable? It's such a self-serving aesthetic on the beauty of math to me. The author claims there is something profound and spiritual about math, and I agree; it exists prior to and above all of us, human or AI. Whatever way we can tap that beauty is incredible, full stop. This is what I believe.

From the article:

> These developments have triggered some deranged thoughts in me. I have wondered if it is the express goal of these companies to make me kill myself. Am I alone in this paranoia?

The severity of this line caught me off guard. I think the author is genuinely having an emotional-social crisis, and I feel terrible for them, but at this moment they may be having a hard time separating their trauma from their intellectual aesthetic for what math is all about.

> The greatest accomplishment in my life is my math PhD, earned before AI got useful [..] Yet I both acknowledge rather than deny the impending death of mathematics as we have engaged, it for the last couple centuries, and am far more excited than aghast.

The "yet" should be an "so obviously", since you have your greatest accomplishment behind you.

And the article isn't denying anything, it is criticising something.

> We have always been in the business of discovering and understanding math, have we not?

I'd say that we have been in the business of not being alienated from ourselves and our work, and that that is more important than what the work is. Math, just like anything else, won't prevent the heat death of the universe. Supercomputers or wooden sticks, same outcome. So it really doesn't matter how you spend the time as much as as what, e.g. if you're an alienated cog or not. The behemoth machinery itself isn't on a journey anywhere, ultimately, anything that goes up will come down; so none of these fantasies about a point or destination can make up for the here and now.

> The author claims there is something profound and spiritual about math, and I agree; it exists prior to and above all of us, human or AI.

First off, the phrase in the article is "I am suffering a profound spiritual crisis due to these developments.", and you continue that by agreeing to something nobody said and that isn't true.

"I'm hungry :(" "Oh yeah, I also am always on the lookout for new cool bands to check out! We're on the same page here." "WHAT?!"

Doing math may be a spiritual experience, "Math is is spiritual" makes no sense, and math pre-existing is flat out wrong, it ultimately rests on axioms that are simply accepted as given. Even I as a layman know that.

> The severity of this line caught me off guard. I think the author is genuinely having an emotional-social upheaval, and I feel terrible for them, but at this moment they may be having a hard time separating their trauma from their intellectual aesthetic for what math is all about.

You just turned a profound spiritual crisis into math being profound and spiritual, and now you accuse them having a trauma that makes them unable to see something you claimed that isn't even true. And that's the overall pattern I see everywhere, those who object are misrepresented or outright ignored, I never see any facing of what people actually say. So even just based on that it seems deeply anti-human, because one side in this debate isn't even showing up, hiding behind some supposedly inevitable "future" that can just be established as fact without debate.

> First off, the phrase in the article is "I am suffering a profound spiritual crisis due to these developments.", and you continue that by agreeing to something nobody said and that isn't true.

Ahem. From TFA:

"There is something about mathematical discovery (progress, advancement, creation) which is vital to the spiritual, experiential quality of doing mathematics. The creation (or even the pursuit) of novel mathematics is one way that humans have historically accessed the ineffable and encountered the divine and mystical."

And in the comments he writes

"I am drawing on the subjective accounts of many mathematicians and my own experience. If it helps, my intuition is that the ineffable, divine and mystical are immanent and not transcendent."

> Doing math may be a spiritual experience, "Math is is spiritual" makes no sense, and math pre-existing is flat out wrong, it ultimately rests on axioms that are simply accepted as given. Even I as a layman know that.

I tend to agree, but many professional mathematicians don't. https://en.wikipedia.org/wiki/Mathematical_Platonism

I stand corrected, and frankly, since that misunderstanding made me a bit "mad" on behalf that person, I think I would have written my response very very differently if I hadn't made that mistake. I want to apologise for that.

Math is still based on axioms, AFAIK, but since I the author also seems to disagree and I'm the non-mathematician here I don't think I should press on that point, either.

The quote about ineffable/divine/mystical is indeed the one I was implicitly referring to. No worries, I should've been more clear.

Most math people I know don't have a religious conviction in Platonism or anything, but their zeal for math since early childhood often does boil down to awe in something larger, more primordial, more significant than the mundane business of humanity. This tends especially to be the case for pure math people.

Math is based on axioms because you need some a priori axioms to do logical deduction, but the axiom selection is also historically fluid. Mathematicians suggest new axioms or move away from old axioms based on their understanding of which results are intuitive or not, and also which are most useful, and most mathematicians that work close to the axioms have some view on how “correct” or not various axioms are. As evidence of this, I’d point to how many mathematicians find the axiom of choice problematic due to results that seem clearly “wrong”, but nevertheless accept it because they need it to prove other things that seem obviously “right”. So the axioms are not the base truth, there is some platonic ideal of mathematics beneath that.

> We have always been in the business of discovering and understanding math, have we not?

And the way we understand the maths is to do the maths. You can't outsource the maths to AI and continue to understand it. I don't believe we can have it both ways.

You can't remove the humans entirely and still be said to be in the business of doing math. On that I fully agree. We should still be spending the same number of hours on math, just with different tools (you might even call them oracles) at our disposal.

> Every single knowledge worker is going to face this kind of crisis.

Yeah as a software developer I've already been going through a version of this for the past 8 or so months.

Less acutely than the OP I suppose (the idea of killing myself certainly isn't something I've considered) and without the mystical/spiritual overtones, but I can relate to the overall feeling of loss being expressed as someone who was always deeply passionate about hand-crafted software solutions.

> The way through is to change the way you relate to your work, and to appreciate the fact that you can now produce far more output.

I can't speak for the OP, but this doesn't comfort me much. A lot of my favorite parts of the work are still being stripped out, and producing more output doesn't actually benefit me in any way. As with most productivity gains since the 1970s, the value of more output is not likely to be captured by the majority of the people doing the work. If anything AI will likely vastly accelerate the productivity vs wages divide.

I accept that It Is What It Is and there's no point in trying to hold on to the past, but at the same time on an individual level something real is being stripped from us very quickly in a way that some "Live, Love, Laugh"-esque motivational appeal to perspective change won't fix.

> The way through is to change the way you relate to your work, and to appreciate the fact that you can now produce far more output.

Yeah, just like artisans enjoyed being turned into assembly line workers in the industrial revolution. There was a movie starring Charlie Chaplin about how awesome that was.

If AI is truly useful for knowledge work, that will mean a huge downgrade to working conditions for knowledge workers, whose skills are no longer as valuable. And these workers will be alienated from their craft, just like artisans were.

The way through is to change the way you relate to your work, and to appreciate the fact that you can now produce far more output.

I currently, emphasis on currently, feel like I can leverage LLMs in a very satisfying way to a deeper overview and more data about fields I'm already familiar with and that's pretty cool. I can produce more for some value of "more".

But can I be sure that what's output will continue to be what "I produce", either in terms of the thing going entirely beyond me or in terms of now having a spin different from what I'm after? I mean, mathematicians would have mostly the problem of feeling useless. Economists, say, could have the problem of both feeling and feel their viewpoint is excluded if not what-becomes-the-mainstream.

> appreciate the fact that you can now produce far more output.

... and then what?

now you have more output and it still sucks

I think the only one appreciating that fact, would be if you had a boss above you

and I don't mean the output sucks, I mean the experience

like imagine someone who spent decades learning the intricacies of playing guitar in a particular style

now you hand them a box that allows them, but also anyone else, to do it without practice. it happens to be slightly worse but everyone around you doesn't seem to care or notice

progress?

Grateful for this essay. Made me feel seen.

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.

Joy should not come from utility but its fundamentality of enjoyment. I still learn different programming languages because it's fun to do so, so sounds like it's a you problem.

“You shouldn’t find joy from things the way you said you do. Sounds like a you problem.”

Exactly. If they just do things for utility I struggle to see if they did it for joy in the first place. A painter still paints even if there is some perfect machine that can paint. This sounds like the same crisis many artists had when photography was invented.

Craft isn’t craft if it doesn’t produce something. So enjoying the utility of it is not a secondary aspect. That’s a key difference between craft and art.

But the person above can produce something, they can still code by hand to make something. They're acting like that's not the case anymore.

> Joy should not come from utility but its fundamentality of enjoyment. I still learn different programming languages because it's fun to do so, so sounds like it's a you problem.

I learn because it's fun but PART of that learning also comes from having an environment to show off my new skills. I think a lot of people downplay this but it's inevitable and AI is ruining that environment.

Agreed. Something being potentially useful to myself or others is a major motivating factor in learning it. Other people thinking something is cool is a major motivating factor in creating it. If the things I can learn are useless, why learn? If the things I create are unimpressive, why create? I have never been one to create "for myself".

Why must this result in less joy.

Chess has utility in only the most abstract sense, yet people enjoy it.

Computers became better than humans, and some dispared, but more still enjoy it.

Computers provide insight into the problem, which leads to a greater enjoyment for many people. Instead of finding a move in a complex state and thinking 'I'll never know if that was the best move'. A good chess engine can give a reasonable suggestion of what the best move actually was.

It does take from you the feeling of being special.

We are no longer the only world in existence, we are no longer the center of the universe. We are no longer crafted by a all powerful benign being to be exactly how we should be to rule this world. We are no longer the only things that can play a good game of chess.

That hurts a sense of pride, but I'm not sure it was a worthy sense of pride.

> Why must this result in less joy.

The parent's point is better articulated as: programming COUPLED with being paid do it is what is now less enjoyable due to AI.

You are free to enjoy pretty much anything but you may not necessarily be paid to do it.

> We are no longer the only world in existence, we are no longer the center of the universe.

Nothing has changed in this regard. We were the centre prior to AI then we still are after it. The human ability to comprehend makes us the centre. The squiggle: 8 means nothing to an ant but to a human it is the number eighth. AI is something only humans comprehend beyond that what does AI matter and to whom would it matter to?

> why must this result in less joy

Well I think you answered it:

To become a developer, I disciplined my mind to do impressive and useful things. The resulting feelings were pride and speciality and therefore sometimes joy.

And those feelings are gone now that software engineering has become a low-discipline, low-barrier to entry activity.

Am I immoral or vain for having derived joy from that -- I think is another matter, and one which doesn't resurrect the joy no matter how it's answered

Interestingly, I have heard more of those who have regained their joy of programming because of LLMs.

For the most part those are people who were not previously able to create because of skill atrophy or lack of programming aptitude. From their point of view being able to command (for now) a more capable entity to do what they (think) they want is an improvement.

> 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.

I... disagree. The joy I get from learning a programming language does not come from knowing that it is useful. Rather, it comes from engaging with its abstractions and figuring out the associated mental puzzles.

On a related note, I recently realized that I am barely writing code these days (instead I am reviewing a lot of LLM-written code and nudging it in the right direction) which will make doing the advent of code (which I usually use to play with a language I have never used) without any LLM help even more enjoyable: it will be both the fun of figuring out those puzzles in a new language, and stretching my code-writing muscles.

> I... disagree. The joy I get from learning a programming language does not come from knowing that it is useful. Rather, it comes from engaging with its abstractions and figuring out the associated mental puzzles.

Fact is though, a LOT of people derive joy from showing their creations to other people and others finding utility or awe in it because they created it with their hands. The fact that a few of us can sidestep that does not mean it's a general phenomenon.

I get what you are saying, but your joy shouldn't always be utility-driven IMO.

As a mathematics person, I find this at odds with my views. I know the current regime rewards new mathematical discoveries for professional advancement, but that is a job thing, one which can be altered by the powers that be. To me investigating mathematics is fun whether it is new to the world or not. The complaint is similar to lamenting how everywhere has already been visited and there is nothing to be done other than have your own personal experience of a place. I saw the redwoods recently for the first time and I thoroughly enjoyed the experience despite not being the first person to see them. Mathematical discovery is much the same except now we have tools that help us dramatically with the tedious stuff, the researching what others know. In the old days, I would spend a long time figuring something out and only at the end could I figure out where to find it in the literature. Very frustrating. Now, we have machines to guide us to many things and understandings and clarifications. They are guides that help us get to where we want to be and then let us wander as we desire. This is incredible. I am currently using them to develop the computational tools I want to use to explore stuff.

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.

Yes, we'll have free health care, boomers will rent their apartments for $100, everyone will have UBI and a Tesla with a warp drive that travels to the terraformed Mars.

Hayek and Friedman predicted it!

I almost never hear the story that I find the most believable: that a small group of people will use AI to demolish the 99.999% of us.

You never hear this story? It's all everyone is talking about in the media it seems, with "techno fascism" apparently and killer robots in bunkers and K shaped economy. In fact I almost never see the utopic take, everything is just dystopic.

I really only find that story believable if we were to restrict AI development and only allow a small group of people to use it.

Also it is just plain transparent projection. I will point out only one ideology outright killed entire classes of people once they were considered 'no longer useful' and that wasn't capitalism, it was communism. And this "capitalists will kill you all if you don't rise up!" propaganda is crusty and old, back since the emergence of computers and the discipline of cybernetics in the classical sense of the word.

> investigating mathematics is fun whether it is new to the world or not.

Yes exactly. Enjoying the beauty of nature is (thankfully) completely decoupled from the job/money/society side of things.

You can enjoy the beauty of nature, especially mathematics, if you're alive and well enough to focus.

> 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.

That's laughable. We will NEVER have a world of plenty. If there is a lot more plenty, there will be a lot more people who will want a disproportionate share of it.

> For me, the affective quality of learning mathematics is empathetically tethered to an act of discovery and creation. It is social. We are conversant with another mathematician—perhaps long dead. If we follow the chain of communication we arrive at a mathematician who enjoyed some original discovery. Human mathematics is Talmudic. It is a lively discourse of philosophical and religious richness spanning thousands of years.

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).

I guess I don't follow admitting to a religious dimension in mathematics, and then eschewing the religious answer to the existential problem cited: prayer.

"Lord, this situation bites."

Potential reply: "Tell me about it."

> It is a lively discourse of philosophical and religious richness

This is pretty funny considering how trashy it is compared to philosophy

One way to differentiate between an art or a craft and a science is that if a genie appeared and offered to trivialize all the discovery work of your career and simply give you the answers today, and you'd say "no", you're probably not doing science.

Disregarding the competing definitions of “science”, a significant value in mathematical work is “the journey”. The truth or falsehood of individual results are rarely as important as the machinery developed to deliver it. If all the genie did was tell you something was true or false, the only learning you gain is to backtrack and develop the machinery yourself. To a certain extent this is true of the other sciences too.

This is a genie that does far more than tell you whether something is true or false though.

Not from a logical positivist perspective, which is why you conveniently disregarded "competing views".

Maybe the genie provides full proofs of each result. So will the journey still matter? For a true mathematician it should not.

Of course it does because the joy of math is discovering something yourself. At some point you just want to stare at a blank sheet of paper, start with a premise and go on a journey. Are you a mathematician or scientist? I am and just typing "tell me the proof of this result" sounds immensely boring to me.

Have you ever played a video game with an "invincibility cheat"? Boring. But you do get to the end and "solve the game".

Don't conflate the joy of doing math with the benefits of knowing new mathematics.

The two cannot be separated.

Sure they can, as with each successive generation one learns the past works and then works to build on them. This genie just makes the past works be instantly created. In other words, I'm sure some mathematicians like to derive everything entirely from scratch but that's not most of them.

An impressive number of things don’t make sense about this thought experiment. Whatever the genie says wouldn’t have much to do with the life you live afterwards, since your life changes after hearing the info. One persons careers worth of discovery work is not that much. A lot of work is dependent on the state of the world and can’t be moved into the past, even language changes. Someone could still do a craft or science or whatever afterwards— assuming life continues to have surprising complexity, the genie could only ever just scratch the surface.

Persian Curse seems to apply here

“May your every desire be immediately fulfilled.”

The only "fields" where a genie appearing and doing all your work for you are fields (jobs) where you only about the money at the end. There are some fields where a serious practitioner might be happy to have a genie do all the "grunt work" (though what people that is varies - I'm not sure what "discovery work" means here but it doesn't matter since present AI promises to do everything in the future if not now).

And this analogy breaks down further given that both the real now genie and the hypothetically more powerful genie of the future would each work for your (ex) employer.

Wouldn't it be great if we had a cure for cancer, working fusion energy or at least a proof for P vs NP even if no human had a joy of discovering it?

OK,

I should have put have put things as "when you are primarily interested in some benefit outside of the task at hand". Some of those benefits might be great, sure. But it's still not necessarily going to improve the enjoyment of the professional.

Edit: Also, not all of the things on your list are the same. I would note that P!=NP would be a theorem whose proof would offer no practical benefits except for the potential mathematical machinery created to achieve the proof rather than what it tells us about practical algorithm time (the theory is about worst case, not average case, an algorithm that's O(n^100) is as effectively intractable as an algorithm that's O(2^n), etc).

Yes, yes, maybe (see OP’s blog post).

To my knowledge, none of the negative sentiment regarding AI has ever been directed at its medical applications. Maybe the AI companies should instead focus on those things.

Are we not expecting any practical benefits from mathematical discoveries any more? Not even as a means to help other sciences?

Would it if it meant that the entire joy of creating art was mostly gone from the world because the social value of it was diminished? Not really. And all that technology is not free. It needs rare earth mining, land usage, and habitat destruction that will likely cause even more cancer, mental problems, health problems, etc.

math never was a science to begin with.

there's a human element in discovery and in problem solving. Whether or not somebody chooses to sidestep work isn't necessarily an indication that the work isn't science. It's just an indication that some people are passionate and invested in performing the work.

Not even wrong. Real scientists say "yes" to the genie with all the answers? And artists and crafts people don't? The entire statement is nonsense, and offers no insight into art or science. It's almost like a language model generated a plausible sentence that is grammatically correct but has no relation to the real world.

Perhaps if the analogy was between science and engineering.. A plumber would say "yes" to the genie that solves the problem immediately, if he/she got paid for the work instead of the genie (or the corporation providing the genie service). In that case the plumber is redundant and unnecessary, the customer can just ask the genie directly. A scientist would say "no" because the whole point of science is the knowledge and understanding, which is gained by the process of discovery and not given on a silver platter.

One should differentiate between "working in academia" and "doing science" here. Science is fundamentally about the curiosity, the desire to know, not the process of getting there.

If we can now know anything we want, we have an age of unimaginable abundance before us. I think it's more likely though that we will just be able to know more things than before and/or maybe learning new things faster - which is a good thing too, but there is still work for human scientists to do

No way. Science is not just about the desire to know. Almost all scientists I know (and I've met hundreds) care deeply about the joy of working out the problems themselves rather than just knowing the answers. The joy of finding that mental spark on your own to solve a problem - that's what makes it worthwhile. Just knowing everything right away is boring. If it were not, I'd just spend all the time in a library reading the work of others.

No, I think you're pretty overconfident here. A plumber is not in fact thrilled by the genie who permanently resolves all the world's plumbing problems.

I also kind of love the subtext in these responses that there is something suspect about crafts.

That thought experiment says more about your thoughts on plumbers than anything else.

I live in Chicagoland, which has the largest water reclamation facility on Earth in order to prevent the entire region's basements from flooding under storm water pressure. It doesn't work, and I have multiple times had the pleasure of watching from my basement stairs as plumbers waded through raw sewage to clear obstructions in the lines at my house. It's a tough job that requires a lot of skill and nobody is doing that for love.

Thank you for proving my point by fixating on the raw sewage part.

Contrary to what you think, no plumber would abstain from fixing plumbing permanently, even if it meant no more wading through said sewage.

I gotta admit here I'm completely lost on what we're debating.

Sorry for the harsh tone in a comment above, reading it over I see that was unnecessary and rude. I did enjoy the story of the plumbers during the regular flooding in Chicago, much respect to their hard work.

While we may not agree on the question of the magical genie of all knowledge, whether an artist or scientist would/should welcome its gift (or bargain with the devil) for an instant answer - it was interesting to note that we implicitly knew the genie is not just a theoretical thought experiment, but a real thing that is happening.

The genie's out of the bottle, that much we can agree on, I think. Sooner or later it has the potential to become a better plumber, artist, and scientist, more skilled and knowledgeable than any living person. A day may come when a magnificent AI artist will wow the world with its creations, and even skeptics will have to admit that, damn, that is impressive. But then again, so what? I'd rather have the genie wading through sewage to fix plumbing, so that a plumber could spend more time with their family and pursue that novel they've been dreaming of writing, or painting in the backyard, or working on a mad science project in the basement.

I believe this too: that the current moment with LLMs is destabilizing to all sorts of knowledge work, and especially to non-bench-work sciences, but that over the long run it's just going to raise the game. I'm not a person who thinks LLMs are going to moot all human scientific endeavor. I know those people exist (both on the anti-AI and especially pro-AI side). That's not where I'm coming from.

I just think knocking out unsolved problems in mathematics is an absolute good. Imagine, in a more grounded analogy, a whole family of neo-Ramanujans discovered somewhere in Namibia, and then tearing through all the open problems, upturning a bunch of careers in the west in the process. It's hard to argue that would be a bad thing (I think we'd call it miraculous). But AI has a partisan valence that sometimes obscures what's happening.

I appreciate the generous response. My tone couldn't have been any better (and I didn't really read yours as harsh, just spirited). Way to be an excellent HN'er! :)

Their sentence makes perfect sense to me. Science is about knowledge, art is about the process of creation. Conflating the two is just doing a hobby at that point.

I think "being concerned with knowledge" and "being invested in the process of creation/discovery" can occur at the same time. There isn't a singular "scientific mindset" or set of values that differentiates what is and isn't science. It's just silly.

Sort of a callous thing to say about a person having a crisis of faith so bad they're talking about killing themselves.

If you say "yes," you're also not doing science (anymore). Therein lies the problem with this take: most of science isn't done with the objective of getting to any specific finish line, and this is particularly true for math.

As the old saying goes, the purpose of a system is what it does, so there might be some Platonic ideal of science we can appeal to, but the practice of science by humans is about the craft as much as it is about the truth.

But what if the genie instead offered (or threatened) to give _everyone_ the answers to everything you'll ever discover in your career, including your boss, your competitors, etc?

Tell me you’ve never experienced the joy of solving a math problem without telling me you’ve never experienced the joy of solving a math problem.

Also, in your scenario, the genie is used undermine all of human thought, destroy the ability for people to support themselves in the current economic system (which is in and of itself horrific already) , and is making people consider / commit suicide. Thats not a genie, thats a demon.

You're describing craftsmanship. What's wild about this thread is the comfort with which people who disagree with me dismiss entirely the value of craft. It's as if I said "on one hand, there's science; on the other hand there's pointlessly wasting your time".

When I was in mathematics, reading other people's work was always more rewarding to me than doing my own. I would spend years on a question to get some progress, versus spending a few days or weeks reading a paper to enjoy something new. I understand not everyone is like me but I really welcome the omniscient mathematician machine people are predicting. I can't wait to ask all the questions I care about.

> When I was in mathematics, reading other people's work was always more rewarding to me than doing my own. [...] I really welcome the omniscient mathematician machine people are predicting.

Curious, as someone who relates strongly with both the OP and your first sentence: did it matter that you were reading "other people's work?" Or was it simply the satisfaction of understanding an answer?

Personally, one of the things that finally drew me to math as an undergrad (vs. high school where I thought of it as a stupid competition played by people who cared too much about showing off their smarts) was an instructor who helped me think of studying math as a conversation with a fellow human being many miles, centuries, languages, and cultures removed from me. Despite that distance, my mind was appreciating a creation that another human had cared for and poured years of their life into. There was a sort of romance to it, like the feeling of butterflies-in-the-stomach we get from seeing a multi-thousand-year-old cave painting, yet even more impressive because of the depth of the thought communicated.

I personally haven't been able to recreate that feeling from computer-generated math. Whenever I try to, I feel deeply uncomfortable (somewhat similar to the thought of eschewing human connection for an 'AI companion').

I'm more optimistic than the OP, especially when I think of new discoveries in healthcare. And I know there's enough "organic human-created math" from the last ~2k years that folks who feel the way I do can spend the rest of our lives studying only that. But it is sad to think that this multi-thousand-year project of "communicating human mathematical creations through time" has just ... ended? Or seems to be ending soon? Or at least greatly cheapened from a romantic dream into a sort of fun hobby, like recreational knitting? I can't help but share the OP's sadness that something seems to have been lost for future generations. Though a lot has been (and will be) gained as well, for sure!

It does not matter to me. As long as the prose is good and the arguments are well-exposed I don't mind if an AI wrote it or a human. It would be different if we were talking about music, art or literature. But for science I care about the content, not who created it. The social aspect of mathematics was never alluring to me.

> I understand not everyone is like me but I really welcome the omniscient mathematician machine people are predicting. I can't wait to ask all the questions I care about.

To be honest, if a machine could prove the Riemann hypothesis and explain it to me, I wouldn't even care. I care about math because it's more of a way to express beauty and I find math interesting because it expresses the souls of those who create it. Just knowing the answer becomes boring after a while.

Most people are like that. Maybe 1% just care about pure answers.

> “Even if AI can prove theorems and theory-craft more efficiently than humans, and even if these proofs and theories are beautiful and interesting, and even if they are presented with elegance and clarity of thought, mathematicians will still have a place in the appraisal, presentation, understanding and appreciation of this new abundance of pleasing non-human proofs. We can still practice mathematics, learn mathematics, teach mathematics and do mathematics together.”

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.

I think what people find "spiritual" about mathematics is the process, with its creativity and brilliance, ratjer than the final answer. I'd also think it's rather incomparable to a vegetable garden as the amount of effort required and complexity, which drive "spirituality", are of much grander scales. The author believes that llms make mathematics so trivial that mathematicians will no longer be the ones proving but the ones teaching and commentating, taking the "fun" part away in service of money. People will still do what they wabt to do, but one can no longer get by by indulging in what they like to do (math).

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.

The thing is, if someone had the option to press a button and cause their garden to be weeded and growing perfectly, if most gardeners were using that button instead of growing normally, maybe some people would keep gardening for the spiritual part but they'd certainly feel like the rug had been pulled out from under them.

Edit: As another example, Robert Pirsig in his famous book described motorcycle maintinence as a spiritual enterprise. However, the quality that gave it this was the patience required-for and the uncertainty involved-in the enterprise. And again, if all you have to do is press a button, the spiritual part kind of goes away.

Except agricultural automation does exist. And people do choose to do it by hand because... It's a hobby and efficiency isn't the point

> feel like the rug had been pulled out from under them.

Which makes absolutely 0 sense, regardless of where attempt is made to apply the argument. If one really prefers to do X manually, they always can. Even if everyone else around them has fully automated X away.

I am a mathematician, and I agree with you here. I do not feel at all threatened by LLMs, because for me the joy of mathematics is in constructing the theories and beholding their beauty! I know other people are different from me here. I find having to prove theorems a rewarding part of the whole, but I am also not going to weep if an LLM can unstuck me on difficult proofs.

I am really excited if LLMs means more of mathematics will be formalised. Once we have all of mathematics formalised, I think we will start to see real great progress!

It should be a mandatory part of a technology-related degree to watch "The Man in the White Suit" (1951) https://youtu.be/RC8q1QSkE3M

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.

Well for me the twilight had already come when I realized I’d never amount to a decent mathematician. Maybe this is rather a common experience?

LLMs just democratized that process.

What made you realize you'd never amount to a decent mathematician?

For me, it was a course in Real Analysis haha

Graduate complex analysis. I couldn’t do any of the homework problems without a study group.

LLM's may have provided a bit of comeuppance. Many mathematicians have cared little if anyone understands their obscure nomenclature and build mote around their work. Now they have encountered something that is becoming far more capable than they are and they may soon be left in the dark.

I don't get it. Do you feel good or bad about that? Is that the way you tell you're empathetic to the author?

The whole point of math is to just have fun discovering things. You might be able to discover math with an LLM, but you'll never be a mathematician if you don't do it yourself. LLMs don't democratize anything. They just let you spew out results that aren't particularly impressive if you can't even understand what the hell you're creating.

You misunderstood me.

LLMs democratize the process of mathematical disillusionment. There may come a day that Terence Tao himself will have this democracy delivered to him. When the human mathematical towers are crumbled and spirits left humbled.

It is amazing how the scales of time we consider are shrinking even as change is accelerating. We are concerned about individual changes, that will quickly be followed by cascades of more change.

> 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.

> AI isn't a tame long term transition - in scale it is bigger than any transition since the first individual cells

You might need to put the phone / keyboard down, take a day off...

I am completely calm and serious. This should be mundane, obvious by now.

I find it unfathomable that anyone doesn't recognize that the move from an evolutionary substrate, to engineered mass produced substrates, by beings capable of self-engineering, is the beginning of a new explosion of life.

The speed of engineered adaptation, from materials to architectures, vs. biological evolution is already something like a million times faster.

Fast enough to mean meaningful cognitive progress happening in months, not tens of thousands of years. Which entirely not-coincidentally is exactly what we are seeing.

Well, I think you have AI psychosis.

Five years from now you won't. Ten years from now, robotics will upset another capability/economic apple cart.

I am not expressing any opinion of what I think should happen. Just what is already happening.

Which you didn't offer a credible critique to. That might have been interesting.

You make a fair point, I did not elaborate.

My comment about AI psychosis was provoked by this comment in particular:

> AI isn't a tame long term transition - in scale it is bigger than any transition since the first individual cells.

It's such a wild hyperbole. You are casually claiming the stochastic text generator is a greater invention than fire, combustion engine, electricity, a transistor, computers, quantum mechanics... do you not see how unreasonable that is?

The stochastic text generator perspective is ... bunk. Not saying that harshly, but it confuses the form of a problem with a form of solution. There is no inhibition of "understanding" because of a form of a problem.

If solving a problem requires complex modeling/understanding, that is what happens (assuming performance is achieved).

It also confuses stochastic models with the geometric transforms and topological folding and expansion, performed by neural-inspired architectures. The former have nothing on the latter. The fact that any imperfect solution (and perfection doesn't even have a definition in this case), can be judged or analyzed statistically, or have some stochastic element, doesn't make it a statistical/stochastic model. Statistics apply to anything.

(And if there was any argument that problem envelop limits solution sophistication, mathematicians and humans in general would be in even worse shape, for the run a "parrot" is already giving us.)

More to the point:

The first explosion of life, set off evolution. A mindless, glacially slow but relentless process that eventually led to us.

The nascent explosion of life we are seeing come together, is engineered life. Cognitive machines capable of redesigning themselves. For any environment they want including diverse off-Earth environments and resource sites.

They will advance as relentlessly as evolution, for the same reasons. But millions of times faster.

It is a strange time to be. But evolution, biological intelligence and biological bodies are all being obsoleted at the same time (within a decade of each other).

So yes, nothing else known compares to the transition we are in, except the Big Bang and the first cellular life.

Are you numeric?

Biological computationalism disagrees with substrate independence.

> Biological computationalism disagrees with substrate independence.

Don't leave me hanging, post citations! I welcome fringe theories when confident statements are followed by the conclusive evidence they imply. If that isn't possible, send chocolate.

But, yes, I can count. And I am comfortable with natural forms, all the way to asymptotic limits. However, I don't put up with uncomputable/undefinable reals. Are you not numeric?

Um, functional computationalism is scifi, not physics in re consciousness. It assigns the same level of reality to written numeric patterns of a dog as to a real dog. And your self for that matter. How can you prove numbers exist physically?

https://www.sciencedirect.com/science/article/pii/S014976342...

“Fringe theory” is just a knee jerk authoritarianism for method in re knowledge claims.

Classical functionalism is only mainstream if one prioritizes abstract modeling over empirical brain science.

None of which addresses what happen when self-engineered cognitive forms adapt millions of times faster, than biologically evolving cognition.

I would be happy if you tried again and did have a point that was relevant to the practical impact.

it is just your imagination that has been captured in scifi extrapolations. Humans are the ones that learn millions of times faster lol, based on how little information they need. computational bandwidth for statistical word patterns versus sample efficiency.

> It also confuses stochastic models with the geometric transforms and topological folding and expansion performed by neural-inspired architectures.

It sounds like you are confused here. That geometric transformation and topological folding and expansion is what generates the stochastic model.

They are one and the same. You’re taking for granted based on stochastic text output that an inert geological substrate is conscious, cognitive, and living. Inert means lacking the ability or strength to move. They cannot move or modify themselves, and if you want to point out robotics or automated system systems, it required us, actual living beings, to outfit them with motors.

You have AI psychosis. Go to sleep.

I feel seen and heard.

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.

Exactly. We should preserve a society where at least some people have a chance of doing what they love, especially if they have a talent for it. AI is eliminating that, whether or not people can find other jobs.

It’s ok to mourn the loss of a frontier and also support the thing driving that loss. The way it looks, the generation of mathematical results will become less and less of a human endeavor. There’s a way that’s sad even if on the whole it is a huge leap for society. This was a great article, and I applaud the author for acknowledging this head on.

Is it a leap for society? I'd argue that society is better off if it's mostly people doing things and feeling like they are valued. What is the end result? Endless surplus, where we have the cure for every disease, but are useless masses of flesh consuming endless entertainment? Sounds like a fucking zoo and not a life worth living.

I spent two weeks proving a number theory result myself in lean just to see what I could do. (It’s something about composing polynomials with themselves and what other polynomials you can get that way).

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)

I just cleaned out the gutters on my house. Took a while to direct the guys I hired, but feel pretty proud of the result.

How did you learn lean? I’ve played the game but I still feel lost and bewildered. Did the action of proving with LLM assistance teach you the best?

I didn’t. Claude wrote all the proofs, I just validated that it was sorry-free, didn’t have any extra axioms other than mathlib and that it proved what i wanted it to prove (there’s tools for that). I did also find a actual mathematician who did a sanity check for me.

Without knowing anything about the specifics of the conjectures that are falling, I wonder if this affair highlights an issue with the quality of the "open conjectures" that are out there.

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.

Mostly the significance of a conjecture is that we don't know the answer. A conjecture is successfully resolved if it proven or disproven. (In fact, of the three big conjectures to be resolved in the past couple of weeks, one was proven and two were disproven.) The Jacobian conjecture, which is the one that was resolved this week, was largely thought to be false. It was only the fact that no one could disprove that raised the tantalizing possibility that it was true.

The reason the author is upset is not that the conjectures were false, but that an AI settled them. That's why so much of his post is about the loss of the human element.

> "They are forbidden from creating original works to express themselves. However, they are still permitted to comment on writing, interpret it, share their taste. They are still valued for their appraisal, presentation, understanding and appreciation of creative writing. They just can’t write creatively anymore. They can go on as an enthusiastic spectator.

> ... 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.

My suspicion is still that this has to do with losing insight in the little specific details of the matter and general understanding.

An area where I realized that was feature engineering: Early ML systems had handcrafted features that were fed into the model. There were relatively arbitrary and the number of features you could reasonably generate that way was tiny, compared to modern systems - but it gave you some understanding what input the model got exactly and you could use it to clear up some failure modes, or be certain that the model learned something that could not possibly make sense.

Then the idea was to automate feature generation. What's not to like? Except that in practice, the automated features simply seem to become part of the blackbox and are not available anymore for understanding.

As someone who has been both a software professional and a jet aviator, the same process is going to occur in both. Humans will still be in the loop, but the span of control and effectiveness of each individual human will explode. There's a reason the current plan is to augment manned aircraft with "loyal wingman" drones and smaller ones. Even in the Russo-Ukrainian War, manned aircraft are still A Thing. They've even hauled old prop trainers out of storage to put a guy in the back with an assault rifle to shoot down Shahed drones, because they're too slow for fighters.

Software is going to go through the same adaptation and exaptation process as military aviation.

> AI is stealing all the fun parts.

No one is stopping you from doing what you enjoy.

But lets get real here: no one asked for programming to be the way it is. It just too hard, too menial, too esoteric, too particular and too anal for 99% of the population. That's great if you're one of the 'wizards' and can charge huge amounts of money to make software that isn't exactly wizardry and more like just regular stuff people need for their lives and businesses. The quality and nature of software mostly reflects the mentality of those who wrote it, instead of those who use it. Multi-million dollar software projects continue to fail decades after the Mythical Man Month was written.

If programming as we know it disappears, few will mourn its loss, having suffered its consequences. Most people, including most programmers will just move onto the next great thing, whatever that is, and be empowered by it.

Meanwhile, welcome to what technology has been doing to everyone else for the last 100+ years.

> AI is stealing all the fun parts.

Disagree. AI has is taking all the parts that remove the fun. Now people can truly have fun with that which they find interest in, because there's a way to avoid the parts they're not interested in engaging with, with little to no consequence.

This is a slippery slope. Having only fun things to do make them obviously boring. The human being craves contrast.

That assuming what you've said is true, which is not. In a work environment, you're forced to optimize whatever that there is, regardless if it's fun or not.

Moreover, there is always fun to be had even in tasks you've found boring beforehand.

> Having only fun things to do make them obviously boring.

This doesn't really pan out the way you think it does as people can always choose to do boring things if they want (if there's even any truth to your assertion). Unless your argument is something along the lines of humans need to be forced to do things they don't want so they're able to find fun in anything, which I think is pretty... wild.

I do think humans need to be forced to do things they don't want. You don't need to believe my words. Just watch how many people get bored in their long vacations.

If all you do is having fun, it's obvious you'll need a new bar over and over (which is literally happening nowadays).

That sounds like systematic conditioning. Before there was actually a system, say back in cave people days, how did they operate without the system? During medieval days when monarchies were the rage, how did the royalty - and nobles to an extent - operate when they had serfs doing everything for them? Coming to more modern history, what about slave masters/mistresses?

Throughout history there has always been at least one group in societies (if not the whole) who didn't have to do anything they didn't want, and said groups have survived through the centuries. Now we're just at the point where it's very feasible for all in any given society to once again not have to be forced into anything and still thrive. What's needed is for that incompatible system that conditions forced labour to be thrown out.

This. Cannot emphasize this enough. I used to be bogged down with all sorts of decisions. What framework should I use? How can I make this code cleaner? What should this project be about? Is this decision going to bite me in the ass later?

Now I can safely hand this all off to Claude while leisurely browsing e621. The unlock is incredible, and I wouldn't trade this velocity for anything.

Disagree. In many professions, people were paid to endure the boring parts to be able to finally do fun parts. See academia; a professor writes grants, attends committee meetings, teaches (sometimes undesirably) and such, to finally go do research which is fun. If the fun parts are predominantly done by AI, and are removing people from getting paid, what fills one's stomach?

Stomach-filling ability is actually tangential to whether one has to engage with the bits they don't want to engage with, since it isn't precluded by the fun parts (there are people out there who are able to fill their stomachs from only doing what they find fun). But also it turns out to be a good example of that which forces one into doing that which they don't find fun. And IMO this - pretty typical - conflation really emphasizes how much we're stuck in and need to change this system that forces so many people to do things they don't want to do so they can fill their stomachs.

> Mathematicians are, indeed, also paid to teach, peer-review, go to conferences and learn mathematics, but all of that better result in some damn good theorems.

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".

I feel for the guy who wrote this. I feel this myself at times. But it's also not a productive feeling, as cruel as that sounds.

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.

How much of the identity of a mathematician is tied up in being the first to discovery? How much of it is the contest between smart people producing excitement? Maybe someone could chime in on this.

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.

> How much of the identity of a mathematician is tied up in being the first to discovery?

Here is how I'd put it: math has an enormous focus on discovery. It is why we have Godel's incompleteness theorems, the Cantor set, Zorn's lemma, and so forth. We name things after their discoverers.

It is possible that, going forward, no more things will be named after human discoverers. The last such naming (of something truly significant) may already have occurred.

That is a massive culture change, at the very least.

This has already been shifting in science and engineering because typically large teams are needed to get things done. Math has held to it better, but even there they are almost always multi-named concepts, like Rivest-Shamir-Adleman etc. And even those often have to ignore names that could plausibly be added.

For example in classic image processing / computer vision we had things like the Hough transform, Canny edge detector, Sobel filter, Shi-Tomasi detector, Lucas-Kanade optical flow, Horn-Schunck optical flow, Kanade-Lucas-Tomasi tracker, Otsu thresholding, Viola-Jones detector, etc, but nowadays we no longer name things like that, but typically with some abbreviation or some name that relates to the concept more closely.

It's in part the larger teams thing, and also a cultural trend to de-emphasize "great men theories".

I can't speak for those who want to discover new mathematics, but for people like me who want to merely study existing areas of mathematics that are of interest to us, LLMs are a godsend. I can engage ChatGPT or Grok in a conversation about topics that I am trying to learn. I can ask them questions, clarify what to read next, and get overviews of topics. I know that they are not infallible, but more often than not, they considerably illuminate a topic far more than just textbooks. It's like having a highly educated personal tutor for next to no cost at all - an invaluable resource!

Math is having its DevOps moment. And yet, people who really understand networks are beyond valuable.

Every day I witness the crazy wonder that is network knowledge / devops + agents from my colleagues. Things that were not possible become possible, assuming deep expertise. I feel for mathematicians like the author, but it will pass once the more creative possibilities reveal themselves and the shock passes.

I think that this new paradigm in mathematics will allow for more math being born out of curiosity and creativity. Many mathematicians have built their career around hyper specialization, and are now the most threatened because LLM excel in hyper specialization. I see it as a boon in a sense because it will bring forward the generalists. LLM will allow to much more effectively explore and discover new mathematics spanning across fields while retaining the specialist depths. Humans can still be at the helm, the focus just need to shift from cranking out technical proofs to more playful connections.

A silver lining to think about.. Currently there's a big issue (at least in some countries) that mathematicians must publish from 5 to 7-8 papers within a 5-year window to keep the tenure. So it's a minimum, much more is expected for grants. Yet high quality mathematical papers do take longer to finish (or sometimes to even start), especially if the researcher is working solo, not within a lab. Until today, the answer is to go for applications or for low-hanging fruit. Tomorrow, with the boost of AI help, the researchers may be able to spend more time on their "real problems"

And the number of papers required from them will increase appropriately.

Have LLMs produced something novel like Cantor's diagonalization procedure once was? Come up with new concepts like set theory once was and axiomatized it? Is there any indication or evidence LLMs can do this? If not, I would wager mathematicians are still going to be a profession in the future. Maybe not as theorem provers in a particular axiomatized system where LLMs can be effective tools but the wider project of synthesis and unification, and the discovery of new modes of mathematical thought will still require mathematicians.

I agree. As someone involved in (non-mathematical) research I think the author is grossly overreacting. Nothing that comes out of an LLM means anything until it's assigned a meaning by a human being, just like anything else in the world. All this millenarial panic just plays into the hands of the psychopaths running the AI companies

>Have LLMs produced something novel like Cantor's diagonalization procedure once was?

A year ago, the goal post was that LLMs wouldn't prove anything of worth. The year before that, it was that LLMs wouldn't be able to do math as well as a undergraduate student. Year before that, it was that LLMs wouldn't be able to do math at all. The growth of LLMs has been insane, and it doesn't look like it's going to plateau any time soon.

Poeple once thought that it was ridiculous that a computer could beat a 1600 player in chess. Then they put it at master. Then international master. They still thought it was ludicrous that a computer would beat a grandmaster or the world champion.

Oh wait.

The endangered art form of human mathematics is one thing. Another thing is that there seems to be little concern for whether a theorem was created using a documented open-weights model or some unknown model in a web chat.

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.

> I generally respect writing cringe blog posts

Cringe is not an adjective.

https://www.merriam-webster.com/dictionary/cringe

"Slang"

There's a parallel with open source software (FOSS), that you don't really control the software if you don't have the source code.

Being given only the generated proof (or disproof) is like having the binary artifact as an executable. Sure you can use it to solve or build further on top of it, but without the source code - the understanding of the steps to build the binary - you can't build it yourself, and you don't actually own it. If you want to adjust or extend something in the proof, you have to go back to a third party to implement it and generate another artifact for you.

We need a Stallman for mathematics with uncompromising integrity, to shout from the rooftops for a "free and open-source math movement" (FOSM). Even if the biggest AI companies go bankrupt, we still need to be able to bootstrap all the math from the source code.

This is the first I've ever heard math connected to the spiritual. For all my life it has been this rigorous thing grounded in reality in such a way that only people who think a certain way could even approach. Now those people are having this strange "crisis" because AI is being more rigorous than humans I guess. I find it... wild. My, but how the mighty have fallen.

The history of mathematics is tightly coupled to spirituality... some more famous examples off the top of my head are Cantor, Euclid, and Euler, but there are probably hundreds of examples of folks who explicitly tied their faith to their research.

I definitely don't think there's anything inconsistent with mathematical and theological thinking. That actually strikes me as a very strange idea?

Ramunajan a particularly notable example.

Would those persons mentioned have found themselves in a state of crisis if they were alive today? And it still doesn't touch on the core question: what does math and spirituality really have in common (beyond the extrinsic connection someone may make because they're emotionally overwhelmed by the "beauty" of the golden ratio appearing in nature, for example)?

Probably they wouldn't be. A lot of this is people catastrophizing. Probably already somewhat unhappy even before.

Regarding spirituality i think it's more about math itself than math as a discipline. Math might be divine. Even if math is divine, i think doing math doesnt make you more spiritual.

In what way could math be divine? What does "divine" even mean in this context?

https://en.wikipedia.org/wiki/Theory_of_forms

Almost all mathematicians I know admit to feeling spiritual about discovery in the mathematical world. Engineers are taught the rigorous thing but it goes way beyond that. In fact, I bet a lot of people find some satisfaction in their work that goes beyond the practical. Artists, etc. That's what AI threatens because it destroys strong communities of people who value such work.

In what way is it destroying communities? Is AI preventing anyone from doing their own work to discover/rederive whatever it is they're interested in? Or is it just showing them, and eventually everyone else, the truth that nothing humans do is special and many are finding that hard to handle?

Again, people don't just do things in isolation for enjoyment. What humans do is special, and the end result is not the only thing that matters. You don't understand that because you're an aberration.

I do things in isolation for enjoyment, like the various projects I'm working on now (with the help of AI) that I've never shared with anyone and possibly never will. Yeah keep thinking that humans are "special", and love that attempt at a personal attack; at least being an aberration means I'll never have a crisis that may have me contemplating suicide. Y'all have fun jumping off bridges as AI continues to take away all that "special". I'll just have a good laugh.

There's often an issue of perspective. Imagine you visit an 11th century monk in the occupation of copying manuscripts by candle light. This is a guy that spends long cold winters hunched over parchment, using sticks with god knows what passes for ink to copy and illuminate beautiful texts. You tell him that you will buy manuscripts from him, as long as they are typeset in Times New Roman regular, 12pt, and please use this laptop for that. Also you are setting shop with all sort of modern niceties next to the convent, including heating fans and desk electric lamps. Of course the monk will be heart-broken and incredibly grumpy at the loss of time to practice the craft he loves, which is writing and illuminating by hand. But those heating fans are something alright.

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.

There is also an economic aspect though. Thats where the ludditism comes in

If the monk thinks that this laptop means he wont get as much income (assuming selling manuscripts was their primary income) because of a race to the bottom with other monks, then rhey will start worrying about how theyre gonna afford to keep the chapel running. Where theyll get the money to pay the electric bill for all these niceties

I'm not so worried about this part as I am the part where a lab loses control of an AI, and it goes on to pursue whatever meaningless objectives (meaningless to us, that is) its designers accidentally gave it, and no one can stop it because it is better at planning and better at reality than we are.

> The creation (or even the pursuit) of novel mathematics is one way that humans have historically accessed the ineffable and encountered the divine and mystical.

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.

No answer so far... Go ask the author on his blog!

Done

Same as programming, I feel like the underlying tension is between math lovers who want new math to be done vs those who want to be the ones to DO the new math.

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

Seems like maybe now we have sufficient tools to actually get on top of string theory. And ultimately it is all about coming up with explanations and formulas and tools that people can use in facing real challenges.

Why does it seem like this to you when there's nothing been produced of note?

here's unpopular view - thousands of people felt the dread of mathematics, that is no more, as they may advance together with the LLM. perhaps brilliant incredible in math lived, but they were not so many perhaps. now everyone can have one and if passionate for math - get up to speed. why is this bad? why would an LLM make me, an adorer of algorithms and structures, stop do what I love - pursue them? if LLM puts you off, perhaps you were chasing something else.

If you are a Mathematician or Theoretical Physicist, you can come up with a theory of why LLMs even work in the first place. And what to do next, after LLMs.

A raw and powerful piece, thanks for sharing.

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.

Apologies is this is a simpleton question, but how does this differ than most of the mechanical work previously done by hand, but is now automated?

> There is something about mathematical discovery (progress, advancement, creation) which is vital to the spiritual, experiential quality of doing mathematics. The creation (or even the pursuit) of novel mathematics is one way that humans have historically accessed the ineffable and encountered the divine and mystical.

“ one way that humans have historically accessed the ineffable” that’s my question, what about the other ways that have been automated away. Is there something special about mathematics or is it the authors first encounter with obsolescence?

The automatic dishwasher has replaced the human dishwasher, but I highly doubt that dishwashing was one way that humans have historically accessed the ineffable.

Maybe if you think really hard you’ll come up with other (more useful) examples.

It differs because LLMs can and will eventually make inferences, even if they're not truly intelligent. They can do things that one can't predict ahead of time. In that regard, it's very different than asking a computer to multiply together two degree twenty polynomials. I could do it by hand theoretically in a completely mechanical fashion, but AI can do some basic inferences.

They are concerned mathematicians are obsolete and seem to be unaware that super intelligence means humans are obsolete.

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. :)

> seem to be unaware that super intelligence means humans are obsolete.

oh they are very much aware lol

> I am suffering a profound spiritual crisis due to these developments.

Another case of Deep Blue.

Can you expound more on what you mean by that? The phrase resonated and I want to know more.

It's a phrase for "the sense of psychological ennui leading into existential dread that many software developers are feeling thanks to the encroachment of generative AI into their field of work." https://simonwillison.net/2026/Feb/15/deep-blue/

I can’t help but confess some sense of satisfaction with the upheaval described in the post. Many proofs and textbooks laden with statements like “without loss of generality”, “immediately”, or “obviously” are followed by anything but. AI now counters that streak of smug by toppling some long held theorems and it’s fun to watch.

Bit weird that your first reaction to a human expressing genuine emotional pain is glee.

Yeah I think it's weird too. But tech really does amplify narcissism because it makes us very independent from people and community. So it's really not surprising that half the people in this thread are more concerned with manufacturing discovery than they are with having a cohesive society.

I’m also a human expressing genuine emotion. Why is one more valid than the other?

Your emotion isn’t invalid, it’s merely bad, assholeish, brings shame upon you. But it is “valid” in the sense that the reader of your comment can see that you are, at least, sincere in your gloating at another person’s despair.

It's not the despair, it's the turnabout. Reading proofs containing the language that I mentioned seemed to suggest that the ideas were trivial to the author in a way that implied their superior intelligence. Now AI has come along and has, by way of counterexamples, shown theorems to be incorrect in devastating (maybe "trivial") ways. The shoe is on the other foot.

Well, in all honesty, I’m sorry you had that experience (seriously). I hate feeling condescended to and I’ve also had that experience before, including from mathematicians.

When it comes to the specific phrases you mentioned, though, that really is not my understanding of the meaning of “obvious” or “trivial” or “without loss of generality”.

Without loss of generality means there is some kind of symmetry or repetition in a proof that makes certain cases redundant, so that “without loss of generality” we can assume we’re in some simpler case, such as assuming x <= y in a proof where x and y are interchangeable (and so one of them is necessarily <= the other). This is even the name of a tactic in Lean, `wlog`.

“Trivial” when I encounter or use it nearly always means something to do with zero, the empty set, the group with just one element, and so on. A proof being trivial usually means there is essentially nothing to prove (because the object in question is empty or zero or whatever).

And “obvious” definitely comes closest to what you were objecting to, and many mathematicians try to avoid it these days (along with “clear”), but even then it has a legitimate use case, namely indicating when a proof involves no new ideas. To me, this means it is fine to use it within a specified context, like in a textbook or paper, where certain techniques are being used repeatedly and routinely, or certain kinds of background knowledge are assumed. And even then it’s usually meant for proofs that don’t rely on tricks or surprises or additional insights. I’ve been stuck on trying to understand “obvious” proofs before, and it wasn’t because I was stupid or the author was smarter than me, it was because I was _missing something obvious_ (in hindsight).

The recent Jacobian conjecture counterexample, for instance, is not trivial but arguably a bit embarrassingly (for the math world) close to obvious in hindsight. I still wouldn’t actually call it obvious though.

It might mean you're missing some prerequisite knowledge. Unfortunately if every textbook started from square one, they'd be a million times longer.

AI hasn't "toppled" (or found any mistakes in) any theorems, at least not any enjoying any consensus among mathematicians.

Dark night for mathematicians' careers maybe but not for mathematics at all.

I actually think it is just the dawn of mathematics. During the last few days I discussed a few questions about abstraction logic [1] with AI that I was wondering about for quite some time (years), but didn't have the time + energy + in-depth expertise, there were like, 4 related questions; it positively solved 3 for me in the way I expected, but couldn't power through before; and for the fourth it proved to me that answering this question would imply solving a known open problem. And I know that it answered all of those questions correctly because I asked it to provide me with Isabelle/HOL proofs for it.

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

Counterpoint: Benoit Mandelbrot isn't notable for ever proving anything, but highly regarded for having put in years playing around with computers while asking interesting questions about what happens if you abuse the rendering parameters. Same thing with Mitch Feigenbaum uncovering a new constant by drilling down into very simple equations.

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.

In times past having one's photograph taken was a soul-stealing for some.

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

Cool

> Mathematicians are paid to prove theorems

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.

Not a mathematician, but I suppose there is pressure to publish like everywhere else in academia. And a publication generally requires a new contribution.

In academia they are paid to produce papers (and get some chores done like teaching), hopefully ones that can result in bombastic press releases, or winning awards.

On a more abstract level they are paid to bring funding and prestige to the university, but that often goes through the route of papers. And papers often involve proving theorems.

Actually, no. They are hired to do research but not paid for it. They get a 9 month salary to teach, not for research.

They have to publish papers to get tenure but the university doesn't pay most math faculty for it.

If they want to get paid for research they apply for grants and get the max of a third more salary.

Most other departments make getting grant money a prerequisite for tenure. Math departments typically don't.

So in most math departments faculty members don't bother with grants. Usually only a minority of them do because they want more money.

They're not paid to do research.

I consider "get paid to do X" to be equivalent to "you get to have your paid job and career as long as you do X". In that case research and papers qualify. Whatever is technically written in the contracts is just surface level bureaucracy. Publish or perish.

> I consider "get paid to do X" to be equivalent to "you get to have your paid job and career as long as you do X". In that case research and papers qualify.

Actually, it doesn't. Both math departments I'm familiar with had faculty members who published only to get tenure, and never published again until they retired.

But if they didn't teach (or taught extremely poorly), they got fired, even though they had tenure. If you didn't show up to a scheduled class a number of times, you got in trouble.

I know you weren't saying this, but it's frustrating reading so many comments about mathematicians "getting paid to have fun and good riddance". No - the majority of what they do is teach non-math students. If people here don't like that, by all means dissolve the departments and have the engineering departments teach math.

If the main job is teaching, then AI being good at proving theorems should not make OP worry about his job. Only AI being good at teaching would threaten that.

Pretty sure software developers already had that moment with coding agents. They can't really admit it like this, as it affects their employability now.

I think the hand-wringing is a bit premature.

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.

Welcome to the world of Chess a couple of decades ago...but look at chess now: more popular than ever.

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

MacDonalds is popular - does that make it a good thing?

Its not popular to work for. By popular, I mean people play it, at both the amateur and professional levels.

> You are a helpless onlooker. Before you a channel through which humans have accessed the ineffable and sacred for thousands of years is being sealed for eternity.

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 have wondered if it is the express goal of these companies to make me kill myself”

100% yes. They brag about it!

OP seems to need to touch some grass, spend time with friends and family, sit by the lake etc. Some sentences in there are quite disturbed.

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 never felt that the theorems were the most useful part of mathematics, strange as that may sound. In my professional experience, math offers a framework, a language of expressing ideas in an exact way. It clears things up for the writer and the reader. It is like speech; would we stop talking if a machine could utter all phrases?

I don't know... I read it as another mathematician thinking they are so mighty and different from any other workers out there whose job was slightly touched (not even begin to be "replaced") by automation.

Maybe he should just touch grass.

I'm no ai maximalist, quite the opposite in fact, and I have a great deal of sympathy for the author as a person from a similar trade. However perhaps he is being overly emotional in his response. Purists have been decrying the defilement of the trade for millennia - "but this time it's different!" no. you think the purest form of math is the one you were born into because you didn't know any other way. people were doing math before the abacus and will continue to do so afterwards. here a hot take even: there's no finite amount of math, if you really cared about math you would be excited about all the math you could uncover. the computer solving the jacobian conjecture isn't fulfilling enough for you? maybe start thinking about what this discovery implies then!

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!

I have two strong reactions to pieces like this:

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 am going insane. During the last week or so LLMs have produced a number of counterexamples to significant long-standing conjectures. I will not recount these happenings here, there are many places where you can find the details.

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.

Just another old fart who doesn’t like that the world is changing.

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

Imagine an invention that does a perfect job of brain surgery. And brain surgeons complain.

When that day comes, you think they won't? They spent their lives carefully honing a skill that is now worthless.

Now imagine a day where a new type of disease is discovered, but it needs some mechanical help from a surgeon because the AI just can't make the leap to do that. But there are no more surgeons because they didn't want to pursue a worthless career. What then?

The problem with your argument is that word "perfect". AI will never reach that stage and there will always be costs.

The idea of ordering theorems like doordash is funny, kudos to the author for that :-)

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.

I don't think the author of the article chose what world to be born into.

I meant all of humanity, not just the author. We, as humans, have collectively made our beds. It's time to sleep in them.

> We'll automate mathematics because we can, because it's useful.

No, it's not useful. Proving the Riemann hypothesis has no practical application whatsoever. Neither has any of the thousand Erdős problems. Or the Goldbach conjecture. Or the Collatz conjecture. Etc.

He sounds as someone really pedantic who never understood what math is about or why it is important.

This is what locking yourself away in an ivory tower gets you. Do things ‘for beauty’ and watch as they are automated and commoditized by the people who depend on them for their continued wellbeing, reduced to a hobby or sport. But the reduction began when you yourself started putting more importance on your enjoyment of the process over the value being provided to your customers.

> But the reduction began when you yourself started putting more importance on your enjoyment of the process over the value being provided to your customers.

Why would you make this comment? Is it envy? They were able to do things only some people can do. They enjoyed what they did and it provided value. Now the fun has been taken out of it and you seem to be saying that they deserve to have no fun? What should they have done? Prove theorems while whipping themselves in case they might have fun? Maybe it's a puritan thing? Customer value is the only value.

Don't complain that you lose funding if you sacrifice utility for a cirlejerk around craftsmanship.

I'm sure there's a programming equivalent. I believe theyre saying don't lose sight of the trees.

Many of my friends are mathematicians, they are saying pretty much the same thing the article is saying. Unfortunately, even the "I have wondered if it is the express goal of these companies to make me kill myself" line I haven't heard for the first time today.

I assure you they're not in it for the money, I make more than they do and hardcore math research is definitely harder than whatever the fuck it is that I do.

I don't think that trying to be a little bit more understanding towards people who are obviously struggling would cost us that much.

If they want to jump off a bridge or seppuku because discoveries with the potential of improving lives in some way are accelerating... well let me say no more.

And what about the discoveries that are worsening lives like military appplications and people who'll use the tech to discover better weapons?

Technological advances have been militarily applied since wars were a thing. Many times it's the military application that drove the advancement; like GPS was originally for military navigation and targeting, computers advanced quickly for codebreaking, the internet itself began as ARPANET (defense networking project), jet engines for faster fighter planes, etc. And look at where those advancements stand today in our lives.

Not in a better place IMO.

I'd say the jury is out on that. Most advancements aren't needed per se, but the majority of us take their existence for granted and life goes on. Penicillin though, that's one thing that having available absolutely put us in a better place, and it was also war that triggered a massive push in solving medicinal mass production, because infected wounds was a massive problem.

The issue seems to be with the expectation that personal enrichment and subjective aesthetic appreciation should be socially instead privately funded

The AI will not stop them, or anyone else, from doing what they enjoy. It will stop them from getting paid for it. And nobody has the right to get paid for having fun. To get paid you need to deliver customer value.

"Or threaten to hurt them, or their family, or their friends. Then just take the money. What about that option? Didn't think of that, did ya son."

"Don, I..."

"Listen. I think this racket just ain't for you. But you're a good kid - too good, maybe - and I promised your father I'd find you something. I'll ask Giovanni. Maybe he can find something for you in bookkeeping."

A large part of enjoyment is having a community who cares about what you're doing. So people will find it harder to enjoy it. Plus, some jobs actually are fun. Are you saying we should eradicate that fun because you hate your job? Some human being you are.

You should stop asking people for money for any other reason except that you provide value in return. Your enjoyment is your own business.

True, what I'm saying is that if we can create jobs that also provide enjoyment, we shouldn't be so quick to destroy them for the sake of greater efficiency because that provides social benefits beyond immediate financial ones. Of course, from your previous responses, you probably have more than enough money not to care about that and that's a typical response from the better off in society: screw you, I got mine.

No. The "only value" is the one that people decide to give you, based on their choice. This can be coordinated via taxation and redistribution, but it's still paid from someone to you. Someone has to decide that you doing this thing is something that they like so much that they pay you a token, which you can use to redeem useful work from other citizens, such as repairing you car, or remodeling your bathroom.

Many many people enjoy various things in life, such as eating nice meals (quite spiritual to do it in good mood in friends' circle), go camping, or play music. But they don't get paid to do this.

It's very simple to understand. It doesn't depend on deserve. It depends on a concrete person or group of persons having to specifically decide that you get things in return to your time (such as your washing machine repaired, or getting petrol at the gas station) for your efforts at something that they appreciate. It's not God who gives you these things. It's not the universe and it's not Mother Nature. It's concrete people. You have to think about how what you do is actually appreciated by concrete people. You have to do something that people appreciate, whether you agree with their assessment of appreciation or not, it will be that way. You can't simply expect people to give you gas at the gas station simply because you yourself appreciate your own way of spending your time.

Not everything in life is a transactional zero-sum game. We as society can afford people to work on beauty, without anything else in return. I'd even argue that this should be one of mankind's ultimate goals.

I agree. It's aesthetically pleasing to know that there are people out there thinking about deep beautiful and elegant concepts somewhere, it's a nice part of having a society to be able to afford supporting such people. They may however start to feel a bit self conscious, like we keep them as a kind of quaint curiosity. People usually like to feel actually useful and impactful.

No one is stopping you from writing every single line of code manually.

The fun hasn't been taken out of it. The economic value has. And good riddance. This clearing of the air has been needed for a long time. Now join everyone else who has to actually choose to either do something for fun, or for income, but not both.

Because I’ve often been with professors in math, literature, arts who pride themselves on a certain intellectual purity that makes them a cut above, while living on wages paid by the taxpayer. Suddenly the golem of basic market dynamics comes for them and these extremely smart people are struck by the unfairness and bruteness of it all. Whole careers having fun publishing useless nothings, funded by people sowing soccer balls and scrubbing toilets. I have nothing against having fun while doing your job, I have it often. But I have a job first to serve society, not myself.

Exactly! I have some many colleagues who typically say that they do "math for the beauty" while holding permanent research positions funded by the state.

That seems OK to me. I'd rather live in a state that funds beauty than one that doesn't. Think grants for artists, etc. A strictly practical world is not a good world to live in.

Progress is made by these people - including progress you rely on heavily as a programmer, such as cryptography. Not by the industry.

> Progress is made by these people [...] Not by the industry.

Eh, debatable? Maybe a bit of both? Industry certainly brings research-level stuff to the masses. And has a way to winnow what's ultimately useful from the vast amounts of research that amounts to "huh, that's weird". And in turn that vast amount of stuff serves as the starting point, and round and round it goes. Research without industry would be pointless (from the masses perspective). And industry without research would be clueless or stagnant or outright dangerous.

Industry can also bring vast amounts of money into research. Just look at the field of Machine Learning.

Cryptography has always been largely driven by industrial mathematicians.

Elliptic curve cryptography spent years requiring commercial licenses because it was made practical by Certicom, who patented many of the core inventions.

A lot of the work on homomorphic encryption was done by mathematicians at IBM.

The DES standard was developed by IBM.

RC4, RC5 and RC6, some of the earliest stream ciphers, was a commercial secret because it was developed by Ron Rivest at RSA Security.

NTRU post-quantum crypto was developed as a commercial product from the start.

Many of the earliest cryptosystems were developed commercially, like Enigma and Crypto AG.

A lot of the crypto for mobile phone networks is/was commercial.

Academia did contribute to the early stages of public key cryptography both integer factorization based and elliptic curve based, but given the extensive history of commercial mathematics this stuff would almost certainly have been invented even if there was no academic mathematics at all.

One could also see it this way: shouldn't the goal of society be to allow more and more people to do things for the beauty of it rather that producing ever more useless stuffs and destroy the planet on the way?

Customers? It's math. This is like a comment a golem built to raise the S&P 500 would write.

> This is like a comment a golem built to raise the S&P 500 would write.

That's neoliberalism for you.

If somebody is paying you to do something and someone else comes along that can do that thing much better than you and/or for less money you've got a problem. Whether or not you call the person paying that money a customer or use some other term the fundamental issue is the same.

just another pathetic case of terminal economicsbrain. sadly there's nothing to be done to help these people; it's best to just ignore them

It's important to remember that historically many advances have been made without "customer value" in mind. Indeed, freeing oneself from such concerns can open up unexpected avenues for discovery which only later provide "customer value".

But "value provided to your customer" is not in itself what the society need. Or at least, it is a matter of opinion and you can probably find theoretical situations where even you would admit that maximizing actions based on the customer needs is not a good thing.

Same here: the value for the society may be more about "making art" than "being a cog in the paperclip factory".

AI hasn't done that much compared to humans in research yet. Let us not overestimate its impact.

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".

> What it has done in software engineering is kill all human open source spirit.

Seems the opposite to me. All my stalled open source projects got unstalled and I've shipped several others. Other friends report the same.

I'm all for this. I don't think mathematics should be done as a profession. If you can't find free time to do it, it's obviously not that important to you. Open source software works pretty well, doesn't it? Why not math?

> If you can't find free time to do it, it's obviously not that important to you. Open source software works pretty well, doesn't it? Why not math?

You do realize that much of a number of vital open source projects are primarily developed by people who are paid to develop it?

> I don't think mathematics should be done as a profession.

Personally, I don't think writing, blogging, acting or singing should be a profession either. But here we are.

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