It happened to chess. Did it kill the game? I don’t think so. There is still plenty left to discover, and machines are actually helping us do that. Chess isn't mathematics, which is vastly more complex.
I think we are entering a new era of human discovery: the AI augmented era. It can be just as fascinating as the previous one, and humans will remain pivotal to it; I am almost certain of that. This new era will come with its fair share of positives. For instance, mechanical correctness and reproducibility will become far easier to achieve. Naturally, there are downsides too, such as unequal access to frontier models, though the strength of the open source movement makes me optimistic on that front.
Adjusting to this new norm is painful, especially if you are a young researcher. Your seniors are lost too (I am one of them), but their seats are much more comfortable than yours. The old days are gone, and that much is clear. The romantic image of the lone mathematician achieving breakthroughs through the raw power of their mind will now feature a mechanical companion. But that doesn't matter; that image was always meant for the cover of a history book. In practice, discovery has always been the cumulative result of countless human efforts, the classic "standing on the shoulders of giants."
All in all, we are entering a period that is fundamentally different from the one before it. This is both exciting and terrifying, as so much is at stake. It is essential that mathematics remains a human endeavor, and that we avoid becoming overwhelmed by AI generated noise. Nonetheless, this is a brave new world, and I still believe there is plenty to be hopeful and optimistic about.
I love the sentiment of your words. Thank you for saying all of this. I think your diagnosis of where some of the pain and problems lie is also on the mark.
I will say of the chess observation: There is an unfortunate societal/economic difference as it currently stands. I think we all see chess as a fun social activity that is intrinsically valuable, and top chess players figure as sportspeople and entertainers in our collective conception.
Mathematicians are currently seen as theorem-provers who advance the state of the art. It has been observed many times over recently that this will have to change.
I also think, to be a little more candid, that your comments contain an implicit critique of my piece which you perhaps chose not to deploy against me. I do engage in a bit of myth-making and romanticism in the article. When I invoke an "age of heroes" I do so against an internal voice that says: "This idealism is cringe and borders on fascistic". But I elected to keep it because it authentically reflects my feelings.
So I am sorry if I have contributed to the silly "romantic image of the lone mathematician achieving breakthroughs through the raw power of their mind". It was not my intention. I hope my comments about math being a Talmudic conversation work against this harmful myth.
What do you think, Pierre? Should I perhaps add an addendum or edit to stop people from getting the wrong idea?
I was going to make the same chess comment, and I agree with your critique of where the analogy breaks down. But here’s another: at some point in every religion people stop writing new scripture, for the most part. Doesn’t kill the priesthood, just means they start devoting their time to exegesis and proselytizing. To even understand what the machines are doing and if it’s right or not is going to require a high degree of mathematical knowledge, and while I know you’re not talking about applied math here figuring out how to take machine findings and use them to solve related problems in other fields or even apply them to related problems in math will still be extremely valuable. Still, if what you value about math is humans finding novel proofs of theorems, then yeah it’s not looking great and I can see why you’d mourn that. Going back to chess it is the case that a certain mystique was lost when even the deepest, most complex moves could be instantly assessed as right or wrong by a computer no more powerful than an iPhone.
The time isn't yet - as the trusty robotic companion will still need a human to provide oversight. But, probably within our lifetimes, there will come a time where the human contribution is no.longer meaningful - chess may be an example.
Indeed, chess is a game, not math, and it lacks this economic and societal dimension. But this is a very recent situation. Universities for the masses are less than one hundred years old. Before, with a few exceptions, math was restricted to rich bourgeoisie who had the time for it. Above, I mentioned that mathematicians will still be around (as a job), just that the way they work will be different.
I have no problem whatsoever with the "Age of Heroes" metaphor. Everybody needs myths and this is what allows us to move on. The romantic figure of Galois and others is one of the motivations to start the journey. Clearly, there will still be smart people to cast the light of the human mind on the world.
@Kirwin do not change anything in your words. I think there is a strong need for such sentiments, for calls of despair. What is at stake is essential. This echoes feelings we all have on the topic but that one does not read much about.
Again, as a community, we need to adjust to a new norm and find new rules to integrate AI into what was until now a human-only craft. What this new framework will be is up to us to build.
Just a side note: AGI / superintelligence is out of the question because the current technology is not providing it. There is already enough on our plate to solve, no need to add extra.
No. The article is beautifully written as is and the romantic sentiment is adequately balanced. You are allowed to express what you feel. Besides that, it's also just sometimes true. Science is cumulative until it isn't. Cue paradigm shifts. This is perhaps less so in mathematics as you guys are in the business of finding eternal truths. But even you have movements like Hilbert's program that rise and fall and sometimes in quite dramatic ways, involving a fair bit of heroism.
I am a late stage PhD, not in maths but an adjacent field. Currently I find myself in the awkward situation that I have to prove something. Up until now I have never tried using AI for the actual intellectual work I am doing. Somehow the idea felt like a violation. But now I wonder, like you, what am I even doing? I subscribed.
Mathematics is not chess. It is not a sport two humans compete in, but a field with content that can be discovered. Importantly it is discovery that matters, not "math skill" of a person. And I don't see why you want humans accompany LLMs. It makes sense now, but in a few years with AGI it would not.
Even if LLMs became perfect theorem provers, the ultimate goal of mathematics would still be that at least some human gets at least a general understanding of what the theorem means, in human terms. This would entail an understanding of its implications for other branches of mathematics as well as, say, for physics and computer science. This job may eventually become automated too, but that doesn’t seem imminent. The AI-human centaur era of mathematics can be very exciting, especially if the bottleneck in making new mathematics moves away from theorem proving into this more fuzzy realm of human integration and meaning making.
It is true that learning Mathematics is the goal of mathemicians. Halmos said there is no difference between an excercise and a research question except that we know answer of one. However, that is not the reason why mathematicians get paid. They are paid to prove theorems not proved before, and to teach engineering students calculus and linear algebra. An AGI would automate both jobs. But separating the income question, discovering new mathematics unknown to others is a way of gaining meaning that is in danger of becoming extinct, and getting meaning from learning math is but an imperfect substitute for that.
It seems unlikely to me that the mathematical landscape can be explored in any depth even if we had trillions of LLMs working continuously to prove new results. Right now the weird spark provided by humans (especially the very quirky humans who like mathematics) interacting with LLMs definitely adds something. Even if that went away, people doing mathematics would still have opportunities to construct new things never seen before. What might go away is the idea that being first to be recognized as having laid claim to a result is notable and should be rewarded with prestige and honors (as most of these would go to the LLMs).
I don’t really know mathematics research enough to have a firm opinion on this, but by analogy to my field (astrophysics) I suspect that researchers will still have plenty to do. An architect has plenty of work even if he doesn’t lay any bricks with his own hands. Lots of new stuff is becoming feasible that we wouldn’t even attempt a few years ago. This just shifts the balance from detail oriented puzzle solvers, who may in fact be partly replaced, to big picture people. For now.
I mean no offense, but nobody’s obligated a career that provides them with the time and focus for deep thinking. The arts has been facing this problem since forever; far more people want to make art for a living than there are people willing to pay for said art. I understand feeling a bit shattered, just as I feel bad for the college athlete that fails to go pro or the law school grad aspiring to be a superior court judge that fails to qualify for a feeder clerkship. But the appropriate response is to pick yourself up and find something else to do with your life that helps other people.
Well if the underlying premise is that life is miserable for the majority not in math or some other rewarding pursuit, then the author is saying "I am an elite with a fulfilling life, it is bad that I shall suffer like the rest of you", and any of us should care?
I do like that. I think it involves some significant changes in lifestyle to bring that about. way more than "stop ai", it's "reform the property relations in society and probably accept significant degrowth"
But bringing it back to the arts, it's simply the case that individual consumers get to choose what to spend their money on, and so the total number of dollars available to, say, novelists is sharply limited. We can increase that number through public subsidy (NEA grants for writing novels or something like that), but those political efforts haven't proved very fruitful so far. Getting mad at the public for not buying enough novels (or not being willing to fund enough NEA dollars) doesn't seem very likely to be helpful.
Life isn’t miserable for those with dreams interrupted. There’s a bit of pain in the immediate aftermath, of course, but you realize it’s not the end of the world.
Please go ahead and pay lots of money for all the people who want to be artists or musicians or authors but who are currently waiting tables or sitting at the cashier at Walmart.
But that's a **game**. I don't do mathematics to play a game and we don't pay mathematicians to play a game.
If it comes to the point where I'm not a useful way to discover new mathematical truths I'm out and society should stop funding human math research. I think that point is farther away than people think but let's not pretend it will 'be ok' because we can do math the way people do crosswords.
it will suck for you just like it sucked for every artisan when they are industrialized out of a profession. Of course, assuming some great replacement, which I doubt-- I think the article is quite alarmist and protectionist.
This situation has brought about the Olympics of terrible analogies. Chess and math are utterly and completely different and the agents involved are too different to compare.
I think chess is very different. If we want to see the best two chess players play, we would probably pick two engines against each other, but we don't want to see engines play. To me, this analogy is just as good as arguing whether mixed martial arts will die because of AI robots.
I feel like math is substantially different in that I don't think it matters to us whether a theorem was created by an engine or a human as long as it is the "best". Whether it will be self-prompted to do so, that may be less obvious and to me the leverage mathematicians have at the moment is that they can direct it a bit more intentionally.
I think it's the direction coding is going in as well right now so I feel like maybe I'm curious if the real benefit of AI harnessed by a mathematician is the amount of new questions that can be created.
I have been dealing with similar feelings and worries lately. Thank you so much for this beautiful and haunting post.
It saddens me to read in the comments variations of "deal with it", "find another hobby", "dedicate yourself to more yourself pursuits" etc. Humans have been practicing some form of mathematics since at least 2000 BC. In all this time, it has been an essentially human activity and part of human flourishing. To quote A. N. Whitehead:
> “I would not go so far as to say that to construct a history of thought without a profound study of the mathematical ideas of successive epochs is like omitting Hamlet from the play which is named after him. That would be claiming too much. But it is certainly analogous to cutting out the part of Ophelia. This simile is singularly exact. For Ophelia is quite essential to the play, she is very charming-and a little mad. Let us grant that the pursuit of mathematics is a divine madness of the human spirit, a refuge from the goading urgency of contingent happenings.”
And indeed, mathematics can play a tremendously important role in a person's life far above and beyond its usefulness. I will never forget that Dunham's "Journey Through Genius" starts with this powerful quote from Bertrand Russell:
> “There was a footpath leading across fields to New Southgate, and I used to go there alone to watch the sunset and contemplate suicide. I did not, however, commit suicide, because I wished to know more of mathematics.”
I don't know where we go from here. I can only hope that Pierre S. is right that mathematics will continue in some form, hopefully with the machines assisting humans rather than replacing them.
I am sorry to hear you are also suffering. Thanks for your response. You've brought some damn fine quotes here as well, a "divine madness of the human spirit" indeed.
I don't know what else to say, other than to echo your hopes and thank you again for your contribution.
I did a BS in pure math, and then entered industry as a software engineer because I needed to earn money. In another life, I might be in your situation now, so I very much resonate with this. The romantic and mystical is a lot of what drew me to math. Passion is good, and human; fuck the people telling you to "chill out" or "get a grip" or whatever.
Humans do a lot of myopic and craven things with our short precious lives. Math is one of the few that is intrinsically valuable. This is encompassed in my piece https://substack.com/@quinoam/p-185273573
At the same time, another part of me reads and accepts this as "well, natural selection comes for us all." Water only runs downhill. LLMs are just the latest expression of replicators maximizing fitness. I wish we had a better world where all that is good and beautiful could be insulated and preserved. But no, nothing is safe. Entropy comes for us all. I have long since accepted this. So I feel calm. I do what I can but this is the way of the world. Most avenues of human life will never be the same, and some have been irrevocably desecrated. So it goes. If we don't all die I hope that we may have more leisure in the future, so all who wish to pursue recreational mathematics have the ability to do so.
Thank you so much for your empathy and thoughts, I'm glad my desperate scream resonated with you. Your piece "my terminal values. like for real" is a brilliant intellectual exercise: Explicitly laying out our base values. I am not very reflective most of the time so I think I should sit with your piece for awhile and try to do something similar for myself.
Intrinsically valuable stuff is what it's all about. And it's so obvious I feel silly saying it.
It is good that you have passed through acceptance and feel calm. I cannot say the same. I still haven't accepted the old realities, like the fact that I will die. Neither mind all these new horrors.
This is true, and I am sorry we are both in this pickle.
Given that you have been steeped in it for longer than the mathematicians, do you have any advice for them on how to get through it? That might be a big ask. I don't supply any advice in the article.
Most of our reactions have flaws - either refusing to use and have anything to do with LLMs, and so likely swamped as time passes. Or embracing them and using the analytical and building aspects of coding to instead supervise AIs to code - so losing part of our soul and not a long term resilient strategy.
I suspect we need some kind of mourning ritual, not sure of the form or timing though. Like a funeral for hand coding small functions. And then maybe more funerals later.
Oh yes, also if you have capacity, look am AI risk campaign such as PauseAI or anything similar. The impact on the crafts we love is (in many future scenarios) but the least impact of this technology on us.
Beautiful piece. Those who tell you to get over it haven't had the experience you describe of their 'work' aligning with their life's moral good. It seems like one of the moral goods that's constitutive of your life so far has been to engage with the ineffable and the past through mathematical discovery. That's made you part of the great human conversation stretching back millennia. How beautiful. That's a meaningful thing and hard to lose. But some people never lose anything as valuable as that because they've never had it in the first place. And some people's moral goods just lie elsewhere, in family life etc.
My point is that if they were familiar with the experience then they would be empathetic.
If you reread my comment, its not about maths but about the ineffable, a sense of the divine broadly conceived or of mystery and truth. Maths is clearly one route there for the author. I'm not into maths but I recognise the feeling and the goal.
Try to actually engage with what this guy is saying.
If I was a Mathematician I would be full of wonder and excitement to know I would live to see a hundred lifetimes of progress. But I can also appreciate your ode to the end of an era.
I suppose, but I’m greatly worried about the shape of the progress. We’re already making progress, first with the planar unit distance problem, then the Jacobian conjecture, then the Dinitz-Garg-Goemans conjecture.
Maybe I am overreacting because I feel so strongly in agreement with the content AND tone of the article, but I feel like AI’s discovery of these results have made them much less exciting. If a mathematician proved them, we would immerse ourselves in the article containing the proof, be emailing the professor, ask them to speak at conferences, etc. Now, even Tao is just trying to investigate this by…asking an AI even more questions about it. This is fair, given that ChatGPT is the “professor” responsible, but we lose the pride in our fellow human in accomplishing something great, and the inkling of hope that maybe, one day, I can be as clever and acclaimed as that other human mathematician. We also become more atomized, one of the main issues targeting society at large today - what is the point of emailing other people or going to a conference if asking an AI the right questions will net you better answers more efficiently?
Like the original author, my thoughts are reactionary and unorganized, and I don’t deny that as an intelligence-assistance technology, AI is here to stay. The only analogy I can really think of is if you were never allowed to look outside again. You would never see anything novel. You lose a quale, a feeling with inherent value. And sure, you can let your AI robot outside, and it can see things and describe everything and synthesize it in even greater detail than the greatest author to ever live, and there is no economic reason for you to go outside if the robot can perform your business and describe it back to you so effectively. This analogy is kind of falling apart, but I hope someone can take my meaning. I’m worried that humanity will lose the ability to create something for a reason other than our own brief and pitiful entertainment, if the AI can create every thought cheaper and better.
another part of that is that if maybe 3-4 companies worldwide control the ai robot that goes outside for you, and they keep it cheap enough for a sufficiently long time to generally degrade most people's skills and demotivate them thoroughly, why the hell would they keep it cheap? as gatekeepers of intelligence, they have every incentive to ramp up the prices exponentially once it's well and truly adopted by most.
It's a hope-killer for those who believe that self-improvement has value. Most seem okay with taking a backseat in life and observing what AI can do like it's just another form of entertainment.
It's my fault. I once stole an entire ream of paper from the math department's office supply room.
In all seriousness, thanks so much for this thought and for sharing around the article Nik. If you read the article above and are now reading this comment you must now go and read Stultiloquy
For me, the problem of evil is (in part) resolved by seeing God as a thing of finite power. If God was arbitrarily powerful the power would appear infinite to us. So, if there is a God, and if God is good, I'm not sure that God could or would chastise us. Though there is no lack of human action and inaction for us to be chastised for presently.
I am not a mathematician but I am one who loves mathematics, and for whom mathematics and theology are frequently co-illuminating. I have pretty much never supposed I would get to do original mathematics - I just get to learn about and try to understand what others have done and discovered. While I have sometimes enjoyed the human stories behind the mathematics, it is the math itself, like walking in the mountains or gazing at the stars (a thing of beauty and glory and wonder in itself), that captivates me.
I have also been a very interested follower over the last 3 years or so of the developments in AI, and a frequent personal user of AI - as a conversational partner, it has been a helpful tool for me to ask clarifying factual questions ("please remind me what primitive recursive means in this context") as well as for exploring ideas ("is the number 3 just the set of relations that pertain to the thing we call 3, or does it have an interiority?" which lead to a paper by Benaceraff, whom I'd never heard of and which I'm currently reading).
Regarding AI overall, I have high hopes and a dreadful expectation of horrible downsides, comparable to, but orders of magnitude greater than, the Internet and then social media. But regarding AI and mathematics, I can't see that I will lose anything (perhaps I may yet repent of such folly). It remains to be seen if AI is as good at proving as at disproving (by discovering a counterexample). Quite possibly it will. (Perhaps the best case is it's only good at discovering counterexamples, and then much less time will be lost trying to prove what isn't true?) But even if AI starts writing all the proofs, while I'm sure that will be felt as a loss (as you yourself express) by many doing real work in mathematics, yet it means too that the beauty of mathematics, and of more and new mathematics, can be brought to millions who otherwise wouldn't have experienced it.
Was the wonder, was the glory, truly in the original discovery (I and no other have now seen this) or in the personal discovery (though others saw it first, it was a wonder to behold)? (Forgive me if I've posed a false dichotomy.) If mathematics is a glorious landscape before us shrouded in undiscovery, I do not now perceive that it matters to me if the ones further illuminating it are men or machines. Only men will glory in the landscape thus revealed.
I've often revisited the thought experiment of the Library of Babel. It in fact does exist (for all practical purposes) - there are websites that will give you a page from a randomly chosen book of this dreadful library, and the program to do the same is easy to write. The knowledge that any such page, in correspondence with the right one time pad, is in fact an encoding of any and every other page that ever has been or will be or could be written, has not vitiated my readings or my writings of meaning.
It seems that your argument following the introduction of the Library of Babel implies that the student learning math, beginning to enjoy it, then to love it, is actually in the same undesirable position you foresee? "I love math!" "Ah, but you'll find it's actually only rewarding once you're doing original work." Didn't you love it when you were an undergrad? Did math only "get good" when you became a grad? Post grad? Perhaps there is some real higher pleasure I have never yet enjoyed - but, trusting you did enjoy math long before then, I would remind you that there are real if humbler pleasures. You say, though, those are inextricably tied to their reality as human discoveries. If, then, mechanical discovery really drains the pleasure of participation, I am sorry. For me, though, the joy (or at least a joy) remains a consequence of objective beauty residing in that which is discovered, not (only) in the subjective joy of the original discovering.
I pray you will find some hope. Despair not of all hope.
Hello again Chris, I'm now safely back in front of this horrid laptop, so here is my longer reply:
There are great pleasures and spiritual fulfillment to be found in the "personal discovery" as you put it. My goodness, to work through Chapter I of Serre's "Trees" and experience the elegance of his arguments while filling in the gaps (he moves quickly sometimes) and wrestling with the exercises. It was was stimulating and joyful. However, I'd like to highlight something in my own example.
Serre is a human with quirks and style and preference. As I worked through Trees I was conversant with him (also John Stillwell, since I was reading the English translation) and felt a part of the culture of mathematics of which Serre is an influential member. Should I have been reading an AI-generated textbook, I feel something of value would be lost. This of course assumes that I am reading LLM prose, and perhaps it would actually be that the proof is AI-generated and the exposition human. In this case I think some of the experience of the personal discovery (an amalgam of many feelings and thoughts) is preserved. So you make a useful distinction, perhaps all is not lost.
All of the above is to say that I did not mean to convey in my piece that only original mathematics, and no other kind, alights the spirit. I concede that point to you completely.
Moving to the other part of your dyad, the "original discovery". I was writing on original discoveries in mathematics, but we can all claim original discoveries I suggest. For instance, if you are the first to notice an interesting and subtle similarity between two people in your personal life, make some sort of deduction about it and then verify it—maybe you suspect they will get along and introduce them—you've made an original discovery. That feels a little special doesn't it? Perhaps it even suggests there is something about you or your relationship to these people, or about the circumstances of your life presently, that allowed you to see it first. I'm not sure. There is a little magic there, no? A little synchronicity? Maybe you append this to the narrative of your life, thinking occasionally with a smile, "I knew they would get along! Imagine if no one ever introduced them."
For the mathematician these two friends might be seemingly unrelated subfields. I think it is ok for the mathematician to want to be the one to acquaint their two favorite subfields (each of which means something special to them and perhaps no one else) instead of merely reporting: "Oh yes, an LLM identified that my two friends might get along and told me to introduce them."
You are forcing me to the edge of my ability to express my feelings on this, Chris. I really have to commend you for that. I'm going to make a final gesture in the direction of my emotions. It's something like this: Generating The Library of Babel seems utterly pathological to me. When I go to the beach I pretty much leave things alone. Maybe I take one remarkable stone if I have some reason to. I don't read others' diaries. I try to take only what I need. Not everything is so universally important. Not everything needs to be done right now. For each thing that can be done in the world, there may be a person for whom doing it would mean the most. Maybe that's not quite it. Maybe it's more like this:
Being the first is healthy food for the ego, and we live in a world that demands that our egos exist. To take away the ability for people to create original things but keep all else unchanged (inequality, suffering, status games, hierarchies, etc.) forces the ego to be and then starves it. It is like bringing a child into the world against their will and telling them they will have to go on without food.
Yes, I think that might be more like it. Thank you for leading me to this realization. I might make a note or post of it later. I am sorry I could not make this more concise. Thanks again so much for your reply, which forced me to work through some of these ideas, and if you read all of this, thanks for sticking with me.
This is such a thoughtful post and I would love to give it a long reply, but I am camping right now! There are some wonderful counterpoints in your response that deserve a proper reckoning with. I think you have brought more clarity to the discussion. For now I will just say thank you so much for your thoughts and I endeavour to give a longer reply when I am not in a tent.
I have written software for 20 years and all of the parts of it I enjoyed have been made both obsolete and valueless this year. I also had various personal upheavals in recent years, but I think even without them, I would have had to reevaluate where I was going and what I was spending my time doing. I'm going to try to go back to school for the things I feel really creatively engaged with and spend time reading about and would keep reading about even if I didn't go to school for it. It doesn't matter what it is -- but you might have something like this.
But even if you don't, and mathematics is your creative outlet, if there is any humanity to math at all, if it really is an art, the artistic aspects of it will never be -- imo, anyway, someone deeply engaged in using AI every week and building things with and around it, even with this effect it's had on my life trajectory -- duplicated by LLMs. They miss the right spark for good writing, or images, or video, for example; and despite using the best and most fancy, expensive models, I regularly run into tasks, usually involving large-scale synthesis, that it is only half-helpful for.
I have recently realized that all my outside-work coding, if I get AI to do it, is really quite awful in any aesthetic sense unless it's a very simple thing (it's able to make simple tools very good looking, and websites, for example) -- so I will keep doing it manually. And when I write the most important parts of a new system, I am realizing it's still better, and justifiable, to do the key parts by hand. I am sure there will be spaces in math that will be elusive to AI, they just might be very unintuitively shaped.
I feel like I got a crash course in what weavers must have in the time of the loom. It is absolutely brutal considering I quite liked coding and was also quite good at it (from what I was told), and now, woops. No one cares. Not even myself, I have started to realize. It's just on tap now. It was February when they got good enough that I was really surprised, and it's been a nauseating roller-coaster from then on.
Sorry. I do agree with one thing another commenter said (though not the rest of it): welcome to the club. We are in it together now. At least you don't have a boss telling you to become a professional manager (of virtual subordinates) now, I hope.
Well written. I've programmed since I was a child, then studied CS and worked professionally as a software developer. At one time, I would've said it was very important to me. A few years after stopping, it surprises me how completely it has faded from my mental space and identity. If anything, I've come to realize that these type of activities are not that important to the core of our humanity and identity.
It's good that you recognize this as a dark night of the soul - a part of your inheritance as a human, a passage alike to dark nights that others have survived. It sucks, it's painful, but you will make sense of it yet. That's one of my favorite phrases of late, "to make sense". People commonly throw it around as a synonym for "I know" but I see it more fundamentally - that there is literally a skillful creation of knowledge, that pays off in the qualia of "aHA" when you have succeeded. My favorite mathtuber (who is generally incomprehensible) said, "How does one make sense? Carefully" - advice for self-identified "bad at math" folks, who tended to metaphorically cut their fingers on the sharp edges of arithmetic, but equally applicable. Take care when you struggle to make sense; know yourself, your motivations, your dreams, your sore spots. Mathematics will not cease to be a bridge to the divine, if you still seek it with all your heart; I know because I see the same transcendence in artists and gamers of all sorts. It's not actually necessary for an undiscovered frontier to lie beyond the ocean; machines might print you a map, but you still have a journey ahead to cross the territory. You still have to feel the rain on your skin, to feel the made-sense of the mathematics, no one else can feel it for you.
Thanks for the reply Kevin, you have a really striking prose style that I enjoy.
Yes, my piece is not an ode to the "aha" as we certainly get to keep that. A useful distinction was clarified by another commenter (Chris Brenan). To quote him:
> Was the wonder, was the glory, truly in the original discovery (I and no other have now seen this) or in the personal discovery (though others saw it first, it was a wonder to behold)?
One fear I have is that we will one day surrender original discovery. If taken at its most general, original discovery encompasses a huge number of human experiences that personally I would like to keep. As I said in my reply to Chris:
> Being the first is healthy food for the ego, and we live in a world that demands that our egos exist.
I'm am not sure if that changes your thoughts on this at all. It was not meant as a counterargument so much as a clarification of what bothers me. It could be that you were already very clear on this distinction and the implications I've observed. Your response to it may even be implicit in your comment, and I've failed to pick it up.
Whatever the case, thank you for acknowledging my pain and thank you for bringing a more constructive and optimistic view of what could be.
Thank you! Both for the compliment, and - I felt a good measure of conviction reading your piece and writing that comment, and I'm glad it was well-received. Hmmm. I don't feel fluent in matters of ego, and so at a certain scale - I probably think and talk around the dysfluency, haha. I think of an idea I bumped into, maybe from a sports psychologist or coach, that high performers tend toward two types: a LOT of ego "of course I'm the best to ever do it," or virtually zero ego, humility built on awareness of imperfection and uncertainty at the edge. I remember the wave of validation washing over me, "oh okay, yeah, that's my Type of Guy" lol, when he named option 2. All that to say, yeah makes sense! I'm not sure I have any more good wisdom though
I signed up a substack account for the first time to make this comment.
I came across a screenshot of your post on twitter and was deeply moved by your emotions. It made me consider the implications of AI in a much more visceral way than any other headline or breakthrough I’ve seen reported.
As someone who’s generally optimistic and excited by the technology, I’ve mostly only ever been concerned by the worst case “doomer” scenarios where AI forcefully ruins human life. I’d never examined the negative impacts AI could have on the human spirit by simply making our passions feel arbitrary.
To me, your message represents ideas that are uniquely human; loss, hopelessness, lack of purpose. Despite the upsetting tone, I found your words to be a beautiful expression of being human, and I think AI will have a very difficult time replicating that.
Thank you for writing this piece. Thank you for being vulnerable. I hope the response that you get for sharing this brings you some comfort, and that you know you aren’t alone.
Thank you so much Austin. I feel very understood in reading your response. You and many others have been very kind to me. There is a lot of vulnerability in my outpouring. And the critiques (for example, that some of my horror is due to a loss of status) have been measured, for the most part, and have only made the discussion more authentic.
Here's another little bit of sweet, sweet cope - the headline making AI contributions (all counterexamples) are the most expensive maths results in history by some obscene factor
This sounds nice but I can't substantiate it. Where are you getting this assessment? I thought at least some of the counterexamples were done in a few short turns with a human, probably less than 1kWh of energy used. (I would like to collect actually expensive projects, it seems like valuable information to track.)
Right, some truly gigantic expenditures there. But can one claim that this makes LLM-assisted counterexamples some of the most expensive mathematics results ever? If this is the case then by the same logic we have to weigh the civilizational costs of earlier mathematics as well (roads for the mail, planting forests for paper) and I don't think they necessarily come out much lower: the costs are lower but the number of people and total economic activity was lower. Maybe ancient mathematics was the most expensive of all as a proportion of overall activity.
Yesterday I was thinking about something similar. "What are we going to do, if machines can think better than us?"
Naturally, I posted this question to a machine and it suggested me to pick up a hobby like pottery, blacksmith, or cooking. It even give me a map of places where I can learn blacksmith near my IP address.
I would implore us to all keep thinking hard, even if we aren't the best thinkers around. But finding refuge in tactile hobbies is also not a bad idea.
The reason why these companies are automating mathematics is that it is very useful to do so. A superhuman mathematician means fewer bridges fall down, better research developments in areas as diverse as marine biology, computer science, and agriculture, etc. It's just such a good and useful thing. I can sympathize with artists far more (after all, art isn't actually very useful to the world, and automating it doesn't make fewer bridges fall down).
> I have wondered if it is the express goal of these companies to make me kill myself.
No, that's not the express goal of these companies. Their express goal is probably much closer to the "country of geniuses in a datacenter" (https://darioamodei.com/essay/machines-of-loving-grace) and unfortunately a side effect of that seems to be killing your will to live. Perhaps you should look towards other means of finding this will through something that doesn't involve having to do mathematics. Maybe it's just seeing the beauty of the world, maybe it's having friendships, maybe it's a dog that you take care of, you know?
But it's what that person is passionate about. Your take is like telling someone 'I'm sorry you won't be able to paint anymore, perhaps you could discover a passion for baking.' That's not what this person is drawn to; it's not who this person is.
Oh boo hoo. Few people can do what they are passionate about as a career. And baking isn't the only alternative -- stop being so intellectually dishonest.
And we're still a long way from AI eliminating professional mathematicians ... finding counterexamples is the lowest hanging fruit.
Deciding your identity is bound to a particular activity is a choice. There is nothing to suggest that such esoteric passions are wired into us biologically. If they developed over time, then new ones can be developed and old ones forgotten.
It's natural to grieve the end of a personal era but not healthy to ruminate in misery indefinitely and not move on.
That is true. I would all suggest that it shows that the identification between who we are and what we do is nothing new. And a great many of us are made for some things much more than others, and none of us are made for machine-occasioned obsolescence
Totally agree with you. I don't celebrate any kind of machine driven obsolescence. It has impacted me in multiple different areas of passion and derailed career plans. I just want to share that, speaking from personal experience, it has not been the end of the world to completely change what I think about and work on. There is an initial adjustment period that is miserable, but I can genuinely say I am happier now after realizing that certain identities I clung to were not necessary. It was freeing to discover that my relative strengths and weaknesses, as well as what I can enjoy and take pride in, are much less rigid than I previously thought.
At least current LLM proofs are not about engineering, but on more pure math side. Analysis, especially methods for optimization at the scale needed for civil engineering, is in need of methods for more complex than to bridge two theorems or inventing counterexamples for some conjecture in algebra. So what LLMs are doing now is picking up useless (from practical standpoint) fruit that can sustain mathematicians, not saving lives with better mathematics.
The only thing I can really agree with here is on getting a dog. The idea that art isn't "useful" because it doesn't keep bridges up makes me wince. We are more than just animals looking for a place to cross into the next grazing field. Paradigmatic changes in the world shouldn't be measure by how useful they are to companies. At some point soon, we are going to have to define how much meaning this technology should be allowed to extract from our existence.
This was heart-breaking, but also beautiful, to read. Thank you.
In considering precisely these things I have found Simone Weil's writing a wonderful balm and resource for looking forward. If you are not already familiar I would commend her to you.
It happened to chess. Did it kill the game? I don’t think so. There is still plenty left to discover, and machines are actually helping us do that. Chess isn't mathematics, which is vastly more complex.
I think we are entering a new era of human discovery: the AI augmented era. It can be just as fascinating as the previous one, and humans will remain pivotal to it; I am almost certain of that. This new era will come with its fair share of positives. For instance, mechanical correctness and reproducibility will become far easier to achieve. Naturally, there are downsides too, such as unequal access to frontier models, though the strength of the open source movement makes me optimistic on that front.
Adjusting to this new norm is painful, especially if you are a young researcher. Your seniors are lost too (I am one of them), but their seats are much more comfortable than yours. The old days are gone, and that much is clear. The romantic image of the lone mathematician achieving breakthroughs through the raw power of their mind will now feature a mechanical companion. But that doesn't matter; that image was always meant for the cover of a history book. In practice, discovery has always been the cumulative result of countless human efforts, the classic "standing on the shoulders of giants."
All in all, we are entering a period that is fundamentally different from the one before it. This is both exciting and terrifying, as so much is at stake. It is essential that mathematics remains a human endeavor, and that we avoid becoming overwhelmed by AI generated noise. Nonetheless, this is a brave new world, and I still believe there is plenty to be hopeful and optimistic about.
I love the sentiment of your words. Thank you for saying all of this. I think your diagnosis of where some of the pain and problems lie is also on the mark.
I will say of the chess observation: There is an unfortunate societal/economic difference as it currently stands. I think we all see chess as a fun social activity that is intrinsically valuable, and top chess players figure as sportspeople and entertainers in our collective conception.
Mathematicians are currently seen as theorem-provers who advance the state of the art. It has been observed many times over recently that this will have to change.
I also think, to be a little more candid, that your comments contain an implicit critique of my piece which you perhaps chose not to deploy against me. I do engage in a bit of myth-making and romanticism in the article. When I invoke an "age of heroes" I do so against an internal voice that says: "This idealism is cringe and borders on fascistic". But I elected to keep it because it authentically reflects my feelings.
So I am sorry if I have contributed to the silly "romantic image of the lone mathematician achieving breakthroughs through the raw power of their mind". It was not my intention. I hope my comments about math being a Talmudic conversation work against this harmful myth.
What do you think, Pierre? Should I perhaps add an addendum or edit to stop people from getting the wrong idea?
I was going to make the same chess comment, and I agree with your critique of where the analogy breaks down. But here’s another: at some point in every religion people stop writing new scripture, for the most part. Doesn’t kill the priesthood, just means they start devoting their time to exegesis and proselytizing. To even understand what the machines are doing and if it’s right or not is going to require a high degree of mathematical knowledge, and while I know you’re not talking about applied math here figuring out how to take machine findings and use them to solve related problems in other fields or even apply them to related problems in math will still be extremely valuable. Still, if what you value about math is humans finding novel proofs of theorems, then yeah it’s not looking great and I can see why you’d mourn that. Going back to chess it is the case that a certain mystique was lost when even the deepest, most complex moves could be instantly assessed as right or wrong by a computer no more powerful than an iPhone.
The time isn't yet - as the trusty robotic companion will still need a human to provide oversight. But, probably within our lifetimes, there will come a time where the human contribution is no.longer meaningful - chess may be an example.
Indeed, chess is a game, not math, and it lacks this economic and societal dimension. But this is a very recent situation. Universities for the masses are less than one hundred years old. Before, with a few exceptions, math was restricted to rich bourgeoisie who had the time for it. Above, I mentioned that mathematicians will still be around (as a job), just that the way they work will be different.
I have no problem whatsoever with the "Age of Heroes" metaphor. Everybody needs myths and this is what allows us to move on. The romantic figure of Galois and others is one of the motivations to start the journey. Clearly, there will still be smart people to cast the light of the human mind on the world.
@Kirwin do not change anything in your words. I think there is a strong need for such sentiments, for calls of despair. What is at stake is essential. This echoes feelings we all have on the topic but that one does not read much about.
Again, as a community, we need to adjust to a new norm and find new rules to integrate AI into what was until now a human-only craft. What this new framework will be is up to us to build.
Just a side note: AGI / superintelligence is out of the question because the current technology is not providing it. There is already enough on our plate to solve, no need to add extra.
No. The article is beautifully written as is and the romantic sentiment is adequately balanced. You are allowed to express what you feel. Besides that, it's also just sometimes true. Science is cumulative until it isn't. Cue paradigm shifts. This is perhaps less so in mathematics as you guys are in the business of finding eternal truths. But even you have movements like Hilbert's program that rise and fall and sometimes in quite dramatic ways, involving a fair bit of heroism.
I am a late stage PhD, not in maths but an adjacent field. Currently I find myself in the awkward situation that I have to prove something. Up until now I have never tried using AI for the actual intellectual work I am doing. Somehow the idea felt like a violation. But now I wonder, like you, what am I even doing? I subscribed.
Mathematics is not chess. It is not a sport two humans compete in, but a field with content that can be discovered. Importantly it is discovery that matters, not "math skill" of a person. And I don't see why you want humans accompany LLMs. It makes sense now, but in a few years with AGI it would not.
Even if LLMs became perfect theorem provers, the ultimate goal of mathematics would still be that at least some human gets at least a general understanding of what the theorem means, in human terms. This would entail an understanding of its implications for other branches of mathematics as well as, say, for physics and computer science. This job may eventually become automated too, but that doesn’t seem imminent. The AI-human centaur era of mathematics can be very exciting, especially if the bottleneck in making new mathematics moves away from theorem proving into this more fuzzy realm of human integration and meaning making.
It is true that learning Mathematics is the goal of mathemicians. Halmos said there is no difference between an excercise and a research question except that we know answer of one. However, that is not the reason why mathematicians get paid. They are paid to prove theorems not proved before, and to teach engineering students calculus and linear algebra. An AGI would automate both jobs. But separating the income question, discovering new mathematics unknown to others is a way of gaining meaning that is in danger of becoming extinct, and getting meaning from learning math is but an imperfect substitute for that.
It seems unlikely to me that the mathematical landscape can be explored in any depth even if we had trillions of LLMs working continuously to prove new results. Right now the weird spark provided by humans (especially the very quirky humans who like mathematics) interacting with LLMs definitely adds something. Even if that went away, people doing mathematics would still have opportunities to construct new things never seen before. What might go away is the idea that being first to be recognized as having laid claim to a result is notable and should be rewarded with prestige and honors (as most of these would go to the LLMs).
I don’t really know mathematics research enough to have a firm opinion on this, but by analogy to my field (astrophysics) I suspect that researchers will still have plenty to do. An architect has plenty of work even if he doesn’t lay any bricks with his own hands. Lots of new stuff is becoming feasible that we wouldn’t even attempt a few years ago. This just shifts the balance from detail oriented puzzle solvers, who may in fact be partly replaced, to big picture people. For now.
There are about 150 professional chess players today (making their living playing chess.
There are approximately half a million professional mathematicians.
The issue isn’t the availability of math or chess, it is the availability of vocations that allow the time and focus for deep thinking about either.
I mean no offense, but nobody’s obligated a career that provides them with the time and focus for deep thinking. The arts has been facing this problem since forever; far more people want to make art for a living than there are people willing to pay for said art. I understand feeling a bit shattered, just as I feel bad for the college athlete that fails to go pro or the law school grad aspiring to be a superior court judge that fails to qualify for a feeder clerkship. But the appropriate response is to pick yourself up and find something else to do with your life that helps other people.
That is a sad response. Nobody is owed anything, but it is a worse world without it.
Your response is basically that life is miserable for most so why not make it miserable for all.
Well if the underlying premise is that life is miserable for the majority not in math or some other rewarding pursuit, then the author is saying "I am an elite with a fulfilling life, it is bad that I shall suffer like the rest of you", and any of us should care?
Or, and hear me out, we should consider why more people should have a satisfying life, don’t cut down the one still asking.
I do like that. I think it involves some significant changes in lifestyle to bring that about. way more than "stop ai", it's "reform the property relations in society and probably accept significant degrowth"
But bringing it back to the arts, it's simply the case that individual consumers get to choose what to spend their money on, and so the total number of dollars available to, say, novelists is sharply limited. We can increase that number through public subsidy (NEA grants for writing novels or something like that), but those political efforts haven't proved very fruitful so far. Getting mad at the public for not buying enough novels (or not being willing to fund enough NEA dollars) doesn't seem very likely to be helpful.
Life isn’t miserable for those with dreams interrupted. There’s a bit of pain in the immediate aftermath, of course, but you realize it’s not the end of the world.
Please go ahead and pay lots of money for all the people who want to be artists or musicians or authors but who are currently waiting tables or sitting at the cashier at Walmart.
Yes!!!
Pretty much that’s the story of a huge portion of historical creativity.
You don’t have to pay gobs of money, just make medical care free and housing cheap.
Please go ahead and use your own money for that.
But that's a **game**. I don't do mathematics to play a game and we don't pay mathematicians to play a game.
If it comes to the point where I'm not a useful way to discover new mathematical truths I'm out and society should stop funding human math research. I think that point is farther away than people think but let's not pretend it will 'be ok' because we can do math the way people do crosswords.
it will suck for you just like it sucked for every artisan when they are industrialized out of a profession. Of course, assuming some great replacement, which I doubt-- I think the article is quite alarmist and protectionist.
This situation has brought about the Olympics of terrible analogies. Chess and math are utterly and completely different and the agents involved are too different to compare.
I think chess is very different. If we want to see the best two chess players play, we would probably pick two engines against each other, but we don't want to see engines play. To me, this analogy is just as good as arguing whether mixed martial arts will die because of AI robots.
I feel like math is substantially different in that I don't think it matters to us whether a theorem was created by an engine or a human as long as it is the "best". Whether it will be self-prompted to do so, that may be less obvious and to me the leverage mathematicians have at the moment is that they can direct it a bit more intentionally.
I think it's the direction coding is going in as well right now so I feel like maybe I'm curious if the real benefit of AI harnessed by a mathematician is the amount of new questions that can be created.
there will be a place for you. godspeed
I have been dealing with similar feelings and worries lately. Thank you so much for this beautiful and haunting post.
It saddens me to read in the comments variations of "deal with it", "find another hobby", "dedicate yourself to more yourself pursuits" etc. Humans have been practicing some form of mathematics since at least 2000 BC. In all this time, it has been an essentially human activity and part of human flourishing. To quote A. N. Whitehead:
> “I would not go so far as to say that to construct a history of thought without a profound study of the mathematical ideas of successive epochs is like omitting Hamlet from the play which is named after him. That would be claiming too much. But it is certainly analogous to cutting out the part of Ophelia. This simile is singularly exact. For Ophelia is quite essential to the play, she is very charming-and a little mad. Let us grant that the pursuit of mathematics is a divine madness of the human spirit, a refuge from the goading urgency of contingent happenings.”
And indeed, mathematics can play a tremendously important role in a person's life far above and beyond its usefulness. I will never forget that Dunham's "Journey Through Genius" starts with this powerful quote from Bertrand Russell:
> “There was a footpath leading across fields to New Southgate, and I used to go there alone to watch the sunset and contemplate suicide. I did not, however, commit suicide, because I wished to know more of mathematics.”
I don't know where we go from here. I can only hope that Pierre S. is right that mathematics will continue in some form, hopefully with the machines assisting humans rather than replacing them.
I am sorry to hear you are also suffering. Thanks for your response. You've brought some damn fine quotes here as well, a "divine madness of the human spirit" indeed.
I don't know what else to say, other than to echo your hopes and thank you again for your contribution.
I did a BS in pure math, and then entered industry as a software engineer because I needed to earn money. In another life, I might be in your situation now, so I very much resonate with this. The romantic and mystical is a lot of what drew me to math. Passion is good, and human; fuck the people telling you to "chill out" or "get a grip" or whatever.
Humans do a lot of myopic and craven things with our short precious lives. Math is one of the few that is intrinsically valuable. This is encompassed in my piece https://substack.com/@quinoam/p-185273573
At the same time, another part of me reads and accepts this as "well, natural selection comes for us all." Water only runs downhill. LLMs are just the latest expression of replicators maximizing fitness. I wish we had a better world where all that is good and beautiful could be insulated and preserved. But no, nothing is safe. Entropy comes for us all. I have long since accepted this. So I feel calm. I do what I can but this is the way of the world. Most avenues of human life will never be the same, and some have been irrevocably desecrated. So it goes. If we don't all die I hope that we may have more leisure in the future, so all who wish to pursue recreational mathematics have the ability to do so.
Thank you so much for your empathy and thoughts, I'm glad my desperate scream resonated with you. Your piece "my terminal values. like for real" is a brilliant intellectual exercise: Explicitly laying out our base values. I am not very reflective most of the time so I think I should sit with your piece for awhile and try to do something similar for myself.
Intrinsically valuable stuff is what it's all about. And it's so obvious I feel silly saying it.
It is good that you have passed through acceptance and feel calm. I cannot say the same. I still haven't accepted the old realities, like the fact that I will die. Neither mind all these new horrors.
You express what many programmers have felt as well - a profession that was one of the earlier casualties of AI.
This is true, and I am sorry we are both in this pickle.
Given that you have been steeped in it for longer than the mathematicians, do you have any advice for them on how to get through it? That might be a big ask. I don't supply any advice in the article.
Most of our reactions have flaws - either refusing to use and have anything to do with LLMs, and so likely swamped as time passes. Or embracing them and using the analytical and building aspects of coding to instead supervise AIs to code - so losing part of our soul and not a long term resilient strategy.
I suspect we need some kind of mourning ritual, not sure of the form or timing though. Like a funeral for hand coding small functions. And then maybe more funerals later.
Oh yes, also if you have capacity, look am AI risk campaign such as PauseAI or anything similar. The impact on the crafts we love is (in many future scenarios) but the least impact of this technology on us.
Thanks for writing your post.
You make it sound as though human software development is dead already. That is far from the truth.
Beautiful piece. Those who tell you to get over it haven't had the experience you describe of their 'work' aligning with their life's moral good. It seems like one of the moral goods that's constitutive of your life so far has been to engage with the ineffable and the past through mathematical discovery. That's made you part of the great human conversation stretching back millennia. How beautiful. That's a meaningful thing and hard to lose. But some people never lose anything as valuable as that because they've never had it in the first place. And some people's moral goods just lie elsewhere, in family life etc.
I feel your pain! And it makes sense to me.
> Those who tell you to get over it haven't had the experience you describe of their 'work' aligning with their life's moral good.
How do you know? This is stupid and intellectually dishonest.
Also, if mathematics is your life's moral good then you're a sociopath.
My point is that if they were familiar with the experience then they would be empathetic.
If you reread my comment, its not about maths but about the ineffable, a sense of the divine broadly conceived or of mystery and truth. Maths is clearly one route there for the author. I'm not into maths but I recognise the feeling and the goal.
Try to actually engage with what this guy is saying.
Aside from a complete failure to engage with what I wrote, there's no such thing as "the divine" or "the ineffable".
> Those who tell you to get over it haven't had the experience you describe of their 'work' aligning with their life's moral good.
Again, HOW DO YOU KNOW? Again, this is a stupid and intellectually dishonest ad hominem.
If I was a Mathematician I would be full of wonder and excitement to know I would live to see a hundred lifetimes of progress. But I can also appreciate your ode to the end of an era.
Most are, and there is some of that in me as well. It is a curious mix of emotions. But I felt that the optimism is well-represented elsewhere.
I suppose, but I’m greatly worried about the shape of the progress. We’re already making progress, first with the planar unit distance problem, then the Jacobian conjecture, then the Dinitz-Garg-Goemans conjecture.
Maybe I am overreacting because I feel so strongly in agreement with the content AND tone of the article, but I feel like AI’s discovery of these results have made them much less exciting. If a mathematician proved them, we would immerse ourselves in the article containing the proof, be emailing the professor, ask them to speak at conferences, etc. Now, even Tao is just trying to investigate this by…asking an AI even more questions about it. This is fair, given that ChatGPT is the “professor” responsible, but we lose the pride in our fellow human in accomplishing something great, and the inkling of hope that maybe, one day, I can be as clever and acclaimed as that other human mathematician. We also become more atomized, one of the main issues targeting society at large today - what is the point of emailing other people or going to a conference if asking an AI the right questions will net you better answers more efficiently?
Like the original author, my thoughts are reactionary and unorganized, and I don’t deny that as an intelligence-assistance technology, AI is here to stay. The only analogy I can really think of is if you were never allowed to look outside again. You would never see anything novel. You lose a quale, a feeling with inherent value. And sure, you can let your AI robot outside, and it can see things and describe everything and synthesize it in even greater detail than the greatest author to ever live, and there is no economic reason for you to go outside if the robot can perform your business and describe it back to you so effectively. This analogy is kind of falling apart, but I hope someone can take my meaning. I’m worried that humanity will lose the ability to create something for a reason other than our own brief and pitiful entertainment, if the AI can create every thought cheaper and better.
another part of that is that if maybe 3-4 companies worldwide control the ai robot that goes outside for you, and they keep it cheap enough for a sufficiently long time to generally degrade most people's skills and demotivate them thoroughly, why the hell would they keep it cheap? as gatekeepers of intelligence, they have every incentive to ramp up the prices exponentially once it's well and truly adopted by most.
It's a hope-killer for those who believe that self-improvement has value. Most seem okay with taking a backseat in life and observing what AI can do like it's just another form of entertainment.
it's more thrilling to feel like an active participant than a passive spectator cheering on the new machine gods.
There is Kierkegaard's severe admonishment - 'it is always edifying to be wrong over against the will of God'
Maybe this is just a chastisement for us all
Stay strong!
It's my fault. I once stole an entire ream of paper from the math department's office supply room.
In all seriousness, thanks so much for this thought and for sharing around the article Nik. If you read the article above and are now reading this comment you must now go and read Stultiloquy
For me, the problem of evil is (in part) resolved by seeing God as a thing of finite power. If God was arbitrarily powerful the power would appear infinite to us. So, if there is a God, and if God is good, I'm not sure that God could or would chastise us. Though there is no lack of human action and inaction for us to be chastised for presently.
I am not a mathematician but I am one who loves mathematics, and for whom mathematics and theology are frequently co-illuminating. I have pretty much never supposed I would get to do original mathematics - I just get to learn about and try to understand what others have done and discovered. While I have sometimes enjoyed the human stories behind the mathematics, it is the math itself, like walking in the mountains or gazing at the stars (a thing of beauty and glory and wonder in itself), that captivates me.
I have also been a very interested follower over the last 3 years or so of the developments in AI, and a frequent personal user of AI - as a conversational partner, it has been a helpful tool for me to ask clarifying factual questions ("please remind me what primitive recursive means in this context") as well as for exploring ideas ("is the number 3 just the set of relations that pertain to the thing we call 3, or does it have an interiority?" which lead to a paper by Benaceraff, whom I'd never heard of and which I'm currently reading).
Regarding AI overall, I have high hopes and a dreadful expectation of horrible downsides, comparable to, but orders of magnitude greater than, the Internet and then social media. But regarding AI and mathematics, I can't see that I will lose anything (perhaps I may yet repent of such folly). It remains to be seen if AI is as good at proving as at disproving (by discovering a counterexample). Quite possibly it will. (Perhaps the best case is it's only good at discovering counterexamples, and then much less time will be lost trying to prove what isn't true?) But even if AI starts writing all the proofs, while I'm sure that will be felt as a loss (as you yourself express) by many doing real work in mathematics, yet it means too that the beauty of mathematics, and of more and new mathematics, can be brought to millions who otherwise wouldn't have experienced it.
Was the wonder, was the glory, truly in the original discovery (I and no other have now seen this) or in the personal discovery (though others saw it first, it was a wonder to behold)? (Forgive me if I've posed a false dichotomy.) If mathematics is a glorious landscape before us shrouded in undiscovery, I do not now perceive that it matters to me if the ones further illuminating it are men or machines. Only men will glory in the landscape thus revealed.
I've often revisited the thought experiment of the Library of Babel. It in fact does exist (for all practical purposes) - there are websites that will give you a page from a randomly chosen book of this dreadful library, and the program to do the same is easy to write. The knowledge that any such page, in correspondence with the right one time pad, is in fact an encoding of any and every other page that ever has been or will be or could be written, has not vitiated my readings or my writings of meaning.
It seems that your argument following the introduction of the Library of Babel implies that the student learning math, beginning to enjoy it, then to love it, is actually in the same undesirable position you foresee? "I love math!" "Ah, but you'll find it's actually only rewarding once you're doing original work." Didn't you love it when you were an undergrad? Did math only "get good" when you became a grad? Post grad? Perhaps there is some real higher pleasure I have never yet enjoyed - but, trusting you did enjoy math long before then, I would remind you that there are real if humbler pleasures. You say, though, those are inextricably tied to their reality as human discoveries. If, then, mechanical discovery really drains the pleasure of participation, I am sorry. For me, though, the joy (or at least a joy) remains a consequence of objective beauty residing in that which is discovered, not (only) in the subjective joy of the original discovering.
I pray you will find some hope. Despair not of all hope.
Chris Brenan
Hello again Chris, I'm now safely back in front of this horrid laptop, so here is my longer reply:
There are great pleasures and spiritual fulfillment to be found in the "personal discovery" as you put it. My goodness, to work through Chapter I of Serre's "Trees" and experience the elegance of his arguments while filling in the gaps (he moves quickly sometimes) and wrestling with the exercises. It was was stimulating and joyful. However, I'd like to highlight something in my own example.
Serre is a human with quirks and style and preference. As I worked through Trees I was conversant with him (also John Stillwell, since I was reading the English translation) and felt a part of the culture of mathematics of which Serre is an influential member. Should I have been reading an AI-generated textbook, I feel something of value would be lost. This of course assumes that I am reading LLM prose, and perhaps it would actually be that the proof is AI-generated and the exposition human. In this case I think some of the experience of the personal discovery (an amalgam of many feelings and thoughts) is preserved. So you make a useful distinction, perhaps all is not lost.
All of the above is to say that I did not mean to convey in my piece that only original mathematics, and no other kind, alights the spirit. I concede that point to you completely.
Moving to the other part of your dyad, the "original discovery". I was writing on original discoveries in mathematics, but we can all claim original discoveries I suggest. For instance, if you are the first to notice an interesting and subtle similarity between two people in your personal life, make some sort of deduction about it and then verify it—maybe you suspect they will get along and introduce them—you've made an original discovery. That feels a little special doesn't it? Perhaps it even suggests there is something about you or your relationship to these people, or about the circumstances of your life presently, that allowed you to see it first. I'm not sure. There is a little magic there, no? A little synchronicity? Maybe you append this to the narrative of your life, thinking occasionally with a smile, "I knew they would get along! Imagine if no one ever introduced them."
For the mathematician these two friends might be seemingly unrelated subfields. I think it is ok for the mathematician to want to be the one to acquaint their two favorite subfields (each of which means something special to them and perhaps no one else) instead of merely reporting: "Oh yes, an LLM identified that my two friends might get along and told me to introduce them."
You are forcing me to the edge of my ability to express my feelings on this, Chris. I really have to commend you for that. I'm going to make a final gesture in the direction of my emotions. It's something like this: Generating The Library of Babel seems utterly pathological to me. When I go to the beach I pretty much leave things alone. Maybe I take one remarkable stone if I have some reason to. I don't read others' diaries. I try to take only what I need. Not everything is so universally important. Not everything needs to be done right now. For each thing that can be done in the world, there may be a person for whom doing it would mean the most. Maybe that's not quite it. Maybe it's more like this:
Being the first is healthy food for the ego, and we live in a world that demands that our egos exist. To take away the ability for people to create original things but keep all else unchanged (inequality, suffering, status games, hierarchies, etc.) forces the ego to be and then starves it. It is like bringing a child into the world against their will and telling them they will have to go on without food.
Yes, I think that might be more like it. Thank you for leading me to this realization. I might make a note or post of it later. I am sorry I could not make this more concise. Thanks again so much for your reply, which forced me to work through some of these ideas, and if you read all of this, thanks for sticking with me.
This is such a thoughtful post and I would love to give it a long reply, but I am camping right now! There are some wonderful counterpoints in your response that deserve a proper reckoning with. I think you have brought more clarity to the discussion. For now I will just say thank you so much for your thoughts and I endeavour to give a longer reply when I am not in a tent.
Which paper by Benaceraff, if I may ask?
What Numbers Could Not Be. I plan to read mathematical truth next. This mathematical philosophy is mostly new to me.
I have written software for 20 years and all of the parts of it I enjoyed have been made both obsolete and valueless this year. I also had various personal upheavals in recent years, but I think even without them, I would have had to reevaluate where I was going and what I was spending my time doing. I'm going to try to go back to school for the things I feel really creatively engaged with and spend time reading about and would keep reading about even if I didn't go to school for it. It doesn't matter what it is -- but you might have something like this.
But even if you don't, and mathematics is your creative outlet, if there is any humanity to math at all, if it really is an art, the artistic aspects of it will never be -- imo, anyway, someone deeply engaged in using AI every week and building things with and around it, even with this effect it's had on my life trajectory -- duplicated by LLMs. They miss the right spark for good writing, or images, or video, for example; and despite using the best and most fancy, expensive models, I regularly run into tasks, usually involving large-scale synthesis, that it is only half-helpful for.
I have recently realized that all my outside-work coding, if I get AI to do it, is really quite awful in any aesthetic sense unless it's a very simple thing (it's able to make simple tools very good looking, and websites, for example) -- so I will keep doing it manually. And when I write the most important parts of a new system, I am realizing it's still better, and justifiable, to do the key parts by hand. I am sure there will be spaces in math that will be elusive to AI, they just might be very unintuitively shaped.
I feel like I got a crash course in what weavers must have in the time of the loom. It is absolutely brutal considering I quite liked coding and was also quite good at it (from what I was told), and now, woops. No one cares. Not even myself, I have started to realize. It's just on tap now. It was February when they got good enough that I was really surprised, and it's been a nauseating roller-coaster from then on.
Sorry. I do agree with one thing another commenter said (though not the rest of it): welcome to the club. We are in it together now. At least you don't have a boss telling you to become a professional manager (of virtual subordinates) now, I hope.
Well written. I've programmed since I was a child, then studied CS and worked professionally as a software developer. At one time, I would've said it was very important to me. A few years after stopping, it surprises me how completely it has faded from my mental space and identity. If anything, I've come to realize that these type of activities are not that important to the core of our humanity and identity.
It's good that you recognize this as a dark night of the soul - a part of your inheritance as a human, a passage alike to dark nights that others have survived. It sucks, it's painful, but you will make sense of it yet. That's one of my favorite phrases of late, "to make sense". People commonly throw it around as a synonym for "I know" but I see it more fundamentally - that there is literally a skillful creation of knowledge, that pays off in the qualia of "aHA" when you have succeeded. My favorite mathtuber (who is generally incomprehensible) said, "How does one make sense? Carefully" - advice for self-identified "bad at math" folks, who tended to metaphorically cut their fingers on the sharp edges of arithmetic, but equally applicable. Take care when you struggle to make sense; know yourself, your motivations, your dreams, your sore spots. Mathematics will not cease to be a bridge to the divine, if you still seek it with all your heart; I know because I see the same transcendence in artists and gamers of all sorts. It's not actually necessary for an undiscovered frontier to lie beyond the ocean; machines might print you a map, but you still have a journey ahead to cross the territory. You still have to feel the rain on your skin, to feel the made-sense of the mathematics, no one else can feel it for you.
Thanks for the reply Kevin, you have a really striking prose style that I enjoy.
Yes, my piece is not an ode to the "aha" as we certainly get to keep that. A useful distinction was clarified by another commenter (Chris Brenan). To quote him:
> Was the wonder, was the glory, truly in the original discovery (I and no other have now seen this) or in the personal discovery (though others saw it first, it was a wonder to behold)?
One fear I have is that we will one day surrender original discovery. If taken at its most general, original discovery encompasses a huge number of human experiences that personally I would like to keep. As I said in my reply to Chris:
> Being the first is healthy food for the ego, and we live in a world that demands that our egos exist.
I'm am not sure if that changes your thoughts on this at all. It was not meant as a counterargument so much as a clarification of what bothers me. It could be that you were already very clear on this distinction and the implications I've observed. Your response to it may even be implicit in your comment, and I've failed to pick it up.
Whatever the case, thank you for acknowledging my pain and thank you for bringing a more constructive and optimistic view of what could be.
Thank you! Both for the compliment, and - I felt a good measure of conviction reading your piece and writing that comment, and I'm glad it was well-received. Hmmm. I don't feel fluent in matters of ego, and so at a certain scale - I probably think and talk around the dysfluency, haha. I think of an idea I bumped into, maybe from a sports psychologist or coach, that high performers tend toward two types: a LOT of ego "of course I'm the best to ever do it," or virtually zero ego, humility built on awareness of imperfection and uncertainty at the edge. I remember the wave of validation washing over me, "oh okay, yeah, that's my Type of Guy" lol, when he named option 2. All that to say, yeah makes sense! I'm not sure I have any more good wisdom though
Curious who's the mathtuber you speak of! Am trying to improve my YouTube algorithm
G++, in a video titled "Mathematics doesn't actually make any sense"
Thank you!
I signed up a substack account for the first time to make this comment.
I came across a screenshot of your post on twitter and was deeply moved by your emotions. It made me consider the implications of AI in a much more visceral way than any other headline or breakthrough I’ve seen reported.
As someone who’s generally optimistic and excited by the technology, I’ve mostly only ever been concerned by the worst case “doomer” scenarios where AI forcefully ruins human life. I’d never examined the negative impacts AI could have on the human spirit by simply making our passions feel arbitrary.
To me, your message represents ideas that are uniquely human; loss, hopelessness, lack of purpose. Despite the upsetting tone, I found your words to be a beautiful expression of being human, and I think AI will have a very difficult time replicating that.
Thank you for writing this piece. Thank you for being vulnerable. I hope the response that you get for sharing this brings you some comfort, and that you know you aren’t alone.
Thank you so much Austin. I feel very understood in reading your response. You and many others have been very kind to me. There is a lot of vulnerability in my outpouring. And the critiques (for example, that some of my horror is due to a loss of status) have been measured, for the most part, and have only made the discussion more authentic.
Here's another little bit of sweet, sweet cope - the headline making AI contributions (all counterexamples) are the most expensive maths results in history by some obscene factor
I'm not surprised. Math is pretty cheap. The overheads are pretty much just chalk, scrap paper and coffee.
What did Renyi say, 'a mathematician is a machine for turning coffee into theorems.'
This sounds nice but I can't substantiate it. Where are you getting this assessment? I thought at least some of the counterexamples were done in a few short turns with a human, probably less than 1kWh of energy used. (I would like to collect actually expensive projects, it seems like valuable information to track.)
I’m talking about the infrastructure and R&D spend which now runs into the many hundreds of billions of dollars
Right, some truly gigantic expenditures there. But can one claim that this makes LLM-assisted counterexamples some of the most expensive mathematics results ever? If this is the case then by the same logic we have to weigh the civilizational costs of earlier mathematics as well (roads for the mail, planting forests for paper) and I don't think they necessarily come out much lower: the costs are lower but the number of people and total economic activity was lower. Maybe ancient mathematics was the most expensive of all as a proportion of overall activity.
I refer you to Renyi - the main cost of maths over the last century has been coffee
Yesterday I was thinking about something similar. "What are we going to do, if machines can think better than us?"
Naturally, I posted this question to a machine and it suggested me to pick up a hobby like pottery, blacksmith, or cooking. It even give me a map of places where I can learn blacksmith near my IP address.
I would implore us to all keep thinking hard, even if we aren't the best thinkers around. But finding refuge in tactile hobbies is also not a bad idea.
The reason why these companies are automating mathematics is that it is very useful to do so. A superhuman mathematician means fewer bridges fall down, better research developments in areas as diverse as marine biology, computer science, and agriculture, etc. It's just such a good and useful thing. I can sympathize with artists far more (after all, art isn't actually very useful to the world, and automating it doesn't make fewer bridges fall down).
> I have wondered if it is the express goal of these companies to make me kill myself.
No, that's not the express goal of these companies. Their express goal is probably much closer to the "country of geniuses in a datacenter" (https://darioamodei.com/essay/machines-of-loving-grace) and unfortunately a side effect of that seems to be killing your will to live. Perhaps you should look towards other means of finding this will through something that doesn't involve having to do mathematics. Maybe it's just seeing the beauty of the world, maybe it's having friendships, maybe it's a dog that you take care of, you know?
But it's what that person is passionate about. Your take is like telling someone 'I'm sorry you won't be able to paint anymore, perhaps you could discover a passion for baking.' That's not what this person is drawn to; it's not who this person is.
Oh boo hoo. Few people can do what they are passionate about as a career. And baking isn't the only alternative -- stop being so intellectually dishonest.
And we're still a long way from AI eliminating professional mathematicians ... finding counterexamples is the lowest hanging fruit.
Deciding your identity is bound to a particular activity is a choice. There is nothing to suggest that such esoteric passions are wired into us biologically. If they developed over time, then new ones can be developed and old ones forgotten.
It's natural to grieve the end of a personal era but not healthy to ruminate in misery indefinitely and not move on.
So why are so many people's surnames, Miller, Smith, Taylor, their family's profession?
The overwhelming majority of people with such surnames today, do not work in the profession suggested by their surname.
That is true. I would all suggest that it shows that the identification between who we are and what we do is nothing new. And a great many of us are made for some things much more than others, and none of us are made for machine-occasioned obsolescence
Totally agree with you. I don't celebrate any kind of machine driven obsolescence. It has impacted me in multiple different areas of passion and derailed career plans. I just want to share that, speaking from personal experience, it has not been the end of the world to completely change what I think about and work on. There is an initial adjustment period that is miserable, but I can genuinely say I am happier now after realizing that certain identities I clung to were not necessary. It was freeing to discover that my relative strengths and weaknesses, as well as what I can enjoy and take pride in, are much less rigid than I previously thought.
Yeah, well, if a painter lost his arm I would say the same. The effect of automation is much like losing an arm in this case.
At least current LLM proofs are not about engineering, but on more pure math side. Analysis, especially methods for optimization at the scale needed for civil engineering, is in need of methods for more complex than to bridge two theorems or inventing counterexamples for some conjecture in algebra. So what LLMs are doing now is picking up useless (from practical standpoint) fruit that can sustain mathematicians, not saving lives with better mathematics.
The only thing I can really agree with here is on getting a dog. The idea that art isn't "useful" because it doesn't keep bridges up makes me wince. We are more than just animals looking for a place to cross into the next grazing field. Paradigmatic changes in the world shouldn't be measure by how useful they are to companies. At some point soon, we are going to have to define how much meaning this technology should be allowed to extract from our existence.
This was heart-breaking, but also beautiful, to read. Thank you.
In considering precisely these things I have found Simone Weil's writing a wonderful balm and resource for looking forward. If you are not already familiar I would commend her to you.
Thanks so much for saying so. I only know the name. Where should I begin with Weil?
I've looked at a number of her books and papers. I'm currently working my way through this anthology and finding it a great collection: https://cominsitu.wordpress.com/wp-content/uploads/2021/09/simone-weil-an-anthology-penguin-books-2005.pdf
Thank you! Added to the backlog lol.