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.
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.
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.
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.
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.
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.
> “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).
> 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.
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.
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.
> "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.
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.
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.
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.
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"
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.
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.
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.
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. :)
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.
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.
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.
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 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.
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".
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 [...] 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.
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 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 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.
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.
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.
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?
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".
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.
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.
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.
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.
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.
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.
> “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).
> 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).
> It is a lively discourse of philosophical and religious richness
This is pretty funny considering how trashy it is compared to philosophy
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."
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.
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...
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.
> "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.
> 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.
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.
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.
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.
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.
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.
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
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.
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.
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.
The issue seems to be with the expectation that personal enrichment and subjective aesthetic appreciation should be socially instead privately funded
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.
Customers? It's math. This is like a comment a golem built to raise the S&P 500 would write.
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.
> This is like a comment a golem built to raise the S&P 500 would write.
That's neoliberalism for you.
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".
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.
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 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.
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.
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.
He sounds as someone really pedantic who never understood what math is about or why it is important.
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.
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.