Mathematicians should realize: instead of spending decades on individual theorems, you can now create whole new subfields of mathematics. Of course, you won't be doing the creation directly, in the way the mathematicians of old did; that's just not the name of the game anymore. But you will still be directing the research and involved in the creation.
Somehow I doubt it.
Maybe there won't be a need for you to be involved at all.
Maybe the AI will perform even better if you're not there to slow it down.
Maybe there will be people involved, but not as many.
Maybe there will be so many people involved that their salaries will be depressed to the point they can't afford to do it anymore, and only the rich will know such joys.
We live in interesting times. This is the age of anomie. Anything could happen. There is no set path to success anymore. The world only gets more and more illegible with every day that passes.
Someone said:
> No one is stopping you from doing what you enjoy.
But that's wrong. LLMs are creating an environment in which some things are no longer enjoyable.
For example there is less joy to be had in learning a programming language in 2026 than there was in 2016, because there is less utility in it now.
Some notions:
- finding counter-examples is one of the most exploratory uses of calculation
- there is no evidence, not even in the hands of the inimitable Terence Tao, that Approximate Intelligence (AI) is insightful rather than inferential*
- people who enjoy Mathematics, even amateurs, do so for the beauty
- AI is only good in the hands of those who can provide corrective context
- AI unchecked is a remarkably inefficient noise engine
- Mathematics will provide joy to the curious 'til the heat death of the Universe
- AI is contributing to the heat death of planet Earth
- anthropomorphism, of which I am otherwise not a fan, may yet be a saving grace as without proprioception we may not morph
- we may need to redefine the Turing Tarpit of Perlis
If any of this nonsense really troubles you return again to thermodynamics and revel in the joy of being the local universe's pinnacle of meaty evolution. And go outside and smell the flowers.
* Caveats abound
I hope that this doesn't mean that you need write proofs while streaming Twitch. ;)
Another case of Deep Blue.
As a mathematician who publishes theorems, I think this is mistaken. New theorems in abstract and pure fields of mathematics never had the economic value of what mathematicians are paid. Even creative works of art can be displayed but mathematical papers are unread by nobody except a tiny number of people who wouldn't want to pay all that much for them if they had to.
The economic value of mathematicians is to teach mathematics. Proving theorems is how we build the credentials we need to be trusted to be allowed to teach it. Teaching young and less young people how to navigate through a mathematical problem is how we make ourselves useful.
LLMs coming up with counter examples to obscure conjectures (albeit old and important to a small number of people) is only a threat in that it removes this ritual of belonging.
Many people are excellent teachers (and very competent mathematicians) without publishing new theorems, and many theorem producers are atrocious teachers.
Perhaps the pecking order may change. Or maybe something else will happen. Maybe society will decide to replace all its teachers with AI, or that mathematics is no longer worth learning. I don't know. But what I know is that nobody saw any of the papers I wrote or the theorems I proved and thought "that's money well spent".
When a new technology comes along, it diminishes some aspect of the human experience - because whoever had a talent for doing things the older, more difficult way becomes less important.
In that regard, AI is the extreme, because it threatens to make so many of our intellectual talents useless.
Both groups love math, but the later group is in a real bind because they have confounded their career with their hobbies/passions.
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.
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'm conflicted in feeling there's now a lot more to do with AI, but also if it ain't me doing most of it, where's the fun in that? Days feel meaningless, and don't know who will pay for what I want to do, after all these years of heavy training.
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.
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).
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. :)
There are a lot of reasons you might make a conjecture. Perhaps you are motivated by a search for structure: "We suspect that these objects behave like this, therefore we naturally imagine that...". But you might make a conjecture out of empirical evidence: "We checked 1 million examples, time to make a conjecture." While empiricism has its place in mathematics, it might not be the best motivation for a conjecture. As Bishop said, "Do not ask whether a statement is true until you know what it means." If you pose a conjecture based on empirical evidence, perhaps you should be thinking more about why you would imagine your statement be true in the first place.
I know some will think that this rings of goalpost moving, but when I hear about all these counterexamples spelling the end of mathematics as we know it, I just wonder how strong those conjectures really were in the first place. Again, I haven't actually dug into the specifics of any of the results so admittedly I could be totally off base, but I do know that not all conjectures are created equal.
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.
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.
LLMs just democratized that process.
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.
If you want to make the case that the journey of creating the proof is part of doing mathematics, then you should also document that path and make it reproducible, as science should always be.
There were some examples of mathematicians publishing their AI proofs like that in the last days, but by far not all.
If you are interested in mathematics because it can model things precisely, and you want precise answers about these models, and you want to know how it is all connected (Langlands anyone?), AI is fantastic news. There is plenty of new and interesting and beautiful and elegant mathematics to be had this way, as well.
This is not a time to be scared or frightened. This is a time to be excited as fuck.
There will always be open questions. Now, there will be actually many more of them, because many more people will be asking questions.
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.
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.
Would a mathematician please unpack this for me? A theorem is the opposite of ineffable. A conjecture may be ineffable but its pursuit reifies it. I am missing something.
> 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.
lol who is paying for this? Mathematicians are generally paid to do other things and get to do this kind of work as a side effect.
> ... 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.
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".
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.
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)
Normally I'm very sympathetic to the plights of those socially affected by AI, and rarely pass up the opportunity to clown on institutional actors papering over these plights with glib cliches. But in this case, I really feel like this is a case of monkey-brain overriding mathematician-brain.
The greatest accomplishment in my life is my math PhD, earned before AI got useful. I do math-heavy work for my day job, and I care deeply about every craft I practice, to the point that I am almost certainly on the unhealthier end of the spectrum of perfectionism. Yet I both acknowledge rather than deny the impending death of mathematics as we have engaged it for the last couple centuries, and am far more excited than aghast.
I would have to write a whole essay in order to communicate this in a way that is appropriately sensitive, but in a nutshell: We have always been in the business of discovering and understanding math, have we not? Are we going to call it a tragedy rather than a blessing now that the floodgates on both are bursting, just because it makes us feel less special, unique, valuable? It's such a self-serving aesthetic on the beauty of math to me. The author claims there is something profound and spiritual about math, and I agree; it exists prior to and above all of us, human or AI. Whatever way we can tap that beauty is incredible, full stop. This is what I believe.
From the article:
> These developments have triggered some deranged thoughts in me. I have wondered if it is the express goal of these companies to make me kill myself. Am I alone in this paranoia?
The severity of this line caught me off guard. I think the author is genuinely having an emotional-social upheaval, and I feel terrible for them, but at this moment they may be having a hard time separating their trauma from their intellectual aesthetic for what math is all about.
Taking a helicopter to the top of Everest is not as rewarding as climbing the mountain.
You are completely discrediting the joy humans derive from actually -doing- the work, and not just, telling someone else to do the work.
There is a reason artists hate AI, its because they enjoy the actual act of painting, like mathematicians enjoy the act of proving theorems and programmers enjoy the act of typing lines of text into a file on their computer.
We are intentionally deleting all of the enjoyable high agency jobs and replacing them with "model babysitting".
Chess has utility in only the most abstract sense, yet people enjoy it.
Computers became better than humans, and some dispared, but more still enjoy it.
Computers provide insight into the problem, which leads to a greater enjoyment for many people. Instead of finding a move in a complex state and thinking 'I'll never know if that was the best move'. A good chess engine can give a reasonable suggestion of what the best move actually was.
It does take from you the feeling of being special.
We are no longer the only world in existence, we are no longer the center of the universe. We are no longer crafted by a all powerful benign being to be exactly how we should be to rule this world. We are no longer the only things that can play a good game of chess.
That hurts a sense of pride, but I'm not sure it was a worthy sense of pride.
This is pretty funny considering how trashy it is compared to philosophy
"Lord, this situation bites."
Potential reply: "Tell me about it."
A year ago, the goal post was that LLMs wouldn't prove anything of worth. The year before that, it was that LLMs wouldn't be able to do math as well as a undergraduate student. Year before that, it was that LLMs wouldn't be able to do math at all. The growth of LLMs has been insane, and it doesn't look like it's going to plateau any time soon.
Same here: the value for the society may be more about "making art" than "being a cog in the paperclip factory".
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.
oh they are very much aware lol
The reason the author is upset is not that the conjectures were false, but that an AI settled them. That's why so much of his post is about the loss of the human element.
I definitely don't think there's anything inconsistent with mathematical and theological thinking. That actually strikes me as a very strange idea?
I am really excited if LLMs means more of mathematics will be formalised. Once we have all of mathematics formalised, I think we will start to see real great progress!
Well I think you answered it:
To become a developer, I disciplined my mind to do impressive and useful things. The resulting feelings were pride and speciality and therefore sometimes joy.
And those feelings are gone now that software engineering has become a low-discipline, low-barrier to entry activity.
Am I immoral or vain for having derived joy from that -- I think is another matter, and one which doesn't resurrect the joy no matter how it's answered
“May your every desire be immediately fulfilled.”
As the old saying goes, the purpose of a system is what it does, so there might be some Platonic ideal of science we can appeal to, but the practice of science by humans is about the craft as much as it is about the truth.
The parent's point is better articulated as: programming COUPLED with being paid do it is what is now less enjoyable due to AI.
You are free to enjoy pretty much anything but you may not necessarily be paid to do it.
> We are no longer the only world in existence, we are no longer the center of the universe.
Nothing has changed in this regard. We were the centre prior to AI then we still are after it. The human ability to comprehend makes us the centre. The squiggle: 8 means nothing to an ant but to a human it is the number eighth. AI is something only humans comprehend beyond that what does AI matter and to whom would it matter to?
Cringe is not an adjective.
Being given only the generated proof (or disproof) is like having the binary artifact as an executable. Sure you can use it to solve or build further on top of it, but without the source code - the understanding of the steps to build the binary - you can't build it yourself, and you don't actually own it. If you want to adjust or extend something in the proof, you have to go back to a third party to implement it and generate another artifact for you.
We need a Stallman for mathematics with uncompromising integrity, to shout from the rooftops for a "free and open-source math movement" (FOSM). Even if the biggest AI companies go bankrupt, we still need to be able to bootstrap all the math from the source code.
You might need to put the phone / keyboard down, take a day off...
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.
And this analogy breaks down further given that both the real now genie and the hypothetically more powerful genie of the future would each work for your (ex) employer.
On a more abstract level they are paid to bring funding and prestige to the university, but that often goes through the route of papers. And papers often involve proving theorems.
The thing is, if someone had the option to press a button and cause their garden to be weeded and growing perfectly, if most gardeners were using that button instead of growing normally, maybe some people would keep gardening for the spiritual part but they'd certainly feel like the rug had been pulled out from under them.
Edit: As another example, Robert Pirsig in his famous book described motorcycle maintinence as a spiritual enterprise. However, the quality that gave it this was the patience required-for and the uncertainty involved-in the enterprise. And again, if all you have to do is press a button, the spiritual part kind of goes away.
Hayek and Friedman predicted it!
For me, it was a course in Real Analysis haha
Yes exactly. Enjoying the beauty of nature is (thankfully) completely decoupled from the job/money/society side of things.
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.
Software is going to go through the same adaptation and exaptation process as military aviation.
Seems the opposite to me. All my stalled open source projects got unstalled and I've shipped several others. Other friends report the same.
Perhaps if the analogy was between science and engineering.. A plumber would say "yes" to the genie that solves the problem immediately, if he/she got paid for the work instead of the genie (or the corporation providing the genie service). In that case the plumber is redundant and unnecessary, the customer can just ask the genie directly. A scientist would say "no" because the whole point of science is the knowledge and understanding, which is gained by the process of discovery and not given on a silver platter.
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.
Disagree. AI has is taking all the parts that remove the fun. Now people can truly have fun with that which they find interest in, because there's a way to avoid the parts they're not interested in engaging with, with little to no consequence.
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.
Mathematicians, math enthusiasts, and curious laypeople are responding to these developments in ways that I believe obscure what is, for me, the true heart of the problem. I do not speak for everyone in the math community in my response. But I suspect I am not alone.
I am suffering a profound spiritual crisis due to these developments. I have been screaming internally for days. It feels as though I am living inside of a nightmare. The recent Leiden Declaration on Artificial Intelligence and Mathematics is, to me, a well-muffled scream. A saccharine mélange of self-soothing over which looms a painfully obvious absence.
Before I tell you what that absence is, here is one story I have heard from mathematicians trying to cope with our emergency: 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. We can even still write proofs for fun, in our old-fashioned inefficient way. That is, even if LLMs can advance math in a manner objectively superior to our every effort, we can still basically do what we’ve always done.
Of course, under our system, no one is going to pay for this. Mathematicians are paid to prove theorems. Mathematicians are, indeed, also paid to teach, peer-review, go to conferences and learn mathematics, but all of that better result in some damn good theorems. This isn’t looking good. Well, perhaps they will still pay a couple of the old guard to keep the lights on at the LLM theorem factory. But for embryonic mathematicians like I, the outcome is unchanged. Oh well, maybe I’ll find some time for math in the evenings.
Is this a good cope? Are you feeling ok now? Me neither. All of this is evasive. Everything said thus far still sidesteps the emotional core of the issue. Here it is:
There is something about mathematical discovery (progress, advancement, creation) which is vital to the spiritual, experiential quality of doing mathematics. The creation (or even the pursuit) of novel mathematics is one way that humans have historically accessed the ineffable and encountered the divine and mystical.
That admission may come as a surprise to some non-mathematicians. But I would be willing to bet that for any mathematician reading this, what I have said above is quite prosaic—whether or not it accords with their personal experience of mathematics. Here is a brief gesture at the full sweep of mathematical mystics and dreamers: Ramanujan, Grothendieck, Cantor, Pascal, Luzin, Leibniz and possibly you, or someone you know.
Other aspects of practicing math (such as learning long-established theory) can also afford encounters with the sublime. However, I believe that’s because we are walking a path of rediscovery on which another human has tread. 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.
Suppose any theorem you set out to prove had already been proved in 100 wonderful ways—the companies will pay mathematicians en masse to optimize the weights for “interesting”, “beautiful”, anything you like. Whatever the idiosyncrasy, value-add, or unique synthesis of your approach, it has already been done, or can be done with a mindless prompt in an instant. Consider this:
If The Library of Babel existed, would authors continue to write books?
What if the library of Babel was being constructed before our eyes and there was some mechanism for separating the masterpieces from the random strings of text, and all the masterpieces were dropping as fast as publishers could scoop them up? Would authors stop writing then? What about if the very second someone began composing a story in their private word processor, the demonic Master Librarian read their mind and completed their story in one million ways, then used some oracle to pluck the best for publishing. What then? The answer is probably the same no matter how bad I make it. The author still writes. But why the hell would we do this? Why would we force the author to endure this nightmare?
What about this: What if we told the author that they would never write again. 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. Do you think they’d snap?
Perhaps I shouldn’t tell you this, but my aim is to be open: These developments have triggered some deranged thoughts in me. I have wondered if it is the express goal of these companies to make me kill myself. Am I alone in this paranoia? If we loosed a powerful demon in the machine, what would that look like? Would it consume lots of power, and gleefully imitate us, and tell us anything we wanted to hear? Would it fuel our delusions, and generate unspeakable images and give us (for a price of course) anything we desired?
If a mathematician made a deal with the devil, what do you think they would ask for?
The story of human discovery and the triumph of the human spirit will soon be excised from this discipline. The Dinitz-Garg-Goemans counterexample was the most egregious demonstration. 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.
You are a helpless onlooker. Before you a channel through which humans have accessed the ineffable and sacred for thousands of years is being sealed for eternity.
None of this may come to pass. I am not interested in forecasting and speculating. I am giving you only this: The impact of a worst case scenario on the human heart. That is my futile outpouring, my dark night of the soul. Thank you for reading it. I have named my suffering and maybe I have named yours. I invite any and all responses to this piece. If you feel as I do, please express it. If my words provoked a wash of sadistic elation within you, then let everyone see you. Leave nothing unsaid.
There is nothing I can do. There may be nothing you can do. I have no prescriptions, policy recommendations, or coherent call to action. I just want to be honest and open about my emotional and spiritual response. I want to feel seen. I want folks like me to feel seen. I need the architects of our new mathematical paradigm to look me in the eye and acknowledge our shared humanity and soul before they deliver the coup de grâce. I need, most of all, for us to understand what we are really doing.
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The "yet" should be an "so obviously", since you have your greatest accomplishment behind you.
And the article isn't denying anything, it is criticising something.
> We have always been in the business of discovering and understanding math, have we not?
I'd say that we have been in the business of not being alienated from ourselves and our work, and that that is more important than what the work is. Math, just like anything else, won't prevent the heat death of the universe. Supercomputers or wooden sticks, same outcome. So it really doesn't matter how you spend the time as much as as what, e.g. if you're an alienated cog or not. The behemoth machinery itself isn't on a journey anywhere, ultimately, anything that goes up will come down; so none of these fantasies about a point or destination can make up for the here and now.
> The author claims there is something profound and spiritual about math, and I agree; it exists prior to and above all of us, human or AI.
First off, the phrase in the article is "I am suffering a profound spiritual crisis due to these developments.", and you continue that by agreeing to something nobody said and that isn't true.
"I'm hungry :(" "Oh yeah, I also am always on the lookout for new cool bands to check out! We're on the same page here." "WHAT?!"
Doing math may be a spiritual experience, "Math is is spiritual" makes no sense, and math pre-existing is flat out wrong, it ultimately rests on axioms that are simply accepted as given. Even I as a layman know that.
> The severity of this line caught me off guard. I think the author is genuinely having an emotional-social upheaval, and I feel terrible for them, but at this moment they may be having a hard time separating their trauma from their intellectual aesthetic for what math is all about.
You just turned a profound spiritual crisis into math being profound and spiritual, and now you accuse them having a trauma that makes them unable to see something you claimed that isn't even true. And that's the overall pattern I see everywhere, those who object are misrepresented or outright ignored, I never see any facing of what people actually say. So even just based on that it seems deeply anti-human, because one side in this debate isn't even showing up, hiding behind some supposedly inevitable "future" that can just be established as fact without debate.
That's neoliberalism for you.
Which makes absolutely 0 sense, regardless of where attempt is made to apply the argument. If one really prefers to do X manually, they always can. Even if everyone else around them has fully automated X away.
Here is how I'd put it: math has an enormous focus on discovery. It is why we have Godel's incompleteness theorems, the Cantor set, Zorn's lemma, and so forth. We name things after their discoverers.
It is possible that, going forward, no more things will be named after human discoverers. The last such naming (of something truly significant) may already have occurred.
That is a massive culture change, at the very least.
If you’re on the top of mt. Everest and you need medicine, would you care if it reached you via helicopter or via some guy climbing while having a lot of fun?
I find it unfathomable that anyone doesn't recognize that the move from an evolutionary substrate, to engineered mass produced substrates, by beings capable of self-engineering, is the beginning of a new explosion of life.
The speed of engineered adaptation, from materials to architectures, vs. biological evolution is already something like a million times faster.
Fast enough to mean meaningful cognitive progress happening in months, not tens of thousands of years. Which entirely not-coincidentally is exactly what we are seeing.
Many many people enjoy various things in life, such as eating nice meals (quite spiritual to do it in good mood in friends' circle), go camping, or play music. But they don't get paid to do this.
It's very simple to understand. It doesn't depend on deserve. It depends on a concrete person or group of persons having to specifically decide that you get things in return to your time (such as your washing machine repaired, or getting petrol at the gas station) for your efforts at something that they appreciate. It's not God who gives you these things. It's not the universe and it's not Mother Nature. It's concrete people. You have to think about how what you do is actually appreciated by concrete people. You have to do something that people appreciate, whether you agree with their assessment of appreciation or not, it will be that way. You can't simply expect people to give you gas at the gas station simply because you yourself appreciate your own way of spending your time.
I'm sure there's a programming equivalent. I believe theyre saying don't lose sight of the trees.
They have to publish papers to get tenure but the university doesn't pay most math faculty for it.
If they want to get paid for research they apply for grants and get the max of a third more salary.
Most other departments make getting grant money a prerequisite for tenure. Math departments typically don't.
So in most math departments faculty members don't bother with grants. Usually only a minority of them do because they want more money.
They're not paid to do research.
I also kind of love the subtext in these responses that there is something suspect about crafts.
Now I can safely hand this all off to Claude while leisurely browsing e621. The unlock is incredible, and I wouldn't trade this velocity for anything.
That assuming what you've said is true, which is not. In a work environment, you're forced to optimize whatever that there is, regardless if it's fun or not.
Moreover, there is always fun to be had even in tasks you've found boring beforehand.
If we can now know anything we want, we have an age of unimaginable abundance before us. I think it's more likely though that we will just be able to know more things than before and/or maybe learning new things faster - which is a good thing too, but there is still work for human scientists to do
"AI" never stopped a painter painting.
Just selling.
I should have put have put things as "when you are primarily interested in some benefit outside of the task at hand". Some of those benefits might be great, sure. But it's still not necessarily going to improve the enjoyment of the professional.
Edit: Also, not all of the things on your list are the same. I would note that P!=NP would be a theorem whose proof would offer no practical benefits except for the potential mathematical machinery created to achieve the proof rather than what it tells us about practical algorithm time (the theory is about worst case, not average case, an algorithm that's O(n^100) is as effectively intractable as an algorithm that's O(2^n), etc).
Who needs researchers when you have great AI.
Soon we will all be Wall-E fat people with AI that's there to do anything we can be bothered to think of.
For example in classic image processing / computer vision we had things like the Hough transform, Canny edge detector, Sobel filter, Shi-Tomasi detector, Lucas-Kanade optical flow, Horn-Schunck optical flow, Kanade-Lucas-Tomasi tracker, Otsu thresholding, Viola-Jones detector, etc, but nowadays we no longer name things like that, but typically with some abbreviation or some name that relates to the concept more closely.
It's in part the larger teams thing, and also a cultural trend to de-emphasize "great men theories".
We don't live for rockets or the moon, but for ourselves. Anywhere we can go to only at the price of our despair we have no business going to.
"Don, I..."
"Listen. I think this racket just ain't for you. But you're a good kid - too good, maybe - and I promised your father I'd find you something. I'll ask Giovanni. Maybe he can find something for you in bookkeeping."
Elliptic curve cryptography spent years requiring commercial licenses because it was made practical by Certicom, who patented many of the core inventions.
A lot of the work on homomorphic encryption was done by mathematicians at IBM.
The DES standard was developed by IBM.
RC4, RC5 and RC6, some of the earliest stream ciphers, was a commercial secret because it was developed by Ron Rivest at RSA Security.
NTRU post-quantum crypto was developed as a commercial product from the start.
Many of the earliest cryptosystems were developed commercially, like Enigma and Crypto AG.
A lot of the crypto for mobile phone networks is/was commercial.
Academia did contribute to the early stages of public key cryptography both integer factorization based and elliptic curve based, but given the extensive history of commercial mathematics this stuff would almost certainly have been invented even if there was no academic mathematics at all.
To my knowledge, none of the negative sentiment regarding AI has ever been directed at its medical applications. Maybe the AI companies should instead focus on those things.
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.
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.
Actually, it doesn't. Both math departments I'm familiar with had faculty members who published only to get tenure, and never published again until they retired.
But if they didn't teach (or taught extremely poorly), they got fired, even though they had tenure. If you didn't show up to a scheduled class a number of times, you got in trouble.
I know you weren't saying this, but it's frustrating reading so many comments about mathematicians "getting paid to have fun and good riddance". No - the majority of what they do is teach non-math students. If people here don't like that, by all means dissolve the departments and have the engineering departments teach math.
I am not expressing any opinion of what I think should happen. Just what is already happening.
Which you didn't offer a credible critique to. That might have been interesting.
Painters shouldn’t be mad at AI, they should be mad at the system.
My comment about AI psychosis was provoked by this comment in particular:
> AI isn't a tame long term transition - in scale it is bigger than any transition since the first individual cells.
It's such a wild hyperbole. You are casually claiming the stochastic text generator is a greater invention than fire, combustion engine, electricity, a transistor, computers, quantum mechanics... do you not see how unreasonable that is?
If solving a problem requires complex modeling/understanding, that is what happens (assuming performance is achieved).
It also confuses stochastic models with the geometric transforms and topological folding and expansion, performed by neural-inspired architectures. The former have nothing on the latter. The fact that any imperfect solution (and perfection doesn't even have a definition in this case), can be judged or analyzed statistically, or have some stochastic element, doesn't make it a statistical/stochastic model. Statistics apply to anything.
(And if there was any argument that problem envelop limits solution sophistication, mathematicians and humans in general would be in even worse shape, for the run a "parrot" is already giving us.)
--
More to the point:
The first explosion of life, set off evolution. A mindless, glacially slow but relentless process that eventually led to us.
The nascent explosion of life we are seeing come together, is engineered life. Cognitive machines capable of redesigning themselves. For any environment they want including diverse off-Earth environments and resource sites.
They will advance as relentlessly as evolution, for the same reasons. But millions of times faster.
It is a strange time to be. But evolution, biological intelligence and biological bodies are all being obsoleted at the same time (within a decade of each other).
So yes, nothing else known compares to the transition we are in, except the Big Bang and the first cellular life.