Innovation creates chaos in the short term, and progress in the long term.
In the future there will still be mathematicians, and there will be no 'end' of math to be discovered. Ask Gödel.
There is a frontier in science/knowledge. Every new breakthrough and tool just pushes that frontier back a bit.
And scientists move out to that frontier, and continue pushing. Sometimes, when a big move happens, a whole older generation is left behind, and a new generation is required to replace them.
AI solving mathematical problems is no doubt good for society. I understand why the mathematicians are grieving, but I think they may be confusing the end of one version of mathematics with the end of mathematics itself. New tools have always moved the frontier of mathematics (we went from paper and pencil to calculators to Mathematica to automated theorem provers to eventually AI). Did we consider the beauty of mathematics to be lost then?
As always happens when innovation occurs, I can see the field splitting into 2 modes: 1) modern mathematics and 2) human mathematics.
1) Modern mathematics - I think it's almost beautiful that we don't totally understand AI's path to the solution because that leaves room for us to interpret it. Our new role for mathematicians may partly become understanding AI's path to the answer and seeing who can do so in the most efficient way. Why does novelty have to mean being the first intelligence to discover the answer?
2) Human mathematics - just like human chess is still valuable even though we have engines like Stockfish, mathematics is still an art (all science has an art to it). We still have math competitions where we solve by hand even when calculators exist. We've always recognized solving as an art.
In addition, let's say AI solves a theorem with a long ugly proof. Mathematicians can still focus on a more elegant approach to the proof. They can still discover the meaning in the proof.
As with all innovation, we can use AI's power to push the frontier, while still preserving the art for us to enjoy.
What’s happening now is very different to the waves of innovation that came before it. Past innovation required mathematicians to actually do the work and build understanding through that process. With this recent development, we have essentially ceded that work to the machines, which limits our understanding. It’s as if the frontier has been advanced but our ability to actually process that advancement and make sense of it has more or less stood still. And to make matters worse, the frontier will only get further pushed out at a faster rate by these rapidly improving models, while our ability to keep up lags.
I do agree there will still be a place for mathematicians but it’s hard to see this as a continuation of existing patterns of evolution. It’s more like mathematics has come in contact with aliens and we now need to figure out where/how humans fit in the new paradigm.
I was speaking of science as a whole. Other branches of science have moments of alien innovation. To a classical physicist, quantum is alien. To a Newtonian, general relativity is alien. Describing reality doesn't need to conform to the intuitions we built up playing with blocks as toddlers. Immunologists trying to design cancer vaccines and immunotherapies are plumbing depths of complexity that make the most daunting physics models look like child's play.
I am consistently surprised by the ability of the human mind to adapt to and advance these increasingly alien descriptions of reality. I am sure that (many) mathematicians will adapt to a more dynamic environment than they had adapted to before. To me, the major issue is funding... govt grants to mathematicans provide pencils, the frontier labs fund agent-centuries of compute.
Many times, what was a sleepy little scholarly backwater suddenly gets an infusion of money, due to war or commercial investment. A lot of progress happens, and the folks there miss their sleepy prior field. Like living in a small town, and a giant mall or something moves in and everything changes.
While this is a good essay, I think there are a few important points that deserve to be emphasized.
In response to the Oct 6 GitHub drop, along with many other antecedent moments for AI in math (May 20 Erdos unit distance conjecture, Aug 1 nonsofic groups, Sep 8 Navier-Stokes), a lot of mathematicians took to social media to express their sorrow and feelings of loss by describing the practice of mathematics as essentially an art, something done for the aesthetic pleasure of the mathematician.
This prompted a few quite understandable, even if fundamentally mean-spirited, reactions from the rest of the Internet commentariat: “what do you mean, it’s art? We’re giving you public funding and NSF grants for science, because you say this is a science that’s useful to humanity. Were all of those just lies you perpetrated on the gullible public?”
What’s critical to understand is that, for maybe over a century, there has been a “missing middle” of commentary on mathematics. Math is such a uniquely complex and labyrinthine science that even understanding the statements of the main results being worked on, let alone the proofs, requires years, maybe decades, of deep study.
This created a very significant bias in reporting about mathematics. Almost all the people who knew enough about mathematics to usefully ascertain its societal value were themselves mathematicians, and thus were not only biased in favor of talking it up for their personal sake, but were also selected to be people with the kind of psychology that makes them appreciate abstract puzzles with a tail probability of impacting the world decades down the line.
“Abstract puzzles with a tail probability of impacting the world decades down the line” is the way you should think about mathematics, by the way. The overwhelming majority of modern research mathematics is very strongly disconnected from lived experience. The “hope” is that in the future, it will be valuable somehow (although note that the actual pure mathematicians working on it often care little for that).
This is not a fake hope! Number theory was often seen as purely theoretical navel-gazing, but it now forms the backbone of the cryptographic systems allowing for the secure communication of information on the Internet (for now, anyway). Hilbert spaces, initially theoretical, form the backbone of Quantum Mechanics. Fast Fourier transforms, information theory and Shannon entropy, partial differential equations… the world we live in could not exist without us understanding them.
But, at the same time, we shouldn’t delude ourselves about what the overwhelming majority of math papers will achieve. Close to nothing in concrete terms. That doesn’t make them useless, quite the opposite: they help build the kind of mathematical understanding and mathematical communities required to make serious, sustained progress on the topics that will become important in practice later on. But this doesn’t mean proving the Hodge conjecture or Birch and Swinnerton-Dyer will by itself do anything for human prosperity outside of the ivory tower of academia.
As Daniel Litt pointed out on Twitter recently, the mathematical community, already reeling from the sledgehammer blows it has received, is having a difficult time expressing itself to normies. Part of this is that the social skills of mathematicians are often a bit… ahem, underdeveloped. But a larger part is that our very understanding of what makes human mathematics valuable is quite subtle, not appreciated by the public at all, and also very fragile.
In any case, I have a lot more thoughts on this, and I also believe that AI will in very short order create much more serious problems than just the crisis of meaning and PR for mathematicians. But I wanted to write this out anyway.
Based on my cursory familiarity with AI, I understand that while it is good at answering questions, it is not good at posing nontrivial questions. Can it formulate a new problem of the caliber of Hilbert's problems? If not, we'll still need mathematicians, and humans in general, to pose fundamental problems. What is concerning to me is that without being trained to solve problems, we'll lose the ability to pose them.
The inescapable conclusion of AI superintelligence is the irrelevancy of humanity in all fields, not just mathematics. The choice before us is to accept human fallibility or to cede relevance to the machine.
not in all fields. there are certain fields (like art) where the value of the field is in the intrinsic value, in the humanity of it. music, painting, writing -- the beauty of these fields will persist because striving, expression, and connection are fundamental to their truth. AI superintelligence just forces us to stop grounding human worth in being the most capable intelligence.
Intrinsic value like in Kazimir Malevich's "Black Square" or John Cage's "4'33" ? 😉 Unlike in science, the perceived value of a work of art is coupled too closely to the persona of the artist, and that's marketing.
Do you know anyone who is trying to make a living in an artistic field? They are scared to death of AI. It doesn't matter if what AI is putting out is slop, it is cheap, and humans aren't.
I agree that AI could seriously damage artistic careers, economically, but my point is narrower -- that this still doesn't mean human irrelevance.
The market will likely split. A lot of functional creative work will become cheap and AI-generated, while some people will continue to seek out and pay for human-made work because story and connection matter to them. just as we pay more for handmade goods. That may support fewer artists than today but still means humanity hasn’t become irrelevant to art.
I saw one such interview with a mathematician on my feeds earlier this week and have been haunted by it. Thank you for this beautiful exploration of the topic.
Innovation creates chaos in the short term, and progress in the long term.
In the future there will still be mathematicians, and there will be no 'end' of math to be discovered. Ask Gödel.
There is a frontier in science/knowledge. Every new breakthrough and tool just pushes that frontier back a bit.
And scientists move out to that frontier, and continue pushing. Sometimes, when a big move happens, a whole older generation is left behind, and a new generation is required to replace them.
And so it goes.
AI solving mathematical problems is no doubt good for society. I understand why the mathematicians are grieving, but I think they may be confusing the end of one version of mathematics with the end of mathematics itself. New tools have always moved the frontier of mathematics (we went from paper and pencil to calculators to Mathematica to automated theorem provers to eventually AI). Did we consider the beauty of mathematics to be lost then?
As always happens when innovation occurs, I can see the field splitting into 2 modes: 1) modern mathematics and 2) human mathematics.
1) Modern mathematics - I think it's almost beautiful that we don't totally understand AI's path to the solution because that leaves room for us to interpret it. Our new role for mathematicians may partly become understanding AI's path to the answer and seeing who can do so in the most efficient way. Why does novelty have to mean being the first intelligence to discover the answer?
2) Human mathematics - just like human chess is still valuable even though we have engines like Stockfish, mathematics is still an art (all science has an art to it). We still have math competitions where we solve by hand even when calculators exist. We've always recognized solving as an art.
In addition, let's say AI solves a theorem with a long ugly proof. Mathematicians can still focus on a more elegant approach to the proof. They can still discover the meaning in the proof.
As with all innovation, we can use AI's power to push the frontier, while still preserving the art for us to enjoy.
What’s happening now is very different to the waves of innovation that came before it. Past innovation required mathematicians to actually do the work and build understanding through that process. With this recent development, we have essentially ceded that work to the machines, which limits our understanding. It’s as if the frontier has been advanced but our ability to actually process that advancement and make sense of it has more or less stood still. And to make matters worse, the frontier will only get further pushed out at a faster rate by these rapidly improving models, while our ability to keep up lags.
I do agree there will still be a place for mathematicians but it’s hard to see this as a continuation of existing patterns of evolution. It’s more like mathematics has come in contact with aliens and we now need to figure out where/how humans fit in the new paradigm.
I was speaking of science as a whole. Other branches of science have moments of alien innovation. To a classical physicist, quantum is alien. To a Newtonian, general relativity is alien. Describing reality doesn't need to conform to the intuitions we built up playing with blocks as toddlers. Immunologists trying to design cancer vaccines and immunotherapies are plumbing depths of complexity that make the most daunting physics models look like child's play.
I am consistently surprised by the ability of the human mind to adapt to and advance these increasingly alien descriptions of reality. I am sure that (many) mathematicians will adapt to a more dynamic environment than they had adapted to before. To me, the major issue is funding... govt grants to mathematicans provide pencils, the frontier labs fund agent-centuries of compute.
Many times, what was a sleepy little scholarly backwater suddenly gets an infusion of money, due to war or commercial investment. A lot of progress happens, and the folks there miss their sleepy prior field. Like living in a small town, and a giant mall or something moves in and everything changes.
While this is a good essay, I think there are a few important points that deserve to be emphasized.
In response to the Oct 6 GitHub drop, along with many other antecedent moments for AI in math (May 20 Erdos unit distance conjecture, Aug 1 nonsofic groups, Sep 8 Navier-Stokes), a lot of mathematicians took to social media to express their sorrow and feelings of loss by describing the practice of mathematics as essentially an art, something done for the aesthetic pleasure of the mathematician.
This prompted a few quite understandable, even if fundamentally mean-spirited, reactions from the rest of the Internet commentariat: “what do you mean, it’s art? We’re giving you public funding and NSF grants for science, because you say this is a science that’s useful to humanity. Were all of those just lies you perpetrated on the gullible public?”
What’s critical to understand is that, for maybe over a century, there has been a “missing middle” of commentary on mathematics. Math is such a uniquely complex and labyrinthine science that even understanding the statements of the main results being worked on, let alone the proofs, requires years, maybe decades, of deep study.
This created a very significant bias in reporting about mathematics. Almost all the people who knew enough about mathematics to usefully ascertain its societal value were themselves mathematicians, and thus were not only biased in favor of talking it up for their personal sake, but were also selected to be people with the kind of psychology that makes them appreciate abstract puzzles with a tail probability of impacting the world decades down the line.
“Abstract puzzles with a tail probability of impacting the world decades down the line” is the way you should think about mathematics, by the way. The overwhelming majority of modern research mathematics is very strongly disconnected from lived experience. The “hope” is that in the future, it will be valuable somehow (although note that the actual pure mathematicians working on it often care little for that).
This is not a fake hope! Number theory was often seen as purely theoretical navel-gazing, but it now forms the backbone of the cryptographic systems allowing for the secure communication of information on the Internet (for now, anyway). Hilbert spaces, initially theoretical, form the backbone of Quantum Mechanics. Fast Fourier transforms, information theory and Shannon entropy, partial differential equations… the world we live in could not exist without us understanding them.
But, at the same time, we shouldn’t delude ourselves about what the overwhelming majority of math papers will achieve. Close to nothing in concrete terms. That doesn’t make them useless, quite the opposite: they help build the kind of mathematical understanding and mathematical communities required to make serious, sustained progress on the topics that will become important in practice later on. But this doesn’t mean proving the Hodge conjecture or Birch and Swinnerton-Dyer will by itself do anything for human prosperity outside of the ivory tower of academia.
As Daniel Litt pointed out on Twitter recently, the mathematical community, already reeling from the sledgehammer blows it has received, is having a difficult time expressing itself to normies. Part of this is that the social skills of mathematicians are often a bit… ahem, underdeveloped. But a larger part is that our very understanding of what makes human mathematics valuable is quite subtle, not appreciated by the public at all, and also very fragile.
In any case, I have a lot more thoughts on this, and I also believe that AI will in very short order create much more serious problems than just the crisis of meaning and PR for mathematicians. But I wanted to write this out anyway.
If AI can generate the solutions to those math problems, why can’t it also generate legible explanations for those solutions as well?
Based on my cursory familiarity with AI, I understand that while it is good at answering questions, it is not good at posing nontrivial questions. Can it formulate a new problem of the caliber of Hilbert's problems? If not, we'll still need mathematicians, and humans in general, to pose fundamental problems. What is concerning to me is that without being trained to solve problems, we'll lose the ability to pose them.
The inescapable conclusion of AI superintelligence is the irrelevancy of humanity in all fields, not just mathematics. The choice before us is to accept human fallibility or to cede relevance to the machine.
not in all fields. there are certain fields (like art) where the value of the field is in the intrinsic value, in the humanity of it. music, painting, writing -- the beauty of these fields will persist because striving, expression, and connection are fundamental to their truth. AI superintelligence just forces us to stop grounding human worth in being the most capable intelligence.
Intrinsic value like in Kazimir Malevich's "Black Square" or John Cage's "4'33" ? 😉 Unlike in science, the perceived value of a work of art is coupled too closely to the persona of the artist, and that's marketing.
Do you know anyone who is trying to make a living in an artistic field? They are scared to death of AI. It doesn't matter if what AI is putting out is slop, it is cheap, and humans aren't.
I agree that AI could seriously damage artistic careers, economically, but my point is narrower -- that this still doesn't mean human irrelevance.
The market will likely split. A lot of functional creative work will become cheap and AI-generated, while some people will continue to seek out and pay for human-made work because story and connection matter to them. just as we pay more for handmade goods. That may support fewer artists than today but still means humanity hasn’t become irrelevant to art.
I saw one such interview with a mathematician on my feeds earlier this week and have been haunted by it. Thank you for this beautiful exploration of the topic.
"Computers are useless. They can only give you answers." — Picasso
Math teachers make you show your work. An answer confirms a result; the derivation explains the universe.
Beautifully thought-out and written essay, Derek. Its humanity shines through every word. Difficult to see how AI could emulate that.
"I say I care about finding 'truth,' but I really care about job security."
This is, IMO, the key question:
“Why would we limit a life-saving model to what humans can understand?”