Incredibly thoughtful. This essay gives that very rare sense of being well reasoned, gods at forest and trees, and sitting atop a shit ton of domain expertise.
Someday, there might be mathematics designed for AI. Mathematics that only a tiny fraction of humans can understand, but a different kind of mathematics might emerge. I wonder if we would still call it mathematics.
What would happen if a non-human layer of mathematics emerged on top of human mathematics? In this article, the distinction between Mathlib and Mathslop might be a precursor to that.
If models advance enough in the future, and new definitions, compressions, and representational forms that are convenient for AI-to-AI communication emerge, what would happen then? Would mathematics split into Human-facing and Machine-facing branches?
This kills me, it is correct, but misses the forest for the trees. Yes, mathematics is a discipline of understanding, but an insular one. The entire field is about trying to understand, but the discipline does not try to be understood. No, that is "your job, not theirs" and that is why this discipline is struggling, struggling in a culture that can barely communicate without emotional morons destroying any constructive communications.
+-- 2mo before by sdfrew
| 4 points / 1 comments
| The Fall of the Theorem Economy
| https://news.ycombinator.com/item?id=47862472
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+-- 2mo before by fuglede_
| 3 points / 1 comments
| The Fall of the Theorem Economy
| https://news.ycombinator.com/item?id=47891494
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+-- 2mo before by mathgenius
| 2 points / 0 comments
| The Fall of the Theorem Economy
| https://news.ycombinator.com/item?id=47909751
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+-- 2mo before by delis-thumbs-7e
| 15 points / 4 comments
| David Bessis on AI destroying mathematics
| https://news.ycombinator.com/item?id=47985962
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+-- 1mo before by magoghm
| 4 points / 0 comments
| The Fall of the Theorem Economy
| https://news.ycombinator.com/item?id=48084737
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+-- 1mo before by cubefox
| 2 points / 0 comments
| The Fall of the Theorem Economy
| https://news.ycombinator.com/item?id=48089716
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+-- 1mo before by cubefox
| 5 points / 0 comments
| The Fall of the Theorem Economy
| https://news.ycombinator.com/item?id=48152469
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+-- 1mo before by tmp10423288442
| 4 points / 1 comments
| The Fall of the Theorem Economy
| https://news.ycombinator.com/item?id=48214866
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`-- this submission by varjag
58 points / 7 comments
The Fall of the Theorem Economy
https://news.ycombinator.com/item?id=48758048
Are we going to see less publicly shared science? With private actors or governments restricting access to AI resources to a few scientists and keeping new knowledge to themselves.
Advancing science in the open was the best strategy when there was real advantage to share the load with every brain on the planet willing to give a try at science, but if a computer can match or surpass the collective output of the entire human scientific community the equation will change.
The core thesis seems to be that the "real value" is not in producing/proving theorems, but in understanding them. AI might be good at producing and proving theorems, but it fails utterly at getting humans to understand them. Even worse, humans have no interest in working on theorems that have already been proven, so we end up with theorems that will never be understood by humans.
I can understand why this is a major concern for mathematicians. They got into their field because they love the beauty of mathematics, and the intellectual satisfaction of understanding non-obvious insights. But to put it crudely, this sounds like a you problem. As someone who isn't a mathematician, the main value I get out of math is its practical applications in science and technology. And their practical applications in human life. I have zero understanding of the math behind cryptography, but I still deeply appreciate the practical value they have provided humanity.
If AI systems start churning out accurate theorem-proofs, and we are able to use those theorems to build things that improve human quality of life, it doesn't bother me one bit that those theorems have not been understood by humans. If this offends your aesthetics, you are certainly entitled to your opinion and your preferences, but that does not make it a societal problem
They got into their field because they love the beauty of mathematics… As someone who isn't a mathematician, the main value I get out of math is its practical applications in science and technology
I have some sad news for you. 99% of the work mathematicians do has no immediate application, nor even an obvious path toward application in the near future. You mentioned cryptography, so for an example consider number theory: no apparent practical applications, going back thousands of years to the time of Euclid and earlier.
It’s been religion, philosophy, and recreation that have provided the motivations to study mathematics all these years, not applications. Applications have almost always followed long after the development of the pure mathematical theory. For number theory, that was the development of cryptography during WW2, millennia after the ancients laid those foundations.
Most unfortunately, it’s the truth value and the understanding which drive applications of mathematics, not the proof work itself. If the AI revolution decapitates the institution of mathematics which produces the understanding, and is unable to replace it, then the applications will cease as well.
I thought it was very interesting, but maybe also incredibly naive politically ? it's like he's re-discovering alienation under capitalism.
A wood-worker could do the same argument, there's the "official" wood-working word of perfect joinery and beautifully finished tables one can buy, but behind it there's the "secret" messy human element, the art, the craft, the mistakes and hard-ships, the elevation of human skills and imagination, the creation of whole new types of wood-working inventions and techniques, the perpetuation of millenia-old traditions, the teaching, the joy of selling to a happy customer, etc.
But now comes techo-capitalism, division of labor, you cut that piece a that piece over and over, you operate that machine, you won't even see the finished table, fuck your human element, we want that profit !
You start with instincts that are more easily ascribed to ethically-neutral or ethically-positive reasoning, and then turn them into a spurious criticism of capitalism. Case in point: in the USSR, the means of producing chairs were 100% state-controlled and not motivated by profit, but the country operated soulless production lines too.
The motivation behind all this is less "haha I want profit" and more "billions of people need chairs, approximately none of them care about the craftsmanship, so it's in our best interest to make furniture in the most resource- and labor-efficient way possible". Even if the state subsidizes the production of handcrafted chairs, the population is the poorer for it on a resource allocation basis, because we now need a million artisanal chair-makers instead of a bunch of factories.
When math is so divorced from science and engineering that there's no conceivable way that it will ever be applied in the real world then it is just a complex puzzle game that a tiny group of people play. It doesn't really matter much. If the 200,000 line Mathslop proof has no real world application and it doesn't help the puzzle solvers then it is double useless.
This is also my stance. The fact that large numbers of people spend large amounts of publicly-funded time exploring what are essentially abstract puzzles is bizarre and not that different from, like, cloistered religious devotees who are supported in spending their time studying scripture and are considered to be the 'source' from which flows a certain kind of universal truth.
Not that it is wrong for them to be doing this---we do want a society where people get to devote their life to what interests them---but it is bizarre because of the framing. For some reason it is ambiently understood in our society that this work is of incontrovertible value, when in fact it is largely not. And the value-producing parts of the work, the parts that end up having applications to other fields, largely run contrary to the actual daily goals of the cloistered devotees: it is mostly the intuition and pedagogy and the compactification and refactoring of knowledge that have value at this point, not the production of esoteric theorems, yet that is expressly not rewarded in the incentive structures.
That latter point is more due to the sorry state of academic incentives in general than to a particular failing of mathematics, though. Were I somehow given the ability to restructure things by fiat I would immediately create journals which publish only useful articles that refactor knowledge, communicate intuition, better explain things, argue for structural improvements to notation and terminology, etc, and this would immediately create an incentive to do that kind of work for working researchers to do work which aligns with the actually-useful output of their fields. I suspect most fields could use something like this. New knowledge is just not that valuable if it is all dumped into a giant pile and unprocessed, and I have seen firsthand a bunch examples where entire subdisciplines are hamstrung in their actual application-heavy work because they don't have easy access to basic tools that are hidden behind hard-to-learn theory.
A crucial caveat: basic research investment runs on the same logic as venture capital investment. We know that most mathematical efforts will be worthless. Our experience has lead us to expect that a very small number of such efforts -- some of them very far removed from applications -- will have payoffs so large that they change the shape of our society.
_We do not know in advance which efforts are going to pay off_. Abstract efforts in topology put us on the road to nuclear energy. Silly number puzzles enabled internet commerce. Non-euclidean geometry gave us synchronized universal GPS.
We should not let our inability to conceive of applications of weird abstract stuff prevent us from making these investments. If our ancestors had fallen in to that trap, we'd be far poorer as a society.
What we can do is ask that people trying new stuff attempt to fail quickly. And that's basically where we are with academia today. Most people who do mathematical work will not have a career in math. They try something new, work for a little while on it, and go do something else when the results turn out to be of modest interest. This leaves behind a messy undigested literature, which is unfortunate. But maybe AI can help us sift that for treasures we missed.
> to come up with a conceptual framework where it became easy to express
Feels a lot like building software from bottom - once you get the building blocks defined right, the action, or the program, are trivial to express. When doing it from the top-down, you write the program using the building blocks you haven't defined yet, and you might end up with overly specific building blocks, needing other blocks for expressing different behaviors.
When you do the bottom-up building blocks right, new behavior is easy to express with them. Essentially, you are building up the language to reach the problem. Or making a DSL, whatever definition you like best.
Greg Egan's description of how mathematics evolves into "truth mining" in his novel Diaspora is seeming more and more prescient. It essentially describes what mathematics would look like after formalization records all theorems discovered so far in a huge, collective database and proof assistants can instantly work out the details of a given proof. What remains of mathematics? According to Egan, visualization, intuition, and insight.
One of the most fruitful approaches in mathematics is to flip back and forth between geometric and algebraic views of a problem. I think this works so well because these are actually handled by two different parts of the brain on a physical level; spatial reasoning is separate from language processing. Cytoarchitecture shows these regions have different "textures;" the local details of the way neurons are wired together are simply different in these different regions of the brain, in the same way a CNN and a transformer have different topologies. Thus, by flipping problems from geometry to algebra and vice versa, we're able to bring an entirely different cognitive style to bear on a problem. For example, the proof of Monge's Theorem by moving to 3D and visualizing not three circles, but three spheres sitting on a table with a book on top of them and then pointing out that the intersection of two planes is a line. What is pages of unintuitive symbol pushing turns into something a child can understand. Going the other way, things like the angle addition formulas or the quadratic formula, which are quite hard to prove geometrically, become quite simple if you use a little algebra.
Current-gen LLMs are still relatively weak at visual reasoning; see the Vision Language Models are Blind paper, for example, or the ARC-AGI benchmark. So that's one way humans can stay ahead of the agents, at least for now.
Spot on! Love Diaspora. This is honestly such a gem of a comment. To some extent, if the AI ever gets "so far ahead" of humans, the most productive aspect will be the frontier visible to humans. We're focused on translating mathematics to lean at the moment, but it'll be as important to translate it to humanese - to the human language of structure, number, geometry. I also completely agree with LLMs being essentially blind to visual reasoning. They really struggle reasoning with Floer Heegard diagrams for example.
I'd be very surprised if there aren't huge areas of undiscovered math that can't be explained with either geometric or algebraic views.
Math is entirely subjective. "Proof" essentially means "Other educated practitioners have the same experience when trying to understand this."
The logical steps that proofs are built on all have that common foundation. Our concept of logic based on our subjective experience of "truth." We've built machines that reproduce our subjective processes mechanically, but there is no sense in which this idea of "true" is truly objective. It happens to be computationally convenient, and it has some relationship to experience, but that doesn't make it an independent reality that all possible observers, human and otherwise, would agree on.
We're really just mapping our own minds through our own experiences.
Animal brains can't abstract like (some of) our brains can. What are the odds our brains are limitless and don't have some similarly crippling limitations from a couple of levels up?
One of the tells for ASI is that it will start reasoning at those levels, using cognitive techniques that are completely incomprehensible - not just because of brute volume, but because our brains won't have the wiring to get a foothold on them.
Some of the products will be reducible to human cognition, in a distorted and simplified form, but many won't.
So - I disagree with Egan. I don't think there's going to be a universal proof library, and even if there were we'd only ever get the Cliff Notes version.
> Our concept of logic based on our subjective experience of "truth."
The idea that we experience objectivity subjectively, thus there is no objectivity, seems like an ultimately nonsensical and self-defeating sleight of hand to me.
You mean they can't come up with mathematical abstractions? That's right of course, but they seem perfectly able to "abstract" in the sense of drawing general rules from specific observations. For example, I noticed recently how my cat friend was happy to exit and enter the house from either of two doors and a window as the opportunity presented itself (i.e. depending on which one was open at the time he wanted to get in or out). E.g. I just happened to open the window and he hopped onto the ledge and out into the mystery of the night, entirely unaided by human hands or voices.
That's an abstraction: one door is like another door and they're both like a window. Both things lead in and out of the house and they can be open or closed at different times. If one is closed another may be open. Something like that, obviously I have no idea what concepts, exactly, he has in his head. And btw, "in", "out", "house", etc are also abstractions, so a house in the UK is like a house in France, or like one in Italy, or one in Greece: that's where the human things are; I guess.
I think it's important to understand this about animal intelligence, that it is very much capable of broad generalisation and abstract thought. Absolutely not to the same degree as humans because we got language and that probably comes with a whole other level of ability for abstraction and reasoning, but I don't believe it's possible for an animal to survive in the physical world if it can't go from the concrete to the abstract, the specific to the general, to some extent.
You mention ASI so I'm comfortable making this comment which is about AI, without feeling I'm hijacking your conversation. I think one only begins to understand how powerful the animal brain is, and how overlooked (in AI discourse) its ability to form such broad, useful abstractions that allow animals to navigate the physical world autonomously while setting and pursuing their own goals, when one tries to reproduce the same behaviour artificially, in computers.
Like you say, yes, maybe when we stop understanding what our computers produce, it will be because they moved a level up from where we are, as ASI. Maybe there is such a level; maybe there isn't. For the time being, there is no artificial system that can spontaneously move in the physical world as easily and effortlessly as my cat friend can (and he's a graceful animal; really, much closer to an African wildcat than its domestic descendant). If you want an AI system to understand the difference between "in" and "out" and that there are doors and windows that connect the two you have to somehow explicitly feed that information into it, either by hand-coding (e.g. the PDDL models used in Planning and Scheduling) or by training on carefully chosen examples of those concepts (as in machine learning), or at least some kind of objective that will lead the AI to develop them, or develop similar concepts (as in Reinforcement Learning). We're still very far from the capabilities of the animal brain.
Also, may I quibble about the fact that our brain, too, is an animal brain?
The ease, extent, and scale at which math integrated with computing along its development makes me wonder if the two fields will effectively enjoin, academically (math and computer programming).
Maybe we will look back on today's math as a kind of arcane, pre-syntactical set of structures that required speakers of the language on both ends interpreting it to make good use of it. No validation or compilation, it can't be applied, just a total wild west - scribbles on a whiteboard and another mathematician making sense of it.
From reading this, it looks like the projected future is mathematicians working more applied to a domain, and the basic research in the academia being severely impacted by the AI companies - who have the money to hire the senior mathematicians from the academia. I guess if some of the biggest universities could come up with their own AI powered programs there could be something to “answer” in a more accessible knowledge, but I don’t see how to properly keep the students motivated to ensure the field keeps producing new people.
People think mathematics is about proving theorems.
I think that's just an accident of history.
When we write software, we very seldom write proofs that our algorithms are correct. We just write tests, and we also run the algorithms and when they fail we know we have a bug and then we proceed to debug, fix, and add new tests (if we are disciplined, but most of us are). In time, by usage and testing, we gain confidence that our battle tested software is correct, mostly. And we tell people that we will never be 100% confident that any software is bug free. But that's a slight lie: if we wanted such confidence, we would start using provers, and create bug-free software. That possibility exists, but it's just extraordinarily expensive.
Well, in math that's the only possibility, and we use it. And it is indeed extraordinarily expensive, but it's also the cheapest among the alternatives. The alternatives are 2: be rigorous and do these proofs, or be sloppy and allow bugs to creep in, and allow an entire school of math to collapse like the Italian school of algebraic geometry [1].
There is one more alternative. If a particular math theorem has some applicability, then you write a program and use it in real life. In time you eliminate the bugs as much as you can, and you get to the steady state of "virtually bug free". At that point you don't have a solid proof that the theorem is correct, but in general you don't really care. Because you feel that a formal proof is just a thing one would pursue for getting academic satisfaction only.
Your example of the Italian school is well taken, although most of their results were later proven to be correct in the right setting. Severi's example is particularly egregious and I think a major reason this became a thing is Severi's refusal to course-correct and accept that some of the results were not correct. It has echoes of Mochizuki, and I fear, once you dig deeper, some issues around the initial declaration of "We've proven the Classification of Finite Simple Groups". There were many genuine gaps, and a lot of lore taken for granted. The sociology around how this happened is interesting - rushing to announce that it was done was the major mistake, it took away almost all incentive to actually write up the proofs and take them through proper peer-review. Genuine mathematical work was falsely reduced to "write up".
It's interesting that mathematics, which is mostly recreational (I received profound disdain at the math department for asking about applications!) has such rigorous standards, but software, which entire civilizations now run on, does not.
I thought this comment would go in a slightly different direction: the body of work that is mathematics has plenty of “bugs”; proofs with mistakes or other human errors. Yet we take the body to be correct (we believe it “works”) in aggregate, partly because the intuition of mathematicians tells us that these bugs are solvable and don’t bring down the whole. Of course the less intuitive/more surprising/more central the statement, the stricter the standard for proof and more eyes that have walked through it.
One thing I've realized after quite a long life of learning and contemplation is that: mathematics and software are essentially the same thing. Add to that that it might be the case that Physics is the same thing too. We'll see on that one, but there are signs...
> When we write software, we very seldom write proofs that our algorithms are correct. We just write tests, and we also run the algorithms and when they fail we know we have a bug and then we proceed to debug, fix, and add new tests
But this is due to a "failure" of programming language progress. We've had formal languages for a long time (see Ada and SPARK) and we've simply failed to use them for most scenarios, instead regressing to imperative manually memory managed languages like C and then deriving from that branch.
> Some prophesy that mathematics will eventually resemble Chess, a sport that a few eccentrics practice with passion and the general public can safely ignore.
How is it not already this? Jon von Neumann was already calling most math this many decades ago. Pull up any random arxiv math paper and it’s abstract nonsense with no applications to the real world.
To be devil's advocate, two things may offer a glimmer of hope:
First, math, generally, is useless. I mean, yes there are of course practical uses of basic thru undergrad-level math, and some beyond that. But for many mathematicians, the sum result of their entire career may lead to exactly zero results that have any real-world value. The entire field they work in may have meaning only to the handful of other individuals on the planet that also work in that field. But to those handful of people, the meaning defines their lives. From a socio-economic perspective, those departments should have been defunded a century ago. Yet they continue. Why? Because it scratches an itch. Not just for those individuals in the field, but also for us as a species. To stop exploring, to eliminate the search for pots of gold that may be buried in some odd corner of sphere packing, or coloring theorems, or Garside categories, and to put a boundary on the limits of our understanding, just because they aren't immediately applicable, is an idea that most humans would not be willing to sacrifice, even if it reduced their tax burden a couple cents. If it was going to happen, it'd have happened already.
The second is, even with AI, it's not free. As the software industry is discovering, far from it. So, given that, who is going to decide what theorems to research and how much it's worth? Congress? Of course not. AI itself? In theory that sounds plausible, but that falls victim to thing 1 above: most math is useless, so AI itself has no value metric it can assign to things, and besides which, without the human element, once the initial curiosity has subsided, there'd be no reason to continue any funding for AI to do it. So no, the only possible owners of this is going to be mathematicians themselves, the ones who care about the field and deeply understand the kwah of their vision.
Combining these, there's a future where, humanistically, "nothing changes". The method changes, the efficiency changes, the scope changes, but the work itself: publishing proofs, remains the domain of professional mathematicians. AI will enable them to be dramatically more daring and broad in their investigations and scope, and will likely write the entirety of the proof. However it will remain the work of the mathematicians to determine, what areas are worth spending limited AI resources on to investigate further, how far to go down rabbit holes, how to prioritize potential connections, and what the ultimate meaning of the findings is. So rather than being an end of mathematics, it could be a dawn of something far greater than anything we've ever seen before.
39 comments
[ 3.2 ms ] story [ 677 ms ] threadWhat would happen if a non-human layer of mathematics emerged on top of human mathematics? In this article, the distinction between Mathlib and Mathslop might be a precursor to that.
If models advance enough in the future, and new definitions, compressions, and representational forms that are convenient for AI-to-AI communication emerge, what would happen then? Would mathematics split into Human-facing and Machine-facing branches?
HN history
Are we going to see less publicly shared science? With private actors or governments restricting access to AI resources to a few scientists and keeping new knowledge to themselves.
Advancing science in the open was the best strategy when there was real advantage to share the load with every brain on the planet willing to give a try at science, but if a computer can match or surpass the collective output of the entire human scientific community the equation will change.
It's a sad outlook.
Yes, but this is when someone reaches ASI and everything changes. For now, a good researcher can build off their discovery in a way their AI can’t.
I can understand why this is a major concern for mathematicians. They got into their field because they love the beauty of mathematics, and the intellectual satisfaction of understanding non-obvious insights. But to put it crudely, this sounds like a you problem. As someone who isn't a mathematician, the main value I get out of math is its practical applications in science and technology. And their practical applications in human life. I have zero understanding of the math behind cryptography, but I still deeply appreciate the practical value they have provided humanity.
If AI systems start churning out accurate theorem-proofs, and we are able to use those theorems to build things that improve human quality of life, it doesn't bother me one bit that those theorems have not been understood by humans. If this offends your aesthetics, you are certainly entitled to your opinion and your preferences, but that does not make it a societal problem
I have some sad news for you. 99% of the work mathematicians do has no immediate application, nor even an obvious path toward application in the near future. You mentioned cryptography, so for an example consider number theory: no apparent practical applications, going back thousands of years to the time of Euclid and earlier.
It’s been religion, philosophy, and recreation that have provided the motivations to study mathematics all these years, not applications. Applications have almost always followed long after the development of the pure mathematical theory. For number theory, that was the development of cryptography during WW2, millennia after the ancients laid those foundations.
Most unfortunately, it’s the truth value and the understanding which drive applications of mathematics, not the proof work itself. If the AI revolution decapitates the institution of mathematics which produces the understanding, and is unable to replace it, then the applications will cease as well.
A wood-worker could do the same argument, there's the "official" wood-working word of perfect joinery and beautifully finished tables one can buy, but behind it there's the "secret" messy human element, the art, the craft, the mistakes and hard-ships, the elevation of human skills and imagination, the creation of whole new types of wood-working inventions and techniques, the perpetuation of millenia-old traditions, the teaching, the joy of selling to a happy customer, etc.
But now comes techo-capitalism, division of labor, you cut that piece a that piece over and over, you operate that machine, you won't even see the finished table, fuck your human element, we want that profit !
The motivation behind all this is less "haha I want profit" and more "billions of people need chairs, approximately none of them care about the craftsmanship, so it's in our best interest to make furniture in the most resource- and labor-efficient way possible". Even if the state subsidizes the production of handcrafted chairs, the population is the poorer for it on a resource allocation basis, because we now need a million artisanal chair-makers instead of a bunch of factories.
To be fair, a number of professional politicians and political scientists don’t understand alienation under capitalism.
Not that it is wrong for them to be doing this---we do want a society where people get to devote their life to what interests them---but it is bizarre because of the framing. For some reason it is ambiently understood in our society that this work is of incontrovertible value, when in fact it is largely not. And the value-producing parts of the work, the parts that end up having applications to other fields, largely run contrary to the actual daily goals of the cloistered devotees: it is mostly the intuition and pedagogy and the compactification and refactoring of knowledge that have value at this point, not the production of esoteric theorems, yet that is expressly not rewarded in the incentive structures.
That latter point is more due to the sorry state of academic incentives in general than to a particular failing of mathematics, though. Were I somehow given the ability to restructure things by fiat I would immediately create journals which publish only useful articles that refactor knowledge, communicate intuition, better explain things, argue for structural improvements to notation and terminology, etc, and this would immediately create an incentive to do that kind of work for working researchers to do work which aligns with the actually-useful output of their fields. I suspect most fields could use something like this. New knowledge is just not that valuable if it is all dumped into a giant pile and unprocessed, and I have seen firsthand a bunch examples where entire subdisciplines are hamstrung in their actual application-heavy work because they don't have easy access to basic tools that are hidden behind hard-to-learn theory.
_We do not know in advance which efforts are going to pay off_. Abstract efforts in topology put us on the road to nuclear energy. Silly number puzzles enabled internet commerce. Non-euclidean geometry gave us synchronized universal GPS.
We should not let our inability to conceive of applications of weird abstract stuff prevent us from making these investments. If our ancestors had fallen in to that trap, we'd be far poorer as a society.
What we can do is ask that people trying new stuff attempt to fail quickly. And that's basically where we are with academia today. Most people who do mathematical work will not have a career in math. They try something new, work for a little while on it, and go do something else when the results turn out to be of modest interest. This leaves behind a messy undigested literature, which is unfortunate. But maybe AI can help us sift that for treasures we missed.
Feels a lot like building software from bottom - once you get the building blocks defined right, the action, or the program, are trivial to express. When doing it from the top-down, you write the program using the building blocks you haven't defined yet, and you might end up with overly specific building blocks, needing other blocks for expressing different behaviors.
When you do the bottom-up building blocks right, new behavior is easy to express with them. Essentially, you are building up the language to reach the problem. Or making a DSL, whatever definition you like best.
"I was in Switzerland", "I was invited to a talk", "I started a machine learning company", look at me bro.
One of the most fruitful approaches in mathematics is to flip back and forth between geometric and algebraic views of a problem. I think this works so well because these are actually handled by two different parts of the brain on a physical level; spatial reasoning is separate from language processing. Cytoarchitecture shows these regions have different "textures;" the local details of the way neurons are wired together are simply different in these different regions of the brain, in the same way a CNN and a transformer have different topologies. Thus, by flipping problems from geometry to algebra and vice versa, we're able to bring an entirely different cognitive style to bear on a problem. For example, the proof of Monge's Theorem by moving to 3D and visualizing not three circles, but three spheres sitting on a table with a book on top of them and then pointing out that the intersection of two planes is a line. What is pages of unintuitive symbol pushing turns into something a child can understand. Going the other way, things like the angle addition formulas or the quadratic formula, which are quite hard to prove geometrically, become quite simple if you use a little algebra.
Current-gen LLMs are still relatively weak at visual reasoning; see the Vision Language Models are Blind paper, for example, or the ARC-AGI benchmark. So that's one way humans can stay ahead of the agents, at least for now.
I think that we're not that far away from AI that can be superhuman at all facets of theorem proving.
I think that we're far away from an AI that can create good abstractions and construct a theory to prove theorems.
Math is entirely subjective. "Proof" essentially means "Other educated practitioners have the same experience when trying to understand this."
The logical steps that proofs are built on all have that common foundation. Our concept of logic based on our subjective experience of "truth." We've built machines that reproduce our subjective processes mechanically, but there is no sense in which this idea of "true" is truly objective. It happens to be computationally convenient, and it has some relationship to experience, but that doesn't make it an independent reality that all possible observers, human and otherwise, would agree on.
We're really just mapping our own minds through our own experiences.
Animal brains can't abstract like (some of) our brains can. What are the odds our brains are limitless and don't have some similarly crippling limitations from a couple of levels up?
One of the tells for ASI is that it will start reasoning at those levels, using cognitive techniques that are completely incomprehensible - not just because of brute volume, but because our brains won't have the wiring to get a foothold on them.
Some of the products will be reducible to human cognition, in a distorted and simplified form, but many won't.
So - I disagree with Egan. I don't think there's going to be a universal proof library, and even if there were we'd only ever get the Cliff Notes version.
The idea that we experience objectivity subjectively, thus there is no objectivity, seems like an ultimately nonsensical and self-defeating sleight of hand to me.
You mean they can't come up with mathematical abstractions? That's right of course, but they seem perfectly able to "abstract" in the sense of drawing general rules from specific observations. For example, I noticed recently how my cat friend was happy to exit and enter the house from either of two doors and a window as the opportunity presented itself (i.e. depending on which one was open at the time he wanted to get in or out). E.g. I just happened to open the window and he hopped onto the ledge and out into the mystery of the night, entirely unaided by human hands or voices.
That's an abstraction: one door is like another door and they're both like a window. Both things lead in and out of the house and they can be open or closed at different times. If one is closed another may be open. Something like that, obviously I have no idea what concepts, exactly, he has in his head. And btw, "in", "out", "house", etc are also abstractions, so a house in the UK is like a house in France, or like one in Italy, or one in Greece: that's where the human things are; I guess.
I think it's important to understand this about animal intelligence, that it is very much capable of broad generalisation and abstract thought. Absolutely not to the same degree as humans because we got language and that probably comes with a whole other level of ability for abstraction and reasoning, but I don't believe it's possible for an animal to survive in the physical world if it can't go from the concrete to the abstract, the specific to the general, to some extent.
You mention ASI so I'm comfortable making this comment which is about AI, without feeling I'm hijacking your conversation. I think one only begins to understand how powerful the animal brain is, and how overlooked (in AI discourse) its ability to form such broad, useful abstractions that allow animals to navigate the physical world autonomously while setting and pursuing their own goals, when one tries to reproduce the same behaviour artificially, in computers.
Like you say, yes, maybe when we stop understanding what our computers produce, it will be because they moved a level up from where we are, as ASI. Maybe there is such a level; maybe there isn't. For the time being, there is no artificial system that can spontaneously move in the physical world as easily and effortlessly as my cat friend can (and he's a graceful animal; really, much closer to an African wildcat than its domestic descendant). If you want an AI system to understand the difference between "in" and "out" and that there are doors and windows that connect the two you have to somehow explicitly feed that information into it, either by hand-coding (e.g. the PDDL models used in Planning and Scheduling) or by training on carefully chosen examples of those concepts (as in machine learning), or at least some kind of objective that will lead the AI to develop them, or develop similar concepts (as in Reinforcement Learning). We're still very far from the capabilities of the animal brain.
Also, may I quibble about the fact that our brain, too, is an animal brain?
Maybe we will look back on today's math as a kind of arcane, pre-syntactical set of structures that required speakers of the language on both ends interpreting it to make good use of it. No validation or compilation, it can't be applied, just a total wild west - scribbles on a whiteboard and another mathematician making sense of it.
"e=mc^2"
And the Lord's people said: "LGTM"
I think that's just an accident of history.
When we write software, we very seldom write proofs that our algorithms are correct. We just write tests, and we also run the algorithms and when they fail we know we have a bug and then we proceed to debug, fix, and add new tests (if we are disciplined, but most of us are). In time, by usage and testing, we gain confidence that our battle tested software is correct, mostly. And we tell people that we will never be 100% confident that any software is bug free. But that's a slight lie: if we wanted such confidence, we would start using provers, and create bug-free software. That possibility exists, but it's just extraordinarily expensive.
Well, in math that's the only possibility, and we use it. And it is indeed extraordinarily expensive, but it's also the cheapest among the alternatives. The alternatives are 2: be rigorous and do these proofs, or be sloppy and allow bugs to creep in, and allow an entire school of math to collapse like the Italian school of algebraic geometry [1].
There is one more alternative. If a particular math theorem has some applicability, then you write a program and use it in real life. In time you eliminate the bugs as much as you can, and you get to the steady state of "virtually bug free". At that point you don't have a solid proof that the theorem is correct, but in general you don't really care. Because you feel that a formal proof is just a thing one would pursue for getting academic satisfaction only.
[1] https://en.wikipedia.org/wiki/Italian_school_of_algebraic_ge...
But this is due to a "failure" of programming language progress. We've had formal languages for a long time (see Ada and SPARK) and we've simply failed to use them for most scenarios, instead regressing to imperative manually memory managed languages like C and then deriving from that branch.
How is it not already this? Jon von Neumann was already calling most math this many decades ago. Pull up any random arxiv math paper and it’s abstract nonsense with no applications to the real world.
First, math, generally, is useless. I mean, yes there are of course practical uses of basic thru undergrad-level math, and some beyond that. But for many mathematicians, the sum result of their entire career may lead to exactly zero results that have any real-world value. The entire field they work in may have meaning only to the handful of other individuals on the planet that also work in that field. But to those handful of people, the meaning defines their lives. From a socio-economic perspective, those departments should have been defunded a century ago. Yet they continue. Why? Because it scratches an itch. Not just for those individuals in the field, but also for us as a species. To stop exploring, to eliminate the search for pots of gold that may be buried in some odd corner of sphere packing, or coloring theorems, or Garside categories, and to put a boundary on the limits of our understanding, just because they aren't immediately applicable, is an idea that most humans would not be willing to sacrifice, even if it reduced their tax burden a couple cents. If it was going to happen, it'd have happened already.
The second is, even with AI, it's not free. As the software industry is discovering, far from it. So, given that, who is going to decide what theorems to research and how much it's worth? Congress? Of course not. AI itself? In theory that sounds plausible, but that falls victim to thing 1 above: most math is useless, so AI itself has no value metric it can assign to things, and besides which, without the human element, once the initial curiosity has subsided, there'd be no reason to continue any funding for AI to do it. So no, the only possible owners of this is going to be mathematicians themselves, the ones who care about the field and deeply understand the kwah of their vision.
Combining these, there's a future where, humanistically, "nothing changes". The method changes, the efficiency changes, the scope changes, but the work itself: publishing proofs, remains the domain of professional mathematicians. AI will enable them to be dramatically more daring and broad in their investigations and scope, and will likely write the entirety of the proof. However it will remain the work of the mathematicians to determine, what areas are worth spending limited AI resources on to investigate further, how far to go down rabbit holes, how to prioritize potential connections, and what the ultimate meaning of the findings is. So rather than being an end of mathematics, it could be a dawn of something far greater than anything we've ever seen before.
Meta:
- Sad to see how well thought out + well written stuff like this only makes it to the HN front page by a fluke. (Below: https://news.ycombinator.com/item?id=48758048#48759391)
- @davidbessis: maybe look for an open alternative to Substack? (avoid bullshit, proprietary, gated access for quality content by humans).