The fall of the theorem economy

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• Hacker News
• I've only started reading it but it looks very good. I'll finish it tomorrow or so.
  by throwaway81523
• I see the AI panic has reached mathematics…
  by khalic
• Is being pompous and on the nose part of being in mathematics?

  "I was in Switzerland", "I was invited to a talk", "I started a machine learning company", look at me bro.

  by asdfsa32
• Being invited to conference talks around the world is a completely normal part of being an active researcher in almost any academic field, so it doesn't register as pompous to other academics.
  by capnrefsmmat
• 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 !

  by Staross
• 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.

  by zerobees
• > I thought it was very interesting, but maybe also incredibly naive politically ? it's like he's re-discovering alienation under capitalism.

  To be fair, a number of professional politicians and political scientists don’t understand alienation under capitalism.

  by TimorousBestie
• I don't think you have it right- the analogy to woodworking and craftsmanship is a category error and probably misses the broad thrust of the essay.

  The goal of a woodworker or craftsman is the production of a finished good. He's arguing that, although it's been convenient to position a mathematician as a "theorem-producer", that's never really been the aim of mathematics, and that the actual products of mathematics are some kind of "mental software"- see his references to neuroplasticity. Basically, he's saying that the goal of mathematics is to create abstract structures that allow humans to reason about increasingly complex concepts, and that the "mathematician as theorem producer" is more like a convenient fiction that mathematicians have allowed to persist for too long, and now threatens to endanger the whole practice of mathematics.

  by throwaway91827
• > 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.

  by codemog
• This might be the most interesting essay on the nature of mathematics I have ever read.
  by cubefox
• It was also about 10x longer than it had to be because an LLM was heavily involved in its writing.
  by pickleRick243
• 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.

  by daxfohl
• Plenty of math showed zero real-world value for centuries until it suddenly did.
  by snackerblues
• 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.
  by a1o
• I built a toy autonomous math research project: https://paulklemstine.github.io/Lean/
  by raver1975
• 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?

  by jdw64
• Science is not about results, it is about the transmission of knowledge. So long as those AI-"sciences" are just inside AI, they are "engineering", not science.

  I am not dismissing engineering (it moves the world we live in), just trying to clarify what science is.

  Applied fluid dynamics works like that: noone has ever really "verified" that the finite-element method applied to some specific model does converge

  by pfortuny
• > 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.

  by rbanffy
• Well, it depends. Sometimes in math you do a lot of chipping away at a problem, and eventually a bigger result falls after all the right foundations are built. That seems to describe bottom-up building.

  On the other hand when a new high-level concept becomes clear and seems to emerge like a revelation, and people start thinking in terms of those new definitions, it seems that a hundred pages worth of smaller results can fall out of it almost effortlessly. This way of describing it is more top-down.

  I don't know that there's an exact parallel with software. Math keeps feeding into itself in a way that software dreams about with our ambitions of code reuse. The old Object Oriented dreams of perfectly encapsulated classes and abstractions partially worked out, but not to the degree that was envisioned.

  The current situation with package managers doesn't look like a tower that keeps growing higher and higher levels of abstractions. It looks like a tower where each person wants to place one tiny brick that they call left-pad, and next year we will rebuild the lower levels instead of going higher. So the top-down and bottom-up building that we do is different. We keep rebuilding the bottom, and we don't very much like when the tower of abstractions get too tall and hard to maintain.

  by tux3
• Well, it depends. Sometimes in math you do a lot of chipping away at a problem, and eventually a bigger result falls after all the right foundations are built. That seems to describe bottom-up building.

  On the other hand when a new high-level concept becomes clear and seems to emerge like a revelation, and people start thinking in terms of those new definitions, it seems that a hundred pages worth of smaller results can fall out of it almost effortlessly. This way of describing it is more top-down.

  I don't know that there's an exact parallel with software. Math keeps feeding into itself in a way that software dreams about with our ambitions of code reuse. The old Object Oriented dreams of perfectly encapsulated classes and abstractions partially worked out, but not to the degree that was envisioned.

  The current situation with package managers doesn't look like a tower that keeps growing higher and higher levels of abstractions. It looks like a tower where each person wants to place one tiny brick that they call left-pad, and next year we will rebuild the lower levels instead of going higher. So the top-down and bottom-up building that we do is different. We keep rebuilding the bottom, and we don't very much like when the tower of abstractions get too tall and hard to reason about.

  by tux3
• 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.
  by guelo
• 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.

  by auntienomen
• 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.

  by ajkjk
• Right; this is my viewpoint too. All the "pure mathematicians" have a bleak future where AI can do all the puzzle solving better and faster. They existed in their own world elevating "theorem proving within a formal system" as the central aspect of "proper" mathematics and everything else as ancillary.

  It always felt wrong to me that while the scientific method iterated starting with the "real world" viz. Observe, Measure, Hypothesize (includes modeling with mathematics), Test and Refine; pure mathematicians lost themselves in the formalization of hypothesizing/modeling and thus lost touch with mapping it to reality. The AI revolution is now showing them up.

  by rramadass
• 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.
  by bsenftner
• Eh? I thought this was the main thrust of the argument: Mathematics has in fact always prized conceptual advancement and understanding over proof, despite presenting itself internally and externally as rewarding the latter. The author calls what he’s proposing “rebranding a plurimillenial project”.
  by twoodfin
• Does the the ⁡(,1) conjecture paper in annnals of Math say 7 years between submission and acceptance? Insane
  by j7ake
• These stories are common in math, e.g. these recently happened to me, a lowly mathematician:

  1) Two and a half years with no reply from a journal (not even to emails I sent that I'd like to retract the paper so I could send it somewhere else). Then suddenly they tell me the paper is accepted.

  2) One year with no reply. Then, my "anxious" collaborator sends them countless emails and gets redirected from person to person and finally an editor tells us that they decided almost immediately to reject our paper but they didn't tell us because "they hate giving bad news".

  These were not top journals like Annals, but decent, prestigious ones, from whom you'd expect some professionalism.

  by bananaflag
• 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

  by whack
• Then you have to make sure that the AIs understand the theorems (sort of build a "world" for that - otherwise how'd there be confidence in the use of said theorems?

  If cryptography didn't exist but the maths did, how'd you use it?

  by RandomLensman