The fall of the theorem economy

davidbessis.substack.com284 pointsby varjag122 comments

Discussion summary

The discussion revolves around an essay on the economy of mathematics, with opinions on its political naivety and its comparison to other crafts. Participants mention the normalcy of academic conferences and the potential future of mathematics resembling a niche sport.

What the discussion says

Some see the essay as naive politically.
Many defend the normalcy of academic conferences.
The analogy to woodworking and craftsmanship is debated.
The essay is considered highly interesting by some.
Concerns about the future of mathematics are expressed.

“It's like he's re-discovering alienation under capitalism.”
— Staross

“Being invited to conference talks is normal for researchers.”
— capnrefsmmat

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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
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.
by chongli
Very good! Thank you!!
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).
by manuelz

Some previous submissions

HN history

  +-- 2mo before by sdfrew
  |   4 points / 1 comments
  |   The Fall of the Theorem Economy
  |   https://news.ycombinator.com/item?id=47862472
  |
  +-- 2mo before by fuglede_
  |   3 points / 1 comments
  |   The Fall of the Theorem Economy
  |   https://news.ycombinator.com/item?id=47891494
  |
  +-- 2mo before by mathgenius
  |   2 points / 0 comments
  |   The Fall of the Theorem Economy
  |   https://news.ycombinator.com/item?id=47909751
  |
  +-- 2mo before by delis-thumbs-7e
  |   15 points / 4 comments
  |   David Bessis on AI destroying mathematics
  |   https://news.ycombinator.com/item?id=47985962
  |
  +-- 1mo before by magoghm
  |   4 points / 0 comments
  |   The Fall of the Theorem Economy
  |   https://news.ycombinator.com/item?id=48084737
  |
  +-- 1mo before by cubefox
  |   2 points / 0 comments
  |   The Fall of the Theorem Economy
  |   https://news.ycombinator.com/item?id=48089716
  |
  +-- 1mo before by cubefox
  |   5 points / 0 comments
  |   The Fall of the Theorem Economy
  |   https://news.ycombinator.com/item?id=48152469
  |
  +-- 1mo before by tmp10423288442
  |   4 points / 1 comments
  |   The Fall of the Theorem Economy
  |   https://news.ycombinator.com/item?id=48214866
  |
  `-- this submission by varjag
      58 points / 7 comments
      The Fall of the Theorem Economy
      https://news.ycombinator.com/item?id=48758048

by octopus143

Tangential but this article got me thinking.
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.
by stackbutterflow
> to share the load with every brain on the planet willing to give a try at science
This is what a lot of scientists love to tell themself or talk about in celebratory speeches.
The truth is: a lot of science is kept behind journal paywalls, so that only "officially approved" (in the sense of: working at a university or an governmental research institute) scientists can easily access it.
by aleph_minus_one
> if a computer can match or surpass the collective output of the entire human scientific community the equation will change
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.
by armchairhacker
It's not even just science. Anyone producing anything digital is now heavily incentivized to keep their wares away from the public internet, in a way we've not seen before. Drop any sample of your unique production online, and the AIs will obsolete you in days. That is a massive loss for people looking for inspiration and guidance.
by toyg
The intelligence is only one part of the story. People need to actually go out there and do experiments. Science is not only theory, but also experimental. The best science happens when experiments show that a previously held assumption was not true. Eg the Michaelson Morley experiment where the assumption of ether was disproved. While AI is an incredibly powerful tool, it does not replace the act of observation. Thus Im sure we will still have scientists in the future and some degree of open science. There are experiments out there that by nature of their complexity need massive public coordination (CERN for instance) which in turn benefits from openness.
What is going to suck though is the ladder for juniors. We dont start out by working on big ticket problems, but usually early career researchers solve really tiny problems in a cheap way. The lowest bar for a cash strapped PhD student would be to contribute to some new theory in some way even if the student doesnt have access to equipment.
by accurrent
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.
by mojosmojo
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.
[1] https://en.wikipedia.org/wiki/Italian_school_of_algebraic_ge...
by credit_guy
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...
by dboreham
> 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.
by satvikpendem
This is a bit reductive about what "proof" actually means in mathematics. Even in math, the kind of formal proofs that tools like Coq can automatically verify are an extreme, and lots of accepted and practiced math is not doing that. Proofs are often more abstract and even occasionally hand-wavy (for example not proving "obvious" statements or minor lemmas).
Mathematicians also occasionally build on top of unproven foundations (e.g. all popular asymmetric encryption schemes are built on top the assumption that certain problems such as integer factoring are hard, for which there is no formal proof), or at least explore both possibilities for statements with unknown truth value (e.g. you can find lots of work that explores the consequences of P = NP and/or P != NP).
However, there is a major separation between math and programs that I think mostly invalidates your proposal - most math we're talking about here is simply not applicable directly to the real world in any way. It's only studied for the interest of mathematicians. There is no real world consequence for Fermat's last theorem, for example - it was just a really interesting to prove theorem. In directly applied math, such as engineering, it is in fact much more common to work with unproven but well tested conjectures.
by simiones
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.
by andai
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.
by christina97