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> However, the declaration argues math is more than a machine for producing correct answers. There might be more to maths than that, but that is definitely the

> However, the declaration argues math is more than a machine for producing correct answers. There might be more to maths than that, but that is definitely the most important part. I love science funding. But not because it's a jobs program for nerds.

bloqs 4mo ago
well put.
19f191ty 4mo ago
No, it's not the most important part. It can be argued that most important part is asking the right questions
- i_am_a_peasant 4mo ago
  I agree with both OP and you
  - psychoslave 4mo ago
    I disagree with everyone, self included, but especially with Cretans.
    - codeduck 4mo ago
      Cretani eunt domo!
      - i_am_a_peasant 4mo ago
        Monty Python fan detected :D love your profile desc btw
- silveraxe93 4mo ago
  Assume someone solves P=NP Do you think Stephen Cook and Leonid Levin deserve more credit than whoever solved it?
  - NotOscarWilde 4mo ago
    That's a bit too simplistic -- if there is a small group that really pushes things forward in a big way, then maybe not, but if this result builds upon decades of prior work, then Cook and Levin might be equally or even slightly more famous than the solver group after the dust settles. But it is a moot point anyway. Cook and Levin are very well known already in TCS, and credit is not directly enumerable like money, so "more than a lot of credit" doesn't make too much sense. For this problem in particular, asking the right kind of question was really important for the field and led to a lot of discoveries even before it will be answered.
  - dchftcs 4mo ago
    If the problem resolves to P=NP, that result would probably be more celebratee than being able to formulate the problem, but being able to formulate the problem and get people interested in it is probably worth more than the average primal dual trick to prove a polylog integrality gap for some integer linear program.
  - jltsiren 4mo ago
    Depends on the solution. If the solution is that P≠NP, the concept of NP-completeness remains the bigger contribution, unless the proof techniques are particularly interesting and lead to other major results. The same applies if P=NP but the proof does not ultimately lead to a practical algorithm. If we get a practical algorithm, the answer is more valuable than the question.
    - redox99 4mo ago
      Do questions like goldbach conjecture, fermats last theorem, etc deserve more credit than whoever answers them?
      - jltsiren 4mo ago
        Fermat's last theorem probably doesn't. While attempts to solve it have led to many mathematical discoveries, the theorem itself feels little more than a piece of trivia. I'm less familiar with the implications of the Goldbach conjecture. In contrast, class NP and NP-completeness quickly became central concepts in theoretical computer science.
delichon 4mo ago
For most engineers a mathemetician is a machine for producing correct algorithms, like a chef is a machine for producing tasty food. In both cases that overlooks the human element, but that's a critical skill for a limited mind with finite resources to grok infinite complexity. You can read that as permission to be an asshole or a neccesary compromise.
fragmede 4mo ago
People need jobs. What's wrong with nerds having jobs via a program?
- psychoslave 4mo ago
  People need many things, there are all kind of theories ready to assess and assimilate if deemed worth it out there. A job is not part of any I’m aware of, though it can encompass some human needs in some cases, or go straight against them in some other case.
  - fragmede 4mo ago
    Every human needs a vocation, better if it's of their choosing.
- RugnirViking 4mo ago
  what's wrong with artists having jobs via a program? whats wrong with struggling alcoholics having jobs via a program? athletes? politicians? there is no inherent virtue in the struggle and effort associated with great mathematical achievement. It may be satisfying and worthwhile for the solver, but not for society at large, any more than any other pleasurable activity. No, as it is, the sole reason for it is in the result itself. In increased understanding, as it flows down into the sciences, and engineering. There are other benefits, recreation and joy as experienced by others, from access to beautiful proofs, though these are never explicit goals of such programs because they are both impossible to quantify and rarely ever remotely relevant compared to the value brought by the practical value brought by maths. Of course, there may be some valid arguments that everyone should have a jobs program in the form of ubi or something similar. But I feel thats very different to arguing for mathematicians specifically
  - fragmede 4mo ago
    The struggle itself is virtuous.
  - mswphd 4mo ago
    for mathematicians, they do a form of fundamental research that is 1. (generally) incredibly cheap to fund, and 2. (occasionally) has extremely out-sized commercial impacts. This is to say that jobs programs for math (and more generally fundamental research) have lead to extremely positive ROI for society, which is the typical justification given for funding them.
psyklic 4mo ago
The most important part of math is advancing human understanding. A correct answer by itself is not as important as understanding why it is correct.
- ragebol 4mo ago
  "What is the answer to the Ultimate Question of Life, the Universe, and Everything" 42
  - lioeters 4mo ago
    The proof is trivial and is left as an exercise for the reader.
- rowanG077 4mo ago
  Once you now something is correct, with a proof. It is MUCH easier to understand why it is correct. Than to start from a slate that you don't even know whether something is correct or not. In that sense AI that can just solve high level math problems is immensely useful. It allows a mathematician to explore ideas at a much more rapid pace.
  - terminalbraid 4mo ago
    Consider that since an LLM is really just an large encoding of data, the "proof" is in there already. All further work on it is effectively only rearranging words. Then all math an LLM is capable of is "done" and we have the "proof" in the LLM which by your definition is now "MUCH easier to understand" and this work is somehow sufficient. Do you see the problem with your reasoning?
    - rowanG077 4mo ago
      You're confusing "contains information" with "has produced a result." A proof being latent in an LLM is no more significant than a proof being latent in a book, a theorem prover, or the axioms themselves. Einstein's papers were latent in the genetic code of his parents and the environment of his time. That doesn't mean general relativity was "already done" before Einstein was born. By your logic, no computation has ever accomplished anything because the output was always implicit in the inputs. The entire purpose of computation is extracting information from representations where it's difficult to see into representations where it's easy to see. So no, this isn't a problem with the original reasoning. It's a problem with yours.
dwroberts 4mo ago
Probably one of the funniest things to read on a site like this, when you consider that eg. Boolean algebra was entirely abstract and had little practical purpose for almost 100 years until Shannon picked it up for use in circuits
- card_zero 4mo ago
  Boole was trying to improve logic for humans, "The Laws of Thought". So it has a connection to human problems, and eventually to practical matters. He could instead have been working on something much more abstract and much less useful. By which I'm trying to make an abstract point about the inevitability of staying somewhat down to earth. I mean "pure" curiosity is great, except it isn't ever really pure, and abstract mathematics isn't ever totally abstract, it's just sort of meta in relation to practical things that humans care about.
  - dwroberts 4mo ago
    > So it has a connection to human problems, and eventually to practical matters. But in relation to parent post I was replying to, it did not provide an answer or solution to anything. It has much closer relation to philosophy than anything. Focusing on only ‘solutions’ in any field is shortsighted because you can’t know how the dots will connect. Someone’s seemingly pointless curiosity or experiment can unlock something unexpected, just like Boole
kleyd 4mo ago
The wording in the declaration may be a bit romanticized. But the points are valid: Is an 80 year old unsolved problem maybe unsolved because it was never prioritized? Some problems stay unsolved because few people consider them worth working on. Who is going to validate the results? Or do we skip that, with the risk of flooding the literature and collective understanding with unverified proofs?
smath 4mo ago
This reminded me of my 11 yr old who, when I give her math problems to solve, is too focused on “getting the right answer”. I’ve told her plainly, I don’t care if you get the right answer right now, I want to see your reasoning. She has yet to understand this.
conformist 4mo ago
> The authors warn the consequences are already becoming visible. AI-generated papers could overwhelm peer-review systems with low-quality work … It seems like a key problem here is that peer-review is expected but not explicitly funded/rewarded while it is probably one of the aspects where humans still add a lot of value. Academia’s incentives are hugely misaligned (… as usual unfortunately).
- armchairhacker 4mo ago
  Math is one field where you can mechanically prove a paper's findings. The only thing that would need to be judged is the (verified) statement's importance.
  - conformist 4mo ago
    Yes in theory, but not yet in practice because not everything is fully formalised.
hyperpape 4mo ago
Even from the most purely instrumental perspective, what we care about is our ability to make use of correct answers, which is quite distinct from the possession of correct answers. There are many theorems that aren't directly interesting, but whose proof requires techniques that are of substantial further interest, that lead to new domains, and/or new practical applications. Simply being handed a proof for those theorems isn't enough--we require the ability to apply those techniques in the real world, or discover further areas of mathematical research that build on that proof or its techniques. It may be that AI can build on its own work for the long-term, but so far, AI does best at exploration in areas that have precisely specified and measurable goals. Actually creating understanding, and making use of mathemtical results outside of pure mathematics is more challenging than simply creating proofs. I think the field will figure out how to make use of AI, and it will be better off for it. But that is not the same as just saying "answers good, grog want more answers."
barrkel 4mo ago
A statement that some proposition is true or false is usually less useful than a new framework for understanding the class of problem. A machine that takes longer and longer to prove propositions in ever more inscrutable ways is hardly useful at all. The machine too needs to produce more generalizable and comprehensible systems, for it to scale up its own conceptualization. Needing to load all the new mathematics in the context window won't be great either.
analognoise 4mo ago
> But not because it's a jobs program for nerds. We’re becoming increasingly embarrassing as a society.
zerobees 4mo ago
Culturally, mathematics is a jobs program for nerds. The field very explicitly takes pride in working on problems that have no obvious applications, and most practitioners are funded publicly or supported by private endowments, with zero pressure to deliver specific results. Of course, this produces useful results every now and then, but it's not like we pursued ruthless efficiency / maximum rate of knowledge advancement before. We just let them do their thing, essentially treating them as artists and letting them pursue the craft for its own sake. If we weren't interested in maximum throughput before, why is that an objective now?
- bdamm 4mo ago
  Except, it was true before and it is still true today that the best "artists" whether graphic or mathematic, are the ones that do somehow manage to cross the chasm of pure research and providing a tangible benefit to their benefactors. That aspect of understanding your customer is not changed by the presence of AI.
- orbital-decay 4mo ago
  Pure mathematics != applied mathematics
- 141205 4mo ago
  Hardy would agree with the viewpoint that you espouse but it would be pushed back against by Arnol'd, Poincare, Gauss, Von Neumann, and even Grothendiek: Arnol'd and Poincare were vituperatively against the division between "pure" and "applied" mathematics; they considered mathematics and physics interchangeable, and Arnol'd lamented that the field had lost a large amount of funding/prestige/relevance due to groups like the Bourbaki that took a purely aesthetic view; Gauss had a critical view of problems like Fermat's last theorem (he felt that you could construct infinitely many such problems, and felt that attempting to prove it was a generally useless endeavor), along with outright calling pure mathematics worthless; but while Von Neumann and Grothendiek were more moderate, both were critical of the field losing motivation/quality as it strayed away from empirical science into—quoting Von Neumann—"abstract inbreeding". Arnold's polemics are perhaps the most infamous and easily found online (see "On Teaching Mathematics"), but the written opinions of Poincare et seq. are also easy to find. Even today the vast majority of research funding for mathematics, at least in the United States, is dolled out for highly applied fields like partial differential equations. The field does not even close to unanimously (contemporarily or historically) "explicitly take pride" in working on problems that have no obvious application, or being a "jobs program for nerds": the notion of such "pure" or "nonapplied" mathematics is at the very least a highly fractious and controversial subject, with a number of big names taking opposing viewpoints (often vehemently). I think your picture of the field is over-represented on the internet, much like the fixation on certain niche fields: Category Theory, Homotopy Type Theory or, worst of all, outright dubious fields like Geometric Algebra; fields with a large number of online promoters, but with much less funding and relevance in the actual academic space. Of course there are reputable people with PHDs that feel this way,—but I can only imagine that there's a legion of tyros, pop math consumers, and undergraduate students who disproportionately promote this viewpoint.
ajs1998 4mo ago
Spoken like someone that has no idea why mathematicians are important.
IshKebab 4mo ago
Maths pretty much is a jobs program for nerds though. It occasionally produces results that are practically useful for society but there's absolutely no way the vast majority of today's maths research falls into that category. There are definitely exceptions, like crypto. I still think it would be pretty silly to stop maths research anyway. And anyway part of the job of maths researchers is to teach maths to undergrads and that's obviously enormously useful to society. But on the scale of "how useful is this research to society" it's dead last after engineering, chemistry, biology, and physics. Well maybe computer science would be last actually!
solid_fuel 4mo ago
>> However, the declaration argues math is more than a machine for producing correct answers. > There might be more to maths than that, but that is definitely the most important part. I love science funding. But not because it's a jobs program for nerds. I can produce an infinite number of verifiably correct papers, if that's all that matters. 1 + 1 = 2 1 + 2 = 3 1 + 3 = 4 1 + 4 = 5 1 + 5 = 6 Shall I continue? Or do you think that choosing which questions to answer might have some level of importance, in addition to getting correct answers?

bloqs 4mo ago

well put.

19f191ty 4mo ago

No, it's not the most important part. It can be argued that most important part is asking the right questions

i_am_a_peasant 4mo ago
I agree with both OP and you
- psychoslave 4mo ago
  I disagree with everyone, self included, but especially with Cretans.
  - codeduck 4mo ago
    Cretani eunt domo!
    - i_am_a_peasant 4mo ago
      Monty Python fan detected :D love your profile desc btw
silveraxe93 4mo ago
Assume someone solves P=NP Do you think Stephen Cook and Leonid Levin deserve more credit than whoever solved it?
- NotOscarWilde 4mo ago
  That's a bit too simplistic -- if there is a small group that really pushes things forward in a big way, then maybe not, but if this result builds upon decades of prior work, then Cook and Levin might be equally or even slightly more famous than the solver group after the dust settles. But it is a moot point anyway. Cook and Levin are very well known already in TCS, and credit is not directly enumerable like money, so "more than a lot of credit" doesn't make too much sense. For this problem in particular, asking the right kind of question was really important for the field and led to a lot of discoveries even before it will be answered.
- dchftcs 4mo ago
  If the problem resolves to P=NP, that result would probably be more celebratee than being able to formulate the problem, but being able to formulate the problem and get people interested in it is probably worth more than the average primal dual trick to prove a polylog integrality gap for some integer linear program.
- jltsiren 4mo ago
  Depends on the solution. If the solution is that P≠NP, the concept of NP-completeness remains the bigger contribution, unless the proof techniques are particularly interesting and lead to other major results. The same applies if P=NP but the proof does not ultimately lead to a practical algorithm. If we get a practical algorithm, the answer is more valuable than the question.
  - redox99 4mo ago
    Do questions like goldbach conjecture, fermats last theorem, etc deserve more credit than whoever answers them?
    - jltsiren 4mo ago
      Fermat's last theorem probably doesn't. While attempts to solve it have led to many mathematical discoveries, the theorem itself feels little more than a piece of trivia. I'm less familiar with the implications of the Goldbach conjecture. In contrast, class NP and NP-completeness quickly became central concepts in theoretical computer science.

delichon 4mo ago

For most engineers a mathemetician is a machine for producing correct algorithms, like a chef is a machine for producing tasty food. In both cases that overlooks the human element, but that's a critical skill for a limited mind with finite resources to grok infinite complexity. You can read that as permission to be an asshole or a neccesary compromise.

fragmede 4mo ago

People need jobs. What's wrong with nerds having jobs via a program?

psychoslave 4mo ago
People need many things, there are all kind of theories ready to assess and assimilate if deemed worth it out there. A job is not part of any I’m aware of, though it can encompass some human needs in some cases, or go straight against them in some other case.
- fragmede 4mo ago
  Every human needs a vocation, better if it's of their choosing.
RugnirViking 4mo ago
what's wrong with artists having jobs via a program? whats wrong with struggling alcoholics having jobs via a program? athletes? politicians? there is no inherent virtue in the struggle and effort associated with great mathematical achievement. It may be satisfying and worthwhile for the solver, but not for society at large, any more than any other pleasurable activity. No, as it is, the sole reason for it is in the result itself. In increased understanding, as it flows down into the sciences, and engineering. There are other benefits, recreation and joy as experienced by others, from access to beautiful proofs, though these are never explicit goals of such programs because they are both impossible to quantify and rarely ever remotely relevant compared to the value brought by the practical value brought by maths. Of course, there may be some valid arguments that everyone should have a jobs program in the form of ubi or something similar. But I feel thats very different to arguing for mathematicians specifically
- fragmede 4mo ago
  The struggle itself is virtuous.
- mswphd 4mo ago
  for mathematicians, they do a form of fundamental research that is 1. (generally) incredibly cheap to fund, and 2. (occasionally) has extremely out-sized commercial impacts. This is to say that jobs programs for math (and more generally fundamental research) have lead to extremely positive ROI for society, which is the typical justification given for funding them.

psyklic 4mo ago

The most important part of math is advancing human understanding. A correct answer by itself is not as important as understanding why it is correct.

ragebol 4mo ago
"What is the answer to the Ultimate Question of Life, the Universe, and Everything" 42
- lioeters 4mo ago
  The proof is trivial and is left as an exercise for the reader.
rowanG077 4mo ago
Once you now something is correct, with a proof. It is MUCH easier to understand why it is correct. Than to start from a slate that you don't even know whether something is correct or not. In that sense AI that can just solve high level math problems is immensely useful. It allows a mathematician to explore ideas at a much more rapid pace.
- terminalbraid 4mo ago
  Consider that since an LLM is really just an large encoding of data, the "proof" is in there already. All further work on it is effectively only rearranging words. Then all math an LLM is capable of is "done" and we have the "proof" in the LLM which by your definition is now "MUCH easier to understand" and this work is somehow sufficient. Do you see the problem with your reasoning?
  - rowanG077 4mo ago
    You're confusing "contains information" with "has produced a result." A proof being latent in an LLM is no more significant than a proof being latent in a book, a theorem prover, or the axioms themselves. Einstein's papers were latent in the genetic code of his parents and the environment of his time. That doesn't mean general relativity was "already done" before Einstein was born. By your logic, no computation has ever accomplished anything because the output was always implicit in the inputs. The entire purpose of computation is extracting information from representations where it's difficult to see into representations where it's easy to see. So no, this isn't a problem with the original reasoning. It's a problem with yours.

dwroberts 4mo ago

Probably one of the funniest things to read on a site like this, when you consider that eg. Boolean algebra was entirely abstract and had little practical purpose for almost 100 years until Shannon picked it up for use in circuits

card_zero 4mo ago
Boole was trying to improve logic for humans, "The Laws of Thought". So it has a connection to human problems, and eventually to practical matters. He could instead have been working on something much more abstract and much less useful. By which I'm trying to make an abstract point about the inevitability of staying somewhat down to earth. I mean "pure" curiosity is great, except it isn't ever really pure, and abstract mathematics isn't ever totally abstract, it's just sort of meta in relation to practical things that humans care about.
- dwroberts 4mo ago
  > So it has a connection to human problems, and eventually to practical matters. But in relation to parent post I was replying to, it did not provide an answer or solution to anything. It has much closer relation to philosophy than anything. Focusing on only ‘solutions’ in any field is shortsighted because you can’t know how the dots will connect. Someone’s seemingly pointless curiosity or experiment can unlock something unexpected, just like Boole

kleyd 4mo ago

The wording in the declaration may be a bit romanticized. But the points are valid: Is an 80 year old unsolved problem maybe unsolved because it was never prioritized? Some problems stay unsolved because few people consider them worth working on. Who is going to validate the results? Or do we skip that, with the risk of flooding the literature and collective understanding with unverified proofs?

smath 4mo ago

This reminded me of my 11 yr old who, when I give her math problems to solve, is too focused on “getting the right answer”. I’ve told her plainly, I don’t care if you get the right answer right now, I want to see your reasoning. She has yet to understand this.

conformist 4mo ago

> The authors warn the consequences are already becoming visible. AI-generated papers could overwhelm peer-review systems with low-quality work … It seems like a key problem here is that peer-review is expected but not explicitly funded/rewarded while it is probably one of the aspects where humans still add a lot of value. Academia’s incentives are hugely misaligned (… as usual unfortunately).

armchairhacker 4mo ago
Math is one field where you can mechanically prove a paper's findings. The only thing that would need to be judged is the (verified) statement's importance.
- conformist 4mo ago
  Yes in theory, but not yet in practice because not everything is fully formalised.

hyperpape 4mo ago

Even from the most purely instrumental perspective, what we care about is our ability to make use of correct answers, which is quite distinct from the possession of correct answers. There are many theorems that aren't directly interesting, but whose proof requires techniques that are of substantial further interest, that lead to new domains, and/or new practical applications. Simply being handed a proof for those theorems isn't enough--we require the ability to apply those techniques in the real world, or discover further areas of mathematical research that build on that proof or its techniques. It may be that AI can build on its own work for the long-term, but so far, AI does best at exploration in areas that have precisely specified and measurable goals. Actually creating understanding, and making use of mathemtical results outside of pure mathematics is more challenging than simply creating proofs. I think the field will figure out how to make use of AI, and it will be better off for it. But that is not the same as just saying "answers good, grog want more answers."

barrkel 4mo ago

A statement that some proposition is true or false is usually less useful than a new framework for understanding the class of problem. A machine that takes longer and longer to prove propositions in ever more inscrutable ways is hardly useful at all. The machine too needs to produce more generalizable and comprehensible systems, for it to scale up its own conceptualization. Needing to load all the new mathematics in the context window won't be great either.

analognoise 4mo ago

> But not because it's a jobs program for nerds. We’re becoming increasingly embarrassing as a society.

zerobees 4mo ago

Culturally, mathematics is a jobs program for nerds. The field very explicitly takes pride in working on problems that have no obvious applications, and most practitioners are funded publicly or supported by private endowments, with zero pressure to deliver specific results. Of course, this produces useful results every now and then, but it's not like we pursued ruthless efficiency / maximum rate of knowledge advancement before. We just let them do their thing, essentially treating them as artists and letting them pursue the craft for its own sake. If we weren't interested in maximum throughput before, why is that an objective now?

bdamm 4mo ago
Except, it was true before and it is still true today that the best "artists" whether graphic or mathematic, are the ones that do somehow manage to cross the chasm of pure research and providing a tangible benefit to their benefactors. That aspect of understanding your customer is not changed by the presence of AI.
orbital-decay 4mo ago
Pure mathematics != applied mathematics
141205 4mo ago
Hardy would agree with the viewpoint that you espouse but it would be pushed back against by Arnol'd, Poincare, Gauss, Von Neumann, and even Grothendiek: Arnol'd and Poincare were vituperatively against the division between "pure" and "applied" mathematics; they considered mathematics and physics interchangeable, and Arnol'd lamented that the field had lost a large amount of funding/prestige/relevance due to groups like the Bourbaki that took a purely aesthetic view; Gauss had a critical view of problems like Fermat's last theorem (he felt that you could construct infinitely many such problems, and felt that attempting to prove it was a generally useless endeavor), along with outright calling pure mathematics worthless; but while Von Neumann and Grothendiek were more moderate, both were critical of the field losing motivation/quality as it strayed away from empirical science into—quoting Von Neumann—"abstract inbreeding". Arnold's polemics are perhaps the most infamous and easily found online (see "On Teaching Mathematics"), but the written opinions of Poincare et seq. are also easy to find. Even today the vast majority of research funding for mathematics, at least in the United States, is dolled out for highly applied fields like partial differential equations. The field does not even close to unanimously (contemporarily or historically) "explicitly take pride" in working on problems that have no obvious application, or being a "jobs program for nerds": the notion of such "pure" or "nonapplied" mathematics is at the very least a highly fractious and controversial subject, with a number of big names taking opposing viewpoints (often vehemently). I think your picture of the field is over-represented on the internet, much like the fixation on certain niche fields: Category Theory, Homotopy Type Theory or, worst of all, outright dubious fields like Geometric Algebra; fields with a large number of online promoters, but with much less funding and relevance in the actual academic space. Of course there are reputable people with PHDs that feel this way,—but I can only imagine that there's a legion of tyros, pop math consumers, and undergraduate students who disproportionately promote this viewpoint.

ajs1998 4mo ago

Spoken like someone that has no idea why mathematicians are important.

IshKebab 4mo ago

Maths pretty much is a jobs program for nerds though. It occasionally produces results that are practically useful for society but there's absolutely no way the vast majority of today's maths research falls into that category. There are definitely exceptions, like crypto. I still think it would be pretty silly to stop maths research anyway. And anyway part of the job of maths researchers is to teach maths to undergrads and that's obviously enormously useful to society. But on the scale of "how useful is this research to society" it's dead last after engineering, chemistry, biology, and physics. Well maybe computer science would be last actually!

solid_fuel 4mo ago

>> However, the declaration argues math is more than a machine for producing correct answers. > There might be more to maths than that, but that is definitely the most important part. I love science funding. But not because it's a jobs program for nerds. I can produce an infinite number of verifiably correct papers, if that's all that matters. 1 + 1 = 2 1 + 2 = 3 1 + 3 = 4 1 + 4 = 5 1 + 5 = 6 Shall I continue? Or do you think that choosing which questions to answer might have some level of importance, in addition to getting correct answers?