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This is not a remark about AI, but there's something funny about mathematics in that every novel result is broadly perceived as a big deal. We attach basically

This is not a remark about AI, but there's something funny about mathematics in that every novel result is broadly perceived as a big deal. We attach basically zero value to writing a new program that hasn't existed before, or a piece of text that hasn't existed before. It's boring, or even a net negative, unless you can show that the result benefits the world in some way. We'd find it weird if OpenAI put out a release saying that an LLM authored an interesting blog post. For mathematics, I think it's really a matter of two things. First, the generation of proof was so severely resource-constrained on the human end that they could actually afford to celebrate every contribution - akin to how software engineering would look like if you had just 200 active SWEs in the entire world. But compounding that, mathematics is basically the only scientific discipline that rejected any notion of utility. It would be fundamentally wrong for you to ask what's the value of solving the Erdős–Hajnal conjecture; the value is that it's solved.

arm32 2mo ago
So I suppose the value is that something like this gets used as a primitive to solve something that actually has impact. Ah, mathematics, never change!
EmilStenstrom 2mo ago
The reason novelty matters for mathematics is that they strictly deduplicate all claims. If someone claim they proved something that we already knew was solved, than that wouldn't be considered novelty. Novelty and deduplication is the combo here. This is not true for blog posts.
dyauspitr 2mo ago
The difference is discovering or proving a universal truth that will go into the corpus of human knowledge forever versus some app to shuttle money around or help people count how long they’re sleeping. It has gravitas unlike some nifty super performant text editor.
not-a-llm 2mo ago
there is no "software" that a lot of people want, yet nobody managed to create yet because they failed too due to it was being hard to implement (excluding AGI/ASI which is not really software)
- kridsdale1 2mo ago
  This is not true. What is the perfect video game that makes the user infinitely happy? What is the perfect economy optimizing program? What algorithm can solve political strife?
  - QuantumFunnel 2mo ago
    As with all things, the answer is always "it depends" based on what is being optimized
- flaburgan 2mo ago
  5 minutes of wikipedia search would give you plenty examples of complicated software engineering problems that would have a big impact on everyone's life.
- zerobees 2mo ago
  As a person who has a number of relatively niche hobbies, I assure you that this is not true. There's a ton of simple things that can be build and will make an immediate difference in the lives of thousands. Watch the workflow any musician, videographer, machinist, etc - they're full of small, weird inefficiencies that AI hasn't really solved for them. It's just that you can't build a billion-dollar company around it. No one could go to a VC and say "we're going to be the Uber of focus stacking and dust removal for microscopy" or "we're the Uber of aligning the beats in two audio tracks".
- addaon 2mo ago
  > there is no "software" that a lot of people want, yet nobody managed to create yet because they failed too due to it was being hard to implement (excluding AGI/ASI which is not really software) What!? I can think of about a billion examples... but for one, I'm still waiting for a good enough CFD/FEM coupled system to model paraglider dynamics across collapse/recovery. And I expect to be waiting quite a while.
andai 2mo ago
>mathematics is basically the only scientific discipline that rejected any notion of utility I think this might depend on the department, but I was at a pure math department last year, and struggling with my Linear Algebra textbook (written by the professor, incidentally, who was not a great communicator). I consulted the machines, and learned, to my great delight, that linear algebra is used in like 20 different fields in the real world. It's "perhaps the most applied branch of mathematics in existence". I complained in the group chat, that our didactic materials, specifically tasked with providing motivation and concrete examples, did not contain a single application, of this most richly applied field. I was promptly pilloried, and shunned. (Apparently that particular department was the wrong one, to ask a question like that!)
- carlmr 2mo ago
  >(Apparently that particular department was the wrong one, to ask a question like that!) Yes, the math department. In any case linear algebra, stochastics, calculus; plenty of engineering and science applications for all these.
- BeetleB 2mo ago
  > I complained in the group chat, that our didactic materials, specifically tasked with providing motivation and concrete examples, did not contain a single application, of this most richly applied field. > I was promptly pilloried, and shunned. Heh. In my day I may have participated in the pillorying. I do think that there is value/merit in professors mentioning real world applications, where they exist. What they're sensitive about are the theorems where there aren't real world applications. They don't want to (and shouldn't) justify them. So even when there are real world applications, the posture is "Who knows if someone is making good use of this in the world somewhere? I don't care. It's not why we learn or teach this!"
  - deleted 2mo ago
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  - arrowsmith 2mo ago
    Knowledge for its own sake is great, but it's worth noting that many "useless" fields of mathematics turned out to be very practical in the long run. Number theory was long thought to have no practical application, but now it's the backbone of cryptography. Boolean algebra was developed in the 19th century (George Boole died in 1864), decades before it was used to build computers. Those "useless" theorems being proved today may turn out to unlock a world-changing technology centuries from now. When the breakthrough comes we'll be grateful for the people who laid the foundations.
redsocksfan45 2mo ago
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tarruda 2mo ago
> It would be fundamentally wrong for you to ask what's the value of solving the Erdős–Hajnal conjecture; the value is that it's solved. I suspect the value is in showing the potential that LLMs have in developing new breakthroughs.
bawolff 2mo ago
We generally do give a lot of credit to programs that do something novel. The first gets a lot of credit. But if its just another CRUD app, nobody cares. Its the same with proofs. First time someone proves something gets a lot of credit. The second proof for the same theorem gets a lot less buzz. But even then, math proofs mostly get buzz when its something famous or at least important. Proving a random lemma usually doesn't get much buzz.
jey 2mo ago
I'm not a mathematician, but I don't think that's true..? It's just that some problems are considered "hard" or known to have been "open" for a long time or that involve some clever/pioneering new technique. There's tons of math papers out there that are in some technical sense a novel contribution but in practice just languish without much attention except maybe from like two other people working in the same subfield.
BeetleB 2mo ago
> It would be fundamentally wrong for you to ask what's the value of solving the Erdős–Hajnal conjecture; the value is that it's solved. No, the value is that Erdos's name is attached to it. Lots of mathematicians prove things they don't publish, or their manuscripts get rejected - not because of a flaw in the proof but because no one cares about the theorem they proved. And I'm sure it'll be the case with LLM models performing proofs. It'll be notable only when the theorem is a known one that people have had difficulty proving.
- dnautics 2mo ago
  > No, the value is that Erdos's name is attached to it. That's unnecessarily reductive. you could have said "most of the value is that erdos' name is attached to it"
ufo 2mo ago
In math, the utility lies in the proof itself. A novel proof of a hard problem usually comes with new insights and abstractions that help solve even more mathematical problems. To go with your analogy, mathematicians care more about the source code of the program than about the result of the program. But I'm afraid that we will see things change with the increase of vibecoded proof slop. A black box proof is not as useful, even if it is correct.
throw310822 2mo ago
> We attach basically zero value to writing a new program What does it mean "new"? And, was it a difficult or trivial accomplishment? A solution to a well known open math problem is both new and non-trivial- you know that many, very smart, very well trained human experts have dedicated time to the problem and haven't been able to solve it, despite good incentives.
- hnisfulomrons 2mo ago
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Wowfunhappy 2mo ago
Biologists celebrate the discovery of new species of fruit fly hidden deep in the Amazon rainforest. Astronomers celebrate the discovery of new giant rocks located zillions of light years away. Neither of these things is immediately “useful” to the world, although they may turn out to be enormously beneficial in ways we can’t immediately predict. To me, these fields also feel central to the human experience—discovering new types of life, or learning more about our place in the universe. I don’t think a mathematical proof is any different.
_the_inflator 2mo ago
Wow, you couldn't be more wrong here. Math is something humans invented and is a model, nothing else. There is no logic per se, but a model that works quite well for us. I studied Math and CS as a very highly gifted and quickly found out, there is no beauty of Mathematical Logic, only humans approval of what they deem most accurate. A good example is set theory. Cantor was not openly welcomed after he introduced his "theory" to others. In fact, he was received quite some pushback and hostility - this doesn't sound like someone received love the mathematical logic's way. In fact, the story of Cantor is really a tragic one. He left math for quite some time, due to the pushback. Only later humans accepted his theory and found it useful. Well, well, what is Mathematical Logic and what not is after all just broad consensus by humans. And if you go deeper, you will hear more of these stories. Math is anything else but logic. Proofs are religious things, often so complicated, they are simply accepted as "approved by a committee". Many profs cannot really explain simple proofs, they refer to the textbook. This doesn't sound like romance nor easily reproducible logic. After all, we deal with human beings.
- ACCount37 2mo ago
  No matter what humans do, it somehow ends up being a popularity contest. It's almost like a twisted mirror of Conway's law.
- throwoutway 2mo ago
  You're also wrong "Math is something humans invented" Majority of mathematicians are platonists and believe arithmetic was existed and was discovered and was not "invented". "There is no logic per se" There is logic to it! Most logicians are mathematicians at heart. See Russel, Godel, Hilbert, etc "no beauty of Mathematical Logic" Mathematicians do focus on beauty. Entire books have been written on this. G.H. Hardy in A Mathematician's Apology even said math MUST be beautfiul "Proofs are religious things" What are you going on about...
  - cfiggers 2mo ago
    Consensus may give a hint to what is or isn't reality. But consensus—even expert consensus—does not determine reality. Experts can be wrong. Most of the experts, even, can be wrong simultaneously. Philosophy is the exercise of testing ideas for oneself in the laboratory of one's own mind. When I test the idea that math is discovered in my own mind, from my own perspective, with my own experience and education brought to bear, I find it unconvincing. When you test the same idea in the laboratory of your mind, with your experience and your education applied, and get a different result, that is interesting. Your result is relevant information to me. If nothing else, it's a good prompt/trigger for me to revisit my earlier conclusion and see if it still holds. But your disagreement—or indeed, the disagreement of a majority of trained mathematicians—does not constitute an automatic reason for me to conclusively determine that you/they are right and I am wrong. I still have my own examination of the concept, with my own supporting and detracting arguments. And the result of my examination continues to be that math being invented is the significantly more persuasive view.
Gtex555 2mo ago
A lot of mathematics often takes 100+ years to find a practical use because we have developed it so much that we have use all the easy maths. Things like CS or SWE are so new that you can still find stuff today that can be used tomorrow. Things like computation and cryptography was all discovered like 100 years before we had a practical use for it. Its an example of late stage scientific discipline. Things like physics, chemistry and biology will get here as well eventually.
anp 2mo ago
It’s far from a perfect analogy but I would imagine that people were pretty hyped about the novelty of the first legitimately useful compiled programs where they didn’t have to allocate their own registers. I wonder how long it took for that novelty to wear off? Or in other words I’d argue novelty is contextual and that these kinds of discoveries’ novelty will eventually wear off too but for right now it’s pretty cool that the “math discovery compiler” works well enough to do this (again imperfect analogy).
hellohello2 2mo ago
This feels mistaken; we develop abstract objects i.e. graphs based on real-world utility or whatever. As we try to improve our understanding of graphs, we value proofs that help us do so, or help other fields of mathematics. We assign 0 value to random proofs about stuff no one cares about... This conjecture had value, simply because some people found it interesting. It is not really different from music, in a sense.
dwohnitmok 2mo ago
> This is not a remark about AI, but there's something funny about mathematics in that every novel result is broadly perceived as a big deal. This isn't true using the level of originality you're implying with your software examples. Technically speaking, many novel mathematics proofs are written all the time (quite a few textbook exercises are actually technically novel problems that have never been posed before they were written in a textbook!) that get absolutely no fanfare. Overwhelmingly though they are not very original or difficult and really just required a fairly routine combination of different pre-existing techniques, even if technically speaking that combination didn't exist before. Those textbook problems are hence easy and therefore not given much public attention even if they are technically novel problems. Indeed over the course of developing a new mathematical result, many many novel results are glossed over to the extent that even their proofs are left out ("as an exercise for the reader") because they are fairly trivial. This is true for the overwhelming majority of new software as well. A new CRUD program may, technically speaking, be novel, but it's almost certainly just a routine combination of different pre-existing things. Mathematics open problems that are actually named are generally problems that have resisted the low hanging fruit of the most obvious combinations of pre-existing problems. When those are solved they are a big deal precisely because they usually require some novelty! Similarly in software, if someone were to create a new kind of database that solves a variety of new classes of problems that current databases fail to solve that would be a big deal! Truly novel software is also perceived as a big deal. Software that is, technically speaking new, but doesn't actually stray far from a fairly obvious remix of pre-existing techniques, isn't really celebrated. In both software and mathematics, the intuitive benchmark is if other practitioners in the field look at the result and would say "Wow! How did you do that?" Professional software developers generally don't look at, e.g. a new blogging platform, and boggle at "Wow! How did they make that?!!"
fragmede 2mo ago
> We attach basically zero value to writing a new program that hasn't existed before We don't? People write new programs that go on to be successful software companies that make millions of dollars! Basic CRUD apps make money for their creators in their niche! There's so much money in software that it's taking over the world. The market is different, you're not getting worldwide household recognition for every little fart or sneeze of programming you output, but how can you say that we attach zero value to new programs when the history of computers is insanely valuable companies making new software and selling it. Windows, Oracle, mongoDB, etc.
homeslice1234 2mo ago
> It would be fundamentally wrong for you to ask what's the value of solving the Erdős–Hajnal conjecture.. I'm not sure about this, TBH I ask myself this quite frequently. In a world where machines are routinely solving very high end math problems every day, producing more proofs than humans would ever really be able to absorb or fully understand.... would that be a good thing? Would that in itself be valueable? It feels like that is a probable future, but I'm not sure that would actually be something we want. I think there's probably more than "value is that it's solved"
calebkaiser 2mo ago
I mean, OpenAI delayed the public release of GPT-2 back in 2019 because it seemed capable of authoring interesting blog posts (that also happened to be untrue). It was a pretty big deal the first time Transformer models were capable of generating that kind of output--no one found it weird. We've just grown to take it for granted that large Transformer models are this capable. The same cycle is happening now for a harder frontier. And proofs represent a pretty good benchmark for model capabilities, so a new model proving a result that a previous model didn't is generally notable in the same way that a model scoring higher on a benchmark is. I'm sure we'll take it for granted in the not-too-distant future.
UltraSane 2mo ago
Proving a novel math theroem now is incredibly hard because all the easy ones have already been proven.
jojva 2mo ago
Isn't it immediately obvious that solving something that humans have been unable to do for decades or more is the most tangible proof of ASI, or at the very least pretty good AGI?
- drdexebtjl 2mo ago
  Is this something humans have been unable to do? There’s only so many people with the necessary skills to solve this. And you need these humans to choose to spend their time solving this, and not something else.
  - famouswaffles 2mo ago
    >Is this something humans have been unable to do? It's a famous open problem so yeah >There’s only so many people with the necessary skills to solve this. And you need these humans to choose to spend their time solving this, and not something else. Sure, but that doesn't mean a lot of very skilled people hadn't attempted and failed to solve this.
    - koe123 2mo ago
      And you’ve known about this problem for how many minutes?
      - famouswaffles 2mo ago
        Long enough to know this has been on the Wikipedia page of open math problems for several years.
QuesnayJr 2mo ago
It's newsworthy because it's a milestone. It was something no human was able to do (despite trying very hard), but a machine did. Humans have written lots of interesting blog posts. The idea that mathematics has rejected any notion of utility is absurd. It's not like topics get picked at random. Conjectures like this are interesting because they are a test of our understanding. The problem sounds easy, but apparently was quite hard.
hyperpape 2mo ago
> rejected any notion of utility. It would be fundamentally wrong for you to ask what's the value of solving the Erdős–Hajnal conjecture; the value is that it's solved. I disagree. Mathematicians care about the utility of a result. It is just that they regard mathematical understanding as a valid type of utility, and that can be arbitrarily far removed from practical utility. But a proof that doesn't help anyone understand anything interesting is not valued. I could go out and define some pointless construction and create proofs about it immediately. It would only matter if I connect it to some other subject of interest within math. I would argue that mathematical understanding is valuable for extrinsic reasons, but it is true that by the time you're a math grad student, you're usually willing to pursue it for no external purpose. Although not a mathematician, Daniel Dennett had a wonderful example about higher order truths of "chmess". https://personal.lse.ac.uk/robert49/teaching/ph445/notes/dennett.pdf https://personal.lse.ac.uk/robert49/teaching/ph445/notes/den...
- roncesvalles 2mo ago
  It seems in mathematics that the utility of a problem is directly correlated with how difficult it is to solve, for some odd reason. If I defined some pointless construction and it turned out to be very difficult to prove, it would automatically over time become considered a "high utility" mathematics problem (again, for some odd reason). Mathematics is largely just smart people working on pointless puzzles, and only by coincidence do these puzzles turn out to have practical applications (it cannot be predicted). Or I guess all the obviously practical problems in mathematics have already been solved -- we're now in a world where math is rarely the limiting factor for human progress (like it was, say, pre-calculus; was FFT the last significant unblock from math?). It's such a waste of the best human minds. Or maybe the best human minds are actually doing something else, maybe we only notice the handful of Terence Taos, not the hundreds of people of equal brilliance who realized pure math is pointless and decided to pursue physics, rocketry, or quantitative finance.
  - ves 2mo ago
    Writing that mathematics is a waste is such a hilariously ignorant comment to make on a programming forum.
breezybottom 2mo ago
Mathematics isn't a scientific discipline.
TacticalCoder 2mo ago
Mathematics is what everything else is built upon. I'm no mathematician but a very good friend of mine is: teacher at a big uni, researcher. Pure math. His entire life he's had --and still has-- to deal with comments like the one you just made, implying that the only value is solving pointless conjecture (if it wasn't pointless, according to your logic, then the value wouldn't be that it is solved). Truth is to be found in this xkcd: https://xkcd.com/435/ https://xkcd.com/435/
dkural 2mo ago
"every novel result is broadly perceived as a big deal" is not at all true. AI companies hype any novel result as proof that AI is good for mathematics, but professional mathematicians write tens of thousands of papers every year, and for 99.99% of them, nobody cares or writes it up. Mathematicians certainly don't go around saying each and every novel proof in their papers are a big deal. Do you have any evidence supporting your statement that it is "broadly perceived" (by whom?) as a big deal?
p1necone 2mo ago
> there's something funny about mathematics in that every novel result is broadly perceived as a big deal. Is this true? Or is it just that mathematics is an isolated enough field that only the results that are a big deal get broadcast widely to the public. I know little of the inner workings of the field of mathematics, but my naive assumption would be that there's probably lots of novel but boring results being discovered/proven all the time and we don't hear about them because no-one outside of the person doing the work and a handful of their colleagues is really that interested in it. Likely a lot aren't published in any way, because they're just stepping stones towards the goal of the actual area/paper/whatever being worked on.
Davidzheng 2mo ago
Sorry but I completely disagree with your statement that "every novel result is broadly perceived as a big deal". Most results certainly are not consider this way (even though the average result has difficulties that are much higher than novel computer program you may have in mind -- no offense)
samus 2mo ago
There is a tension between applied and theoretical mathematics, and it's as old as the whole science itself. Mathematics arose to solve practical problems (land surveying and division, as well as trade) and recognizing the underlying principles make it possible to abstract that knowledge. That might lead to centuries of ivory tower activity and what could be regarded as a purely artistic pursuit until somebody figures out how to apply a theory to a new practical problem. Or relations to another theory are discovered, and suddenly there is a new approach to previously intractable questions. A good example is be number theory, which is the foundation of modern cryptography.

arm32 2mo ago

So I suppose the value is that something like this gets used as a primitive to solve something that actually has impact. Ah, mathematics, never change!

EmilStenstrom 2mo ago

The reason novelty matters for mathematics is that they strictly deduplicate all claims. If someone claim they proved something that we already knew was solved, than that wouldn't be considered novelty. Novelty and deduplication is the combo here. This is not true for blog posts.

dyauspitr 2mo ago

The difference is discovering or proving a universal truth that will go into the corpus of human knowledge forever versus some app to shuttle money around or help people count how long they’re sleeping. It has gravitas unlike some nifty super performant text editor.

not-a-llm 2mo ago

there is no "software" that a lot of people want, yet nobody managed to create yet because they failed too due to it was being hard to implement (excluding AGI/ASI which is not really software)

kridsdale1 2mo ago
This is not true. What is the perfect video game that makes the user infinitely happy? What is the perfect economy optimizing program? What algorithm can solve political strife?
- QuantumFunnel 2mo ago
  As with all things, the answer is always "it depends" based on what is being optimized
flaburgan 2mo ago
5 minutes of wikipedia search would give you plenty examples of complicated software engineering problems that would have a big impact on everyone's life.
zerobees 2mo ago
As a person who has a number of relatively niche hobbies, I assure you that this is not true. There's a ton of simple things that can be build and will make an immediate difference in the lives of thousands. Watch the workflow any musician, videographer, machinist, etc - they're full of small, weird inefficiencies that AI hasn't really solved for them. It's just that you can't build a billion-dollar company around it. No one could go to a VC and say "we're going to be the Uber of focus stacking and dust removal for microscopy" or "we're the Uber of aligning the beats in two audio tracks".
addaon 2mo ago
> there is no "software" that a lot of people want, yet nobody managed to create yet because they failed too due to it was being hard to implement (excluding AGI/ASI which is not really software) What!? I can think of about a billion examples... but for one, I'm still waiting for a good enough CFD/FEM coupled system to model paraglider dynamics across collapse/recovery. And I expect to be waiting quite a while.

andai 2mo ago

>mathematics is basically the only scientific discipline that rejected any notion of utility I think this might depend on the department, but I was at a pure math department last year, and struggling with my Linear Algebra textbook (written by the professor, incidentally, who was not a great communicator). I consulted the machines, and learned, to my great delight, that linear algebra is used in like 20 different fields in the real world. It's "perhaps the most applied branch of mathematics in existence". I complained in the group chat, that our didactic materials, specifically tasked with providing motivation and concrete examples, did not contain a single application, of this most richly applied field. I was promptly pilloried, and shunned. (Apparently that particular department was the wrong one, to ask a question like that!)

carlmr 2mo ago
>(Apparently that particular department was the wrong one, to ask a question like that!) Yes, the math department. In any case linear algebra, stochastics, calculus; plenty of engineering and science applications for all these.
BeetleB 2mo ago
> I complained in the group chat, that our didactic materials, specifically tasked with providing motivation and concrete examples, did not contain a single application, of this most richly applied field. > I was promptly pilloried, and shunned. Heh. In my day I may have participated in the pillorying. I do think that there is value/merit in professors mentioning real world applications, where they exist. What they're sensitive about are the theorems where there aren't real world applications. They don't want to (and shouldn't) justify them. So even when there are real world applications, the posture is "Who knows if someone is making good use of this in the world somewhere? I don't care. It's not why we learn or teach this!"
- deleted 2mo ago
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- arrowsmith 2mo ago
  Knowledge for its own sake is great, but it's worth noting that many "useless" fields of mathematics turned out to be very practical in the long run. Number theory was long thought to have no practical application, but now it's the backbone of cryptography. Boolean algebra was developed in the 19th century (George Boole died in 1864), decades before it was used to build computers. Those "useless" theorems being proved today may turn out to unlock a world-changing technology centuries from now. When the breakthrough comes we'll be grateful for the people who laid the foundations.

redsocksfan45 2mo ago

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tarruda 2mo ago

> It would be fundamentally wrong for you to ask what's the value of solving the Erdős–Hajnal conjecture; the value is that it's solved. I suspect the value is in showing the potential that LLMs have in developing new breakthroughs.

bawolff 2mo ago

We generally do give a lot of credit to programs that do something novel. The first gets a lot of credit. But if its just another CRUD app, nobody cares. Its the same with proofs. First time someone proves something gets a lot of credit. The second proof for the same theorem gets a lot less buzz. But even then, math proofs mostly get buzz when its something famous or at least important. Proving a random lemma usually doesn't get much buzz.

jey 2mo ago

I'm not a mathematician, but I don't think that's true..? It's just that some problems are considered "hard" or known to have been "open" for a long time or that involve some clever/pioneering new technique. There's tons of math papers out there that are in some technical sense a novel contribution but in practice just languish without much attention except maybe from like two other people working in the same subfield.

BeetleB 2mo ago

> It would be fundamentally wrong for you to ask what's the value of solving the Erdős–Hajnal conjecture; the value is that it's solved. No, the value is that Erdos's name is attached to it. Lots of mathematicians prove things they don't publish, or their manuscripts get rejected - not because of a flaw in the proof but because no one cares about the theorem they proved. And I'm sure it'll be the case with LLM models performing proofs. It'll be notable only when the theorem is a known one that people have had difficulty proving.

dnautics 2mo ago
> No, the value is that Erdos's name is attached to it. That's unnecessarily reductive. you could have said "most of the value is that erdos' name is attached to it"

ufo 2mo ago

In math, the utility lies in the proof itself. A novel proof of a hard problem usually comes with new insights and abstractions that help solve even more mathematical problems. To go with your analogy, mathematicians care more about the source code of the program than about the result of the program. But I'm afraid that we will see things change with the increase of vibecoded proof slop. A black box proof is not as useful, even if it is correct.

throw310822 2mo ago

> We attach basically zero value to writing a new program What does it mean "new"? And, was it a difficult or trivial accomplishment? A solution to a well known open math problem is both new and non-trivial- you know that many, very smart, very well trained human experts have dedicated time to the problem and haven't been able to solve it, despite good incentives.

hnisfulomrons 2mo ago
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Wowfunhappy 2mo ago

Biologists celebrate the discovery of new species of fruit fly hidden deep in the Amazon rainforest. Astronomers celebrate the discovery of new giant rocks located zillions of light years away. Neither of these things is immediately “useful” to the world, although they may turn out to be enormously beneficial in ways we can’t immediately predict. To me, these fields also feel central to the human experience—discovering new types of life, or learning more about our place in the universe. I don’t think a mathematical proof is any different.

_the_inflator 2mo ago

Wow, you couldn't be more wrong here. Math is something humans invented and is a model, nothing else. There is no logic per se, but a model that works quite well for us. I studied Math and CS as a very highly gifted and quickly found out, there is no beauty of Mathematical Logic, only humans approval of what they deem most accurate. A good example is set theory. Cantor was not openly welcomed after he introduced his "theory" to others. In fact, he was received quite some pushback and hostility - this doesn't sound like someone received love the mathematical logic's way. In fact, the story of Cantor is really a tragic one. He left math for quite some time, due to the pushback. Only later humans accepted his theory and found it useful. Well, well, what is Mathematical Logic and what not is after all just broad consensus by humans. And if you go deeper, you will hear more of these stories. Math is anything else but logic. Proofs are religious things, often so complicated, they are simply accepted as "approved by a committee". Many profs cannot really explain simple proofs, they refer to the textbook. This doesn't sound like romance nor easily reproducible logic. After all, we deal with human beings.

ACCount37 2mo ago
No matter what humans do, it somehow ends up being a popularity contest. It's almost like a twisted mirror of Conway's law.
throwoutway 2mo ago
You're also wrong "Math is something humans invented" Majority of mathematicians are platonists and believe arithmetic was existed and was discovered and was not "invented". "There is no logic per se" There is logic to it! Most logicians are mathematicians at heart. See Russel, Godel, Hilbert, etc "no beauty of Mathematical Logic" Mathematicians do focus on beauty. Entire books have been written on this. G.H. Hardy in A Mathematician's Apology even said math MUST be beautfiul "Proofs are religious things" What are you going on about...
- cfiggers 2mo ago
  Consensus may give a hint to what is or isn't reality. But consensus—even expert consensus—does not determine reality. Experts can be wrong. Most of the experts, even, can be wrong simultaneously. Philosophy is the exercise of testing ideas for oneself in the laboratory of one's own mind. When I test the idea that math is discovered in my own mind, from my own perspective, with my own experience and education brought to bear, I find it unconvincing. When you test the same idea in the laboratory of your mind, with your experience and your education applied, and get a different result, that is interesting. Your result is relevant information to me. If nothing else, it's a good prompt/trigger for me to revisit my earlier conclusion and see if it still holds. But your disagreement—or indeed, the disagreement of a majority of trained mathematicians—does not constitute an automatic reason for me to conclusively determine that you/they are right and I am wrong. I still have my own examination of the concept, with my own supporting and detracting arguments. And the result of my examination continues to be that math being invented is the significantly more persuasive view.

Gtex555 2mo ago

A lot of mathematics often takes 100+ years to find a practical use because we have developed it so much that we have use all the easy maths. Things like CS or SWE are so new that you can still find stuff today that can be used tomorrow. Things like computation and cryptography was all discovered like 100 years before we had a practical use for it. Its an example of late stage scientific discipline. Things like physics, chemistry and biology will get here as well eventually.

anp 2mo ago

It’s far from a perfect analogy but I would imagine that people were pretty hyped about the novelty of the first legitimately useful compiled programs where they didn’t have to allocate their own registers. I wonder how long it took for that novelty to wear off? Or in other words I’d argue novelty is contextual and that these kinds of discoveries’ novelty will eventually wear off too but for right now it’s pretty cool that the “math discovery compiler” works well enough to do this (again imperfect analogy).

hellohello2 2mo ago

This feels mistaken; we develop abstract objects i.e. graphs based on real-world utility or whatever. As we try to improve our understanding of graphs, we value proofs that help us do so, or help other fields of mathematics. We assign 0 value to random proofs about stuff no one cares about... This conjecture had value, simply because some people found it interesting. It is not really different from music, in a sense.

dwohnitmok 2mo ago

> This is not a remark about AI, but there's something funny about mathematics in that every novel result is broadly perceived as a big deal. This isn't true using the level of originality you're implying with your software examples. Technically speaking, many novel mathematics proofs are written all the time (quite a few textbook exercises are actually technically novel problems that have never been posed before they were written in a textbook!) that get absolutely no fanfare. Overwhelmingly though they are not very original or difficult and really just required a fairly routine combination of different pre-existing techniques, even if technically speaking that combination didn't exist before. Those textbook problems are hence easy and therefore not given much public attention even if they are technically novel problems. Indeed over the course of developing a new mathematical result, many many novel results are glossed over to the extent that even their proofs are left out ("as an exercise for the reader") because they are fairly trivial. This is true for the overwhelming majority of new software as well. A new CRUD program may, technically speaking, be novel, but it's almost certainly just a routine combination of different pre-existing things. Mathematics open problems that are actually named are generally problems that have resisted the low hanging fruit of the most obvious combinations of pre-existing problems. When those are solved they are a big deal precisely because they usually require some novelty! Similarly in software, if someone were to create a new kind of database that solves a variety of new classes of problems that current databases fail to solve that would be a big deal! Truly novel software is also perceived as a big deal. Software that is, technically speaking new, but doesn't actually stray far from a fairly obvious remix of pre-existing techniques, isn't really celebrated. In both software and mathematics, the intuitive benchmark is if other practitioners in the field look at the result and would say "Wow! How did you do that?" Professional software developers generally don't look at, e.g. a new blogging platform, and boggle at "Wow! How did they make that?!!"

fragmede 2mo ago

> We attach basically zero value to writing a new program that hasn't existed before We don't? People write new programs that go on to be successful software companies that make millions of dollars! Basic CRUD apps make money for their creators in their niche! There's so much money in software that it's taking over the world. The market is different, you're not getting worldwide household recognition for every little fart or sneeze of programming you output, but how can you say that we attach zero value to new programs when the history of computers is insanely valuable companies making new software and selling it. Windows, Oracle, mongoDB, etc.

homeslice1234 2mo ago

> It would be fundamentally wrong for you to ask what's the value of solving the Erdős–Hajnal conjecture.. I'm not sure about this, TBH I ask myself this quite frequently. In a world where machines are routinely solving very high end math problems every day, producing more proofs than humans would ever really be able to absorb or fully understand.... would that be a good thing? Would that in itself be valueable? It feels like that is a probable future, but I'm not sure that would actually be something we want. I think there's probably more than "value is that it's solved"

calebkaiser 2mo ago

I mean, OpenAI delayed the public release of GPT-2 back in 2019 because it seemed capable of authoring interesting blog posts (that also happened to be untrue). It was a pretty big deal the first time Transformer models were capable of generating that kind of output--no one found it weird. We've just grown to take it for granted that large Transformer models are this capable. The same cycle is happening now for a harder frontier. And proofs represent a pretty good benchmark for model capabilities, so a new model proving a result that a previous model didn't is generally notable in the same way that a model scoring higher on a benchmark is. I'm sure we'll take it for granted in the not-too-distant future.

UltraSane 2mo ago

Proving a novel math theroem now is incredibly hard because all the easy ones have already been proven.

jojva 2mo ago

Isn't it immediately obvious that solving something that humans have been unable to do for decades or more is the most tangible proof of ASI, or at the very least pretty good AGI?

drdexebtjl 2mo ago
Is this something humans have been unable to do? There’s only so many people with the necessary skills to solve this. And you need these humans to choose to spend their time solving this, and not something else.
- famouswaffles 2mo ago
  >Is this something humans have been unable to do? It's a famous open problem so yeah >There’s only so many people with the necessary skills to solve this. And you need these humans to choose to spend their time solving this, and not something else. Sure, but that doesn't mean a lot of very skilled people hadn't attempted and failed to solve this.
  - koe123 2mo ago
    And you’ve known about this problem for how many minutes?
    - famouswaffles 2mo ago
      Long enough to know this has been on the Wikipedia page of open math problems for several years.

QuesnayJr 2mo ago

It's newsworthy because it's a milestone. It was something no human was able to do (despite trying very hard), but a machine did. Humans have written lots of interesting blog posts. The idea that mathematics has rejected any notion of utility is absurd. It's not like topics get picked at random. Conjectures like this are interesting because they are a test of our understanding. The problem sounds easy, but apparently was quite hard.

hyperpape 2mo ago

> rejected any notion of utility. It would be fundamentally wrong for you to ask what's the value of solving the Erdős–Hajnal conjecture; the value is that it's solved. I disagree. Mathematicians care about the utility of a result. It is just that they regard mathematical understanding as a valid type of utility, and that can be arbitrarily far removed from practical utility. But a proof that doesn't help anyone understand anything interesting is not valued. I could go out and define some pointless construction and create proofs about it immediately. It would only matter if I connect it to some other subject of interest within math. I would argue that mathematical understanding is valuable for extrinsic reasons, but it is true that by the time you're a math grad student, you're usually willing to pursue it for no external purpose. Although not a mathematician, Daniel Dennett had a wonderful example about higher order truths of "chmess". https://personal.lse.ac.uk/robert49/teaching/ph445/notes/dennett.pdf https://personal.lse.ac.uk/robert49/teaching/ph445/notes/den...

roncesvalles 2mo ago
It seems in mathematics that the utility of a problem is directly correlated with how difficult it is to solve, for some odd reason. If I defined some pointless construction and it turned out to be very difficult to prove, it would automatically over time become considered a "high utility" mathematics problem (again, for some odd reason). Mathematics is largely just smart people working on pointless puzzles, and only by coincidence do these puzzles turn out to have practical applications (it cannot be predicted). Or I guess all the obviously practical problems in mathematics have already been solved -- we're now in a world where math is rarely the limiting factor for human progress (like it was, say, pre-calculus; was FFT the last significant unblock from math?). It's such a waste of the best human minds. Or maybe the best human minds are actually doing something else, maybe we only notice the handful of Terence Taos, not the hundreds of people of equal brilliance who realized pure math is pointless and decided to pursue physics, rocketry, or quantitative finance.
- ves 2mo ago
  Writing that mathematics is a waste is such a hilariously ignorant comment to make on a programming forum.

breezybottom 2mo ago

Mathematics isn't a scientific discipline.

TacticalCoder 2mo ago

Mathematics is what everything else is built upon. I'm no mathematician but a very good friend of mine is: teacher at a big uni, researcher. Pure math. His entire life he's had --and still has-- to deal with comments like the one you just made, implying that the only value is solving pointless conjecture (if it wasn't pointless, according to your logic, then the value wouldn't be that it is solved). Truth is to be found in this xkcd: https://xkcd.com/435/ https://xkcd.com/435/

dkural 2mo ago

"every novel result is broadly perceived as a big deal" is not at all true. AI companies hype any novel result as proof that AI is good for mathematics, but professional mathematicians write tens of thousands of papers every year, and for 99.99% of them, nobody cares or writes it up. Mathematicians certainly don't go around saying each and every novel proof in their papers are a big deal. Do you have any evidence supporting your statement that it is "broadly perceived" (by whom?) as a big deal?

p1necone 2mo ago

> there's something funny about mathematics in that every novel result is broadly perceived as a big deal. Is this true? Or is it just that mathematics is an isolated enough field that only the results that are a big deal get broadcast widely to the public. I know little of the inner workings of the field of mathematics, but my naive assumption would be that there's probably lots of novel but boring results being discovered/proven all the time and we don't hear about them because no-one outside of the person doing the work and a handful of their colleagues is really that interested in it. Likely a lot aren't published in any way, because they're just stepping stones towards the goal of the actual area/paper/whatever being worked on.

Davidzheng 2mo ago

Sorry but I completely disagree with your statement that "every novel result is broadly perceived as a big deal". Most results certainly are not consider this way (even though the average result has difficulties that are much higher than novel computer program you may have in mind -- no offense)

samus 2mo ago

There is a tension between applied and theoretical mathematics, and it's as old as the whole science itself. Mathematics arose to solve practical problems (land surveying and division, as well as trade) and recognizing the underlying principles make it possible to abstract that knowledge. That might lead to centuries of ivory tower activity and what could be regarded as a purely artistic pursuit until somebody figures out how to apply a theory to a new practical problem. Or relations to another theory are discovered, and suddenly there is a new approach to previously intractable questions. A good example is be number theory, which is the foundation of modern cryptography.