3 ms·

To further this assertion, there is almost no value to deeply esoteric math that is technically correct, but completely inapplicable to any scientific reality,

by datsci_est_2015 5mo ago

To further this assertion, there is almost no value to deeply esoteric math that is technically correct, but completely inapplicable to any scientific reality, and completely unintelligible to humans. Consider these findings deep, dark corners in the unfathomably large hyperspace of mathematics. My guess is AI will be incredibly adept at identifying these types of findings, and it will be exceedingly difficult for humans to identify what is meaningful and what is not in the slop.

yaris 5mo ago
Works of Shinichi Mochizuki immediately come to mind. He is not AI but provides very good examples of math that is useless because it is incomprehensible by (other) humans.
- seanmcdirmid 5mo ago
  Do AIs produce answers whose work is incomprehensible to humans? It seems like you could just have the AI elaborate multiple times until you were satisfied with the explanation and documentation of what went into figuring out the answer. It’s not like the AI is one shotting the answer in a single opaque query anyways.
  - datsci_est_2015 5mo ago
    Like other commenters, I think you’re also underestimating the complexity of esoteric higher level math. Consider the “Magnus Carlsen” of mathematics, who is more capable of understanding mathematics than any other human. But then also realize that that individual has probably devoted their entire career into a specific subdomain of mathematics. Within other deep recesses of mathematics, this Magnus equivalent will be less capable than their peers without years of rewiring their brain to understand the esoteric concepts and properties within that other subdomain. LLMs will be able to dig deeper and broader than any human mathematician, and find results that are completely useless to humans because it would take more than an entire lifetime to “speak the language” of the concepts the LLMs have produced. The only way those results can become useful to humans is if then the LLM itself finds a way for it to be practical to humans once again. So, no, I don’t think this represents the “democratization” of mathematics where mathematicians are no longer necessary because anyone can just prompt the LLM to explain it. The bar for entry level mathematics is lower, for sure, but research level mathematics will continue to be unapproachable for anyone who hasn’t devoted their career to it.
    - seanmcdirmid 5mo ago
      I don't get it. LLMs don't have ego, they don't have the ability to say "no, this should be obvious, I'm not going to explain further", they are just token predictors, and given context, they can generate more tokens. If you don't understand how the answer was derived? You just ask more questions and it isn't going to get bored or annoyed, it will just try to answer the questions. Is that what is offending you so much?
      - datsci_est_2015 5mo ago
        No, it doesn’t sound like you get it. It has nothing to do with the properties of LLMs and everything to do with the complexity of mathematics. Have you ever been exposed to concepts that are so complex that you feel like you could devote your entire lifetime to trying to understand it and still fall short? It’s a very humbling experience, especially if you have classmates who pick it up effortlessly. Without a human holding the reins, consider an LLM a rudderless superboat speeding erratically towards the horizon, finding and proving meaningless theorems that not even your most talented classmate could ever begin to understand. My point is the human is a critical piece to the puzzle, but not just any human, a career mathematician.
        bawolff 5mo ago
        > Have you ever been exposed to concepts that are so complex that you feel like you could devote your entire lifetime to trying to understand it and still fall short? It’s a very humbling experience, especially if you have classmates who pick it up effortlessly. > Without a human holding the reins, consider an LLM a rudderless superboat speeding erratically towards the horizon, finding and proving meaningless theorems that not even your most talented classmate could ever begin to understand. This feels like a little bit of a jump to me. AIs arent actually alive so of course someone is going to have to pose the question. They arent going to just do stuff on their own. And of course mathmaticians are going to need to interpret the results if we are to glean anything beyong if the conjecture is true or false. But you seem to be suggesting that mathematicians will have to micromanage every step. That seems like a bit of a jump which i dont see much evidence for.
  - epgui 5mo ago
    It’s easy to imagine this being a problem both in quality and in volume. Verifiable work is less valuable than verified work. And noise is always costly.
- pfdietz 5mo ago
  It's not that it's incomprehensible, it's that it appears to be wrong.
psychoslave 5mo ago
Esoterism is mostly a social tool to keep those not initiated excluded from the private club. Most of the time mathematics becomes tricky less due to unfathomable intrinsic complexity, and more due to the way it’s communicated. LLMs don’t give a shit about social side effects, leave alone on unconscious level, because they are void of any intention. At most they are tuned on their thin edge layer to lean toward this or that kind of output, but that’s it. Now the landscape shift as it’s sold (I guess) is that anyone can take a postdoc gibberish infused with the hard gained academic winks and subtle references and turn it into a ELI5 "does it have any applicability for my concrete issue at stake, prove it through Lean, good let’s deploy".
- datsci_est_2015 5mo ago
  When I use the word “esoteric”, I mean it at an absolutely hyperbolic level. Like exploring new-but-basically-useless axiom spaces, and creating concepts for which there exists no clean metaphor in time-space - like quantum mechanics on steroids. And then creating multiplicatively more complex concepts by combining those concepts together. There’s no way to “ELI5” this type of complexity. I’m talking about concepts exponentially more esoteric than quantum mechanics, and even within quantum mechanics there is nothing to ELI5 for a concept like “spin”. The best you can do is say that it’s a property of a particle. But imagine the words “property” and “particle” are also completely meaningless to you because they’re built on even more layers of conceptual mathematical abstraction.
jmorenoamor 5mo ago
Sorry but I couldn't agree less. Deep esoteric research and trivial looking boring research can be as useful as state of the art trending areas. "Jobs for nerds" as has been stated, has given surprising and unexpected advances, or leveraged incredible advancements. An standard and boring bacteria in a specific Spanish biome, gave us CRISPR-Cas. There ar hundreds of examples. True knowledge is, and will be, a human endeavor, deiven by human curiosity. Promoting curiosity is the sign of a developed society.
- datsci_est_2015 5mo ago
  > Sorry but I couldn't agree less. > … > True knowledge is, and will be, a human endeavor, deiven by human curiosity. Promoting curiosity is the sign of a developed society. Unless I misunderstand, it sounds like you do agree? My point is that without human mathematicians LLM output is meaningless, and without human mathematicians holding the reins, LLMs would probably quickly devolve into “proving” things that are not only completely unintelligible by humans, but have no utility. Your examples of esoteric mathematical concepts are anecdata. The vast majority of esoteric mathematics does not have utility. Mathematics is an incredibly large space of concepts. Consider the number of provable theorems in number theory alone, perhaps even related to specific subsets and sequences of numbers. The vast majority of the findings in that domain will not be isomorphic to some real world problem, they will be trivia. We will need mathematicians to separate the signal from the noise.
canjobear 5mo ago
Your model of what AI is good at is wrong. Generative AI is not good at wandering off into novel esoteric abstract corners while maintaining correctness, it is good at things that are close to its training data. I suspect that humans will long outperform AI in the domain of "novel esoteric abstract useless math" whereas AI will outperform humans in the domains of (1) making connections between already-well-understood concepts, things that seem obvious in retrospect but which no human figured out just because of the accidents of what people happened to focus on, and (2) proving things that require long, tedious, intellectually unsatisfying calculations, which would cause a human mathematician to give up for boredom.
- datsci_est_2015 5mo ago
  My understanding is that we’re talking about “tool-assisted” proof generation, which provides some guard rails but would still allow significant creativity. Tools like Lean, Coq, etc.
sunshowers 5mo ago
Elliptic curves over reals and the complex numbers had some physical/scientific meaning, but elliptic curves over finite fields had none before cryptography.

yaris 5mo ago

Works of Shinichi Mochizuki immediately come to mind. He is not AI but provides very good examples of math that is useless because it is incomprehensible by (other) humans.

seanmcdirmid 5mo ago
Do AIs produce answers whose work is incomprehensible to humans? It seems like you could just have the AI elaborate multiple times until you were satisfied with the explanation and documentation of what went into figuring out the answer. It’s not like the AI is one shotting the answer in a single opaque query anyways.
- datsci_est_2015 5mo ago
  Like other commenters, I think you’re also underestimating the complexity of esoteric higher level math. Consider the “Magnus Carlsen” of mathematics, who is more capable of understanding mathematics than any other human. But then also realize that that individual has probably devoted their entire career into a specific subdomain of mathematics. Within other deep recesses of mathematics, this Magnus equivalent will be less capable than their peers without years of rewiring their brain to understand the esoteric concepts and properties within that other subdomain. LLMs will be able to dig deeper and broader than any human mathematician, and find results that are completely useless to humans because it would take more than an entire lifetime to “speak the language” of the concepts the LLMs have produced. The only way those results can become useful to humans is if then the LLM itself finds a way for it to be practical to humans once again. So, no, I don’t think this represents the “democratization” of mathematics where mathematicians are no longer necessary because anyone can just prompt the LLM to explain it. The bar for entry level mathematics is lower, for sure, but research level mathematics will continue to be unapproachable for anyone who hasn’t devoted their career to it.
  - seanmcdirmid 5mo ago
    I don't get it. LLMs don't have ego, they don't have the ability to say "no, this should be obvious, I'm not going to explain further", they are just token predictors, and given context, they can generate more tokens. If you don't understand how the answer was derived? You just ask more questions and it isn't going to get bored or annoyed, it will just try to answer the questions. Is that what is offending you so much?
    - datsci_est_2015 5mo ago
      No, it doesn’t sound like you get it. It has nothing to do with the properties of LLMs and everything to do with the complexity of mathematics. Have you ever been exposed to concepts that are so complex that you feel like you could devote your entire lifetime to trying to understand it and still fall short? It’s a very humbling experience, especially if you have classmates who pick it up effortlessly. Without a human holding the reins, consider an LLM a rudderless superboat speeding erratically towards the horizon, finding and proving meaningless theorems that not even your most talented classmate could ever begin to understand. My point is the human is a critical piece to the puzzle, but not just any human, a career mathematician.
      - bawolff 5mo ago
        > Have you ever been exposed to concepts that are so complex that you feel like you could devote your entire lifetime to trying to understand it and still fall short? It’s a very humbling experience, especially if you have classmates who pick it up effortlessly. > Without a human holding the reins, consider an LLM a rudderless superboat speeding erratically towards the horizon, finding and proving meaningless theorems that not even your most talented classmate could ever begin to understand. This feels like a little bit of a jump to me. AIs arent actually alive so of course someone is going to have to pose the question. They arent going to just do stuff on their own. And of course mathmaticians are going to need to interpret the results if we are to glean anything beyong if the conjecture is true or false. But you seem to be suggesting that mathematicians will have to micromanage every step. That seems like a bit of a jump which i dont see much evidence for.
- epgui 5mo ago
  It’s easy to imagine this being a problem both in quality and in volume. Verifiable work is less valuable than verified work. And noise is always costly.
pfdietz 5mo ago
It's not that it's incomprehensible, it's that it appears to be wrong.

psychoslave 5mo ago

Esoterism is mostly a social tool to keep those not initiated excluded from the private club. Most of the time mathematics becomes tricky less due to unfathomable intrinsic complexity, and more due to the way it’s communicated. LLMs don’t give a shit about social side effects, leave alone on unconscious level, because they are void of any intention. At most they are tuned on their thin edge layer to lean toward this or that kind of output, but that’s it. Now the landscape shift as it’s sold (I guess) is that anyone can take a postdoc gibberish infused with the hard gained academic winks and subtle references and turn it into a ELI5 "does it have any applicability for my concrete issue at stake, prove it through Lean, good let’s deploy".

datsci_est_2015 5mo ago
When I use the word “esoteric”, I mean it at an absolutely hyperbolic level. Like exploring new-but-basically-useless axiom spaces, and creating concepts for which there exists no clean metaphor in time-space - like quantum mechanics on steroids. And then creating multiplicatively more complex concepts by combining those concepts together. There’s no way to “ELI5” this type of complexity. I’m talking about concepts exponentially more esoteric than quantum mechanics, and even within quantum mechanics there is nothing to ELI5 for a concept like “spin”. The best you can do is say that it’s a property of a particle. But imagine the words “property” and “particle” are also completely meaningless to you because they’re built on even more layers of conceptual mathematical abstraction.

jmorenoamor 5mo ago

Sorry but I couldn't agree less. Deep esoteric research and trivial looking boring research can be as useful as state of the art trending areas. "Jobs for nerds" as has been stated, has given surprising and unexpected advances, or leveraged incredible advancements. An standard and boring bacteria in a specific Spanish biome, gave us CRISPR-Cas. There ar hundreds of examples. True knowledge is, and will be, a human endeavor, deiven by human curiosity. Promoting curiosity is the sign of a developed society.

datsci_est_2015 5mo ago
> Sorry but I couldn't agree less. > … > True knowledge is, and will be, a human endeavor, deiven by human curiosity. Promoting curiosity is the sign of a developed society. Unless I misunderstand, it sounds like you do agree? My point is that without human mathematicians LLM output is meaningless, and without human mathematicians holding the reins, LLMs would probably quickly devolve into “proving” things that are not only completely unintelligible by humans, but have no utility. Your examples of esoteric mathematical concepts are anecdata. The vast majority of esoteric mathematics does not have utility. Mathematics is an incredibly large space of concepts. Consider the number of provable theorems in number theory alone, perhaps even related to specific subsets and sequences of numbers. The vast majority of the findings in that domain will not be isomorphic to some real world problem, they will be trivia. We will need mathematicians to separate the signal from the noise.

canjobear 5mo ago

Your model of what AI is good at is wrong. Generative AI is not good at wandering off into novel esoteric abstract corners while maintaining correctness, it is good at things that are close to its training data. I suspect that humans will long outperform AI in the domain of "novel esoteric abstract useless math" whereas AI will outperform humans in the domains of (1) making connections between already-well-understood concepts, things that seem obvious in retrospect but which no human figured out just because of the accidents of what people happened to focus on, and (2) proving things that require long, tedious, intellectually unsatisfying calculations, which would cause a human mathematician to give up for boredom.

datsci_est_2015 5mo ago
My understanding is that we’re talking about “tool-assisted” proof generation, which provides some guard rails but would still allow significant creativity. Tools like Lean, Coq, etc.

sunshowers 5mo ago

Elliptic curves over reals and the complex numbers had some physical/scientific meaning, but elliptic curves over finite fields had none before cryptography.