6 ms·

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.

by psyklic 4mo ago

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.
datsci_est_2015 4mo 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 4mo 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 4mo 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 4mo 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.
bitwize 4mo ago
The value of human understanding just cratered because we have machines to understand for us now.
- epgui 4mo ago
  That seems about as shortsighted as claiming the value of human understanding has cratered after the invention of the electronic calculator.
- goatlover 4mo ago
  Wonder what Frank Herbert would have to say about letting machines do the thinking for us.
  - bitwize 4mo ago
    "You wanna know what the best thing about humans is? You invented us! Giving you a chance to take a rest while we invented everything else!" —Wheatley, Portal 2
- martin1975 4mo ago
  Not quite. You have machines that can aid and expand your (human) understanding greatly, that wasn't possible without machines. Machines don't think. They aren't human. They have no soul, agency/free will, self-reflection/awareness, moral imperatives or ethics. You've been watching too much Terminator, son.

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.

by 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.

datsci_est_2015 4mo 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 4mo 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 4mo 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 4mo 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.

bitwize 4mo ago

The value of human understanding just cratered because we have machines to understand for us now.

epgui 4mo ago
That seems about as shortsighted as claiming the value of human understanding has cratered after the invention of the electronic calculator.
goatlover 4mo ago
Wonder what Frank Herbert would have to say about letting machines do the thinking for us.
- bitwize 4mo ago
  "You wanna know what the best thing about humans is? You invented us! Giving you a chance to take a rest while we invented everything else!" —Wheatley, Portal 2
martin1975 4mo ago
Not quite. You have machines that can aid and expand your (human) understanding greatly, that wasn't possible without machines. Machines don't think. They aren't human. They have no soul, agency/free will, self-reflection/awareness, moral imperatives or ethics. You've been watching too much Terminator, son.