Proofs and refutations
Proofs and Refutations is for those interested in the methodology, philosophy and history of mathematics.
Stats for nerds
34recommend
55mentions
45people
2008–2026years
76% recommend22% neutral2% critical
Mentions per year
What Hacker News says
- > everyone agrees that it's the result that counts.
I don't know if I agree with that. That presumes that the entire point of mathematics is to produce results. I don't believe that at all.
I believe that the entire point of mathematics is to produce human understanding and mental models that are useful or beautiful to humans. (I also believe there's a mysteriously deep connection between those two notions, but that's a different discussion.)
At any rate, I think that the old process, where humans had to cut the paths themselves before the clearing emerged, is one that led to understanding first, then a distilled result. And then the secondary task of distilling the understanding down to a clean proof is the icing on the cake. Some mathematicians are very good at keeping the intuition and understanding clear when presenting proofs. Some are not.
However, very often, the end result and proof is not where the real understanding lies, and that is one of the reasons why mathematics has been such a social endeavor. Often you can't glean the "hidden truths" behind austere proofs before you engage with people who can help you see the picture behind it. Many people who went through Baby Rudin as an undergrad can maybe relate to what I'm saying here. For a very insightful analysis about how intuition and discourse lead the discovery of mathematical truth, I highly recommend Imre Lakatos' amazing book "Proofs and Refutations."
What the models are producing is not what we had before. They are producing the cargo cult equivalent: Equations and words on paper which by means of tracing through the steps might take you from A to B. And after a while, humans might discover the true facts behind why that path from A to B exists, and then understanding emerges. And only then.
But when the process is done through AI, no one (no human) learns how to make discoveries and connections on their own. We are not letting people exercise the cognitive apparatus which is used to gain intuition and understanding. My fear is that after a while, fewer human mathematicians are able to gain the understanding mathematicians before them did.
It's as if someone removed all the barbells from a gym and made automated hydraulic jacks do all the lifting. After a while, people would wonder why our power lifters aren't as strong anymore. "But weights get lifted! That's the entire point, right?!"
- Thanks for the recommendation!
Proofs and Refutations: The Logic of Mathematical Discovery - Imre Lakatos
I appreciated your defense of mathematics as a communal process of developing human understanding, not just a way to solve problems and generate proofs. While the latter is important and valuable, replacing mathematicians on an industrial scale will destroy the cultural context that cultivated the field over generations.
- BTW, counterexamples in mathematics are really important and often help to refine definitions and sharpen proofs.
1) I recommend the wonderful 1976 book Proofs and Refutations by Imre Lakatos.
2) There is a considerable list of books dedicated to counterexamples, e.g. in topology, probability, analysis, etc.
- Excluding supergeniuses, pure mathematics—even at a very basic, undergraduate level—simply can't be understood passively. Even with an infinitely patient AI teacher who could answer any question on-demand, it'd still require a massive amount of work to actually understand anything in research-level mathematics. Basically every single word in a mathematical definition is a term of art, and (IME) if one doesn't grok each of those words at a fairly deep level, the new definition never really makes too much sense. And this applies recursively: each of the words has some thoroughly inscrutable definition of their own.
Of course it'd be super helpful to have, say, a teacher who could tailor explanations to anyone's precise background (e.g. where possible, using examples that come from the student's field of study when explaining some abstract concept). Or, if some definition comes with some precondition that has no obvious purpose, perhaps an omniscient teacher could explain why it's there with concrete counterexamples.[0] But even granting all this, I think that mathematical intuition is necessarily based on a lot of hard work actually exploring definitions on one's own, with pencil-and-paper and a lot of thought. That is to say, even though the process could probably be sped up a lot with a nigh-omniscient teacher[1], I doubt that a student wouldn't still need years of training to even have a clue what's going on.
(I'm saying all this, by the way, as someone who is terrible at all this and has very little mathematical maturity[2]—I'm speaking from my own frustrating experience....)
[0] c.f. Lakatos' excellent book Proofs and Refutations
[1] without the "curse of knowledge," or else we're back to square one of "answers that are correct but useless"
[2] e.g. the "post-rigorous stage" described in https://terrytao.wordpress.com/career-advice/theres-more-to-...
- Proofs and Refutations is a great book
- Proofs and Refutations is a great book.
