3 ms·

Mathematicians might not like the insinuation, but the best reason to formalize mathematics is indeed to check its correctness: https://www.andrew.cmu.edu/user

by t_mann 3y ago

Mathematicians might not like the insinuation, but the best reason to formalize mathematics is indeed to check its correctness: https://www.andrew.cmu.edu/user/avigad/meetings/fomm2020/slides/fomm_buzzard.pdf https://www.andrew.cmu.edu/user/avigad/meetings/fomm2020/sli... tl;dr: contradictory stuff even gets published in the likes of Annals of Mathematics, and even top journals are reluctant to retract published proofs that have later been shown to be false (which in itself shouldn't be possible). Like it or not, the reasons for formalizing mathematics abound, and they have a lot to do with the fact that humans are fallible, but mathematics shouldn't be. The incredibly tedious work of tracking down a proof's correctness to the last axiom is, while overwhelming for humans, exactly the kind of work that classical computers excel at.

staunton 3y ago
Of course there's mistakes in papers. It's not that avoiding them isn't the primary motivation for formalization because they don't happen. It's not the primary motivation because mistakes in math papers happening sometimes is OK. If some result is wrong (as opposed to just details in a proof which could be fixed) and it turns out to be an important result, the mistake is found eventually. This fairly rare occurrence wastes some time for people who maybe used the result and build upon it, but arguably a lot less time than it would take to formalize all papers with existing tools. This might change in the coming decades as formalization becomes easier. The perspective that this is not the main motivation is also the position of people working on Lean's Mathlib. E.g. Kevin Buzzard has said this at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/Array.2EanyM.20is.20inconsistent/near/396178821 https://leanprover.zulipchat.com/#narrow/stream/270676-lean4...
- t_mann 3y ago
  Well, it appears that Kevin Buzzard hasn't fully made up his mind on this point (from the presentation by him that I linked): > If my work in pure mathematics is neither useful nor 100 percent guaranteed to be correct, it is surely a waste of time. So I have decided to stop attempting to generate new mathematics, and concentrate instead on carefully checking "known" mathematics on a computer I also remember him saying sth along the lines that his concern about how much of recent math is reliable being a major inspiration for his initial involvement in an interview (could have been for Quanta Magazine or so), but couldn't find it now, so can't verify.
  - staunton 3y ago
    That's actually quite interesting. I hadn't read the entire presentation, to be honest. Maybe it's a case of communicating priorities differently to different stakeholders... After all, it's about what the primary motivation is. Aside from that, the statement you're quoting is deeply puzzling to me. No work is ever "100% guaranteed to be correct", not even something formally verified by a computer, so that statement is either false, vacuous concerning correctness or he didn't mean it literally... (Maybe 100% really means 99.99999% of the time, or something)
    - t_mann 3y ago
      Not clear to me from when the chat statement is, but given that the presentation is from 2020, it would actually be funny if he went from 'this is going to get us 100% certainty' to 'well, we might never be fully sure, but at least it's a cool library'. Jokes aside, I'm pretty sure he was aware of the limitations at all times, and I'm reading this statement in the context of his examples in the presentation. There's a nice story in a Quanta interview about how he was approached by Peter Scholze to help formalize part of a proof. Scholze had asked whether they'd gone through his recent work, and when he said Yes, Scholze asked whether they'd carefully checked theorem 9.1. Buzzard said no, they didn't have enough time left when they got to chapter 9. And Scholze said, see, that's the problem, is anyone in the world really going to check 9.1? And then there are the stories in the presentation, where entire published proofs are put in question, not because of any direct mistake by the authors, but because of a problem in a lemma they used from previous work. This kind of transitive dependency chain could go down multiple papers, and there's really no guarantee that the later results are salvageable. So it seems to me that formalization would be a great practical tool to mitigate those problems. And then yes, it's also super cool in its own right (imho), just the fact of collecting the knowledge, and then the related work might also prove useful for formal verification (of hard/software systems), which has practical relevance, and mathlib might also just happen to become a great enabler for developing math-focused AI. So lots of good reasons to do it, but correctness might still be the most direct payoff: Terry Tao has a different blog post about how he found a (fixable) error in a recent proof of his, thanks to Lean. If that happens to him on one of the first occasions where he uses the tool seriously, then we can only guess what's waiting to be fixed (or thrown out) in the whole body of published mathematics.

by t_mann 3y ago

staunton 3y ago

Of course there's mistakes in papers. It's not that avoiding them isn't the primary motivation for formalization because they don't happen. It's not the primary motivation because mistakes in math papers happening sometimes is OK. If some result is wrong (as opposed to just details in a proof which could be fixed) and it turns out to be an important result, the mistake is found eventually. This fairly rare occurrence wastes some time for people who maybe used the result and build upon it, but arguably a lot less time than it would take to formalize all papers with existing tools. This might change in the coming decades as formalization becomes easier. The perspective that this is not the main motivation is also the position of people working on Lean's Mathlib. E.g. Kevin Buzzard has said this at https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/Array.2EanyM.20is.20inconsistent/near/396178821 https://leanprover.zulipchat.com/#narrow/stream/270676-lean4...

t_mann 3y ago
Well, it appears that Kevin Buzzard hasn't fully made up his mind on this point (from the presentation by him that I linked): > If my work in pure mathematics is neither useful nor 100 percent guaranteed to be correct, it is surely a waste of time. So I have decided to stop attempting to generate new mathematics, and concentrate instead on carefully checking "known" mathematics on a computer I also remember him saying sth along the lines that his concern about how much of recent math is reliable being a major inspiration for his initial involvement in an interview (could have been for Quanta Magazine or so), but couldn't find it now, so can't verify.
- staunton 3y ago
  That's actually quite interesting. I hadn't read the entire presentation, to be honest. Maybe it's a case of communicating priorities differently to different stakeholders... After all, it's about what the primary motivation is. Aside from that, the statement you're quoting is deeply puzzling to me. No work is ever "100% guaranteed to be correct", not even something formally verified by a computer, so that statement is either false, vacuous concerning correctness or he didn't mean it literally... (Maybe 100% really means 99.99999% of the time, or something)
  - t_mann 3y ago
    Not clear to me from when the chat statement is, but given that the presentation is from 2020, it would actually be funny if he went from 'this is going to get us 100% certainty' to 'well, we might never be fully sure, but at least it's a cool library'. Jokes aside, I'm pretty sure he was aware of the limitations at all times, and I'm reading this statement in the context of his examples in the presentation. There's a nice story in a Quanta interview about how he was approached by Peter Scholze to help formalize part of a proof. Scholze had asked whether they'd gone through his recent work, and when he said Yes, Scholze asked whether they'd carefully checked theorem 9.1. Buzzard said no, they didn't have enough time left when they got to chapter 9. And Scholze said, see, that's the problem, is anyone in the world really going to check 9.1? And then there are the stories in the presentation, where entire published proofs are put in question, not because of any direct mistake by the authors, but because of a problem in a lemma they used from previous work. This kind of transitive dependency chain could go down multiple papers, and there's really no guarantee that the later results are salvageable. So it seems to me that formalization would be a great practical tool to mitigate those problems. And then yes, it's also super cool in its own right (imho), just the fact of collecting the knowledge, and then the related work might also prove useful for formal verification (of hard/software systems), which has practical relevance, and mathlib might also just happen to become a great enabler for developing math-focused AI. So lots of good reasons to do it, but correctness might still be the most direct payoff: Terry Tao has a different blog post about how he found a (fixable) error in a recent proof of his, thanks to Lean. If that happens to him on one of the first occasions where he uses the tool seriously, then we can only guess what's waiting to be fixed (or thrown out) in the whole body of published mathematics.