4 ms·

Formalism and its “derivatives” died a long time ago already. Even Hilbert couldn’t do much about it. That being said, constructive math is still a great idea.

by rq1 3y ago

t_mann 3y ago
Not sure what you mean by that - what I meant by formalization was to really deduce every statement in a proof from basic axioms and rules. So basically what Prof Tao did in this blog post (except that the hard rote labor has all been abstracted away and handed off to a machine). This kind of formalization is hardly dead I'd say, in fact it's just starting to gain traction in mainstream mathematics, as Prof Tao's recent work shows (if it's not too much work for a Fields medal winner to formalize their proofs, then what excuse do mere mortal mathematicians have?). So what did you mean? Edit: it's clear that the original four-color proof had nothing to do with this kind of formalization, it was just the debate that got me thinking about it. And tools like Lean4 aren't limited to constructive math afaik.
- ykonstant 3y ago
  >And tools like Lean4 aren't limited to constructive math afaik. In fact, Lean is strongly geared towards classical logic: most of its tactics assume and use classical logic, especially the excluded middle.
- staunton 3y ago
  > what excuse do mere mortal mathematicians have? As an ardent proponent of formalizing proofs, I still have to disagree with this framing. Mortal mathematicians don't need any excuse not to formalize their stuff. They rather need a reason to do it. One can give lots of reasons but mathematicians often don't know them or don't agree with them. An important point is, for example, to understand that most formalizing effort is not primarily motivated by a desire to make sure there are no errors in the math.
  - trevyn 3y ago
    >most formalizing effort is not primarily motivated by a desire to make sure there are no errors in the math. Is this because the proof author is generally confident that the proof is essentially correct?
    - staunton 3y ago
      Here's people who are working on Lean's Mathlib discussing this 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
    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...

Formalism and its “derivatives” died a long time ago already. Even Hilbert couldn’t do much about it. That being said, constructive math is still a great idea.

by rq1 3y ago

Formalism and its “derivatives” died a long time ago already. Even Hilbert couldn’t do much about it. That being said, constructive math is still a great idea.

t_mann 3y ago

Not sure what you mean by that - what I meant by formalization was to really deduce every statement in a proof from basic axioms and rules. So basically what Prof Tao did in this blog post (except that the hard rote labor has all been abstracted away and handed off to a machine). This kind of formalization is hardly dead I'd say, in fact it's just starting to gain traction in mainstream mathematics, as Prof Tao's recent work shows (if it's not too much work for a Fields medal winner to formalize their proofs, then what excuse do mere mortal mathematicians have?). So what did you mean? Edit: it's clear that the original four-color proof had nothing to do with this kind of formalization, it was just the debate that got me thinking about it. And tools like Lean4 aren't limited to constructive math afaik.

ykonstant 3y ago
>And tools like Lean4 aren't limited to constructive math afaik. In fact, Lean is strongly geared towards classical logic: most of its tactics assume and use classical logic, especially the excluded middle.
staunton 3y ago
> what excuse do mere mortal mathematicians have? As an ardent proponent of formalizing proofs, I still have to disagree with this framing. Mortal mathematicians don't need any excuse not to formalize their stuff. They rather need a reason to do it. One can give lots of reasons but mathematicians often don't know them or don't agree with them. An important point is, for example, to understand that most formalizing effort is not primarily motivated by a desire to make sure there are no errors in the math.
- trevyn 3y ago
  >most formalizing effort is not primarily motivated by a desire to make sure there are no errors in the math. Is this because the proof author is generally confident that the proof is essentially correct?
  - staunton 3y ago
    Here's people who are working on Lean's Mathlib discussing this 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
  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...