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Hopefully as theorem provers improve and become more expressive, we can begin to state these theorems and even formalize proofs of them, letting a computer veri

Hopefully as theorem provers improve and become more expressive, we can begin to state these theorems and even formalize proofs of them, letting a computer verify the proof for us. Check out Lean[1] if this sounds interesting to you. [1]: https://leanprover.github.io/ https://leanprover.github.io/

gpm 7y ago
I've been checking out lean in my spare time over the last week. While it seems like a nice formal proof checker, it doesn't seem to have any automated their theorem proving abilities yet? I can't imagine trying to do math where I have to manually supply a proof of every trivial statement... Something like Isabelle and it's sledgehammer automated proof finder seems more reasonable to me (which I hope to find time to dip my toes into this week). Am I missing something?
- bonoboTP 7y ago
  I assume it could be done by having a "library" of such trivial statements. It's like in programming: we wouldn't want to deal with trivialities like how to decode utf8 or deal with files in a high level app, so we use libraries/APIs. You have to memorize them to some extent, but there can be autocomplete, documentation etc, just like in programming. Surely it needs a different mentality compared to old fashioned idealist type math people who may dislike such mechanization as taking the "soul and art" out of pencil and paper math, but I think the new generation of math people do have more overlap with CS-like thinking and they would be onboard.
  - gpm 7y ago
    Trivial statements of the kind I'm talking about are often incredibly proof specific, because they are often of the form "take the proof context and rephrase it so a pre proven theorem applies". Not the type of thing that can be enumerated in a library.
    - bonoboTP 7y ago
      Well, I'm not a mathematician, but it seems like if it's really trivial, it shouldn't be hard to formalize on a high level why it's true, like almost in natural language and then compile that high-level expression into a lower level argument. And if it's hard to even do that, then is the statement really trivial, or is it perhaps something that seems intuitive, but may not be sure (perhaps some pathological edge case may apply)?
      - gpm 7y ago
        So here's a dumb example I just came up with. Let's say we know that x = 3^a 5^b. It's a trivial fact at this point that x isn't divisible by 2 (by the fact that prime factorization is unique). Disclaimer: I'm not actually good enough with lean to prove this sort of statement quickly... I could be under or even over selling lean here, hence why I phrased my original comment as a question about whether I was missing something. If I was trying to prove this formally, I'd have to say something like (in computer speak): Suppose x is divisible by 2, x = 2 * y, y has a prime factorization p by <theorem in library>, and 2 * p is therefore a prime factorization of x (ouch, already not sure how to specify that statement). 2 * p = x = 3^a 5^b, so 2 * p = 3^a 5^b. Both are prime factorizations, prime factorizations are unique by <theorem>, therefore both expressions should have the same number of 2 terms, but the one on the left has at least one, and the one on the right has 0, so they don't. This is a contradiction, so x is not divisible by two. You see why I'd like a computer to fill in the long formal proof instead of doing it myself? It's not because the statement might be false, it's because it's a pain to phrase everything in terms of the re-usable result about unique prime factorization. While I'm disclaiming things, disclaimer 2: I'm a hobbyist, not a real mathematician.
        bonoboTP 7y ago
        I didn't mean that the necessary high level tools exist today. It may very well be that computerized formal proof systems are at the stage where programming was when only assembly existed. It was very tedious to express what you wanted. I think the analogy is quite close:in both cases you're trying to express an idea very precisely, for an uncompromisingly literal-minded computer. I was formulating a vision with ergonomic tooling, helpful language constructs that haven't been invented yet, etc.

gpm 7y ago

I've been checking out lean in my spare time over the last week. While it seems like a nice formal proof checker, it doesn't seem to have any automated their theorem proving abilities yet? I can't imagine trying to do math where I have to manually supply a proof of every trivial statement... Something like Isabelle and it's sledgehammer automated proof finder seems more reasonable to me (which I hope to find time to dip my toes into this week). Am I missing something?

bonoboTP 7y ago
I assume it could be done by having a "library" of such trivial statements. It's like in programming: we wouldn't want to deal with trivialities like how to decode utf8 or deal with files in a high level app, so we use libraries/APIs. You have to memorize them to some extent, but there can be autocomplete, documentation etc, just like in programming. Surely it needs a different mentality compared to old fashioned idealist type math people who may dislike such mechanization as taking the "soul and art" out of pencil and paper math, but I think the new generation of math people do have more overlap with CS-like thinking and they would be onboard.
- gpm 7y ago
  Trivial statements of the kind I'm talking about are often incredibly proof specific, because they are often of the form "take the proof context and rephrase it so a pre proven theorem applies". Not the type of thing that can be enumerated in a library.
  - bonoboTP 7y ago
    Well, I'm not a mathematician, but it seems like if it's really trivial, it shouldn't be hard to formalize on a high level why it's true, like almost in natural language and then compile that high-level expression into a lower level argument. And if it's hard to even do that, then is the statement really trivial, or is it perhaps something that seems intuitive, but may not be sure (perhaps some pathological edge case may apply)?
    - gpm 7y ago
      So here's a dumb example I just came up with. Let's say we know that x = 3^a 5^b. It's a trivial fact at this point that x isn't divisible by 2 (by the fact that prime factorization is unique). Disclaimer: I'm not actually good enough with lean to prove this sort of statement quickly... I could be under or even over selling lean here, hence why I phrased my original comment as a question about whether I was missing something. If I was trying to prove this formally, I'd have to say something like (in computer speak): Suppose x is divisible by 2, x = 2 * y, y has a prime factorization p by <theorem in library>, and 2 * p is therefore a prime factorization of x (ouch, already not sure how to specify that statement). 2 * p = x = 3^a 5^b, so 2 * p = 3^a 5^b. Both are prime factorizations, prime factorizations are unique by <theorem>, therefore both expressions should have the same number of 2 terms, but the one on the left has at least one, and the one on the right has 0, so they don't. This is a contradiction, so x is not divisible by two. You see why I'd like a computer to fill in the long formal proof instead of doing it myself? It's not because the statement might be false, it's because it's a pain to phrase everything in terms of the re-usable result about unique prime factorization. While I'm disclaiming things, disclaimer 2: I'm a hobbyist, not a real mathematician.
      - bonoboTP 7y ago
        I didn't mean that the necessary high level tools exist today. It may very well be that computerized formal proof systems are at the stage where programming was when only assembly existed. It was very tedious to express what you wanted. I think the analogy is quite close:in both cases you're trying to express an idea very precisely, for an uncompromisingly literal-minded computer. I was formulating a vision with ergonomic tooling, helpful language constructs that haven't been invented yet, etc.