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> Theorems and conclude that in every logic there are statements which are true, but not provable, or provable, but not true. And this is the ultimate problem w

by freyrs3 13y ago

> Theorems and conclude that in every logic there are statements which are true, but not provable, or provable, but not true. And this is the ultimate problem with type systems: in their quest to reject "bad" programs, they must reject "good" programs as well because they cannot prove their "goodness". This really is a gross misuse of Gödel's theorem. Taken to it's logical end his argument is that any field that has logical foundations should value human judgement over rigor and proof because of incompleteness. Should we just throw out all of mathematics as well because there are true but unprovable theorems?

jeapostrophe 13y ago
We should not throw out mathematics, but we should recognize that most mathematical proofs are not based on logic, but human intuition in the first place. Human intuition and wisdom are valuable and lay the foundation for all that we do and there's no reason to ignore them when we're programming and insist that all valid programs have to follow some particular set of rules.
- freyrs3 13y ago
  > mathematical proofs are not based on logic Mathematical proofs are explicitly logical statements, if they aren't based in logic they aren't proofs by definition. Mathematical intuition is what leads us to ideas but it's only within the framework of formal reasoning that we "know" anything within mathematics. Poincare said it nicely: "It is by logic we prove, it is by intuition that we invent."
  - jeapostrophe 13y ago
    If you really believe this, you should try to translate any contemporary (published) mathematical proof into a real logic, like Coq. It is hard and people earn PhDs doing this because most proofs are so intuition heavy in the first place.
    - freyrs3 13y ago
      I have spent several months doing proofs in Coq, as well as in Isabelle. I contest conflating Coq with "real logic", it is perfectly possible to do rigorous proofs without the aid of a computer-aided proof assistant, if that wasn't the case then there wouldn't be any reason to trust the authors of the proof assistant in the first place.
      - jeapostrophe 13y ago
        Certainly, but most math proofs aren't "rigorous proofs" by any stretch.

by freyrs3 13y ago

jeapostrophe 13y ago

We should not throw out mathematics, but we should recognize that most mathematical proofs are not based on logic, but human intuition in the first place. Human intuition and wisdom are valuable and lay the foundation for all that we do and there's no reason to ignore them when we're programming and insist that all valid programs have to follow some particular set of rules.

freyrs3 13y ago
> mathematical proofs are not based on logic Mathematical proofs are explicitly logical statements, if they aren't based in logic they aren't proofs by definition. Mathematical intuition is what leads us to ideas but it's only within the framework of formal reasoning that we "know" anything within mathematics. Poincare said it nicely: "It is by logic we prove, it is by intuition that we invent."
- jeapostrophe 13y ago
  If you really believe this, you should try to translate any contemporary (published) mathematical proof into a real logic, like Coq. It is hard and people earn PhDs doing this because most proofs are so intuition heavy in the first place.
  - freyrs3 13y ago
    I have spent several months doing proofs in Coq, as well as in Isabelle. I contest conflating Coq with "real logic", it is perfectly possible to do rigorous proofs without the aid of a computer-aided proof assistant, if that wasn't the case then there wouldn't be any reason to trust the authors of the proof assistant in the first place.
    - jeapostrophe 13y ago
      Certainly, but most math proofs aren't "rigorous proofs" by any stretch.