5 ms·
No. Constructivism is strictly weaker.
by monochromatic 8y ago
No. Constructivism is strictly weaker.
- danharaj 8y agoStrictly weaker in a weak sense, every classically true proof can be translated into a correct constructive proof of a similar theorem that is classically equivalent. So is it weaker or is it more fine-grained? That's a philosophical question.
- monochromatic 8y agoI haven’t heard this. What do you mean by “a similar theorem that is classically equivalent”?
- danharaj 8y agohttps://en.wikipedia.org/wiki/Double-negation_translation https://en.wikipedia.org/wiki/Double-negation_translation
- AnimalMuppet 8y agoCan it? Here's a proof that requires the law of the excluded middle. If you don't have that, you don't have a proof of that theorem. What's the correct constructive proof of a similar theorem "that is classically equivalent"? Or, to say it differently, aren't there things that are true in "classical" (non-constructivist) math that are not true in constructivist math? My impression was that what is provably true in classical math is a strict superset of what is provably true in constructivist math.
- danharaj 8y agoIf you use LEM to prove A, you can translate that proof into intuitionistic logic so that it proves Not (Not A). This is classically equivalent to A, but not intuitionistically. So the real difference between the two is that intuitionistic logic makes a finer distinction between propositions that classical logic says are the same.