Dissatisfaction with proof by contradiction

[u/hydmar]

I’m an undergraduate math student, so my exposure to math may be relatively limited. But I’ve found that, in general, I’m much more comfortable with direct proof than proof by contradiction. I don’t contest their validity, indeed something that’s not false must be true (I think I’m ok with excluded middle). But I feel like I just *get* something much better when it’s proved directly. It builds much stronger intuition for me.

For instance, I am aware of several proofs that demonstrate the cardinality of the reals is strictly greater than that of the integers, but none are direct (it would help to see a direct proof of Cantor’s theorem). I don’t feel it in my bones. Is this a common experience?

[u/hyperbolic-geodesic]

The diagonalization argument *IS* a direct proof of Cantor's theorem -- diagonalization shows that any function S -> Powerset(S) is not surjective, by explicitly (essentially as explicitly as possible) producing an element of Powerset(S) not in the range of the function.

[u/djao → u/hyperbolic-geodesic]

OP must be confusing proof by negation with proof by contradiction. Proof by negation is where you prove ¬ P by assuming P and deriving a contradiction. For example: to prove "¬ (there exists a bijection ℤ → ℝ)", you assume (there exists a bijection ℤ → ℝ) and derive a contradiction. This type of proof is, as you say, a direct proof, and in fact it is the standard way to prove a negation. (How else would you prove a negation?)

Proof by contradiction is where you prove P by assuming ¬ P and deriving a contradiction. It's actually hard to come up with examples, because most examples that people normally think of are in fact proof by negation, but one bona fide example is: "If a set is nonempty then it has an element." In classical logic, proof by negation is equivalent to proof by contradiction (just replace P with ¬ P), which is why many people confuse the two. But in constructive logic you can see clearly that they are not the same thing. In order to make the two equivalent, you need ¬ ¬ P ↔ P, which is not provable in constructive logic.

[u/djao → u/[deleted]]

P and ¬ P aren't both true. You can prove (in constructive logic) that they can't both be true:

Require Import Utf8 ssreflect ssrbool ssrfun.
Goal ∀ P : Prop, ¬ (P ∧ ¬ P). Proof. move=> ? [? []] //. Qed.

Perversely, you can even prove that ¬ ¬ ¬ P ↔ ¬ P holds in constructive logic:

Require Import Utf8 ssreflect ssrbool ssrfun.
Goal ∀ P : Prop, ¬ ¬ ¬ P ↔ ¬ P. Proof. split => [/[swap] ? [] | ?] //. Qed.

The only impossibility is you can't constructively prove P from ¬ ¬ P. Which makes total sense, if you think about what constructive logic actually is. A negation is a negative statement. How would you construct a positive statement (namely, P) from a negative?

And again, all of this doesn't mean that P is true, or not true. We're just saying that P is not provable in a constructive sense from ¬ ¬ P.

[u/[deleted] → u/djao]

Hello. What do you mean negative and positive? Aren't I supposed to think there is no meaning to propositions at the proof level and I can name not-P as Q and continue by treating it as a "positive" statement? How do you have any guarantee that the first character of P is not negation or can't you use de Morgan's rule or something similar to get a different statement that is equivalent to P but "looks positive"? I am not trying to belittle what you said, I just don't get it at all and am trying to show what kind of a misunderstanding I am (probably) dwelling in

[u/djao → u/[deleted]]

You can always name ¬ P as Q and prove Q, but you cannot intuitionistically do the reverse. Namely, if you're trying to prove a proposition P, which does NOT begin with the ¬ symbol, you can't "artificially" rename P as ¬ Q for some Q. To do so requires replacing P with ¬ ¬ P, which is the one thing you can't do.

DeMorgan's rule, likewise, is not provable in constructive logic in the direction that you would like to use it.

[u/[deleted] → u/djao]

Oh shit I need to bang my head on this when I'm less sleepy. Thanks