[zozbot234]

As others have explained, there is indeed nothing wrong with "proof by contradiction" of a negative statement. Intuitionistic logic does not view statements phrased in the "positive" and in the "negative" as fully equivalent, as classical logic does. (There are ways of formalizing this point of view quite rigorously, such as in so-called "ecumenical logics" where classical and constructive/intuitionistic reasoning can in fact coexist and interoperate, but statements derived from classical reasoning can only be translated in the negative as seen within constructive reasoning.)

[AnimalMuppet]

So I asked specifically for you to assume that I don't already agree with intuitionist logic. Now, assuming that, what is wrong with proof by contradiction of a positive statement?

Both you and johnnyjeans gave me answers that already assumed intuitionism. I don't assume that. Can you give me any reasons that start from where I am and show my why I should adopt intuitionism?

[auggierose]

You should not. Platonism is the only defensible mathematical philosophical position.

[AnimalMuppet]

That doesn't make platonism a "half-assed academic hack", though (which was the original claim I was questioning).

[auggierose]

I fully agree. Platonism isn't a hack, it is the only philosophy of math that makes sense to me. Something is either true, or it isn't. There is no third case. This is because mathematical objects are real, not just something our minds make up. To quote Arnold: Mathematics is the part of physics where experiments are cheap.

That said, I don't have a problem with intuitionistic logic, but I think about it as a platonist, for example via Kripke models. I also don't have a problem thinking about it in terms of a certain restricted class of proofs, that people call constructive proofs.