2
votes
1answer
126 views

What if 'proof by contradiction' is not a valid method of proof?

I've just been reading this question about the existence (or lack thereof) of contradictions in maths. I've been wondering: What if 'proof by contradiction' is not a valid method to (dis)prove a ...
2
votes
1answer
44 views

Are there statments which do not have a constructive proof?

I understand that a lot of statements are just non-nonconstructive in nature (like negative statements), and I understand that a lot of statements are not provable without the axiom of choice. ...
4
votes
1answer
115 views

Curry-Howard Correspondence (Proof Theory)

As you all know, the Curry-Howard correspondance provides a link between type theory and predicate logic. Concepts featured in the former, such as $\Pi$-type and $\Sigma$-type can, by the ...
6
votes
4answers
391 views

Are there “essentially non-constructive” statements?

There exist constructive and non-constructive proofs. Sometimes, for a mathematical statement, we can have both non-constructive and a constructive proof. However, are there statements for which ...
11
votes
6answers
1k views

Aren't constructive math proofs more “sound”?

Since constructive mathematics allows us to avoid things like Russell's Paradox, then why don't they replace traditional proofs? How do we know the "regular" kind of mathematics are free of paradox ...