One of my math teacher did not like proof by contradiction. He mentioned that it is a indirect way to prove something and not a genuine way to prove a statement.
In elementary math class, there is a following statement: "there is no rational number whose square is 2." I have only seen a proof by contradiction so far.
Is there a direct proof for there is no rational number whose square is 2?
Also I want to hear your opinion about proof method: proof by contradiction. Is this really bad? Should I avoid using this method whenever possible?
Best Answer
As another answer indicated, proof by contradiction is really only objected to by constructivist mathematicians, who use intuitionistic logic. But typically constructivists will accept the usual proof of the irrationality of $\sqrt 2$. If I'm right in assuming that your professor is a constructivist, then I'd imagine that they would accept that proof. This might be confusing. As you say, the irrationality of $\sqrt 2$ seems like a proof by contradiction, so to explain this I'll explain constructivism to the best of my knowledge first.
In intuitionistic logic, one refuses the law of excluded middle:
$$ P \vee \neg P $$
which can be thought of as saying that $\neg (\neg P) \iff P$ for all propositions $P$. In case you're unfamiliar with the jargon, $\neg$ means "not" and $\vee$ means "or." Now, a proof by contradiction is something of the form:
$$ \text{Suppose } \neg P \text{ and derive a contradiction. Then, we must have } P. $$
But there is a very similar proof method called proof by negation that is constructively valid:
$$ \text{Suppose } P \text{ and derive a contradiction. Then, we must have } \neg P. $$
These appear almost identical (and in classical logic they are identical), but the crux is that proof by contradiction uses the law of excluded middle whereas proof by negation does not. Proof by negation essentially comes from the fact that $\neg P$ is the same as $P \implies \text{false}$. This is often taken as the definition of $\neg P$. From this perspective, proof by negation is an application of the deductive principle that to prove $P \implies \text{false}$, one can suppose $P$ and derive $\text{false}$.
The irrationality of $\sqrt 2$ is a proof by negation, not by contradiction. The proof I'm aware of starts with "Suppose $\sqrt 2 = \frac{p}{q}$..." and ends with "...so $p$ and $q$ must both be even, which is a contradiction. Hence, $\sqrt 2$ is irrational." This is a proof by negation and is therefore constructively valid.
Now, as for proof by contradiction, I personally see no issue with it. I accept the law of excluded middle in my work and don't think twice about using proof by contradiction. I think the majority of mathematicians would agree with me on this. However, constructivism is a very real viewpoint and it's worth understanding it. I can't do it much justice myself, but I highly recommend an article written by Andrej Bauer entitled "5 Stages of Accepting Constructive Mathematics." There are many very technical things in this article, but I think you could still learn a lot about constructivism and why (I assume) your professor dislikes proof by contradiction.