I stumbled across article titled "Proof of negation and proof by contradiction" in which the author differentiates proof by contradiction and proof by negation and denounces an abuse of language that is "bad for mental hygiene". I get that it is probably a hyperbole but I am genuinely curious about what's so horrible in using those two interchangeably and I struggle to see any difference at all.

**to prove $\neg\phi$ assume $\phi$ and derive absurdity**(*proof by negation*)

and

**to prove $\phi$ suppose $\neg\phi$ and derive absurdity**(*proof by contradiction*)

More specifically the author claims that:

The difference in placement of negations is not easily appreciated by classical mathematicians because their brains automagically cancel out double negations, just like good students automatically cancel out double negation signs.

Could you provide an example where the "difference in placement of negations" can be appreciated and make a difference?

The author later use two cases: the irrationality of $\sqrt{2}$ and the statement "a continuous map [0,1) on $\mathbb{R}$ is bounded" but I can't see the difference. If I massage the semantics of the proof a little bit I obtain two valid proofs as well using negation/contradiction.

Can we turn this proof into one that does not use contradiction (but still uses Bolzano-Weierstrass)?

Why would we want to do that if both proof methods are equivalent?

I feel that the crux of the article is the following sentence:

A classical mathematician will quickly remark that we can get either of the two principles from the other by plugging in ¬ϕ and cancelling the double negation in ¬¬ϕ to get back to ϕ. Yes indeed, but the cancellation of double negation is precisely the reasoning principle we are trying to get. These really are different.

I have done some research and it seems that $\neg\neg\phi$ is the issue here. To quote Wikipedia\Double_Negation on that:

this principle is considered to be a law of thought in classical logic,2 but it is disallowed by intuitionistic logic

I should probably precise that my maths background is pretty limited so far as I am finishing my first year in college. This is bugging me and I would really appreciate if someone could explain it to me. Layman terms would be great but feel free to dive deeper as well. That will be homework for the summer (and interesting for more advanced readers)!

**Are proofs by contradiction and proofs of negation equivalent? If not, in which situation do differences matters and what makes them different?**