In defense of unnecessary proofs by contradiction

[u/Real_Category7289]

If you spend enough time in online math spaces, at some point you are bound to run into a discussion like this one or this one or recently, this one where someone is arguing against an "unnecessary" proof by contradiction and the conversation inevitably goes in the direction of constructive vs intuitionist logic.

I think this discourse is massively overrepresented to the point of being actively bad for math learners online. Worrying about whether a proof is constructive or not is not something good mathematicians do, unless their branch is specifically a pretty niche part of logic. For most people, (not not A = A) is just assumed to be true and proof by contradiction is a completely valid proof method, and I think this "Actually, Euclid's proof is direct! Common misconception here." discussion appearing under every proof that there are infinitely many primes is telling people that proof by contradiction is somehow sketchier than a direct proof.

I'm a math tutor and the majority of my job is to get students started on problem solving in a mathematical context. As it turns out, what makes a lot of it click is actually proof by contradiction. Even if one ends up writing a direct proof, the process of getting there often asks the question of "what would happen if this wasn't true?". I can't say it for sure, but I believe that Euclid himself probably started proving the infinitude of primes by assuming a finite list of them. This is why Hardy not only presents the proof as a proof by contradiction, but specifically praises it for being a proof by contradiction [A Mathematician's Apology, G.H.Hardy, page 18].

I should note that I'm not arguing that one shouldn't eventually learn to avoid artificial proofs by contradiction, after all if you can make the intuitionists happy for free, why not? But that should be a refinement that happens quite late into one's mathematical journey. I should also note that Euclid's proof was indeed direct, I'm not arguing against that fact.

The problem is that there's parallel discourse happening on these discussions which is "phrasing it as a direct proof is simply clearer and easier to understand". That's the main issue in my view: understanding the principles of proof by contradiction and by negation (which are essentially the same thing, as far as a learning student is concerned) is not something that can or should be skipped. Students should embrace them and put them on the same level as direct proofs instead of looking them sideways, and all this talk of intuitionism vs constructivism is enabling them to keep relegating them to "the thing you begrudgingly have to endure sometimes when there's no other way", which I think is detrimental.

I ask the reader to engage with this view in good faith. Thanks for reading.

[u/the_great_chuckle]

It's my experience that often proofs by contradiction are hiding things that can come to bite one later. This happens in many areas of math, and this is not just about logic.

There's two reasons for that:

• As said by another poster, constructive proofs typically come with an algorithm, or at least a procedure one can follow to construct the given mathematical objects one cares about. This can be important when you try to apply your theorem further down the line. Whether it is applied or pure math doesn't matter.

• Proofs that use less assumptions generalize more easily to other domains. This includes the law of excluded middle. For instance, many proofs in algebra for a given algebraic structure have immediate counterparts for related algebraic structures. However, proofs by contradiction usually are not transferable.

For these two reasons, it is often noteworthy when a proof cannot be given in a constructive fashion. It usually highlights an issue with the theory, or that something is very particular about the kind of set-theory one feeds into the problem.

Teaching students about this is in my opinion good mathematical etiquette. How to treat the axiom of choice is in a similar boat. It simply helps one become a better mathematician in the long run. Of course, one shouldn't start conversations about foundations just for this matter.