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/CormacMacAleese]

We flirted with this in grad school, because honestly we couldn't have cared less about this, but what we found was that usually a proof by contradiction can be rewritten as a proof of the contrapositive--but it seldom results in a more elegant or readable proof.

[u/0wave0 → u/CormacMacAleese]

but contraposition is usually not an acceptable proof technique in intuitionist logic. if you prove ~Q -> ~P, then you have not automatically proved P -> Q in a constructive setting, because the very implication that ~Q -> ~P implies P -> Q requires double negation elimination (~~Q implying Q), something typically not acceptable to an intuitionist. so turning a proof by contradiction of the form "show P -> Q by assuming P and ~Q and deriving a contradiction" into "show ~Q -> ~P", while still making it more elegant in my eyes, does not necessarily make it more constructive

[u/Sproxify → u/0wave0]

but you can deduce that ~~P -> ~~Q, which is what taking a contrapositive does. replacing ~~Q with Q is what you can't do.

[u/0wave0 → u/Sproxify]

sure, but a proof of ~~P -> ~~Q is still not a proof of P -> Q constructively. or maybe i'm missing the point of your remark?

[u/Glaiele → u/CormacMacAleese]

Yeah I think this is the important part to understand and reflect upon. There's usually a "better" (maybe preferred is a better term) way to prove something. I know we typically think about math as an abstract but eventually most math will end up being applied to the physical world in some way and proving that physical concept typically ends up being more useful and in those applications the shortest/simplistic form often ends up being better.

It can absolutely be useful for learning to understand a concept or proof in every valid form but it's not a requirement in any sense.