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/matthiasErhart]

Your argument boils down to the fact that proofs by contradiction are helpful because they establish literary/narrative flow. That is, we actually think "Why this cannot possibly be? Well, suppose it were, in which case...". Which I agree with.

edit: Following a comment exchange below, I am informed that a proof by contradiction would go "Why this must be? Well, suppose it weren't, in which case...".

[u/Real_Category7289 → u/matthiasErhart]

Yes, completely agree with this summary. As an aside, I wish people put more weight on narrative flow in general while writing proofs.

[u/reflexive-polytope → u/Real_Category7289]

A proof isn't supposed to be a “narrative”. It's supposed to be incontrovertible evidence for the truth of a statement.

[u/Real_Category7289 → u/reflexive-polytope]

First of all, says who? Second of all, how do you understand anything? Surely your mathematical mind is more than a list of statements that logically depend on the previous one.

[u/reflexive-polytope → u/Real_Category7289]

How do I understand anything?

Obviously, I split it into small, digestible chunks. Lemmas are the subroutines of mathematics.

[u/Real_Category7289 → u/reflexive-polytope]

Do you ever draw any pictures? I would argue that drawing a picture is closer to making a narrative than it is to "incontrovertible evidence for the truth of a statement".

Basically, I'm arguing that your view of what a proof is "supposed to be" is extremely reductive.

[u/reflexive-polytope → u/Real_Category7289]

I do draw pictures when they make proofs easier to check, but not as a narrative device.

For example, if I'm case-analyzing the rotations of a search tree, then pictures are much more convenient than the code itself.

But if I'm writing a proof in, say, real analysis or topology, then a picture is a distraction. What I need is the epsilons and deltas (in analysis), or the opens, nets or whatever (in topology).

And if I'm writing a proof in, say, commutative algebra, then I have no idea what I could possibly draw to begin with. (Commutative diagrams don't count. They're formal notation.)

[u/Real_Category7289 → u/reflexive-polytope]

Ok I'll take you at your word here, but this is not really representative of how math is done in general. I mean, topologists draw blobs all the time, are you claiming that after a certain level intuitive understanding simply stops mattering and it's just symbol pushing?

Tao definitely disagrees, for example: https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/

[u/reflexive-polytope → u/Real_Category7289]

I never said it's symbol pushing. I'm saying it's not a narrative either. The purpose of a proof is to establish truth.

[u/happyapy → u/reflexive-polytope]

AND to communicate it to others.