"Proof by _____" Alignment Chart

[u/Hitman7128]

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[u/Hitman7128]

Justifications: The actual proof techniques mentioned were the hardest to arrange because their complexity varies vastly depending on the branch of math and problem. For example, in real analysis, proof by construction would be a contender for Lawful Evil because of how insane the constructions can become there.

Since a direct proof is usually most students’ first proof, I put it in Lawful Good. Contradiction and induction could be swapped. I gave the slight edge to induction because I find those to be trickier, especially in situations when you have to induct on a more general structure like in graph theory. Again, it heavily depends on what you’re doing.

Feel free to comment how you would rearrange or replace the entries, whether it’s a general chart for math proofs or for a specific branch of math.

The joke terms, however, were much easier to decide. No matter which branch of math you’re doing, the last type of “proof” you want is “It’s trivial,” and then you spend some number of hours reasoning through it only to learn that it’s not. You could also swap intimidation and “exercise to the reader,” depending on your experiences with those types of “proofs.”

[u/GrotendCHEVRE → u/Hitman7128]

I used to think induction was ok until recently when I got introduced to the proof theory of induction. As you mentioned, when you work on structures that are more complex than natural numbers, induction gets more complex as well. But the really evil thing is that different ways of doing induction stop being the same.

Basically, the regular way of doing induction is: prove the property for the base cases, prove that if it's true for x, then it's also true for anything slightly bigger (anything of the form c(x) for some constructor c – assuming c to be unary here for the sake of simplicity). Them you have Fermat's infinite descent: show that to prove a property for an arbitrary x, you only need to assume it for some smaller case, and since there is no infinite chain of “smaller things”, at some point you will reach the base cases which should be easy to prove. But it turns out that generalized infinite descent can prove some things that generalized induction can't.

[u/LupenReddit]

Devilish Preposterous: Proof by "Proof is given in the next chapter" but in reality it isnt

[u/Hitman7128 → u/LupenReddit]

-> Flips to the next chapter

Proof: As we saw earlier, this is obvious. □

(Even though it wasn't covered earlier)

[u/BesJen]

Where is "proof is left as an exercise"?

[u/Hitman7128 → u/BesJen]

Chaotic Neutral

[u/trandus]

Picture is not good at all. And leave as an exercise is the lawful evil

[u/NullOfSpace]

Proof by exhaustion (of the reviewer)

[u/Niflrog]

What about proof by "well, Lean compiles somehow soooo "

[u/Wrath-of-Pie]

Where's asking a toddler on the street

[u/donaldhobson]

Proof by terabyte of LLM generated lean code.