Is it bad form to write mysterious proofs without explaining what one intends to do? [closed]

Often when doing assignments, I find myself deliberately writing in a “mysterious” way. By this I mean that the reader usually will not understand what exactly is going on and what for, until the very end where all the things come together.

A simple example is if I wish to prove that S is true by showing that it is equivalent to s being true, and then proving that s is true. Often I will find myself doing this by writing something that reads like …

“Consider s, this seemingly random object that I present to you out of
nowhere. Let us work on this for the next 1-2 pages, don’t ask me why
….. [1-2 pages later] …. and that proves that s is true. Now by noticing this and that, we see that this implies S is true. Surprise! QED”

Is this bad form? It also seems a bit pretentious to be, because the reason I think I sometimes do this is because these are the proofs I have mostly met in textbooks. Rarely is the proof sketched before it’s given, very often new, foreign, confusing objects are introduced without introductions and motivations, and it’s usually near the end of the proof that I would get my “aha” moment. The problem with this is of course that if one does not know the why behind some of the steps during the first reading, then one will have a harder time remembering how the pieces fit together throughout the proof.

But of course my instructors are not students reading (undergaduate) textbooks, and therefore they can perhaps deal with these sorts of mysterious proofs? Maybe they even prefer them, rather than having to waste time on reading me informally writing a few sentences prior to the proof outlining my ideas, giving a proof sketch, etc? I also do not wish to run the risk of sounding patronising or arrogant: “look at me and my geniusly complicated proof that I will now explain to you step by step”.

Answer

The purpose of proofs is communication. If your proof is obscure, then you have failed to communicate.

Strive to be as clear as possible, including motivation for complicated arguments, if necessary.

Attribution
Source : Link , Question Author : Jaood , Answer Author : lhf

Leave a Comment