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

Question

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 1

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.

Answer 2

As an instructor I want to see proofs that are as transparent as possible - English where words do the job, complex notation only when necessary. Sometimes words in advance about the general structure of the proof will be helpful; sometimes they're unnecessary. I understand that the balance is a judgment call, and my students and I may differ about the balance. When I read homework I try to teach style as well as check for correctness.

Deliberately obscure writing wastes my time.

If you've mostly met obscure proofs in your textbooks then I think your instructors should have chosen better books.

Answer 3

There is absolutely no reason to deliberately write mysterious arguments and test the patience of your instructors and their ability to follow your "mysterious" arguments if you can write things clearly. If a certain mental picture or idea lead you somewhere, then by writing it down explicitly and getting feedback you will learn much more than by intentionally obfuscating your argument. When you have to grade 50 assignments on a tight schedule, there is nothing more annyoing that reading a two-page argument that is ridden with mistakes/inaccuracies when you have no idea why the person that wrote it bothered to go into that direction in the first place.

Let me give you a silly example. Often in real analysis text books when you prove that some limit is equal to $L$, the proof starts with something like "Let $\delta := \min \{ \frac{\varepsilon}{12}, 4 \}$. Then...". While this is fine from a strict point of view, I wouldn't suggest people that encounter the material for the first time to try and reproduce such proofs. The reason is that if one writes a wrong proof in this format, it is often very difficult to identify and point out the origin of the error. If, on the other hand, a student writes "Given $\varepsilon > 0$, we want to find $\delta > 0$ such that .... By manipulating ... we see that if $\delta$ satisfies ... then we will have ..." then if something goes wrong, it is much easier to understand what was the wrong step and give a constructive feedback instead of just marking an X. If the proof is correct, then there's no problem but then there's usually not much need for feedback. But if the proof is wrong and you have to spend ten minutes to understand why the person chose $\min \{ \frac{\varepsilon}{12}, 4 \}$ and not $\min \{ \frac{\varepsilon}{24}, 1 \}$ and the text gives you no clue whatsoever, then you just won't do it.

Answer 4

The problem here is really that the impeccable "logical" order isn't always the clearest way of explaining something. There's a couple of ways to get around this. Suppose, for example, that your argument is of the form $$\varphi_0 \rightarrow \varphi_1 \rightarrow \cdots \rightarrow \varphi_{389} \qquad \therefore \varphi_0 \rightarrow \varphi_{389}$$

Ask yourself: how did you actually come up with this argument? Well, you probably made an educated guess that $\varphi_0 \rightarrow \varphi_{87}$, that $\varphi_{87} \rightarrow \varphi_{217}$, and that $\varphi_{217} \rightarrow \varphi_{389}.$ Then you went about filling in the details.

So there's these special intermediate $\varphi$'s that are especially simple, or interesting, or natural-looking steps toward your actual goal.

Approach 0. Tell the reader about these special intermediate steps ahead of time.

For example:

The proof will proceed in three sections. In the first section, we will demonstrate that $\varphi_0 \rightarrow \varphi_{87}$. In the second section...

Then put "Section X" at the beginning of each section and be sure to remind the reader of what you're doing.

Approach 1. Alternatively, try reducing the argument to fewer lines by interjecting technical lemmas into the proof whenever you need them, and then proving them afterward. For example:

Assume $\varphi_0$. It's clear that $\varphi_1$, from which we deduce $\varphi_2$. Hence using:

Lemma 0. You're beuatiful.

we deduce $\varphi_{87}$. Multiplying both sides by your chest hair demonstrates that $\varphi_{88}$. Now use:

Lemma 1. If two people have identical chest hair, they're equal.

to see that $\varphi_{217}$.

(etc.)

Then go and prove the lemmas afterward.

Addendum. Rereading my original answer, I notice it has a major deficiency that I'd like to address here. In particular, the above answer fails to mention the importance of the word "show" and phrases like:

• "we're trying to show that..."
• "hence, it is enough to show that..."

Appropriate use of such phrases is a key tool for writing clearly, not only because they clarify what it is you're doing, but also because they give you a means of beginning at the end of the proof and working backwards, which is often a lot clearer.

Let's take a look at your example:

"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"

You could rewrite this as:

Our goal is to show $S$. By Theorem (whatever), it suffices to show $s$. Rewriting to make $x$ the subject, we see that it suffices to show $s'$. Therefore ..... [1-2 pages later] .... Hence it suffices to show that $1 > 0$. But this is trivial.

Be sure not to say "we need to show ." You never need to show a goal, because it's not a necessary step for completing the proof, rather it's sufficient for completing the proof. Better to say: "it's enough to show that," etc.

Here's a bigger example of how to write in this way that you might find helpful.

Problem. Show that $$\forall n \in \mathbb{N} : \sum_{i = 0}^{n-1}(i+1)^2 = \frac{n(n+1)}{2}.$$

We proceed by induction. For the base case, our goal is to show that $$\sum_{i = 0}^{0-1}(i+1)^2 = \frac{0(0+1)}{2}.$$ That is, we're trying to show that $0 = 0.$ But this is trivial.

For the inductive step, assume $$\sum_{i = 0}^{n-1}i+1 = \frac{n(n+1)}{2} \tag{$*$}.$$ Our goal is to show the following $$\sum_{i = 0}^{n}i+1 = \frac{(n+1)(n+2)}{2}.$$ Since the function $t \mapsto t - a$ is injective for all $a \in \mathbb{R}$, hence by subtracting $(*)$ from both sides, we see that it is enough to show that $$\sum_{i = 0}^{n}i+1 - \sum_{i = 0}^{n-1}i+1 = \frac{(n+1)(n+2)}{2} - \frac{n(n+1)}{2}.$$

Simplifying, we deduce that it's enough to show that $$n+1 = \frac{(n+1)(n+2)}{2} - \frac{n(n+1)}{2}.$$ But this is true, by elementary algebra. This completes the proof.

Answer 5

tl;dr: Your instructors tolerate it to teach you the value of a correct proof. Later on in your career you will need to learn the value of a clear, well-communicated proof as well.

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Instructors give full credit because at this phase of education it is very important to emphasize that correctness comes from the mathematical content of the proof alone. The instructor's feelings are irrelevant. If your proof is correct, no matter how silly you are or how much of a dork you choose to be when writing a proof, we accept it. As a grader I've given full credit to a proof written with every variable as a meme. I hated it; but this was the contract between instructor and student and I honored it and despite the nuisance I feel this was good pedagogy (and well-intentioned enough).

This will not fly if you are the instructor or you are the writer. If you want your students to be confused, give them exercises. Confusion is important and exercises are where it should happen.

It will also not fly if you are the writer. Your editors or proofreaders will expect you to write clearly and informatively.

Academic papers often value terseness, but that is not something that derives from delightful obfuscation. So you won't be doing that.

Answer 6

I find this practice irksome for several reasons: it makes it tough on the reader; it may indicate a lack of understanding by the writer; and it contributes to a perception of mathematics as being abstruse and rife with legerdemain.

The practice seems to arise once formal proof writing rears its head, although less so in, say, an early course on Euclidean geometry. One of the first places I notice it is when $\varepsilon$ finds its way into a mathematical discussion. To this end, I thought I would point out a proof that I think is a good example of how to prove a proposition in a textbook. The book is:

Bear, H. S. (2001). A primer of Lebesgue integration. Academic Press.

Here is the proof (p. 11) followed by a few comments:

The proof begins by telling you what it will be similar to in terms of earlier proofs. It does not go through each of the four parts, but rather picks one of them to show "by way of illustration" how they can each be proved.

Bear explicitly connects the assumed limit with both the notation $x_\alpha \rightarrow \ell$ and the equivalent $x_\alpha - \ell \rightarrow 0$; I think many textbooks would elide over this equivalence. Moreover, he writes out the proof to show that, for any fixed $\varepsilon > 0$, the desired bound of $3\varepsilon$ can be achieved. Certainly some care should be taken in introducing this approach, but I find it easier to read: Yes, he could have gone back and tinkered with things to achieve the bound of "$< \varepsilon$" instead. But once one has some understanding of how these proofs go, I find that those careful selections in reworking of proofs feel a bit like Mad Libs. My own preference, although I do not think this convention is widely shared, is to allow a bound of a constant multiplied by epsilon, as Bear does here. I find it easier reading, and less mysterious; both of which, to answer the OP, are laudable goals.

Answer 7

Is this bad form?

Yes

I wonder why you are asking this - how could it not be bad form?

What you describe is common. Our Discrete Structures prof used exactly that style. His first act on day 1 of DS101 was not to say "Good day", but to write "Definition 1.0.0.1" on the board, and give the formal definition of whatever he started with. Then "Axiom 1.0.0.2" ... "Theorem 1.0.0.3" ... "Proof 1.0.0.4" and so on ad infinitum.

Everything was clearly labeled, logical, correct, orderly, properly cross-referenced and absolutely useless. Nobody had a clue about what the purpose of this class was (unless they picked up a book, or already knew what discrete mathematics was about). It certainly did not do anything to teach how to use all this mess on your own, or how to get some kind of intuition. Frankly, as you had to get a textbook anyway, I found it hard to see what the point of his lecture was. Yes, using textbooks in addition to lectures is common, but the lecture should not be a totally opaque mess.

Compare this to some full lectures that are available online by well-known universities (for example, and this is a nice coincidence, Stanford Discrete Structures Lecture Notes). One thing they usually have in common is that the prof goes to great lengths motivating what he is teaching. Of course this is the good form. Of course you want to do that in your proofs, or in whatever you do. No matter whether you are targeting students or your seniors.

Answer 8

Humm... $1$-$2$ pages is quite short. :)

Now honestly, there are so many ways of writing mathematics that each of us writes it in our own way. It also depends a lot on the audience and on the topic. Specialists often know what is going on and a single sentence may be sufficient, someone who is studying it for the first time "deserves" a different approach.

Do try to read what you wrote thinking that someone else is reading it! This is a very good exercise that often leads to improvement and self-awareness of what is involved. Experience will come with time.

By the way, it is really wonderful (and atypical?) that you are taking this into consideration.

Answer 9

Since all answers agree that this is a bad idea, I'm tempted to add a rare case for the opposite.

Say you have been working for 7 years on a famous conjecture that has tortured mathematicians for 300+ years without anyone else having even come close to proving it. But you have managed to finally tackle it, in a 150 pages paper.

In that case, it seems like a good idea to have a "mysterious" proof - and provide it in, say 3 days lectures - that escalate into proving the unexpected and exciting result.

At Last, Shout of 'Eureka!'