# Organization of a paper -- completely treat simple case first, remark on harder case after, or unify presentation?

I have a question about how to organize a paper, and completely disagree with the way my advisor wants it done. The field is operations research, and involves mostly a bunch of mathematical theorems. For the purpose of this discussion, say there are two sections: Preliminaries and Results -- the former establishes some basic theorems and properties of certain functions which are then used in the results section. The conclusion of the Results section is how to solve certain optimization problems.

There are also two natural cases of the problem, call them Easy and Hard, that we should look at (i.e., any reader would be confused if one case was covered and the other wasn't). Most of the content, but not all, in "preliminary Results" and "Results" is considerably simpler for the Easy problem, and a bit more complicated for the Hard problem. But, they admit a more-or-less unified treatment, with much (but not all) of the Easy problem being a special case of the Hard problem.

The way I have organized this paper is to discuss both cases together -- in Preliminaries I discuss the Easy and Hard problems and show how they are unified, as well as providing separate statements / proofs for the preliminary theorems. Then, in the Results section, I provide one statement that solves both the Easy and the Hard problem together. There is also a bunch of statements about how to interpret some results which say things like "in the Easy case, we have the result A, but in the Hard case, it is similar to A, but for this slight difference". The organization in this way makes sense to me because the reader can have in mind both examples in their mind, see how they are unified together, get the complete picture. It also leads to a shorter paper.

However, my advisor believes that I should ignore the Hard Problem in both the Preliminaries and Results section, giving only the Easy results. Then, add another section, The Hard Problem (say), and show how everything can be extended into this case. I see some merit in doing this, as (most) things are a bit easier for the Easy case. However, shouldn't like ideas be considered together? With this organization, you basically need to read the paper twice, once for each case. As well, I feel like my advisor is telling me to write the paper as if the reader is some fool that knows nothing of the field and can't manage to understand both examples at once.

Any general advice for what may constitute a preferable organization for such a case?

• I myself do always like to see an easy case first... :) Commented Apr 19, 2022 at 17:49
• Suppose you were to give a 20 minute talk. Would you talk about the hard case at all? What about a 10 minute talk? Commented Apr 19, 2022 at 17:55
• "As well, I feel like my advisor is telling me to write the paper as if the reader is some fool that knows nothing of the field and can't manage to understand both examples at once." What's wrong with writing a paper that can be read more easily by people who "know nothing of the field" (an exaggeration, I assume)? Conversely, what is the benefit of only writing for experts in the field, except obtaining a shorter paper? Your job as a writer is to make the reader's job as easy as possible (even if they can understand both examples at once), not to write as short a paper as possible. Commented Apr 19, 2022 at 18:06
• Your advisor publishes in that particular area, and therefore is almost undoubtedly the one to listen to in terms of organization. Commented Apr 19, 2022 at 18:47
• "As well, I feel like my advisor is telling me to write the paper as if the reader is some fool that knows nothing of the field and can't manage to understand both examples at once." That's in general excellent advise for writing understandable papers. Commented Apr 20, 2022 at 12:29

## 6 Answers

I'm under the impression that your perspective is influenced by a number of misconceptions which are very common for mathematicians writing their first papers. So I'll try to clarify those misconceptions in what follows.

(My apologies in case I am sometimes a bit too blunt - but I think it is important that somebody makes you aware of these points.)

I feel like my advisor is telling me to write the paper as if the reader is some fool that knows nothing of the field

I think you might be overestimating the abilities of your readers: You have been dealing with the subject of your paper for probably quite some time and in a lot of detail. In this situation it is easy to get the impression that everybody else understands everything in your paper just as easily and clearly as you do. But they don't.

In addition, it is, generally speaking, an advantage rather than a disadvantage if your paper can also be understood be people who are not experts for specifically this topic.

Moreover, people who really know the topic very well just skim through the easy part of the paper and then quickly get to the hard part.

and can't manage to understand both examples at once.

It is irrelevant that your readers might be capable of managing to understand both examples at once. Your goal should be to make your results and arguments as easily and clearly accessible as possible. Just because your readers could also cope with a more complicated presentation of the topic, this does not mean that you should force them to do that.

It also leads to a shorter paper.

No, most likely it doesn't. The reason is that you're measuring "length" in the wrong unit. Numbers of pages are not a relevant quantity. Instead, you should try to minimize the time that your readers need to find and understand the information they are looking for.

Thus, clearer and easier presentation typically makes for a shorter paper (in terms of the time investment required from readers) rather than for a longer one.

[At this point someone will typically point out that numbers of pages are actually relevant in some cases since some journals have page limits. However, I stand by what I said above. If a clear and easy-to-follow presentation leads to a paper that exceeds the page limit of journal X, the appropriate conclusion is that journal X is not the right journal for this specific paper.]

The organization in this way makes sense to me because the reader can have in mind both examples in their mind

As I said above, please try not to overestimate the capabilities of your readers. Keeping several things in mind at the same time requires considerable cognitive load, even for people who are used to think about complicated things.

From personal experience I can tell you that the only situation in which I manage to keep more than one complex thing in my mind at once is when I develop things for myself in my head. Whenever I just follow a presentation (no matter whether it is a talk or a written paper), the requirement that I keep multiple things in mind quickly exceeds my mental capabilities.

The solution to this problem is, of course, that I do not simply read results and proofs when I read a paper, but that I try to actively "develop" parts of the theory when I read a paper - and whenever I get stuck, I will check in the paper how to proceed. Now the point is: What will make it easier for me to try to "develop", myself, parts of the theory that I am reading? A very general exposition which contains a lot of information and special cases at once, or a more incremental exposition which starts off with an easy case and treats the hard case only afterwards?

see how they are unified together, get the complete picture.

For the same reasons as explained in the previous paragraph, it is often much easier to get the complete picture if this picture is developed step by step, starting with the easiest case, rather then if it presented all at once.

However, shouldn't like ideas be considered together?

Not necessarily. There are various ways to consider and compare similar ideas. One such way is to treat them together, as you suggest. But it can also be done by treating them consecutively.

With this organization, you basically need to read the paper twice, once for each case.

Ok, the following point might sound a bit surprising at first, but it is important: Most mathematicians do not read papers. I'm serious, so once again in bold: Most mathematicians do not read papers.

They do something else instead, namely: they read parts of papers.

Most researchers are very busy (or at least they like to claim that they are busy - for instance, I often make this claim, but if it were really true, I probably wouldn't be writing this lengthy answer right now). Thus, they try to use their time in an efficient way. Once you have a certain amount of experience in mathematical research, reading an entire paper cover-to-cover is, in many cases, no longer the most efficient way to access the information in the paper that you would like to retrieve.

Instead, many people will, for instance, briefly skim through a paper, find a result that they find interesting, and then try to understand the result. This can mean very different things, depending on the precise situation. For instance, they might try to understand the statement and to check whether it generalizes a result that they know. Or they might compare it to their favourite one-fits-all counterexample and check why it doesn't yield a contradiction. Or, in some cases, they might even read the proof. Or they might put the paper away, pull it out once again two months later and then read the proof.

This observation has two important corollaries:

• Papers should be written in a way which makes them easily readable for people who do not read the entire paper (or do not read it at once).

• They notion of a reader who sits there and "reads the paper" is not a good leitmotif for writing a paper. Thus, the idea that "you basically need to read the paper twice" approaches the task from the wrong perspective because it is not consistent with how most people read scientific literature.

[Just in case anybody feels offended because they do read entire papers cover-to-cover: I am not claiming that nobody does this. I am just saying that in my experience the majority of people do not do this, or at least do not do this often.]

• +1 for creating that warm fuzzy feeling of being understood; "often make this claim, but if it were really true" Commented Apr 20, 2022 at 7:14
• Relevant xkcd for the first point: xkcd.com/2501 Commented Apr 21, 2022 at 13:30
• I read only parts of this answer, while pretending that I am very busy. Still I completely agree with it. :-) Commented Apr 21, 2022 at 15:53

Summarizing the comments so far, with which I agree:

Your advisor is right. Be as kind as possible to your readers - you will get more of them that way. The experts will not think you are treating them as fools. Tell them all about the easy problem. If they're interested enough they will read on to the extension.

I feel like my advisor is telling me to write the paper as if the reader is some fool that knows nothing of the field

According to Polya (who cites somebody else), Zermelo mockingly preached to conference speakers:

I. You cannot overestimate the stupidity of your audience.

II. Insist on the obvious and glide nimbly over the essential.

Polya comments as follows on these "Zermelo's rules":

Zermelo’s personal remarks were often witty; very unjust on the whole, but striking and revealing about some particular point. So was the criticism implied by the two rules; I had to laugh and I could not forget the rules. Years later I realized that these rules, suitably interpreted, give often applicable sound advice.

Any general advice for what may constitute a preferable organization for such a case?

From the general writing perspective (without making any assumptions on how smart a potential reader may or may not be): if ideas are easily separable, then they should be separated. After all, this is the reason why we have sections, subsections, paragraphs, chapters, etc.

Without seeing your draft paper, I would have a general tendency to defer to your advisor simply on the basis that they are more familiar with the details than a reader of your post. In the present case that is augmented by the fact that your advisor's suggestion is consistent with what I would suggest doing --- start with the easy case and then abstract to the harder case after this.

The main advantage of doing things the way your advisor suggests is that it make things a lot easier for your reader to understand. The easy case acts as a segue for the harder case and allows the reader to ease into the problem and its solution. By first presenting the easier problem and its solution, you give the reader something simple to start with, and then when you present the harder case the reader is already partly familiar with the details and the general method of proof. This is much less taxing for the reader and it is likely to make your paper much more useful.

I feel like my advisor is telling me to write the paper as if the reader is some fool that knows nothing of the field

Our fields are so narrow these days that even most Mathematicians are probably "some fools who know nothing of your field", and certainly Computer Scientists, Physicists and Engineers of various disciplines. For example, the fool writing this answer likes to (try and) read mathematicians' work occasionally, so please cater to him :-)

In other words: Treat the simple case first.