I am writing a mathematics paper for publication which frequently:

  1. Uses special functions the average reader would not be aware of.
  2. Uses my own notation.

Within the first section of my paper (introduction/preliminaries), I define the relevant special functions. Should I also define my own notation within the preliminaries, or wait until its use naturally arises within the paper to introduce it? My notation can be understood without context, though its usefulness will obviously not be clear.

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    Have you got a specific journal in mind? If so, have a look at that journal's submission guidelines. It varies from journal to journal. – Earlien Feb 17 at 4:45
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    I am not aware that such guidelines are used in mathematics and for me this would be a sufficient reason not to publish in such a journal. It’s just to much a question of personal style and there are good reason for doing one or the other (or even doing a hybrid solution). – Sebastian Bechtel Feb 17 at 7:56
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    Have you discussed best writing practices (for your discipline) with your adviser? Probably the best qualified person to talk about the structure of a manuscript (from a short communication through journal papers to thesis) is the senior academic in your chosen field guiding and supervising you on best research practices - your adviser? – penelope Feb 17 at 15:19

There is no general rule. Different authors have different opinions about this. To my knowledge no mathematics journal has guidelines about this sort of thing.

A rule of thumb could be that if your notation is going to be used throughout the paper, then it should be introduced at the beginning in a special section called "notations", "conventions", "background", or however you want to say this. If the notation is only going to be used locally in some section or even one proof, then I would introduce it at that point.

Another possibility is to make a categorized table of notations at the beginning or an alphabetical list of notation with references to where the object is defined, if your paper is long and complex enough to justify it. This can of course be combined with the previous rule of thumb.

I would recommend against making the "background" section as part of the introduction. The introduction is supposed to be something that can be read by almost everyone slightly interested in the paper, to know what the main results are and how they fit with the general literature, and in case of stars aligning, whether the paper is worth reading. (The abstract is to decide whether you'll even glance at the paper at all.) Keep it short and simple and avoid introducing too many things at once. In my opinion it is a mistake made by many younger, and also more experienced authors to make the introduction way too technical.

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    A couple personal thoughts: (1) it's okay to put a notation section at the end of the introduction like acknowledgements if it's short, particularly if some of your later sections are more-or-less independent. (2) What I define in the notation section at the beginning are things like quasi-standard notation, things that are easy to define and frequently used, and notational remarks that will be useful for someone skimming the paper. For bjects that require some exposition or notation used only locally/infrequently, I typically define them as they arise. – Kimball Feb 17 at 14:32
  • There's also the issue of definitions that rely on some intermediate results to make sense (if you're defining the doohicky of x to be the smallest real number such that something holds, the proof that there is a smallest real number with that property better be either before or immediately after that definition). – user3482749 Feb 17 at 17:14
  • @user3482749 I don't know, that seems easy enough to finesse. Instead of "The doohicky of x is the smallest real number r such that P(x, r).", something like "If there is any real number r such that P(x, r), then the doohicky of x is the smallest such r." should work fine. Later it can be a nice surprise that every x has a doohicky, but it needn't appear in the definition. I think it's usually possible to pull such shenanigans. – Daniel Wagner Feb 18 at 1:03
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    @DanielWagner The question is whether there's an actual benefit to doing so. – user3482749 Feb 18 at 12:00

What you should prioritize in any paper, mathematics or otherwise, is readability. Organize the paper so that your reader can comfortably follow your argument without a lot of jumping around.

In a short, relatively "flat" paper, almost any organization will probably be ok. Flat in the sense of later parts not depending fundamentally on earlier parts.

But in a longer paper it becomes an issue, for me at least. Things should be introduced with some context: Why is this being introduced here. If I see a definition, I expect to be able to easily grok why it is here from some prior context or by being used quite soon in the development. A definition or piece of notation introduced without context is just irritating.

As an extreme example, imagine a calculus or abstract algebra textbook in which all of the book's definitions are in the first chapter along with every notation that will be used.

I'll note that the very purpose of defining things in mathematics is to give us something to think (and write) about. Defining something with no context is just noise. We define rational number or Abelian group for example, because we want to say things about them. If you define them, but don't soon discuss them, the reader has no context.

So, first think about the flow of the paper from the standpoint of the reader. I think that in most work with any significance the "just in time" organization will probably work better. I'll admit there may be exceptions. But it is the readability that should be prioritized, not some abstract concept of an "ideal" organization.

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    Yes! But I would recommend both. It is at least equally annoying to be skimming a paper only to find a piece of information/definition that I could only figure out by essentially reading the entire paper up to that point to find where it is talked about, but could have been a bullet point in an appendix/preliminary section. And using the text book comparison again, they also solve this by doing both. – whn Feb 17 at 19:20
  • @whn internal cross references. – paul garrett Feb 18 at 1:29

Everything you write in a paper should be the answer to the question that is in the reader's mind at that point. So the title needs to answer the question: "Do I bother with this at all?". The abstract, as @user119516 says, answers "do I glance at this?", and the introduction answers "do I read this?".

If you follow that principle then it is obvious that you do not divorce your definitions or novel notation from the points in the paper where they are used, because until the reader reaches those points they have no question in their mind about them.

If your paper is very long then it might help the reader if you collect all your notation and maybe your definitions into some helpful appendix, but don't interrupt the flow of the paper with such stuff.

Fashion plays a part here. At one time, and maybe still in some sub-disciplines, it was considered reasonable for the first sentence of a pure mathematics paper to be something like : "Let A be set ...". No-one writing with the reader in mind would do so, surely.

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