In writing mathematical papers, what qualifies as a "remark" and not just some discussion

Question

Can someone enlighten me as to what qualifies as a "remark" in mathematical writing. What is the difference between a remark, and just some discussion.

Context: I introduce a lemma, and then a theorem, and then I want to say something about the theorem. Would that count as a remark (with a remark section) or would it count as just a discussion (where no remark section is needed).

For example:

Theorem 1.2: All function $\phi_\pi$ is analytic

Remark 1.2: For $\pi = 3.1415926$, the function is well known to be real analytic.

Do I need a remark section for this? In other words, would it be acceptable to just say

Theorem 1.2: All function $\phi_\pi$ is analytic

For $\pi = 3.1415926$, the function is well known to be real analytic, see references.

Answer 1

This isn't the sort of thing there are official rules for. There are not any circumstances under which a Remark label is needed, and some authors don't use them at all.

Some reasons you might choose to label something as a Remark include:

• If you want to refer to it again later, so you want to be able to say something like "By Remark 1.2,...".
• You want to call attention to this particular remark.
• The remark is tangential to the main focus of your discussion, so you want to isolate it as a discrete "Remark" that is separate from your main discussion.

Basically, just use common sense about how labelling it as a Remark will affect the flow of your writing.