I have one question for my masters thesis. If I have A, which is also known as B or C, can I write:
A (also known as B or C)
(or similar) or can I use a foonote like this:
A¹
¹ Also known as B or C
Or more in general: When do I add extra information (like synonyms) between parenthesis, and when do I add them as footnote?
This is mostly about the reader's flow. Something that is parenthesized is something you expect the reader to actively read, but you're signalling that it's secondary information. The main drawback to parentheses is that if the text in them becomes too long, the reader has to work very hard to remember the main point you're making. You're essentially talking about a lot of unimportant stuff while you have an unfinished, important sentence going. So make sure you put only short, simple things in parentheses.
If you put something in a footnote, you're signalling that the reader should skip it on first reading and they should only investigate if something is unclear, or if it's a second reading and they need all the details. The drawback to footnotes is that they are often more tantalising than they should be. While the reader should ignore them, they are too curious, and break their concentration to look up the footnote. This pulls them out of the text for something they were supposed to ignore.
On the whole, try to avoid both as much as you can. In your example, ie. this method is also known as X, parentheses should be fine and a footnote is probably overkill. However, you could also consider finishing your main point first, and moving the parenthesized statement to the end of the paragraph as a full sentence. That way the reader can finish absorbing the primary information unencumbered by details, and gloss over the aka's once the hard work is finished.
To a large degree, it all depends on your chosen writing style guide and, perhaps, your institution's and/or advisor's recommendations. Having said that, I would suggest using the following heuristic (rule of thumb) to determine potential use of footnotes versus text in parentheses: if information in question is short, such as "also known as B or C", it is preferred to use that text in parentheses, otherwise (for a longer text) consider one of the two alternative options, as follows.
The first alternative option is to use a footnote; however, using footnotes is discouraged by major writing style guides (i.e., by APA Style Guide) or advised to be limited, especially for explanatory, non-bibliographic notes (i.e., by MLA Style Guide). That aspect is likely to be very field-dependent, therefore, you can consider the second alternative option: simply adding explanatory sentence(s) after the text that requires such explanations or clarifications.
Looking at my PhD thesis (a little over 200 pages) I have 11 footnotes. 7 of these could be in parentheses with no rewording, the other 4 need at least a full sentence. Picking a few pages at random I often have 2-3 pieces of extra information in parentheses per page. Parenthetical commas aren't so easy to count, but I'd assume there are at least as many as parentheses. Explanatory sentences are also hard to count and too numerous, but can be a good way to inset this information. In fact you may find in proofreading (by you or someone else) that your parentheses get edited into new sentences to avoid run-on sentences.
I think there's a hierarchy:
in order of decreasing relevance to the main flow of the text (not decreasing importance).
It could easily be argued that I've used all these options too much but it suggests that (in my writing style at least) footnotes are a last resort. Where I have used them it's generally because I have a line of reasoning in which a point needs to be mentioned for avoidance of doubt, but to mention it inline would break the thread of the argument. That's the aim at least.
Incidentally, because I used a numeric-superscript citation style (a common one in my field, and I had the choice), counting was easier. I chose to use lower case alphabetical footnote keys reset per page (easy in LaTeX) in case I had a lot, and to avoid confusion with numbered footnotes or other uses of (e.g.) asterisks.
In mathematics, footnotes aren't very common nowadays (and used to be used for providing references). The differentiation is of course a choice of personal style, but one approach is: treat footnotes like annotations (as if they are written in another voice by another person--say author's comments to the reader). Another is: never use footnotes.1
1(Personally I like footnotes2, but I use them sparingly in academic papers but semi-often in less formal things like online notes.)
2(and parentheses)
Given that people may be searching for B or C, I think using the parenthesis is better. That way a search takes them to the text, also when Google shows the context of the search result, it is more likely to be useful.
Foot notes are also hard to read where you a using a device with a small screen that has issues displaying pages. Text with parenthesis is a lot easier to reformat for different display sizes etc (reading mode on IPhone for example).
I just browsed through my dissertation. I use quite a lot of footnotes: I count 45 in 200 pages.
Looking through them, I now find that many contain the words "strictly speaking". These are aimed at pedants such as myself who think they've spotted a mistake or inaccuracy: they explain how to deal with technicalities, why an abuse of notation is justified, etc. As explained in other answers, having these remarks in the main text would distract from the normal flow of reading, and the reader can often do quite well without them.
In your example, I'd use parentheses. I found one similar instance in my dissertation where I actually use both:
... chordal (also ambiguously* called triangulated)...
Here, I used a footnote to describe the other meaning of "triangulated", because it's not really in scope for the text, whereas it is a relevant fact that the term "triangulated" is sometimes also used.
Looking through my thesis, I was able to find another use case for footnotes, which has not been mentioned so far. Sometimes I define a concept or a symbol in a way which is slightly different from some other sources. I then include a footnote which describes how the symbol is used elsewhere. This could be important if the reader is trying to compare my result with other results or to follow a chain of citations which use different terminologies.
I think that in this case, including the alternate definition in the main text would be dangerous. For example, saying "we define $X = a + b$ (other sources define $X = a + b + 1$.)"
The reader would get confused between which is my definition and which is the alternate definition. By keeping the alternate in a footnote there is a much stricter separation between these two.
That depends mainly on the citation style required by your publisher, or requested by your supervisor. Some of them are parenthetic (Smith, 2015) some are footnotes (1: Smith, 2015)
On a more particular note, generally Citations refer to the source from which you take your assertion [i.e.: "This is cool" (Smith, 2015)] whereas footnotes just expand a concept without referring to a source [i.e.: "This is cool"1 1: "The term cool varies, but we use it here in its more common meaning"]
Your example is quite vague though, so my answer aims to be general but might not fit to your case.
