Punctuation and Other Rules for Variables and Their Verbal Definitions in Math Narrative [closed]

Question

To better understand what I'm asking about, let's immediately define some examples. Imagine that you are writing some paper which involves a lot of math narrative. And you have a term, say, computing unit (processor). Then at some point you introduce a mathematical variable, which you would use in formulas and text, $m$ and you define it as number of computing units (verbal definition).

Question #1: When I want to refer to $m$ in the text, I can do it in 3 ways:

1. One could observe linear speedup given increasing number of computing units $m$ in the system.
2. One could observe linear speedup given increasing number of computing units in the system.
3. One could observe linear speedup given increasing $m$ in the system.

Is there any general stylistic guideline on which one to prefer, when, and why? If not, then I'd still like to hear your personal opinion as long as it is backed up by healthy reasoning and/or authority.

I personally prefer #1 because as soon as $m$ is defined, I believe that it is easier for reader to follow and constantly stay in context of narrative if I refer to this quantity with verbal definition (which reminds reader of its purpose) and with variable (which assures reader that indeed I refer to these computing units and not other ones and that formulas involving $m$ presented before or after are indeed connected with current sentence/statement). Can you think of any case where this method #1 would be harmful/irritating/misleading?

Question #2: While the first question was about general guideline, this one is about positioning of verbal definition with respect to variable. So let's assume that we praise method #1 from previous question, then there are 2 ways to write it:

1. One could observe linear speedup given increasing number of computing units $m$ in the system.
2. One could observe linear speedup given increasing number $m$ of computing units in the system.

Any general guideline? Pros and cons? You personal opinion?

Maybe that example was not so good at demonstrating possible confusion, but here is another one:

1. Consider average velocity of stream $v$.
2. Consider average velocity $v$ of stream.
3. Consider average stream velocity $v$.

Notice how #1 confuses about whether $v$ refers to "stream" or "average velocity of stream". #2 and #3 read better. Would you even omit word "stream" in narrative if you've already defined $v$ as "average velocity of stream" once?

Question #3: Punctuation. The general guideline which is well-known among technical writers, I suppose, is: a text (narrative) involving math is still a text in the first place and has to obey the same punctuation rules as any other text not related to math. Well, then consider the following sentence:

Our valuable employee, Alexander, was granted permission to access this room.

Notice how "Alexander" is surrounded with commas. Following this rule, shouldn't we apply it to math narrative then? For example:

Average stream velocity, $v$, is a very important quantity in this context.

Can you feel the analogy between two? What are your thoughts about it?

Answer 1

I hope that this question stays open because it (question 1 in particular) deals with something that is a big problem in many math papers. Whether the problem is specific to mathematics, I can't say.

The answer to the first question is that alternative #1 is almost always preferable. The author should help the reader to remember what sort of object or quantity is denoted by the symbol $m$. The only exception to this that I can think of is when it was mentioned in the previous sentence, or previously in the same sentence. Why risk wasting a few minutes of the reader's time (used to find the definition of $m$) in order to save a few words? Moreover, the use of redundant words can help to remove ambiguity between similar symbols: it is much easier to confuse $m$ with $M$ than it is to confuse the number $m$ with the model $M$, for example.

For question 2, I agree with Matt F. that alternative #1 ("velocity of [the] stream $v$") is not good because the reader might think that $v$ denotes the stream. I think that the comparison between #2 ("velocity $v$ of [the] stream") and #3 ("stream velocity $v$") is more subtle and depends on the details of your example, namely whether "stream velocity" is a set phrase in your field.

For question 3, I agree that it is okay to treat the symbol "$v$" like the name "Alexander." In either case, it should be enclosed in commas if and only if it is an inessential part of the sentence (that is, it could be removed from the sentence without changing the meaning.) In your particular example this would depend on whether you were saying something specifically about the average stream velocity $v$ (as opposed to the average velocity $w$ of some other stream) or merely reminding the reader of the symbol that you have been using everywhere to denote average stream velocity.

Answer 2

For Q2, I prefer "2. One could observe linear speedup given increasing number $m$ of computing units in the system", and also "2. Consider average velocity $v$ of stream." In both cases that placement minimizes confusion about what the variable represents.

Note: This principle can be taken to extremes.

For Q3, I prefer no commas. The punctuation is fine in "Our valuable employee Alexander was granted permission." It's fine with the variable too.

Answer 3

No matter what you write, someone will misinterpret it. Your goal should be to minimize this misinterpretation. In this spirit, let me suggest some alternatives that you may find helpful.

For question 1, set up the premise that you will talk about a class of systems of similar configuration, but with adjustable parameters m,p, and q, where m is the number of computing units, and p and q bear some other characterizing relationship with the system S(m,p,q).

You can then say things like: "holding other things equal, systems with larger m finish sooner/have higher throughput/are more responsive/are faster". Of course, it might be better to use "larger number of computing units" than "larger m"; the point of the notation is to abbreviate: most readers capable of retaining context will make the translation "larger m" to "larger number of computing units" themselves, and likely can also do "have higher throughput" to "are faster" on their own.

In order for this to work well, some transition between wordy and brief which reinforces the intended translation is recommended, e.g. "One can see operating temperature in the graph rising according to m (number of computing units)." Usually one or two instances near the definition are enough for transition, but if you are using this thirty times or more, an occasional reminder can be useful.

Regarding positioning of variable, you need to ensure consistency. This might be done the following way, which I will use for my interpretation of each of your three sentences:

• Given stream v, consider v's average velocity.

• Picking a stream and determining its average velocity using method M, let us call this average velocity v.

• Notice the generality of method M in computing the average velocity of a stream. Let us denote the result of this method v (which depends on the stream), while later results using method N we will call w.

Note that as the notion of v changes, the description for it gets more complex. In all cases, there is a short sentences or phrase saying "This is what I will mean when I use v".

Note that the positioning of v inside the phrase/sentence is not as important as the existence of the phrase "v means this" or "let (descriptive phrase) be called v". If the phrase itself does not carry enough definitive punch, repeat with a clarification, e.g. "note that v is a result of method M in this chapter: next chapter we will use method N and the result w and provide contrasting statistical analyses of v and w".

Regarding punctuation, it is used according to certain conventions to alter meaning in certain ways. If you don't know how to use it, don't use it. Alternatively, when you do use it, provide an alternative that reinforces the intended interpretation. I will add two supporting statements for the case of one or two commas:

• (one comma " v,") "While v can vary from stream to stream, we find the relationship momentum M = mv to hold for all streams studied."

• (two commas ", v,") "As we will see, the numerical value v holds for all streams whose source is at the same altitude and have a common terminus at sea-level."

Gerhard "What Do These Quotes Mean?" Paseman, 2014.05.09