Using typographical variants of the same letter as mathematical symbols in a paper

Question

Sometimes I struggle in choosing notations for a mathematical paper. There are probably no explicit rules, but unwritten conventions. In a discussion, is it proper to use different typographical variants of the same letter to denote different variables?

For example, is it acceptable to use an italic K (𝐾), an upright K (K), a blackletter K (𝔎) and a calligraphic K (𝒦) for four different variables in one section?

Answer 1

Enhancing on some of the other answers: while this is OK, and I have done this and worse in extremis (one of my papers is known as "the one with the four types of arrow"), your paper will be difficult to read.

You need to be aware of the fact that many readers will have a hard time tracking the differences between symbols. I recommend:

1. Doing everything in your power notationally to avoid this situation in the first place
2. If you must, first choose things that are easy to tell apart: e.g. capital vs. lowercase vs. mathcal. Upright vs. italic or symbol vs. symbol-bar are much harder to tell apart at a glance.
3. Most important, in any symbol-heavy paper, include a "cheat sheet" table up front that gives the definitions of all important symbols or symbol-classes

Answer 2

Just an opinion: I think four variants of the same letter in one section is probably too much. In particular, I think (non-bolded) upright letters don’t work well for typesetting mathematics.

Also you say ”for four different variables”; this suggests that maybe you want to denote four of the same kind of thing in four different variants. (If I'm wrong, ignore the rest.) That is definitely to be avoided. As @Wrzlprmft says, the variant should reflect the kind of object you’re talking about, and related objects of different kinds should be denoted by the different variants of the same letter. The first example that comes to mind is a Lie group G and its Lie algebra 𝔤 (blackletter g).

Here’s an apposite quote from Littlewood’s Miscellany, via Milne:

It is said of Jordan's writings that if he had 4 things on the same footing (as a, b, c, d ) they would appear as a, M₃', ε₂, ∏"_1,2.

Bad Jordan!

Answer 3

Yes, it is acceptable and I have done so in publications myself.

However, you should not use these variants randomly but follow some system, e.g., you could use a calligraphic K do denote the set that contains K₁, K₂, K₃, … and a blackletter K do denote some transformation of that set. You should also check whether there are some generally accepted conventions in your field (and follow them), e.g., in some fields, vectors are indicated by using upright boldface letters while the corresponding normal italic letters are used for their components.

(In general, I prefer to use one typographic variant for symbols representing similar structures, e.g., lowercase italic letters for natural numbers, lowercase greek letters for real numbers, uppercase italic letters for countable sets of natural numbers, calligraphic uppercase letters for uncountable collections of natural numbers, etc.)

Answer 4

One thing to keep in mind is that readers may want to work through some of your derivations while they read it, using pen and a separate piece of paper. Bold symbols can be handwritten using doublestruck letters, and calligraphic letters are generally fine, but distinguishing between italic and upright Ks in handwritten notes is a killer and it is likely to discourage exactly the sort of attentive reader that you want to really read your paper in depth.

This suggests, then, the rule of thumb that if you can consistently use those symbols in handwritten notes you will generally be fine. What symbols did you use when you developed the material? That is a good starting point, and beware of trying to cram extra concepts onto the same symbols after you've moved off the pen-and-paper and onto the computer screen.

Answer 5

In mathematics, the typeface says what your variable is. It is like writing pCounter in a computer program to make clear that the variable holds a pointer.

For instance, one of the many common notations in linear algebra is that capital letters like $A$ represent matrices, lowercase letters like $a$ represent vectors, Greek letters such as $\alpha$ represent scalars, script letters such as $\mathcal{A}$ represent subspaces.

The actual letter that you use often is a "default letter" for that kind of mathematical object: for instance, $D$ often represents a diagonal matrix, $H$ a Hermitian one, and so on.

Of course, the choice isn't always obvious, and often notations for different fields clash (for instance, you may want to use $\delta_{ij}$ as the Kronecker delta, but $\delta$ as the prototypical calculus "small positive number").

Using the same letter in different typefaces is often reserved for related quantities: for instance, you could call $d$ a vector, $D$ the diagonal matrix with diagonal entries $d$, and $\mathcal{D}$ its spanned subspace (just the first example that crossed my mind). Or vectors and their containing subspaces: $u\in \mathcal{U}$, $v\in\mathcal{V}$.

If you use the same letters for different things, I assume that they are either "default letters" or related quantities. Otherwise, you are just making life harder for your readers.

TL;DR: It depends on what your variables represent.