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

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?

• Follow the style of other papers in your field. Do you mean a paper in the field of mathematics, or a paper in some other field that uses more math than usual? – Oswald Veblen Nov 27 '14 at 13:54
• @OswaldVeblen, I mean a paper in the field of mathematics. – Zuriel Nov 27 '14 at 15:04
• On Mathematics Educators StackExchange, you may be interested in matheducators.stackexchange.com/questions/4437/… on using script or calligraphic letters in mathematics. – J W Nov 27 '14 at 20:35
• Technically, it's OK, since those K's are four different symbols. Practically, though, you're asking the reader to keep in mind that they're different and to keep in mind what each of them means, which makes it more difficult to pay attention to the information you're trying to convey. I would worry especially about using both the italic and upright versions together. Enough authors accidentally forget to italicize a symbol occasionally, so many of us habitually overlook that distinction. – Andreas Blass Nov 28 '14 at 2:57
• In typeset maths all maths all variables are in italics and functions are straight, so using both of those will probably confuse people. The blackletter and calligraphic are also easily confused, since we don't regularly write either. On the other hand, you can use lower-case, upper-case, calligraphic and blackboard-bold, and possibly also bold. At some point though it might be better to have some bars, hats, stars, dashes, subscripts, superscripts... – Jessica B Feb 6 '15 at 7:54

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
• I am not sure you work in math. The practice from the question is common in mathematics papers; a "cheat sheet" up front is not. I can't think of any example off the top of my head of a math paper I've read that does have such a table. – Oswald Veblen Nov 27 '14 at 13:52
• @OswaldVeblen While I am not "in math," in the areas in which I do work I often write papers with significant mathematical content. I will strongly argue that "cheat sheet" tables make a big difference if you want to communicate with people who are not themselves abstract mathematicians. I also find that building the table is a good means of self-criticism for notational clarity. – jakebeal Nov 27 '14 at 14:08
• @OswaldVeblen: it's true this isn't common, but that doesn't make it a bad idea. In fact when I am reading a paper closely, I often find myself writing up such a table. Why don't we do it more often? (By the way, one example of a paper with such a table is the Feit-Thompson paper, but that is something of a special case!) – potentially dense Nov 27 '14 at 14:24
• @jakebeal: What is an "abstract mathematician"? Note that the OP indicated that he is working in the field of mathematics, so if your advice is intended to be followed everywhere outside of mathematics, that should be clarified in the answer itself. On the other hand, I think Artie makes a good point: it is true that we usually do not include symbol lists in math papers unless they are book-length. But that doesn't imply that it's a good idea. – Pete L. Clark Nov 27 '14 at 16:27
• @PeteL.Clark I think that a good table of symbols is always a net gain for readability. The only disadvantages are 1) it takes space, which can be a problem if your manuscript has a strict page limit, and 2) maybe some scientific communities think less of you or your manuscript for "showing weakness" in this way? The second point doesn't apply to any community I participate in, but maybe there are math communities that operate that way... – jakebeal Nov 28 '14 at 13:56

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.

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.)

• So I guess you don't use many calligraphic uppercase letters :) – Tobias Kildetoft Nov 27 '14 at 12:26
• @TobiasKildetoft: That’s what you get for keeping the examples simple. It’s uncountable collections of natural numbers now. – Wrzlprmft Nov 27 '14 at 13:18
• @TobiasKildetoft, I am not familiar with it; I didn't even quite get your joke (am I right in assuming that it is a joke?). Could you please clarify a bit? – Zuriel Nov 27 '14 at 15:02
• Zuriel: I first wrote “uncountable sets of natural numbers” in the last paragraph. The set of all natural numbers (ℕ) is countable, hence every set of natural numbers is a subset of the set of all natural numbers and thus countable. Thus there exist no uncountable sets of natural numbers. Thus I would have no reason to use calligraphic uppercase letters. There exist uncountable collections of natural numbers though, as they do not exclude repetition. – Wrzlprmft Nov 27 '14 at 15:08
• @Wrzlprmft Of course this is only a tangent to your answer, but whenever I have heard "collection" in mathematics it has been a synonym for "set". Using it to mean "multiset" strikes me as very non-standard. – Trevor Wilson Feb 6 '15 at 2:18

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.

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.