In which areas of mathematics is it traditional to use script letters, such as $\mathcal{ABCDEFG}$, and is there a pedagogical advantage to doing so?

Update: Charles Wells has written a post on this topic on his Gyre&Gimble blog.

  • The Laplace transform traditionally uses a script $\mathscr{L}$. – Joel Reyes Noche Sep 20 '14 at 13:33
  • 2
    Also, the letters $\cal{ABCDEFG}$ are calligraphic (the LaTeX command is \cal). The letters $\scr{ABCDEDG}$ are script (the LaTeX command is \scr). – Joel Reyes Noche Sep 20 '14 at 13:35
  • I'm using the term script informally, as in Using script fonts in LaTeX, but thanks for pointing it out. – J W Sep 20 '14 at 13:47
up vote 4 down vote accepted

When using categories it is common (though not universal) to denote the categories by script letters; see http://www.staff.science.uu.nl/~ooste110/syllabi/catsmoeder.pdf and http://wwwhome.ewi.utwente.nl/~fokkinga/mmf92b.pdf (page 8) for two examples. [The examples are not chosen very specifically, they just came up early on in a search for examples.]

Another example where a script letter is common is for denoting the powerset of a set $S$, as $ \mathcal{P}(S)$.

Regarding the pedagogical advantage, I would say that it can be useful if one has several 'layers' or 'types of things' to distinguish them by different fonts. For example, it is common to denote sets by upper-case and their elements by lower-case letters.

If there is a third layer it can make sense to ressort to a third font, and thus use script letters. Like, categories $\mathcal{C}, \mathcal{D}$ have objects $C_1, C_2$ and $D_1, D_2$, respectively; and let $f$ and $g$ denote a morphism of $C_1, C_2$ and $D_1, D_2$, respectively.

The question now splits a bit as there are at least two separate questions:

  1. Does it make sense to use several fonts in this way?

  2. Is script a good choice for a font?

For 1. I would say that it can help to structure things and to have a permanent reminder what is what. It is (with different fonts) also very wide spread, for example, Greek letters for angles. Yet, one should also not overdo this. There is a risk that students become so fixated on some specific notation that they are completely off track when then confronted with a different set of notational conventions.

For 2. I would also say, yes, in some cases. What is good about the font is that it is not too different but still distinguishable. By contrast, Fraktur (as somewhat frequently used for ideals or the symmetric group and related things) can be difficult to read and write. For instance, I am under the impression that for not few students the notations $\mathfrak{S}_n$ and especially $\mathfrak{A}_n$ for the symmetric and alternating groups are not at all intuitive, since they do not really "see" an "S" and an "A" in a different font but just some unknown symbols.

What is not so good is that it can be a bit 'heavy' or 'voluminous' (for lack of better words). So, it seems better to use it for things that do not occur that frequently and/or are rather 'high' in the hierarchy, like, element $a$, in set $A$, in a collection of sets $\mathcal{A}$.

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