This is a draft of an addition to the entry Alphabets in abstractmath.org.
Mathematicians use the word script to refer to two rather different styles. Both of them apply only to uppercase letters.
Script
| $A$: $\scr{A}$ | $H$: $\scr{H}$ | $O$: $\scr{O}$ | $V$: $\scr{V}$ |
| $B$: $\scr{B}$ | $I$: $\scr{I}$ | $P$: $\scr{P}$ | $W$: $\scr{W}$ |
| $C$: $\scr{C}$ | $J$: $\scr{J}$ | $Q$: $\scr{Q}$ | $X$: $\scr{X}$ |
| $D$: $\scr{D}$ | $K$: $\scr{K}$ | $R$: $\scr{R}$ | $Y$: $\scr{Y}$ |
| $E$: $\scr{E}$ | $L$: $\scr{L}$ | $S$: $\scr{S}$ | $Z$: $\scr{Z}$ |
| $F$: $\scr{F}$ | $M$: $\scr{M}$ | $T$: $\scr{T}$ | |
| $G$: $\scr{G}$ | $N$: $\scr{N}$ | $U$: $\scr{U}$ |
Calligraphic
| $A$: $\cal{A}$ | $H$: $\cal{H}$ | $O$: $\cal{O}$ | $V$: $\cal{V}$ |
| $B$: $\cal{B}$ | $I$: $\cal{I}$ | $P$: $\cal{P}$ | $W$: $\cal{W}$ |
| $C$: $\cal{C}$ | $J$: $\cal{J}$ | $Q$: $\cal{Q}$ | $X$: $\cal{X}$ |
| $D$: $\cal{D}$ | $K$: $\cal{K}$ | $R$: $\cal{R}$ | $Y$: $\cal{Y}$ |
| $E$: $\cal{E}$ | $L$: $\cal{L}$ | $S$: $\cal{S}$ | $Z$: $\cal{Z}$ |
| $F$: $\cal{F}$ | $M$: $\cal{M}$ | $T$: $\cal{T}$ | |
| $G$: $\cal{G}$ | $N$: $\cal{N}$ | $U$: $\cal{U}$ |
Using script
- In LaTeX, script letters are obtained using “\scr” and calligraphic using “\cal”. For example, “{\scr P}” gives ${\scr P}$. The file Script fonts for LaTeX shows how to get variations other than the ones shown above.
- Both script and calligraphic are used to provide yet another type style for naming mathematical objects.
- One of the most common uses is to refer to the powerset of a set $S$: ${\scr P}(S)$, ${\scr P}S$, ${\cal P}(S)$, ${\cal P}S$.
- There may be some tendency to use script or cal to name objects that are in some way high in the hierarchy of objects or else a space that contains a lot of the stuff you are talking about. In most of the paper I found in a cursory exam of Jstor shows only a couple of exceptions (in Lie algebra). This is one of many places in abmath where I throw out examples of usages in math that I have found but have not verified through serious lexicographical research.
- The names of categories are commonly denoted by script or calligraphic. Some authors have trouble because they want to put names of categories such as “Set” and “Grp” in cal or scr but don’t have lower case letters in those styles. In Toposes, Triples and Theories the online version went through several changes over the years. Category Theory for Computing Science uses bold for category names.
- I have never run across a paper that used both script and calligraphic to mean two different things.
Acknowledgments
Thanks to JM Wilson for suggesting this topic and to the various people on Math Stack Exchange and Math Educators Stack Exchange who discussed script and cal.
References
- Writing mathcal letters by hand.
- Using script letters in handwritten math.
- Advantages of using script letters.
- Script fonts for LaTeX.
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I will produce another short note like this one about sans serif.