This post is a revision of the part of the abmath article on alphabets concerning the fraktur typeface, followed by some corrections and remarks. A revision of the section on the Greek alphabet was posted previously.
Fraktur
In some math subjects, a font tamily (typeface) called fraktur, formerly used for writing German, Norwegian, and some other languages, is used to name math objects. The table below shows the upper and lower case fraktur letters.
| $A,a$: $\mathfrak{A},\mathfrak{a}$ | $H,h$: $\mathfrak{H},\mathfrak{h}$ | $O,o$: $\mathfrak{O},\mathfrak{o}$ | $V,v$: $\mathfrak{V},\mathfrak{v}$ |
| $B,b$: $\mathfrak{B},\mathfrak{b}$ | $I,i$: $\mathfrak{I},\mathfrak{i}$ | $P,p$: $\mathfrak{P},\mathfrak{p}$ | $W,w$: $\mathfrak{W},\mathfrak{w}$ |
| $C,c$: $\mathfrak{C},\mathfrak{c}$ | $J,j$: $\mathfrak{J},\mathfrak{j}$ | $Q,q$: $\mathfrak{Q},\mathfrak{q}$ | $X,x$: $\mathfrak{X},\mathfrak{x}$ |
| $D,d$: $\mathfrak{D},\mathfrak{d}$ | $K,k$: $\mathfrak{K},\mathfrak{k}$ | $R,r$: $\mathfrak{R},\mathfrak{r}$ | $Y,y$: $\mathfrak{Y},\mathfrak{y}$ |
| $E,e$: $\mathfrak{E},\mathfrak{e}$ | $L,l$: $\mathfrak{L},\mathfrak{l}$ | $S,s$: $\mathfrak{S},\mathfrak{s}$ | $Z,z$: $\mathfrak{Z},\mathfrak{z}$ |
| $F,f$: $\mathfrak{F},\mathfrak{f}$ | $M,m$: $\mathfrak{M},\mathfrak{m}$ | $T,t$: $\mathfrak{T},\mathfrak{t}$ | |
| $G,g$: $\mathfrak{G},\mathfrak{g}$ | $N,n$: $\mathfrak{N},\mathfrak{n}$ | $U,u$: $\mathfrak{U},\mathfrak{u}$ |
- Many of the forms are confusing and are commonly mispronounced by younger mathematicians. (Ancient mathematicians like me have taken German classes in college that required learning fraktur.) In particular the uppercase $\mathfrak{A}$ looks like $U$ but in fact is an $A$, and the uppercase $\mathfrak{I}$ looks like $T$ but is actually $I$.
- When writing on the board, some mathematicians use a cursive form when writing objects with names that are printed in fraktur.
- Unicode regards fraktur as a typeface (font family) rather than as a different alphabet. However, unicode does provide codes for the fraktur letters that are used in math (no umlauted letters or ß). In TeX you type "\mathfrak{a}" to get $\mathfrak{a}$.
- In my (limited) experience, native German speakers usually call this alphabet “Altschrift” instead of “Fraktur”. It has also been called “Gothic”, but that word is also used to mean several other quite different typefaces (blackletter, sans serif and (gasp) the alphabet actually used by the Goths.
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I have been doing mathematical research for around fifty years. It is clear to me that mathematicians' use of and familiarity with fraktur has declined a lot during that time. But it is not extinct. I have made a hasty and limited search of Jstor and found recent websites and research articles that use it in a variety of fields. There are also a few citations in the Handbook (search for "fraktur").
- It is used in ring theory and algebraic number theory, in particular to denote ideals.
- It is use in Lie algebra. In particular, the Lie algebra of a Lie group $G$ is commonly denoted by $\mathfrak{g}$.
- The cardinality of the continuum is often denoted by $\mathfrak{c}$.
- It is used occasionally in logic to denote models and other objects.
- I remember that in the sixties and seventies fraktur was used in algebraic geometry, but I haven't found it in recent papers.
Acknowledgements
Thanks to Fernando Gouvêa for suggestions.
Remarks about usage in abstractmath.org
The Handbook has 428 citation for usages in the mathematical research literature. After finishing that book, I started abstractmath.org and decided that I would quote the Handbook for usages when I could but would not spend any more time looking for citations myself, which is very time consuming. Instead, in abmath I have given only my opinion about usage. A systematic, well funded project for doing lexicographical research in the math literature would undoubtedly show that my remarks were sometimes incorrect and very often, perhaps even usually, incomplete.
Corrections to the post The Greek alphabet in math
Willie Wong suggested some additional pronunciations for upsilon and omega:
Upsilon: $\Upsilon ,\,\upsilon$ ŭp'sĭlŏn; (Br) ĭp'sĭlŏn. (Note: I have never heard anyone pronounce this letter, and various dictionaries suggest a ridiculous number of different pronunciations.) Rarely used in math; there are references in the Handbook.
Omega: $\Omega ,\,\omega$: ōmā'gɘ, ō'māgɘ; (Br) ōmē'gɘ, ō'mēgɘ. $\Omega$ is often used as the name of a domain in $\mathbb{R}^n$. The set of natural numbers with the usual ordering is commonly denoted by $\omega$. Both forms have many other uses in advanced math.
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Fraktur was still used a lot in algebraic number theory when I was in grad school, and maybe still is. When it is written on the board, cursive forms of the letters are used, and they don't quite look like the printed forms. I found and learned the cursive forms in a little book on basic German. It might be useful to include those in this article.
I modified the post to include algebraic number theory.
In grad school I learned algebraic number theory and Lie algebra from Leonard Carlitz and Joe Schonfield respectively, and I remember the fraktur in printed texts, but I have no recollection of how they wrote them on the board. Schonfield provided dittoed notes, but I no longer have them. *Sigh*, another lexicographical citation lost to history.
I find it extremely hard to read papers with Fraktur. I don’t have any word to put in my mind like theta. And some of the typefaces are even more strange looking than how you drew. One letter (I think G) can look almost Arabic in some books. It’s like trying to/ think of Prince as “squiggle”.
One thing you said brings up something I have noticed: Some people think more in words, others think more in pictures. I think more in pictures, so when I see a fraktur symbol I don’t need a word to pronounce it with. This does not make me better (or worse) than you, because it has a flip side: when I suddenly understand something (in math or when trying to repair some device or something) I have to stop and think up sentences that allow me to explain it to someone else.