# Have there been efforts to introduce non Greek or Latin alphabets into mathematics?

As a physics student, often I find when doing blackboard problems, the lecturer will struggle to find a good variable name for a variable e.g. "Oh, I cannot use B for this matrix, that's the magnetic field".

Even ignoring the many letters used for common physical concepts, it seems most of the usual Greek and Latin letters already have connotations that would make their usage for other purposes potentially confusing, for instance one would associate $p$ and $q$ with integer arguments, $i,j,k$ with indices or quaternians, $\delta$ and $\varepsilon$ with small values, $w$ with complex numbers and $A$ and $B$ with matrices, and so forth.

It then seems strange to me that there's been no effort to introduce additional alphabets into mathematics, two obvious ones, for their visual clarity, would be Norse runes or Japanese katakana.

The only example I can think of offhand of a non Greek or Latin character that has mainstream acceptance in mathematics would be the Hebrew character aleph ($\aleph$), though perhaps there are more.

My question then, is have there been any strong mainstream efforts, perhaps through using them in books, or from directly advocating them in lectures or articles, to introduce characters from other alphabets into mathematics? If there have been, why have they failed, and if there haven't been, why is it generally seen as unnecessary?

Thank you, and sorry if this isn't an appropriate question for math.stackexchange.com, reading the FAQ made it appear as if questions of this type were right on the borderline of acceptability.

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I do not think that we are lacking letters. What you describe with the $B$ field happens if people are not focussed. I think (and may be completely wrong here :-) the greek and latin letters are in use cause in the 19th century that was what was usually known to 'educated' people. Nowadays esp. the usage of greek alphabet causes additional learning efforts for most students (as would the use of Norse Runes, Japanese katakane, Arabian letter, Russian symbols not used in the Latin character set...). Even in Germany we usually do not make use of the three special characters we have, ä, ö, ü. –  user20266 Jul 1 '12 at 17:20
I think that the reader should be able to distinguish between some classical constant or a simple variable. For example, the golden number is denoted by $\varphi$. But we use this Greek letter few times, everyday, in many contexts. It's clear when we are talking about that number or not. –  Sigur Jul 1 '12 at 17:45
+1 for suggesting katakana. –  cactuar Jul 1 '12 at 18:02
@Thomas: the German umlauts are unsuited especially for physics simply because you would confuse ü with $\ddot{u}\equiv \tfrac{\partial^2 u}{\partial t^2}$, but that's not an issue for most non-Latin-derived letters. — IMO, the real solution would be to abandon the habit of giving every variable just one letter as an identifier, rather than a whole word as is common in programming. But that's not likely to happen, since multiletter identifiers are awkward unless you use clever functional or object-oriented tricks to avoid symbol repetition. –  leftaroundabout Jul 1 '12 at 19:43
–  dtldarek Jul 1 '12 at 19:55

In advanced number theory arithmeticians have introduced the russian letter Ш, pronounced "shah".
But this is very localized.

Apart from the Greek alphabet, the only different alphabet I know of used in a Latin environment is Fraktur, popularly known as Gothic.
It is massively used in algebra for ideals in rings.
Actually, essentially all standard references in commutative algebra and algebraic geometry make use of Fraktur: Atiyah-Macdonald, Dieudonné-Grothendieck's EGA, Görtz-Wedhorn, Hartshorne, Jacobson, Matsumura, Qing Liu, Shafarevitch, Zariski-Samuel,...

Edit The $\LaTeX$ command for Fraktur is $\text {\mathfrak}$. For example:

Let $\mathfrak p$ be a prime ideal, $\mathfrak q$ a primary ideal and $\mathfrak a,\mathfrak b, \mathfrak c \:$ arbitrary ideals of the ring $A$, then...

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Fraktur is not a different alphabet; it is a different typeface. –  Théophile Jul 1 '12 at 17:57
Isn't fraktur more of a font variant of the Latin alphabet? –  alex.jordan Jul 1 '12 at 18:00
I've also seen Ш used for the Dirac comb. –  mrf Jul 1 '12 at 19:10
Fraktur also sees a fair bit of use in set theory, where $\mathfrak{c}$ is used for the cardinality of the reals and the cardinal invariants of the continuum (see mathoverflow.net/questions/8972#9027 ) seem to typically be written with Fraktur ($\mathfrak{b}, \mathfrak{d}$, etc.). –  Steven Stadnicki Jul 1 '12 at 19:20
Here is the complete Sütterlin alphabet. Of these I’ve definitely used at least A, B, G, M, N, U, and V in mathematical writing, though not in many years. –  Brian M. Scott Jul 1 '12 at 21:37

To add to some of the letters and alphabets mentioned, in set theory, the Hebrew letters $\aleph$ and $\beth$ are used.

It is common to, as you mention, use specific variables letters for specific purposes. More obscure, foreign letters are probably seldom used simply because they have no need to be introduced. Mathematicians already they have two alphabets to choose variables from!
However, for things that have specific purposes, like constants or special functions, cannot be given the same variable letter without causing some confusion.

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We also use $\gimel$. –  Andres Caicedo Jul 1 '12 at 22:30
@AndresCaicedo Thanks. It seems Latex also has $\daleth$. Asaf also pointed out that Cantor used $ת$. –  Argon Jul 2 '12 at 14:35

In the 1960's, a fellow at IBM by the name of Kenneth Iverson created a new mathematical notation that originally held the name "Iverson's Better Math".

He published it in a book called A Programming Language, and since IBM wasn't too keen on the internal nickname, the notation itself came to be known as APL. (Iverson didn't mean programming language in the computer programming sense, though an interpreter was in fact soon implemented, and you can now execute APL on a computer.)

You can see the symbol set used in this on-line APL interpreter, and you'll note that there aren't too many greek or latin characters at all. (The iota generates a vector of sequential integers , rho reshapes the rows and columns of an n-dimensional matrix, alpha and omega are used for defining functions of one or two variables).

APL relies heavily on function composition, and often variable names are not needed at all. The number we call pi is represented by a circle (which also can be used for all the trigonomic functions), and one of those symbols does the work of e and log.

It also uses an explicit multiplication sign, which means any word at all can be used to represent a variable, whenever you actually do need one.

Iverson's book is online at http://www.jsoftware.com/papers/APL.htm if you're interested. There's also a shorter article called Notation as a Tool of Thought, which I believe was his Turing Award lecture.

This was a serious effort to reform mathematical notation, and it was actually quite popular at the time. You could even get a typewriter with the APL characters (in fact the character set was partially chosen based on what you type on an IBM typewriter). But the commercial book publishers of the day had trouble with all those new characters, and of course it has a rather steep learning curve.

People still use APL today, especially in the financial markets, along with modern variations like J and K that stick to ascii symbols while managing to remaining just as cryptic.

I'm not sure that's exactly mainstream acceptance, but there you go :)

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I think it's wonderful that you posted this, since prior to being a student of physics, I came from a computer background, and thinking about the lack of brevity in computer science notation while studying for exams led me directly to Iverson's article Notation as a Tool of Thought, which in turn led me to the J language (which I am still a fan and advocate of) and also probably indirectly pushed me towards my current field of study, via Math. And then, my thoughts on notation, specifically the Iverson/APL/J school of thought made me wonder what mathematicians thought about new notation. –  Mark Allen Jul 1 '12 at 23:32

The letter eth (ð), present in the Old English, Icelandic, Faroese, and Elfdalian alphabets, "is sometimes used in mathematics and engineering textbooks as a symbol for a spin-weighted partial derivative", according to Wikipedia — i.e. it's used as a variation on the usual partial derivative symbol ∂.

(Which makes me wonder if Kip Thorne has ever used the letter thorn (þ)... if not, someone should persuade him to!)

In a similar vein, I've also just learnt that ħ, which in quantum physics represents the reduced Planck constant (i.e. the Planck constant h divided by 2π) is "used in Maltese for a voiceless pharyngeal fricative consonant". (I'm ashamed to say I didn't even know Maltese was a language!)

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Someone should also persuade Lenhard Ng to find a use for the letter eng (ŋ) if he hasn't already... Ok, I'll stop now! –  onestop Jul 1 '12 at 19:43
@Kenny: It is exactly the same as the Maltese letter, as far as I can tell. I don't think onestop is claiming that h-bar was derived from the Maltese letter (or indeed vice versa). –  TonyK Jul 1 '12 at 21:11
ħ is not only used in Maltese; I have seen it used for transliterations of the letter het in Semitic Abjads. –  Argon Jul 1 '12 at 21:13
Thorn (þ) is used in Model theory. –  Andres Caicedo Jul 1 '12 at 22:31
The h-bar symbol is also in the Serbian variant of the Cyrillic alphabet; where it makes a sound a little like "ch", but with the tongue against the top teeth. –  user22805 Jul 2 '12 at 3:55

Runes would be an exceptionally bad choice: too many of them already look like something else. The $t$ rune is $\uparrow$, the $l$ rune is $\upharpoonright$, the $h$ rune is $\huge{\ast}$, the $i$ rune is $\mid$, and the $r$ and $b$ runes are R and B. The $m$ rune looks like a stick-figure $\Psi$, the $u$ rune is easily confused with a handwritten lower case $n$, one of the two versions of the $s$ rune is a slightly skewed N, the $f$ rune looks a lot like a sloppily written F, and the $n$ and $a$ runes are like plus signs with the horizontal bar tilted one way or the other. See NORWEGIAN RUNES AFTER CA. 600 AD and NORWEGIAN-DANISH RUNES FROM THE 800'S here.

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The letter $\varnothing$ is actually a Dansk-Norsk letter.

In addition to $\aleph$ and $\beth$ which were mentioned by Argon, there is also $\gimel$ (Gimel, the third letter in Hebrew) and Cantor used Tav (the last letter) but that one didn't stick.

I should add that it became a convention in mathematics to name variables with a certain type of letter. Of course that $n$ being a free variable can be anything, but it alerts the reader that the variable is a positive integer; similarly $\varepsilon$ is a very small quantity.

Think about it for a moment, what is $0$ if not a convention to the additive neutral element? You can see this going further and $0,1$ are used as the neutral elements in rings and groups which have little to do with the real numbers. Why? Because it alerts the reader that this is a special element.

Similarly the fonts can indicate things, in the course in measure theory the professor told us that "lowercase is for elements, capital for sets, and cursive for collections of sets". You often see people use $\cal F$ for a filter, even if $F$ was not used before because this is the font for filters.

Due to these convention it is often hard to find letters when you have used the basic ones already (sometimes in various fonts).

(I actually heard a story about one of the professors which used $x,{\rm x}, X, {\scr X}$ and after a few minutes just wrote a huge $x$ symbol because he ran out of letters.)

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I find the symbol $\varnothing$ clearly different from the letter Ø/ø. It is topologically similar, but such similarities does not make a Greek "nu" the same as a "v", nor a zero the same as the letter O/o. –  Henning Makholm Jul 1 '12 at 23:25
@Henning, I can actually say the same on $\gimel$ and ג which is closer to $\lambda$ actually (and thus a source of great confusion for freshmen!). I suppose that $\emptyset$ may appear closer. –  Asaf Karagila Jul 1 '12 at 23:32
@HenningMakholm: According to jeff560.tripod.com/set.html it did start out as an Ø (introduced by Weil), just like the set membership symbol started out as an $\epsilon$ (epsilon). The stylized math versions have evolved later. –  Hans Lundmark Jul 2 '12 at 6:43
As I mentioned in a comment to my post, it seems Latex has the letter Dalet, $\daleth$, as well. –  Argon Jul 2 '12 at 14:37

The Cyrillic Л, which is analogous to L, is the first letter in the name Lobachevsky and has been used in hyperbolic geometry for the Lobachevsky function $$Л(\theta) = -\int_0^\theta \log|2\sin t|\,dt.$$ This notation was introduced by Milnor. See

1) chapter 7 of Thurston's "Geometry and Topology of 3-manifolds" (written by Milnor),

2) the appendix to Milnor's "Hyperbolic Geometry: the first 150 years" in the 1982 AMS Bulletin,

3) Zagier's "Hyperbolic Manifolds and Special Values of Dedekind Zeta-functions" in Invent. Math. 83 (1986), 285--301.

4) Ratcliffe's "Foundations of Hyperbolic Manifolds" (section 10.4).

Milnor used the lower-case л while Zagier and Ratcliffe used the capital Л.

Often this function is written with the Greek letter $\Lambda$, perhaps because it's not as easy to produce Л without special effort.

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Let $X$ be a quotient of a bounded symmetric domain by an arithmetic group. There are a bewildering number of different natural compactifications of $X$: $2^{r-1}$ Satake compactifications, a whole family of toroidal compactifications, Borel-Serre, ... papers that need several of these often use both $\overline X$, $\widetilde X$, $X'$, etc.

Namikawa tried to popularize the notation $X^サ$ for the Satake compactification. サ is katakana, the first initial of Satake. It did not stick.

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I've seen the Russian letter shah in Topology books as well. A shah-space.

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too long for a comment:

• Greek letters are precariously overloaded. There is both the Dirichlet eta $\eta(s)$ and the Dedekind eta function $\eta(\tau)$. "$\pi$" can occasionally mean a permutation or a prime in a field of characteristic $k$ as well as the ratio of the circumference to the diameter of a circle and the prime counting function $\pi(n)$ "$\theta$" is used for both an angle and for the Jacobian theta functions. Given the half period ratio, using $\tau$ to denote $2\pi$ is not a good idea. (aside: there are definite issues that one proving theta function identities runs into when dealing with lots of adjacent $\pi$, $\tau$, $n$, and $i$ just given their resemblance of shape).

• Mathematicians are generally not typographers, and historically, installing custom typefaces in $\LaTeX$/$\TeX$ has been a very tricky procedure. Besides a few delightful exotic symbols here and there (mentioned in other answers, the Tate-Shafarevich group, the Lobachevsky function, and the Beth numbers), I don't think there have been any Bourbaki-scale two-dimensional symbolic language reform movements which appropriately address the user-interface problems which overreliance on limited symbolic alphabet repetoire cause: $n$ is natural or an integer, $z$ is on the complex plane, $|q|<1$, except in cases like the Rogers-Ramanujan continued fraction, $\mathbb{Z}_{n}$ is a group but $\mathbb{R}^{n}$ is many different things depending on context, but $\mathbb{C}_{n}$ isn't sensible. And sometimes $d/dq$ is awkward with $q$-hypergeometric identities because $d$ is a parameter. (Chapter 4 of Gasper's /Basic Hypergeometric Series/ for instance, eqn. 4.1.3) (aside: I actually want to use Mayan glyphs for certain functors, and I have the infrastructure to make this happen) Very pointedly: a limited symbol repetoire leads to overloaded symbols with confusing and inconsistent meanings, and the process of disambiguation has measurable costs in terms of time-from-exposure-to-comprehension. Finding efficient long term solutions is at right angles to scope of m.se, so I won't say anything more here.

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