# Are there widely accepted math symbols using non-Latin alphabets or characters other than Greek and Hebrew?

We have $$\pi$$ and $$\aleph_0$$ borrowed from Greek and Hebrew alphabets. Are there widely accepted math symbols using non-Latin alphabets or characters other than Greek and Hebrew?

• Not "widely accepted" at all, but C. Chevalley's notes on classfield given in Japan use many Japanese symbols... – paul garrett Dec 29 '19 at 20:44
• Maybe you could change "non-English" to "non-Latin"? – fdb Dec 30 '19 at 1:13
• Wikipedia lists some under List of letters used in mathematics and science. Russian Ш (Sh) for the Tate-Shafarevich group and shuffle product, and Japanese よ (Yo) for Yoneda embedding. – Conifold Dec 30 '19 at 6:50
• I can't find it in the Selecta, but I could have sworn that in one of his expository papers ("How to write mathematics" is a natural one, and well worth a (re)read even if it doesn't contain this quote!) Halmos bemoaned the lack of interest in alternative alphabets in mathematics notation. – LSpice Dec 30 '19 at 20:05
• @fdb Not very well. Sorry. – bmargulies Dec 31 '19 at 3:54

The letter Ш (sha) of the Cyrillic alphabet is widely accepted in theoretical computer science as the symbol for the shuffle product, which gives the shuffle algebra. The same letter is also used to denote the Tate-Shafarevich group, but I'm not sure if it's really a standard (the letter was introduced by Cassels only in 1990 in 1962 instead of TS, see below the KCd's comment).

Sometimes, see for example here, the Hiragana symbol よ is used to denote the Yoneda embedding.

• Nice to know that Japanese characters are involved!! – modnar Dec 29 '19 at 14:17
• The letter Ш was introduced by Cassels long before 1990! He used it in his 1962 Stockholm ICM talk (see p. 238 of mathunion.org/fileadmin/ICM/Proceedings/ICM1962.1/…) and in his 1962 paper "Arithmetic on Curves of Genus 1. IV" (Crelle volume 211, see the first page). In his 1962 paper "Arithmetic on curves of genus 1. III" (Proc. LMS volume 12) he used the earlier notation "TS" (essentially), so I think 1962 can be considered the birth year of the Ш notation by Cassels. – KCd Dec 30 '19 at 16:21

It is sometimes asserted that $$\varnothing$$ for the empty set was introduced by Bourbaki using a Danish and Norwegian letter.

EDIT:

The source is the Weil autobiography, cited in Jeff Miller's collection of the origins of mathematical expressions:

André Weil (1906-1998) says in his autobiography that he was responsible for the symbol:

Wisely, we had decided to publish an installment establishing the system of notation for set theory, rather than wait for the detailed treatment that was to follow: it was high time to fix these notations once and for all, and indeed the ones we proposed, which introduced a number of modifications to the notations previously in use, met with general approval. Much later, my own part in these discussions earned me the respect of my daughter Nicolette, when she learned the symbol Ø for the empty set at school and I told her that I had been personally responsible for its adoption. The symbol came from the Norwegian alphabet, with which I alone among the Bourbaki group was familiar.

The citation above is from page 114 of André Weil's The Apprenticeship of a Mathematician, Birkhaeuser Verlag, Basel-Boston-Berlin, 1992.

• André Weil says in his autobiography that he took $\emptyset$ from ø, having seen it in Norway. Of course, he did not know that the Norwegians got it from Danish. – Robert Furber Dec 29 '19 at 18:25
• This answer would be improved by citing a source or sources that support this claim, rather than just saying "it is sometimes asserted". – V2Blast Dec 31 '19 at 3:04

There are several non-alphabetic symbols, the best known is the integral sign $$\int$$ and the Weierstrass $$P$$-function $$\wp$$. To be sure their origins are letters of Latin alphabet, but they are special stylized symbols, and as far as I know there is no computer code for them in the standard sets of computer characters. Strictly speaking they do not belong to any alphabet. $$\wp$$ imitates Weierstrass handwriting. Symbols $$\partial$$ for the partial derivative and $$\infty$$ also belong to this list.

There are also mathematicians who use Cyrillic letters, for example John Milnor used Л for some standard function is hyperbolic geometry, but this can hardly be called "widely accepted". Л is the first letter in Lobachevski's name (Лобачевский). I've seen other Cyrillic letters used, but again this is not "widely accepted".

An interesting case is the "letter" $$\nabla$$ usually used for the gradient. This is a apparently not a letter from any alphabet, and it is called "Nabla" which is the Greek word for some Phoenician musical instrument. See Nabla symbol. Unlike $$\wp$$ it is included in Unicode.

• Another widely accepted non-alphabetic symbol is $\sqrt{}$. – modnar Dec 29 '19 at 13:33
• Yes, there are many non-alphabetic symbols, $+,-,=,\sim,\approx,\equiv$ etc. But I suppose that no letter from a real alphabet, other than Latin and Greek, is "widely accepted". Recently I was shown Arabic letters used as math symbols, but only in some countries where people are familiar with them. – Alexandre Eremenko Dec 29 '19 at 13:37
• Unicode has ℘ which approximates $\wp$ well enough for many, – Jasen Dec 30 '19 at 0:30
• @Jasen why "approximates"? It was intended to be Weierstrass elliptic function, see here (search for U+2118). Quoting: "Should have been called calligraphic small p or Weierstrass elliptic function symbol, which is what it is used for. It is not a capital "P" at all. A formal alias correcting this to WEIERSTRASS ELLIPTIC FUNCTION has been defined." – Ruslan Dec 30 '19 at 8:25
• Unicode also has multiple glyphs for integrals, including dedicated ones for multiple integrals and other special forms. – Austin Hemmelgarn Dec 31 '19 at 2:28

The cyrillic letter Ш (sha)is -- for obvious reasons when looking at the graph) also used to denote the "function" (well, it is a distribution if you want to be picky) given by the sum of integral displacements of the Dirac-delta function, see https://en.wikipedia.org/wiki/Dirac_comb

• Yes, it's true, but in this context the letter Ш it's not widely accepted and often it's used (for the same reason) the letter ω (see e.g. arxiv.org/pdf/math/0203030.pdf). Moreover I've noticed that the wikipedia article in English uses III, while the Russian one Δ !! – user6530 Dec 31 '19 at 9:12