Mathematical objects whose name is a single letter

(Not research-level, but perhaps not easily answered elsewhere — you decide if MO can afford the innocent fun. If so, it should likely be “community-wiki” i.e. one object per answer.)

I am seeking stories of mathematical objects that, in context, eat out namespace because their (most usual) name literally is a letter (e.g., in calculus, $e$).

Per discussion in the comments, please rather exclude letters that are frozen out by being merely common notation ($e$ in group theory, $g$ in Riemannian geometry, whole alphabets in semisimple Lie theory), and not really the name of any single object. But include: how such (poor?) practice came about; what did or didn’t help reclaim letters (new names, new typography,...); or any good story.

Wikipedia’s disambiguation pages can suggest many ($c$, $e$, $i$, $j$, $k$, $o$, $q$, $t$, $F$, $G$, $J$, $K$, $L$, $O$, $P$, $W$, $Y$, $\mathcal O$, $\wp$, $\delta$, $\zeta$, $\eta$, $\vartheta$, $\varkappa$, $\lambda$, $\xi$, $\pi$, $\sigma$, $\tau$, $\chi$, $\mathrm B$, $\Gamma$, $\mathrm H$, $\Xi$, $\Omega$,...), but I am sure that is not all.

• $i$ or $j$ for $\sqrt{-1}$? – jeq Dec 31 '17 at 4:28
• Ш (en.wikipedia.org/wiki/Tate%E2%80%93Shafarevich_group) -- but I've seen that used also for the "row of deltas" that is its own Fourier transform by Poisson summation. – Noam D. Elkies Dec 31 '17 at 5:30
• $\aleph$ is owned by set theory, and if you allow font based answers then Lie theory owns the celtic fonts - $\mathfrak{g}$, $\mathfrak{h}$, etc. – Paul Siegel Dec 31 '17 at 5:35
• $\emptyset$ - the 28th letter of the alphabet – Bjørn Kjos-Hanssen Dec 31 '17 at 5:52
• I don't quite follow the intent about a letter being frozen out vs. just being a common notation: a paper in number theory might use $\Sigma$ for a set of primes while also using it for summation, or $\pi$ for $3.1415\ldots$ and for an irreducible representation. Context or a few extra words serve to make it clear what means what and neither $\pi$ nor $\Sigma$ is frozen. – KConrad Dec 31 '17 at 6:30

A lovely story of $\nabla$ can be found on Wikipedia

Introduced by Hamilton in 1837, his initial notation was ◁. Quotes from Wikipedia:

The name comes, by reason of the symbol's shape, from the Hellenistic Greek word νάβλα for a Phoenician harp, and was suggested by the encyclopedist William Robertson Smith to Peter Guthrie Tait in correspondence.

(...)

After receiving Smith's suggestion, Tait and James Clerk Maxwell referred to the operator as nabla in their extensive private correspondence; most of these references are of a humorous character. C. G. Knott's Life and Scientific Work of Peter Guthrie Tait:

It was probably this reluctance on the part of Maxwell to use the term Nabla in serious writings which prevented Tait from introducing the word earlier than he did. The one published use of the word by Maxwell is in the title to his humorous Tyndallic Ode, which is dedicated to the "Chief Musician upon Nabla," that is, Tait.

• Wonderful point, thanks for sharing this! – guest Jan 3 '18 at 20:21

Amir Alexander's 2014 book Infinitesimal says that John Wallis introduced the symbol $\infty$ for infinity (p.280). I asked on HSM, "Where did John Wallis get the idea for $\infty$?," and user Conifold said that the source might be "the Egyptian ouroboros symbol, snake biting its tail":

(Image from mythologian.net.)

• $\infty$ is not a letter, so this is not an answer to the question. – Matt F. Jan 3 '18 at 16:01
• Correct, not a letter. Another theory is that $\infty$ arose as a calligraphed Greek $\omega$... – Joseph O'Rourke Jan 3 '18 at 18:17

To get the ball rolling with actual answers...

$\mathfrak{c}$ is the Cardinality of the continuum.

• The question asks for stories, e.g. how the name came about, and what did or didn’t help reclaim letters. So far this provides a letter and a link, but not a story. – Matt F. Jan 3 '18 at 13:30
• @MattF., good point! Unfortunately, I don't know the history of this notation well enough to comment intelligently about it. But this is community wiki, so if anyone else can, please feel free! – Igor Khavkine Jan 3 '18 at 14:28
• See "Why did Cantor (and others) use $\mathfrak{c}$ for the continuum?." The scholarly answers show that Cantor never used "c" for the continuum. – Joseph O'Rourke Jan 3 '18 at 14:56

How about the plethora of symmetric functions and generalizations?

This is just a small overview of the various generalizations of Schur polynomials, and related bases of symmetric (or quasi-symmetric) functions. Furthermore, the notation is not completely standardized.

It does not exactly answer OP's question, but some of these at least are standardized (say the $e_\lambda$, $m_\lambda$). I myself wonder a bit of the practice in this area.

Imagine how different would be our mathematical discourse if students struggled with $\alpha$-$\beta$ proofs rather than $\epsilon$-$\delta$ proofs!

From an earlier MO question: "Why do we use $\epsilon$ and $\delta$?," an answer quoting Judith Grabiner:

$\epsilon$ corresponds to the initial letter in the word "erreur" (or "error"), and Cauchy in fact used $\epsilon$ for "error" in some of his work on probability."

As @RyanBudney summarizes, "So it's $\epsilon$ for error in the answer, and presumably $\delta$ is in reference to difference in the input variables." Or, as @SimonRose says, "Or possibly that $\delta$ is just the next letter over..."

• Despite Cauchy's sometimes using $\epsilon$ for error, I believe we also owe to him the use of $h$ (as in $\lim_{h \to 0} \frac{f(x + h) - f(x)}h$). I always explain this to my students by some variant of the jocularity "we need a letter to stand for 'a little bit', so we choose $h$ since it doesn't occur anywhere in the phrase." – LSpice Jan 1 '19 at 3:06

In set theory, $V$ is universally known as the universe of all sets.

Similarly, meanwhile, $L$ is the constructible universe.

• What are the stories behind these choices? I've always thought that V was chosen to suggestively depict the cumulative hierarchy, with an empty set at the bottom and spreading out by taking a power set of what one had previously at a successor ordinal (and collecting all previous stages at limit ordinals). But I can't guess where L comes from. – Todd Trimble Jan 3 '18 at 21:29
• $V$ (or V) was introduced by Peano in his "Arithmetices principia. Nova methodo exposita" from 1889. He uses V for verum and $\Lambda$ for falsum, and the symbols were adopted to denote the universe of all sets, and the empty set, accordingly (Peano also uses V explicitly for "the class of all individuals"). See pp. viii and xi here. – Andrés E. Caicedo Jan 4 '18 at 6:32
• To $V$ and $L$ one should add $K$, the core model. – Andrés E. Caicedo Jan 4 '18 at 6:33
• In his 1939 "Consistency proof for the generalized continuum hypothesis", Gödel uses $M$ to denote the constructible universe. He switches to $L$ in his 1940 monograph, "The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory". No reason is given for the choice of symbol in either case. – Andrés E. Caicedo Jan 4 '18 at 6:41
• And $K$ is for kernel. It was introduced (or, at least, one of its versions was) in MR0611394 (82i:03063). Dodd, A.; Jensen, R. The core model. Ann. Math. Logic 20 (1981), no. 1, 43–75. – Andrés E. Caicedo Jan 4 '18 at 6:45

Trivial example, but not to be omitted, the Greek and Latin numerals. E.g.:

• Yes, and our hexadecimal too. Thanks Pietro for exactly the sort of answer I was hoping for — I added an image of the first trigonometric table to illustrate. Do we know how they were verbalizing the numbers? – Francois Ziegler Jan 4 '18 at 0:09
• If you mean pronouncing the digit's symbols as letters, I do not have any knowledge about it. I guess a trace of it, if any, could be found in dramatic literature, which is one important source of information about the spelling of classic languages. After all it could be, like anglophones sometimes do when spelling $0$ as "oh" instead of "zero". It is also true that digits and numeric codes enter in our life in a very massive way, that did not exist in ancient times, when on the contrary written names were more used than symbols even in formulas. – Pietro Majer Jan 4 '18 at 13:36
• It would be funny to know of Roman boys greeting each other with "da mihi v" – Pietro Majer Jan 4 '18 at 13:37

Given that there is not much difference between a letter and a symbol (especially in view of the two most upvoted answers), I propose:

• The letter "+" is used for the mathematical object commonly called addition.

• The letter "-" is used for the mathematical object commonly called subtraction.

• The letter "×" is used for the mathematical object commonly called multiplication.

Etc.

• Besides, the symbol "+" comes from a Latin et, after the contraction of "e", so it's indeed the single letter "t". – Pietro Majer Jan 3 '18 at 23:15
• @PietroMajer So in a sense $+$ is $\&$ then :D – მამუკა ჯიბლაძე Jan 4 '18 at 4:48