I'm not sure if this has been asked. I'll explain the question by an example.
Fields are often denoted by the letter k, which comes from the German word Körper, meaning body (like corpse, corporeal).
Most mathematical symbols relate directly or indirectly to the English names, so what other exceptions are there?
(Yes, this is inspired by the other post about languages in math)
As an undergraduate, I was told that $V$ is often used to denote a neighborhood because the French translation is voisinage. Anyone else hear this?
This one is pretty well-known: the notation $e$ for the identity of a group comes from the German word Einheit, meaning unit.
I'd be willing to bet that the notation $G$ for a group also comes from German... but we don't notice, because the German word for group is Gruppe!
Here's a fun one: the notation $Z$ for a topological quantum field theory comes indirectly from the notation $Z$ for a partition function in statistical mechanics, which comes from the German word Zustandssumme, meaning state sum. I said "indirectly" because partition function in quantum field theory isn't a statistical-mechanical partition function... it just looks like one after you Wick rotate! (Then again, maybe there's a deeper sense in which the QFT partition function really is a statistical-mechanical partition function. Does anybody know?)
I've been told that the notation $\mathcal{O}$ for the structure sheaf of a scheme/variety/whatever comes from the Italian word "olomorfo/olomorfa" for "holomorphic".
I should note that I don't have any evidence for this claim beyond "I heard it somewhere from somebody". It would be great if anybody could corroborate this.
The notation $\mathcal{F}$ for sheaves comes from the French word "faisceau" meaning "bundle".
Also "gerbe" means "sheaf" in French.
I've heard that the "$K$" of $K$-theory comes from the German word "Klasse(n)" meaning "class(es)", but I don't have any concrete evidence for this.
In homological algebra, one sometimes uses Z and B to denote cycles (or closed form) and boundaries (or exact forms), respectively. Z must be for Zycle.
$\mathbb{N}$ comes from the German "Natürliche Zahlen"=natural number
$\mathbb{Z}$ comes from the German "ganZe Zahl"=integer numbers
$\mathbb{Q}$ comes from the Latin "Quotient"= result of a division
$\mathbb{R}$ comes from the German "Reelle Zahl"=real numbers
$\mathbb{C}$ comes from the French "nombre Complexe"=complex numbers
The letter $T$ in the names for the separation axioms $T_1$, $T_2$, etc in point set topology comes from "Trennungsaxiom" in German. http://de.wikipedia.org/wiki/Trennungsaxiom
I asked a while ago about the etymology of the name conductor. Often the conductor of an order in a number field is denoted by $\mathfrak f$. This comes from the original German name Führer given by Dedekind.
Oh, but of course $\emptyset$ comes from Bourbaki. Interestingly, so does $\Rightarrow$ to denote implication, and $\in$ instead of $\varepsilon$. The "Dangerous bend" comes from Bourbaki as well.
However, my all time favorite is the set of associated primes of a module M. $Ass(M)$ is in fact called the assassinator of $M$, and its elements are called assassins.
There is a "classic" book about the history of mathematical notations by Florian Cajori though there has been some "revision" of his work by more recent scholars.
$x,y,z$, and in particular that $x$ is the independent variable and $y$ the dependent variable, are due to Descartes, if I'm not mistaken.
Utile erit scribit ∫ pro omnia. (It is useful to write ∫ instead of omnia) – Leibniz (1675-10-29)
(Source for this quotiation: Eriksson, Estep, Hansbo, Johnson: Computational differential equations, end of Ch. 3)
In response to some comments: omnis means “all”. Compare omnivore. Here endeth the Latin lesson.
F for a closed set comes from the French ferme (=firm, cf. fermer=to close).
What about G for an open set? Is this also an example of the next-letter phenomenon? (as in Michael's comment to this answer to the question.)
In design theory we talk of a t-(v,k,λ). I think v originally meant "varieties", but I don't know if any of the other symbols meant anything; it would be nice to find out that they did. λ seems an odd choice for an integer... in many other contexts it gets used as a real number.
I'm not sure how relevant this is outside of Ireland, but while doing basic mechanics, if you ever see acceleration denoted as $f$, as it is in the "log tables" here, as in $v=u+ft$, the $f$ in this case stands for the Latin for acceleration, festinatio (with festino meaning "I hurry", so festinatio would very roughly and more literally translate as "hurriedness"), which is funny because adcelero is the Latin for "I speed up" which looks a lot more like acceleration.
Similarly, displacement denoted by $s$ as in $s=ut+\frac12 at^2$ is from the Latin for displacement, summoveo (with moveo meaning "I move [something]").
And, of course, velocitas, the Latin for speed. I can imagine u being used for velocity as well since the Romans actually pronounced "v" as "u", so the two are pretty much interchangeable.
Wolfram has nice a little paragraph on the history of the term "Ring" right after the list of ring axioms.
