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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)

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On a somewhat unrelated note, I only learned recently that the words manifold and variety are synonymous. The former is German, and the latter is French (I might be wrong). The French would call differential manifold "variété différentielle," while algebraic variety is just "variété algébrique." –  liuyao Dec 9 '09 at 2:49
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Let me add to this question: what is the origin of the word "ring"? –  Mariano Suárez-Alvarez Dec 9 '09 at 2:58
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You should community wiki this. –  Kevin H. Lin Dec 9 '09 at 3:03
    
Interesting! I had always assumed that K stood for something like "Kampen"... which, as I just found out, is not actually a German word. :P –  Vectornaut Dec 11 '09 at 19:37
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Some cognate of (English word) "variety" is "manifold" in all the Romance languages. "Manifold" comes from Germann, I believe actually from Riemann. –  Harrison Brown Dec 18 '09 at 19:43
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22 Answers

$\mathbb{Z}$ comes from the German "Zahlen" which means "numbers".

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Center of a group is denoted Z, from German word Zentrum

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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?

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Yes, I think this is true. And U is for 'Umgebung', the German word for neighborhood. –  user717 Dec 9 '09 at 10:21
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And U and V are consecutive letters, making them especially convenient for using to represent two neighborhoods that occur together. One often sees, say, a function f: U -> V where U and V are open subsets of A and B, respectively. –  Michael Lugo Dec 9 '09 at 13:47
    
@Arminius: I hope I don't double post. Was having trouble posting earlier, but I did not know that. Thanks. –  MLevi Dec 10 '09 at 0:22
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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.

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Great! I always wondered where this comes from. –  user717 Dec 9 '09 at 10:17
    
This is new for me!! cool!! –  Yuhao Huang Dec 11 '09 at 4:13
    
finally I know why this letter is used :) –  Martin Brandenburg Jan 22 '10 at 21:40
    
I used to believe that the O came as the geometrical approximation of a physical ring, from the local ring idea. –  ogerard May 16 '10 at 8:54
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Um, can someone point to an actual reference justifying this? Why would a letter from an Italian word be used here when the topic was systematically developed by the French? It feels like an artificial etymology (and Kevin is conceding he has no source on this, but people's willingness to believe it surprises me). I once heard O was in honor of Oka. Hmm... Perhaps O is related to the very long (since 1870s) tradition of using O for rings in number theory, which was based on Dedekind's term for a ring: order, or rather Ordnung in German. He wrote fraktur o a lot in his work. –  KConrad Aug 13 '10 at 4:59
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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?)

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Who was the first to use $Z$ for a partition function? –  Kevin H. Lin Jan 6 '10 at 3:10
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The notation $\mathcal{F}$ for sheaves comes from the French word "faisceau" meaning "bundle".

Also "gerbe" means "sheaf" in French.

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It's in fact the same etymology for "fascism". Go look up your Roman history for why "bundles" have anything to do with government. –  Scott Morrison Dec 9 '09 at 6:45
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Indeed wikipedia has a brief explanation: en.wikipedia.org/wiki/Fascism#Etymology –  Kevin H. Lin Dec 9 '09 at 14:16
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Surely a native French speaker will read this? I'm not one, but I thought the French noun gerbe meant "spray", as in a spray (bouquet) of flowers. This also explains why gerber is slang for "to vomit". –  Tom Leinster Apr 12 '10 at 20:04
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In French, the word "gerbe" commonly refers to an arrangement of wheat like this upload.wikimedia.org/wikipedia/commons/thumb/0/06/… –  François G. Dorais Apr 13 '10 at 0:16
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I can't think of an English word that exactly matches the common usage of faisceau in French; I could use any of cone, beam, ray, spray, jet, stream, or sheaf (thanks to Tom) depending on context. The best description I can come up with is: things tied together in a directed way. It has lots of uses from light ray (faisceau lumineux) to muscle fibres (faisceau musculaire). –  François G. Dorais Apr 13 '10 at 1:08
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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.

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Now I'm wondering why class field theory was not called K-field theory... –  Yuhao Huang Dec 11 '09 at 4:21
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It is stated in the first paragraph of this introduction by Karoubi: arxiv.org/ftp/math/papers/0602/0602082.pdf –  Jose Brox Dec 12 '09 at 0:17
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I think this is also mentioned at the beginning of Rosenberg's book on K-theory. –  Dan Ramras May 3 '10 at 2:07
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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

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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.

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Z is for "Zykel" which means cycles, and B is for "Bilder" which means "images". (I'm german and that is what I learned in my german algebraic topology class) –  Konrad Voelkel Dec 9 '09 at 8:40
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Probably one reason for using the first letters of the German words here is that in English 'cycle' and 'chains' have the same first letter. –  Lennart Meier Dec 9 '09 at 9:06
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Also, boundary is Rand in german, with boundaries translating to Ränder. –  Mikael Vejdemo-Johansson Dec 10 '09 at 20:48
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$\mathbb{N}$ 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

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maybe \mathbb R comes from the french "nombres reelles" or the english "real numbers" instead? how do you know? For complex numbers, I would even argue for a german/latin origin, since germans used "complex" instead of "komplex" at the time of Gauss. –  Konrad Voelkel Dec 11 '09 at 20:47
    
I was under the impression that Q, R, C are all from French (Quotient, Réel, Complexe). I doubt the Romans needed a symbol for Q... –  François G. Dorais Apr 13 '10 at 0:21
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Apparently Z, Q, R are all eventually due to Bourbaki and stand for the German Zahlen, Quotient, Reelle. However, they were all randomly used by someone else before... jeff560.tripod.com/nth.html –  François G. Dorais Apr 13 '10 at 1:34
    
The word Quotient is actually a Latin word, inherited by many modern languages. –  psihodelia Apr 27 '10 at 10:41
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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.

http://en.wikipedia.org/wiki/Florian_Cajori

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Why did you put the word revision in quotes? –  Kevin H. Lin Apr 12 '10 at 23:50
    
Some but not all have taken issue with some of Cajori's scholarship. –  Joseph Malkevitch Apr 20 '10 at 13:11
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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.

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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.

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In particular $\emptyset$ is due to André Weil, yet in comparison with other marks he has left on mathematics, it is the most vacuous. –  Zavosh Apr 12 '10 at 22:35
    
I enjoyed the wordplay, Zavosh. Cheers! –  Harry Gindi Apr 12 '10 at 22:41
    
Oh, I had always assumed that "dangerous bend" was due to Knuth! But you are correct: en.wikipedia.org/wiki/Bourbaki_dangerous_bend_symbol –  Kevin H. Lin Apr 12 '10 at 23:54
    
$Ass$ doesn't sound polite in modern English though... –  Qfwfq Apr 15 '10 at 8:25
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$x,y,z$, and in particular that $x$ is the independent variable and $y$ the dependent variable, are due to Descartes, if I'm not mistaken.

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Did he choose these letters more or less at random, or did he have some particular reason for the choice? –  Kevin H. Lin Dec 9 '09 at 17:35
    
Well, now I'm not sure. Wikipedia en.wikipedia.org/wiki/Ren%C3%A9_Descartes credits Descartes with suggesting $x^2$ for "$x$ times $x$". Leibniz invented the words "coordinate, abscissa, ordinate". –  Theo Johnson-Freyd Dec 9 '09 at 23:14
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I once heard that the typesetter of Descartes' book on geometry asked him whether any particular choice of letters is important and when Descartes replied that it is not, he suggested to use x and y because they are rarely used in French and so he will not run out of them when he typesets the book. –  Dmitri Pavlov Dec 10 '09 at 15:26
    
Dmitri: Interesting! I wonder if anybody can corroborate this story. –  Kevin H. Lin May 4 '10 at 12:34
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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.

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What is "omnia"? –  Kevin H. Lin Dec 9 '09 at 14:03
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Also, I think the $\int$ symbol is supposed to be an elongated "S", which is supposed to stand for "summa"(?) in Latin(?), meaning "sum". –  Kevin H. Lin Dec 9 '09 at 14:14
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I guess integrating is stretching the notion of sum :) –  José Figueroa-O'Farrill Dec 9 '09 at 15:07
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It's meant to be Greek to Latin: $\Sigma\to\int$ $\Delta\to d$ –  liuyao Dec 9 '09 at 15:47
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I'm assuming omnia (literally all, I think) is used to mean something along the lines of "sum". –  Cory Knapp Dec 9 '09 at 16:28
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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.)

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According to Wikipedia (en.wikipedia.org/wiki/G%CE%B4_set) it comes from the German word Gebiet. –  Qiaochu Yuan Dec 11 '09 at 21:01
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You're confusing fermé (closed) with ferme (firm, rigid); the accent makes a big difference! –  François G. Dorais Apr 15 '10 at 17:29
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Touche ;) –  Konrad Swanepoel Apr 27 '10 at 18:53
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You mean touché . Touche without accent is not an adjective but the word for a piano key, the equivalent of the english "touch" as in "the painter's delicate touch" or a hit on a target. –  ogerard May 16 '10 at 8:45
    
No, I don't; note the emoticon. (On the other hand, ferme was a mistake.) –  Konrad Swanepoel May 17 '10 at 11:30
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Pat Ballew's blog Math Words has interesting stuff.

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$E$ is sometimes used for vector spaces, from the French word "espace"="space".

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Could also be Euclid. –  François G. Dorais Apr 15 '10 at 14:24
    
@François: that's more likely for Euclidean space $\operatorname{E}^n$. –  Qfwfq Apr 15 '10 at 17:52
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Wolfram has nice a little paragraph on the history of the term "Ring" right after the list of ring axioms.

Ring (from Wolfram Mathworld)

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Yeah, I had seen that. That does not explain why 'ring' was chosen, though. –  Mariano Suárez-Alvarez Dec 9 '09 at 4:37
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I think it mentions on there that "ring" comes from the cyclic structure you get in many rings, e.g. successive powers of the same element in $\mathbb{Z}/k\mathbb{Z}$. –  REDace0 Dec 13 '09 at 19:23
    
Certainly thanks to the chosen word (ring, anneau in french) I have always thought of a ring as a torus, like the product of one operation (+) by the other (x) and closed in these two dimensions. I guess that I would also like to see sub-rings and modules as small rings with the initial ring like a thread passing in their holes. –  ogerard May 16 '10 at 8:51
    
This is a false etymology. Everyone thinks it's from cycling in modular arithmetic (I once did), but please look at mathoverflow.net/questions/35286/… for the correct history of the term ring. –  KConrad Aug 13 '10 at 4:47
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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.

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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.

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