# German mathematical terms like “Nullstellensatz”

There are quite a few german mathematical theorems or notions which usually are not translated into other languages. For example,

Nullstellensatz, Hauptvermutung, Freiheitssatz, Eigenvector (the "Eigen" part), Verschiebung.

For me, as a German, this is quite entertaining. Do you know other examples? Please one per answer, please give a reference for the term or a short explanation of what it means.

It would be great to see an explanation why there is no translation.

EDIT: Some more examples can be found at Wikipedia: Ansatz, Entscheidungsproblem, Grossencharakter, Hauptmodul, Möbius band, quadratfrei, Stützgerade, Vierergruppe, Nebentype.

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Does Eigenvalue count as an answer...? – Abel Stolz Apr 19 '11 at 9:05
Hauptidealsatz (sometimes) – KConrad Apr 19 '11 at 9:07
The notation $\mathbb Z$ comes from "Zahlen". – Roland Bacher Apr 19 '11 at 9:09
By the way, there are also non-mathematical words in English that are simply taken over from German, e.g. kindergarten, gesundheit, doppelgänger, ... – Tom De Medts Apr 19 '11 at 11:57
@Micheal Lugo: I assume the second German should be English. That is, "band" is a perfectly reasonable English word. – user9072 Apr 19 '11 at 17:35

Führerdiskriminantenproduktformel.

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I had to google this to convince myself it was not a joke... – Marcel Bischoff Apr 19 '11 at 9:22
Can't resist to post this drawing by Mark Twain who was a special friend of the german language which is famous for its long words...gutenberg.org/files/5788/5788-h/5788-h.htm#p612 – user5831 Apr 19 '11 at 13:09
Dear unknown, concatenation is non-assocative. Proof: Mädchen(handelsschule) $\neq$ (Mädchenhandels)schule – Georges Elencwajg Apr 19 '11 at 16:03
Dear Georges Elencwajg, yes, and even if the decomposition is actually unique one can cause confusion. Some people like to tease kids by pronouncing the following wrong and asking them whether they know what it means: Blumento|pferde (suggesting Blumento-horses which do not exist) instead of Blumentopf|erde (flower-pot soil); same for Palat|schinken (suggesting it is Palat-ham which does not exist) which is actually just one word Palatschinken (some form of crepe, derived from Czech, Hungarian, or Romanian). – user9072 Apr 19 '11 at 16:32

Ansatz. Although I suppose it is used more in physics than in mathematics. I don't know why the translation is not used often, but I guess it has to do something with the fact that in the beginning of the 20th century German was used much more than English in the scientific literature, I believe.

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Which translation? – Tom Goodwillie Apr 19 '11 at 10:57
Well maybe not a literal translation, but in most cases "educated guess" can be used as well. – Pieter Naaijkens Apr 19 '11 at 11:16
@Pieter: Maybe that this is really the closest english expression, but personally I would never use it, because an "Ansatz" has a very different feeling than a "Schuss ins Blaue" / "educated guess". If you have an Ansatz, you have an idea of what is going on or should be going on. Maybe you have physical reasons to believe that the solution of your equation should have a particular nice form or something like that. An educated guess on the other hand is ... well, guessing. And that is a very different kind of approach I think. – Johannes Hahn Apr 19 '11 at 15:13
@Johannes Hahn, while I tend to agree that 'Ansatz' is not really an 'educated guess' I would say 'Schuss ins Blaue' is neither; as this I believe is more a 'shot in the dark' so a 'wild guess'. – user9072 Apr 19 '11 at 15:26

I've seen schlicht-function for functions $f(x)=x +a x^2 + b x^3 + ...$ for powerseries without constant term and $f'(0)=1$. But I do not really know, whether this is really the german word schlicht (=simple) or only some coincidence.

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It certainly is the german word schlicht (maybe innocuous would be a more adequate translation than simple). Conformal mapping was dominated by the German school (Koebe, Bieberbach) before the Finnish school took over. However, univalent seems to be the preferred term nowadays. – Theo Buehler Apr 19 '11 at 9:20

Verlagerung. Sometimes translated as the transfer.

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This is a notation rather than a term, but the wide use of the letter $K$ to denote a field in Algebra refers to the German word Körper.

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And $\mathfrak O$ comes from Ordnung. – Chandan Singh Dalawat Apr 19 '11 at 9:10
While the customary $U$ for subspaces comes from Unterraum. – Alex B. Apr 19 '11 at 9:53
And the $U$ for a neighbourhood comes from Umgebung. – Chandan Singh Dalawat Apr 19 '11 at 11:09
Surfaces are denoted $F$ in some of Mumford's books because of the German word Fläche. – Chandan Singh Dalawat Apr 19 '11 at 12:04
I always assumed $U$ was for subspaces because you call a topological space $T$ for topological space, and when you take a subspace of it, you just take the next letter. – Max Apr 20 '11 at 17:14

Jugendtraum (Kronecker).

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Was it Avner Ash who said "I'm too old to have a Jugendtraum"? – Laurent Berger Apr 19 '11 at 11:40
Last but not least, I appreciate the proposal of Friedrich Hirzebruch to translate Alterstraum in the title as "midlife crisis". ---Manin  – Chandan Singh Dalawat Apr 19 '11 at 12:09

Zahlbericht (Hilbert), Klassenkörperbericht (Hasse), Das blaue Hasse (Zahlentheorie, Akademie-Verlag, Berlin).

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Ganzstellensatz.

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Nebentypus, Positivstellensatz.

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Apparently the term K-theory comes from the German word "Klasse", according to Wikipedia and http://arxiv.org/abs/math/0602082

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You're probably right, but the truth should never interfere with a good story. What I heard is that the (contravariant functor) K comes from grothendiecK. There is also a covariant G. – Donu Arapura Apr 19 '11 at 11:19
That the K of K-theory comes from Klasse is explained by Grothendieck himself in the first issue of the journal K-theory. He first wanted to use C, from the French word "classe", but being an analyst he feared that it would cause a confusion with $C(X)$, the continuous functions on $X$. So he decided to use the initial of the translation of "classe" in his native German, Klasse. – Georges Elencwajg Apr 19 '11 at 16:28

Verschränkungsoperator is the (perhaps even original) german version of "intertwiner" which I really like. But I've not seen that very much ;)

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The word idele ultimately comes from the abbreviation "id. ele." for ideales Element.

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The word idele often is written with an accent on the first e, hinting at its French origin (Chevalley introduced ideal elements, and Hasse suggested the word idel; see p. 91 in Emil Artin und Helmut Hasse: die Korrespondenz 1923 - 1934, available online). – Franz Lemmermeyer Apr 19 '11 at 14:50
It is true that the word idèle was introduced by Chevalley but it was based on the German contraction "id. ele." for ideales Element. If it has been based on a French contraction, it would have been éléïde. – Chandan Singh Dalawat Apr 20 '11 at 2:33
Non, non, c'est pas du verlan. L’étymologie du mot d’« idèle » est bien documentée. – Chandan Singh Dalawat Mar 15 '12 at 3:28

Umkehr map (pushforward map).

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The term "Umkehrabbildung" (Abbildung being the german word for map) usually refers to and is translated as "inverse map". Do you know why are pushforwards are named like this? I mean $f_\ast$ is usually not invers to $f$ or even to $f^\ast$... – Johannes Hahn Apr 19 '11 at 15:05
@Johannes: Usually you use the term Umkehr map when a map going in the other direction is much easier to define, hence the “reversal”. – Dmitri Pavlov Apr 19 '11 at 18:56
Strangely enough, many topologists in Germany tend to call umkehr maps "pushforward". Maybe because otherwise they could be confused with inverse maps; maybe "pushforward" just sounds more faashionable. – Johannes Ebert Apr 20 '11 at 8:23

Gentzen's Hauptsatz (cut elimination theorem) : This is a fundamental result in structural proof theory, and is at the heart of Gentzen's consistency proof of elementary number theory. It is very funny that the word literally means "main theorem," with no reference to the subject domain, yet it is standard in logic in English to use just the word "Hauptsatz" to refer to this (family of) theorem(s) in proof theory.

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Here's another one: Hauptvermutung

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In GR (and other branches of mathematical physics) one uses vierbein (tetrad) and more often these days also vielbein, for local orthonormal frames in a (pseudo-)riemannian manifold.

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Die Vierergruppe.

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I've also heard it called "The Group of Fear". – Allen Knutson Apr 27 '11 at 3:37

Klein bottle

which has probably started out as:

Kleinsche Fläche (=Klein surface)

Kleinsche Flache (lost umlaut in English print)

Klein bottle (translation of Flasche instead of Flache)

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In German I never heard this called anything else than 'Kleinsche Flasche' (I thus somewhat doubt there was the development you sketch). – user9072 Apr 19 '11 at 11:33
It was then retranslated to German as Kleinsche Flasche which I omitted above because the question asks about usage in English. So unless you are 120 years old, you have no chance to have heard Kleinsche Fläche. – user11235 Apr 19 '11 at 11:35
Here the commenting unknown again: sorry for my initial doubts. What you write certainly agrees with the German Wikipedia entry, though that entry is a bit vague (as it is more or less reportes a rumour). Perhaps I will try to find some more information. If this turns out to be true it is a quite fun development. So, +1. – user9072 Apr 19 '11 at 11:43
After a bit of googling, I found the term "Kleinsche Fläche in an old German book: books.google.com/… – Tara Brough Apr 19 '11 at 11:47
Very interesting! I wonder which came first---the term "Klein bottle," or the standard bottle-shaped immersion of the Klein bottle? To me, the picture of the Klein surface in the book Tara Brough linked is quite striking---I don't think I've ever seen the Klein surface drawn that way, even though it's a perfectly natural way of doing it. – Vectornaut Apr 19 '11 at 21:41

Viergeflechte, the original German name for 2-bridge knots, still occasionally used in an English context. In his Mathematical Review of Schubert's 1956 paper "Knoten mit 2 Bruecken" Fox explicitly notes that "Viergeflecht" is untranslatable.

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The $\int$ symbol is a german S introduced by Leibniz and stands for Summe (Sum)

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There is nothing German about the glyph $\int$ for the letter S. It can be found in almost all French and English books of the time. – Chandan Singh Dalawat Apr 19 '11 at 11:50
Even because Leibnitz wrote in Latin. – Pietro Majer Apr 19 '11 at 12:07
Several aspects of the typography stand out to modern eyes. Most noticeable of these is the use of the 'long s', visually resembling (but not pronounced as) a modern 'f' (as in the word 'Goſpel'). The modern form of the letter 's' was only used at the end of words, and in a few other specific circumstances. The 'long s' persisted in English print until the late 1700s, and survives in mathematics today as the symbol to denote an integral ('s' to denote a sum of infinitesimals). bl.uk/onlinegallery/sacredtexts/kingjames.html  – Chandan Singh Dalawat Apr 22 '11 at 7:07

The notation $G_\delta$ is from German, G for "Gebiet", and $\delta$ for "Durchschnitt". Strangely enough, the notation for the co-sets, $F_\sigma$, is from French " fermé" and "somme".

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I did not know that, it's very cool! Well, I suspected that the F stood for "fermé", but then you tend to automatically assume that G was used because it's the next letter available... So far, this has to be my favorite answer. – Thierry Zell Apr 19 '11 at 15:30

And what about the Wiedersehen metric?

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Isn't it auf Wiedershehen metric, with auf and all? – Mariano Suárez-Alvarez Apr 19 '11 at 17:05

"Urelement" is used in set theory as a fancy name for an atom, i.e., something that can be a member of a set but is not itself a set.

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Ur- as a prefix is frequently used in ordinary English, though; see sense 3 at dictionary.reference.com/browse/ur . – Qiaochu Yuan Apr 20 '11 at 6:03

This is an answer to the part of the question about why these terms are not translated into English. The reason is that words such as "nullstellensatz", "Schadenfreude" and so on that you mistakenly think are German are in fact perfectly good English words and so do not need translation. (Look up Schadenfreude in the Oxford English Dictionary if you do not believe it is an English word, though they have not yet caught up with nullstellensatz.) The point is that unlike languages such as French and German that try to remain pure, English has been happily looting terms from other languages for centuries, and the only difference between "nullstellensatz" and "house" is that "house" was stolen so long ago that we have forgotten about it.

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English has been happily looting... For example the word loot (लूट) from us Indians. – Chandan Singh Dalawat Apr 19 '11 at 14:26
Whether this is done more 'happily' in English, I don't know. But, it is certainly also done in German, with certain regional variations in the extent (three of the italic words might not be common everywhere, but the others are universal several even without common alternative): "Von der Trafik aus, flanierte ich ueber das Trottoir, einen salutierenden Offizier mit Pistole und einen Portier nonchalant passierend, ins Souterrain." And, in France you might well wish your collegues after a 'planning' on friday afternooon 'bonne week-end'. – user9072 Apr 19 '11 at 15:07
@Chandan: one can imagine Indians using that word a lot in talking with/about the Englsh, initially :) – Mariano Suárez-Alvarez Apr 19 '11 at 17:04
Sorry, but "house" most certainly did not come into English from German. The word hus occurs in every one of the oldest Germanic languages. By standard sound-changes, the long "u" shifted to "ou" in English, just as "ut" became "out", "mus" became "mouse", etc. – Lubin Apr 20 '11 at 15:03
Bof -- the French loot as many words as anglophones, they just frenchify them quickly. My favourite current regular -er verb is "liker", snaffled from facebook: je like, tu likes, il like, nous likons, ... – J.J. Green Mar 13 '12 at 20:01

Plastikstufe = a certain higher dimensional analogue of an overtwisted disk in contact geometry. This is not a real German word. It is a compound of the German words for "plastic" and "step", but this does not have any obvious relevance to its mathematical meaning. There is a funny story about where this word came from which however is not appropriate for this forum.

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How can it be non appropriate to this forum if it is the origin of a mathematical term, in the context of a discussion of mathematical terms?! – Mariano Suárez-Alvarez Apr 19 '11 at 17:01

Einheit = word for unit in algebra. Hence, some use the notation $e\in G$ to denote the element of a group such that $ex = xe = x , \forall x \in G$. Unit is the appropriate translation, yet some algebraist still use the letter $e$ to denote the identity element in a group.

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Well, sometimes it's also accidents of language that force this: e.g. "identity" is a good word to describe the unit, but the letter i was not really available any more, was it? – Thierry Zell Apr 19 '11 at 16:31
According to Cajori, $i$ was first used by Euler in 1777 in a memoir which was not printer until 1794, after his death. It apparently did not appear anywhere else until 1801, when Gauss started to use it systematically. – Mariano Suárez-Alvarez Apr 19 '11 at 19:15

The Verschiebung morphism.

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deck transformation?

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@Mariano: Is it? I thought the "deck" in "deck transformation" comes from "Überdeckung" (=covering). – Johannes Hahn Apr 19 '11 at 20:13

There's Soergel's Endomorphismensatz and Struktursatz.

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