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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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    $\begingroup$ Does Eigenvalue count as an answer...? $\endgroup$
    – Abel Stolz
    Commented Apr 19, 2011 at 9:05
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    $\begingroup$ Hauptidealsatz (sometimes) $\endgroup$
    – KConrad
    Commented Apr 19, 2011 at 9:07
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    $\begingroup$ The notation $\mathbb Z$ comes from "Zahlen". $\endgroup$
    – Roland Bacher
    Commented Apr 19, 2011 at 9:09
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    $\begingroup$ By the way, there are also non-mathematical words in English that are simply taken over from German, e.g. kindergarten, gesundheit, doppelgänger, ... $\endgroup$
    – Tom De Medts
    Commented Apr 19, 2011 at 11:57
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    $\begingroup$ "Möbius band" isn't really German. "Möbius" is a name and "band" is a perfectly reasonable German word. Oddly enough, the space seems to make the difference here; "Möbiusband" would feel much more German to me. I'm a native speaker of English; I have a mere smattering of German, enough to find the German Wikipedia article for this object and see what it's called in German. $\endgroup$
    – Michael Lugo
    Commented Apr 19, 2011 at 17:25

60 Answers 60

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Führerdiskriminantenproduktformel.

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    $\begingroup$ I had to google this to convince myself it was not a joke... $\endgroup$
    – Marcel Bischoff
    Commented Apr 19, 2011 at 9:22
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    $\begingroup$ en.wikipedia.org/wiki/Conductor-discriminant_formula $\endgroup$
    – user5831
    Commented Apr 19, 2011 at 12:53
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    $\begingroup$ 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 $\endgroup$
    – user5831
    Commented Apr 19, 2011 at 13:09
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    $\begingroup$ Dear unknown, concatenation is non-assocative. Proof: Mädchen(handelsschule) $\neq$ (Mädchenhandels)schule $\endgroup$
    – Georges Elencwajg
    Commented Apr 19, 2011 at 16:03
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    $\begingroup$ 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). $\endgroup$
    – user9072
    Commented Apr 19, 2011 at 16:32
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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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    $\begingroup$ 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. $\endgroup$
    – Thierry Zell
    Commented Apr 19, 2011 at 15:30
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    $\begingroup$ I always thought this was the origin of $F_\sigma$, but it looks like the notation originated with Hausdorff and there's no evidence he had anything French in mind (although I guess there's no proof he didn't). See mathoverflow.net/questions/74004/…. $\endgroup$
    – Henry Cohn
    Commented Oct 3, 2011 at 23:18
  • $\begingroup$ Are you sure $\sigma$ is for French somme and not German Summe? $\endgroup$
    – Gerald Edgar
    Commented Aug 21, 2022 at 12:04
  • $\begingroup$ Yes, $\sigma$ may come from any language where the term for “sum” comes from Latin; but I think $F$ for “closed” can only be from French. I learnt the information about $G_\delta$ and $F_\sigma$ from the book “Real Analysis” by E.Stein and Rami Shakarchi (footnote on page 23). $\endgroup$
    – Pietro Majer
    Commented Sep 5, 2022 at 10:59
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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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    $\begingroup$ Which translation? $\endgroup$
    – Tom Goodwillie
    Commented Apr 19, 2011 at 10:57
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    $\begingroup$ Well maybe not a literal translation, but in most cases "educated guess" can be used as well. $\endgroup$
    – Pieter Naaijkens
    Commented Apr 19, 2011 at 11:16
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    $\begingroup$ @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. $\endgroup$
    – Johannes Hahn
    Commented Apr 19, 2011 at 15:13
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    $\begingroup$ @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'. $\endgroup$
    – user9072
    Commented Apr 19, 2011 at 15:26
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    $\begingroup$ I would put Ansatz closer to Postulate than to Educated Guess: you make an educated guess as to what is going on, but then you stick with that to all its consequences and only afterwards see if you got it right or not, and what kind of solid (a posteriori) evidence you can collect. $\endgroup$
    – Emilio Pisanty
    Commented Jun 6, 2012 at 16:36
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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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    $\begingroup$ English has been happily looting... For example the word loot (लूट) from us Indians. $\endgroup$
    – Chandan Singh Dalawat
    Commented Apr 19, 2011 at 14:26
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    $\begingroup$ 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'. $\endgroup$
    – user9072
    Commented Apr 19, 2011 at 15:07
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    $\begingroup$ @Chandan: one can imagine Indians using that word a lot in talking with/about the Englsh, initially :) $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Apr 19, 2011 at 17:04
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    $\begingroup$ 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. $\endgroup$
    – Lubin
    Commented Apr 20, 2011 at 15:03
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    $\begingroup$ 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, ... $\endgroup$
    – J.J. Green
    Commented Mar 13, 2012 at 20:01
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All this should be compiled in a Festschrift.

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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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    $\begingroup$ And $\mathfrak O$ comes from Ordnung. $\endgroup$
    – Chandan Singh Dalawat
    Commented Apr 19, 2011 at 9:10
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    $\begingroup$ While the customary $U$ for subspaces comes from Unterraum. $\endgroup$
    – Alex B.
    Commented Apr 19, 2011 at 9:53
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    $\begingroup$ And the $U$ for a neighbourhood comes from Umgebung. $\endgroup$
    – Chandan Singh Dalawat
    Commented Apr 19, 2011 at 11:09
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    $\begingroup$ 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. $\endgroup$
    – Max
    Commented Apr 20, 2011 at 17:14
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    $\begingroup$ The Z for the center of a group/algebra/etc comes from Zentrum $\endgroup$
    – Bin Gui
    Commented Sep 21, 2017 at 2:46
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Jugendtraum (Kronecker).

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    $\begingroup$ Was it Avner Ash who said "I'm too old to have a Jugendtraum"? $\endgroup$
    – Laurent Berger
    Commented Apr 19, 2011 at 11:40
  • $\begingroup$ @Laurent: Certainly not Manin. See his Real multiplication and noncommutative geometry (ein Alterstraum) in The legacy of Niels Henrik Abel, Springer, Berlin, 2004. $\endgroup$
    – Chandan Singh Dalawat
    Commented Apr 19, 2011 at 11:55
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    $\begingroup$ Last but not least, I appreciate the proposal of Friedrich Hirzebruch to translate Alterstraum in the title as "midlife crisis". ---Manin $$ $$ $\endgroup$
    – Chandan Singh Dalawat
    Commented Apr 19, 2011 at 12:09
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The word idele ultimately comes from the abbreviation "id. ele." for ideales Element.

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    $\begingroup$ 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). $\endgroup$
    – Franz Lemmermeyer
    Commented Apr 19, 2011 at 14:50
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    $\begingroup$ 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. $\endgroup$
    – Chandan Singh Dalawat
    Commented Apr 20, 2011 at 2:33
  • $\begingroup$ ...unless it's verlan! [en.wikipedia.org/wiki/Verlan]. $\endgroup$
    – JeffE
    Commented Mar 14, 2012 at 15:39
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    $\begingroup$ Non, non, c'est pas du verlan. L’étymologie du mot d’« idèle » est bien documentée. $\endgroup$
    – Chandan Singh Dalawat
    Commented Mar 15, 2012 at 3:28
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An indirect answer:

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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    $\begingroup$ In German I never heard this called anything else than 'Kleinsche Flasche' (I thus somewhat doubt there was the development you sketch). $\endgroup$
    – user9072
    Commented Apr 19, 2011 at 11:33
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    $\begingroup$ 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. $\endgroup$
    – user11235
    Commented Apr 19, 2011 at 11:35
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    $\begingroup$ 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. $\endgroup$
    – user9072
    Commented Apr 19, 2011 at 11:43
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    $\begingroup$ After a bit of googling, I found the term "Kleinsche Fläche in an old German book: books.google.com/… $\endgroup$
    – Tara Brough
    Commented Apr 19, 2011 at 11:47
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    $\begingroup$ 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. $\endgroup$
    – Vectornaut
    Commented Apr 19, 2011 at 21:41
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Verlagerung. Sometimes translated as the transfer.

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  • $\begingroup$ In Corps locaux, Serre uses $\mathrm{Ver}$ for the transfer map, noting that it is "called transfer (Verlagerung in German)." $\endgroup$
    – LSpice
    Commented Dec 31, 2018 at 23:49
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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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    $\begingroup$ 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. $\endgroup$
    – Donu Arapura
    Commented Apr 19, 2011 at 11:19
  • $\begingroup$ @Donu: I've never heard that one, but I like it. At least it's a good mnemonic. $\endgroup$
    – Johannes Hahn
    Commented Apr 19, 2011 at 15:07
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    $\begingroup$ 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. $\endgroup$
    – Georges Elencwajg
    Commented Apr 19, 2011 at 16:28
  • $\begingroup$ Georges, OK, you have me convinced. But the alternate explanation, that I think I learned from Grayson, had sounded preposterous enough to be true. $\endgroup$
    – Donu Arapura
    Commented Apr 19, 2011 at 17:19
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Umkehr map (pushforward map).

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    $\begingroup$ 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 $f^\ast$... $\endgroup$
    – Johannes Hahn
    Commented Apr 19, 2011 at 15:05
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    $\begingroup$ @Johannes: Usually you use the term Umkehr map when a map going in the other direction is much easier to define, hence the “reversal”. $\endgroup$
    – Dmitri Pavlov
    Commented Apr 19, 2011 at 18:56
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    $\begingroup$ 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. $\endgroup$
    – Johannes Ebert
    Commented Apr 20, 2011 at 8:23
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Here's another one: Hauptvermutung

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  • $\begingroup$ I already mentioned this in (the first version of my) question. But don't mind :-) $\endgroup$
    – Martin Brandenburg
    Commented Apr 20, 2011 at 8:21
  • $\begingroup$ Ah, sorry -- I looked for it, but didn't see it. $\endgroup$
    – José Figueroa-O'Farrill
    Commented Apr 20, 2011 at 8:23
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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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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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    $\begingroup$ 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?! $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Apr 19, 2011 at 17:01
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    $\begingroup$ I remember having heard the story a few years back :D Plastikstufe is a real german word, though: In german words can be composed like this. A Plastikstufe is a step (in a stair) made out of plastic. $\endgroup$
    – Gerrit Begher
    Commented Aug 6, 2013 at 9:24
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The Verschiebung morphism.

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Größencharakter. http://en.wikipedia.org/wiki/Hecke_character

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Zugzwang - a sort of Nash Equilibrium. This terminology is specifically used in Chess.

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  • $\begingroup$ I'm familiar with the chess term, but didn't know it had been used formally in game theory. Is the definition exactly as in chess? $\endgroup$
    – Yemon Choi
    Commented Apr 20, 2011 at 1:06
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    $\begingroup$ @Yemon: apparently yes. From wikipedia: "The term finds its formal definition in combinatorial game theory, and it describes a situation where one player is put at a disadvantage because he has to make a move when he would prefer to pass and make no move." $\endgroup$
    – Thierry Zell
    Commented Apr 20, 2011 at 1:50
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Die Vierergruppe.

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    $\begingroup$ I've also heard it called "The Group of Fear". $\endgroup$
    – Allen Knutson
    Commented Apr 27, 2011 at 3:37
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deck transformation?

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    $\begingroup$ Isn't that just good ol' boring English? (It was probably Germanic a few hundred years ago, of course :) ) $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Apr 19, 2011 at 19:12
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    $\begingroup$ @Mariano: Is it? I thought the "deck" in "deck transformation" comes from "Überdeckung" (=covering). $\endgroup$
    – Johannes Hahn
    Commented Apr 19, 2011 at 20:13
  • $\begingroup$ @JHahn: well, "deck" is an English word, isn't it? $\endgroup$
    – Qfwfq
    Commented Mar 11, 2018 at 10:11
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    $\begingroup$ I had thought the name "deck transformation" was thought of like shuffling a "deck" of cards. I'm not a native speaker either, so is this too far fetched? $\endgroup$
    – Klaus Niederkrüger
    Commented Aug 10, 2019 at 15:43
  • $\begingroup$ @KlausNiederkrüger it is good mnemonic rule, but historically (IMHO) it comes from German. $\endgroup$
    – Anton Petrunin
    Commented Aug 11, 2019 at 14:07
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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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I've seen schlicht-function for functions $f(x)=x +a x^2 + b x^3 + \cdots$ 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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    $\begingroup$ 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. $\endgroup$
    – Theo Buehler
    Commented Apr 19, 2011 at 9:20
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    $\begingroup$ In complex function theory, "schlicht" is usually a synonym for one-to-one. $\endgroup$
    – Andreas Blass
    Commented Apr 19, 2011 at 14:02
  • $\begingroup$ Could you tell me what is meant by "schlicht annular region" or by "schlicht domain"? $\endgroup$
    – Igor Belegradek
    Commented Apr 19, 2011 at 15:06
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Some famous book published in about 1950 says that for lack of an English word for the concept the word Faltung is used. In recent decades, the adapted Latin word convolution has served.

Paul Halmos tried unsuccessfully to expunge the words eigenvector and eigenvalue from the language, using the terms proper vector and proper value in his book Finite-dimensional Vector Spaces.

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"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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Stufe (=level) of a non-real field (wikipedia.de). It is the least number of squares $a_i^2$ such that $\sum_i a_i^2 = -1$, $\infty$ if no such sum exists.

In this paper, the level of a subgroup of $SL_2(\mathcal{O})$ is defined ($\mathcal{O}$ a number field), as the generalisation of the stufe of a field, so the term has been translated, but only in a shift of context.

To pick a random paper, try The stufe of number fields.

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    $\begingroup$ Whoops, I saw KConrad got this one in the comments, and Martin wanted him to post it as an answer. $\endgroup$
    – David Roberts ♦
    Commented Apr 19, 2011 at 23:40
  • $\begingroup$ Allow me to add the following quote from Gödel's 1940 book about the relative consistency of GCH and AC with ZF: The latter proof requires the singling out of one element in every non empty class, which however can be accomplished by considering, in every class, the subset of elements of lowest "Stufe" (in the sense of J. v. Neumann [...]). Here "Stufe" = rank, in the sense of the cumulative hierarchy. $\endgroup$
    – David Roberts ♦
    Commented Aug 22, 2022 at 2:02
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There's Soergel's Endomorphismensatz and Struktursatz.

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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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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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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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    $\begingroup$ 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? $\endgroup$
    – Thierry Zell
    Commented Apr 19, 2011 at 16:31
  • $\begingroup$ yes, i agree it is easy to use up the alphabet, but in this case, some people actually would use i to denote identity, cf. jeff560.tripod.com/i.html [scroll down to identity] All the people I have read in the olden days, would write things like $\sqrt{-4} = 2\sqrt{-1}$ instead of $\sqrt{-4} = 2i$. I do not know of the first use of the symbol $i$ as a solution to $x^2 -1 =0$. $\endgroup$
    – Andrew Stout
    Commented Apr 19, 2011 at 17:49
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    $\begingroup$ 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. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Apr 19, 2011 at 19:15
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