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.

• Does Eigenvalue count as an answer...? Apr 19, 2011 at 9:05
• Hauptidealsatz (sometimes) Apr 19, 2011 at 9:07
• The notation $\mathbb Z$ comes from "Zahlen". Apr 19, 2011 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, ... Apr 19, 2011 at 11:57
• "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. Apr 19, 2011 at 17:25

The practice to use Gothic letters sometimes for ideals ($\mathfrak{a}$, $\mathfrak{b}$, ...) and often for Lie algebras ($\mathfrak{g}$, $\mathfrak{h}$, ..) seems to be of German origin.

Also to use the lesser known "kernel" instead of the better known "core" seems to stem from the German "Kern".

• "Kernel" and "nucleus" seem to me to be the most obvious options for English translation of German "Kern", although "core" would have been possible. Apr 20, 2011 at 12:30
• The choice of "core" for Kern would have led to "cocore" for Cokern. Apr 22, 2011 at 6:12
• I don't know what to think about the gothic letters: yes, it is German in origin because it was the alphabet used in German back then. But borrowing symbols wouldn't rank it the same way as borrowing a word, I think. Apr 23, 2011 at 15:10

I would like to mention a handful of examples that may be considered passé nowadays, but were prominent at some point in time.

• Schlicht: I dare to address this once again because I consider that the feedback in the comments below Gottfried's entry is kind of misleading. About this one, Boas says that (see [1, page 97]):

... When I was an undergraduate, there was no regular colloquium at Harvard, but there was a Mathematical Club, whose meetings were regularly attended by faculty. Once somebody gave a talk on schlicht functions. After the talk, Julian Lowell Coolidge asked plaintively whether there was an English word for 'schlicht'. Osgood replied, "Well, you could call them univalent functions, and everybody would know that you meant 'schlicht'". You need to know that Osgood had been trained in Germany, wrote his treatise on complex analysis in German, and was apt to tell German jokes to his classes.

It has to be noted that in practice univalent and schlicht are not perfect synonyms. For instance, on Function theory of one complex variable by Greene and Krantz, we can read this (my emphasis):

A holomorphic function $$f$$ on the unit disc $$D$$ is usually called schlicht if $$f$$ is one-to-one. We are interested in such one-to-one $$f$$ that satisfy the normalizations $$f(0)=0$$ and $$f^{\prime}(0)=1.$$ In what follows, we restrict the word schlicht to mean one-to-one with these normalizations.

What is more, several online sources include right from the start those normalizations in their definition of schlicht, e.g., planetmath.org, Wikipedia, and Wolfram MathWorld.

• Aussonderungsaxiom: Of all axioms of Zermelo, I have noticed that, for some godforsaken reason, in some books/papers written in English (and even in Spanish) this one is (or was) occasionally called by its German name.

• Limes: That's right... It was not a typo in Ahlfors's text on Complex Analysis. I recently came across this one in another book, but I just can't recall which one it was.

EDIT: According to Gerald Edgar "limes" is a Latin word. Yet, I will leave it here because I believe that it is a loan word in German which made it to other languages due to the influence of treatises written originally in German.

• Drehstreckung: Tristan Needham recalls this one when he apologizes for the coinage of the term 'amplitwist'. More specifically, he writes

To the expert reader I would like to apologize for having invented the word 'amplitwist' ... as a synonym (more or less) for 'derivative', as well the component terms 'amplification' and 'twist'. I can only say that the need for some such terminology was forced on me in the classroom: if you try teaching the ideas in this book without using such language, I think you will quickly discover what I mean! Incidentally, a precedence argument in defence (sic) of 'amplitwist' might be that a similar term was coined by the older German school of Klein, Bieberbach, et al. They spoke of 'eine Drehstreckung', from 'drehen' (to twist) and 'strecken' (to stretch).

Last but not least, in several works of old (z.B., Perron's Die Lehre von den Kettenbrüchen, Knopp's Theory and Application of Infinite series, Khinchin's Continued Fractions), there appears the following notation for general continued fractions:

$$\underset{j=1}{\overset{\infty}{\LARGE\mathrm K}}\frac{a_j}{b_j} = \cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\cfrac{a_3}{b_3+\ddots}}}.$$

Guess what the $$\mathrm{K}$$ stands for...

References

[1] Lion Hunting & Other Mathematical Pursuits: A Collection of Mathematics, Verse and Stories by Ralph P. Boas Jr.

• Do you happen to know the meaning of the term "schicht domain" or "schicht region"? I have seen this in a paper of Huber published in the 50s and I am still unsure what it means. Mar 31, 2012 at 0:05
• Unfortunately, I don't. Anyway, thanks a lot for taking a look at my entry. Apr 14, 2012 at 6:16
• I thought "limes" is from Latin. Jun 6, 2012 at 14:38
• @Igor Belegradek: Why don't you take a look at this thread? mathoverflow.net/questions/114190/schlicht-domain Nov 22, 2012 at 22:39
• Indeed, @GeraldEdgar, limes is good Latin, plural is limites. Feb 27, 2020 at 22:15

Spiegelungssatz. The meaning of this theorem is briefly discussed in the article: Iwasawa theory and $p$-adic deformations of motives [MR1265554 (95i:11053)] by Ralph Greenberg.

Let $p$ be an odd prime, and $K_\infty=\mathbf{Q}(\mu_{p^\infty})$. Let $L_\infty$ denote the maximal unramified abelian pro-$p$ extension of $K_\infty$, and $M_\infty$ the maximal abelian pro-$p$-extension of $K_\infty$ that is unramified outside the primes above $p$. Let $Y_\infty={\rm Gal}(L_\infty/K_\infty)$ and $X_\infty={\rm Gal}(M_\infty/K_\infty)$. We can decompose ${\rm Gal}(K_\infty/\mathbb{Q})\cong\Delta\times\Gamma$, where $\Delta={\rm Gal}(\mathbf{Q}(\mu_p)/\mathbf{Q})$ and $\Gamma\cong\mathbf{Z_p}$. Both $Y_\infty$ and $X_\infty$ have a natural structure of $\Lambda$-modules ($\Lambda=\mathbf{Z_p}[[\Gamma]]$) coming from the action of ${\rm Gal}(K_\infty/\mathbf{Q})$ by inner automorphisms. The latter action gives in particular an action of $\Delta$, and hence we can decompose $Y_\infty=\bigoplus_{i=0}^{p-2}Y_\infty^{\omega^i}$ and $X_\infty=\bigoplus_{j=0}^{p-2}X_\infty^{\omega^j}$ as $\Lambda$-modules, where the superscript denotes isotypical component under the action of $\Delta$, and $\omega:\Delta\rightarrow\mu_{p-1}$ denotes the mod $p$ cyclotomic character. The spliegelungsatz is then described by Greenberg in loc. cit. as an argument using Kummer theory and class field theory that allows to relate the structures of $X_\infty^{\omega^j}$ and $Y_\infty^{\omega^i}$ for $i+j\equiv 1\pmod{p-1}$ as $\Lambda$-modules.

• +1. You beat me to it by two hours. Apr 20, 2011 at 2:39

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.

• 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? Apr 19, 2011 at 16:31
• 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 $x^2 -1 =0$. Apr 19, 2011 at 17:49
• 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. Apr 19, 2011 at 19:15

In topology the separation axioms $T_0$ , $T_1$ .. etc, where the $T$ stands for Trennungsaxiom

• I used to think that T stood for Tikhonov. Jun 6, 2012 at 15:45
• well, at least according to Munkres it stands for "Trennungsaxiom" (see page 211 Topology 2nd ed.) and wikipedia agrees, though whoever wrote the wikipedia article might have read it from munkres Jun 6, 2012 at 16:07
• I don't think that letters are really interesting ... Jun 6, 2012 at 16:41

And what about the Wiedersehen metric?

• Isn't it auf Wiedershehen metric, with auf and all? Apr 19, 2011 at 17:05
• Aw -- man, I was gonna respond with a comment just saying "Auf". Apr 19, 2011 at 23:27

One that is similar in spirit "eigenvalue" in that it mixes the two languages is $$\text{umkehr map}$$

• Thanks John! I couldn't really believe this, but in fact there are even papers titled "Umkehr maps" ;-) Jun 6, 2012 at 21:24

The Hegelian term Aufhebung has been appropriated by Lawvere to refer to relations between essential subtoposes of a cohesive topos, with a view to doing abstract homotopy theory. See the nLab for more.

The following theorem is known as Kugelsatz:

Let $$X$$ be an open set in $$\mathbb{C}^n, \quad n \geq 2$$ and $$K \subset X$$ a compact subset such that $$X\setminus K$$ is connected. Then the restriction map $$\rho: \mathcal{O}(X) \mapsto \mathcal{O}(X \setminus K)$$ is an isomorphism of $$\mathbb{C}$$-algebras (this version after: Volker Scheideman, Introduction to Complex Analysis in Several Variables, Birkhäuser 2005).

The first result of this kind is due to Hartogs, with $$X$$ and $$K$$ being concentric euclidean balls, hence the name (Kugel=ball). Many textbooks in several complex variables have been written by German-speaking authors (Grauert+Fritzsche, Kaup brothers are other examples), so the German name stuck even in the English version. The theorem is also referred to as "tomato can principle".

Bew (short for beweisbar, introduced by Gödel's incompleteness paper) is still used as a provability predicate in some mathematical logic papers.

In physics and other subjects (not so much in math) we hear about plenty of Gedankenexperiments.

Don't forget Hilbert's Satz 90, anomalous because of the "90" and not just the "Satz".

There are also French words like étale cohomology.

• Maybe I'm a little slow today, but I don't find it as easy to come up with French mathematical terms. There is the unfortunately named gerbe, "étale" is a weird one because it looks like it somehow lost its last accent... Can't think of much more right now. I wonder if the concatenation property of German is what makes is so attractive for math, given that many example exhibit this feature. Apr 19, 2011 at 21:37
• Etale with only the first e marked is a French word. I have seen it on the package of a light bulb. Apr 19, 2011 at 21:56
• A distinguished mathematician once referred to an assertion he was making in a conference talk as a "Theorem 90". He went on to explain that he was he was 50% sure of the proof--and that he had explained to a colleague, who was 40% sure. Apr 19, 2011 at 21:58
• @Thierry Zell, easy as child's play to find one more :) ... dessin d'enfant
– user9072
Apr 20, 2011 at 0:41
• @Tom: étale does exist; it's just obsolete in the common language, hence it's "weird" connotation. Apr 20, 2011 at 0:46

There is Ahlfor's scheibensatz in complex function theory, which is a generalization of Ahlfors five islands theorem

There's a kind of combinatorial design called a gerechte design - essentially it's a Latin square with additional block constraints. (I gather there's been a fad in recent years for newspapers to print partial gerechte designs of a certain kind for readers to complete.) As a technical term, the word comes from the following paper:

W. U. Behrens (1956). Feldversuchsanordnungen mit verbessertem Ausgleich der Bodenunterschiede. Zeitschrift für Landwirtschaftliches Versuchs- und Untersuchungswesen, 2, 176–193.

Behrens' gerechte designs were 'fair' in how they apportioned plots of land to different treatments in an agricultural trial.

If you think of the symbols, you can also see Gothic, alternatively called German, letters. Also, in algebraic topology, it is common to show the cycles by $Z$, which is the first letter of Zykel.

Also, many words that are Latin or Greek, in terms of the ingredients, were first coined and used in German, like Topologie which used to be called Analyse Situs.

It was common to show curvature by $K$, which stands for Krummung. Also, it was common to show a domain by B, for Bereiche. Or in riemannian geometry, the metric tensor is represented by $g$, which stands for Gravit\"at Also, Faltung used to be common in English before the word convolution took over.

I can also add Umlaufssatz in the differential geometry of surfaces.

There are so many more...

• There is also the inverse tendency that the German terms tend to be forgotten, now that English has become so prevalent. Many German students will happily use "Konvolution" when they read it in a paper before I teach them to use "Faltung". Similarly, "bottleneck" like in "bottleneck objective function" tends to be sometimes literally translated into "Flaschenhals" instead of "Engpass" (meaning narrow pass, which is (or used to be) the usual term in this situation). A case which I particularly deplore is the thoughtless translation of "line segment" into "Liniensegment" instead of "Strecke". Mar 1, 2013 at 11:27
• Gothic is ambiguous (as a term describing a typeface); Fraktur is probably what you meant, although it is a special case of blackletter (as opposed to whiteletter) fonts. Feb 27, 2020 at 16:44

There is also the Quermassintegral (mixed volumes of the form $V(K,K,\ldots,B,B)$ where $B$ is the unit ball, see Wikipedia), which I'm not even sure is German (not a lot of Qs in German usually).

• Yes, German has Q's, and pronounces "qu" as "kv" as in "kvetch" (which indeed is from German "quetschen"). Mar 31, 2012 at 3:18

Schubfachprinzip ("drawer principle" or "shelf principle" or "Dirichlet's box principle"). It is now easy to guess we are talking about P-H P.

Ganzstellensatz.

Zusammenstellung. Means "compilation" or "survey". Can be used in the first section of a paper, as one starts compiling "preliminary facts" to refer to later in the paper. That's the way I've seen it used in a paper by Raoul Bott.

I believe Albrecht Frölich uses the german term beweis, instead of the english proof, in his chapter of the classic "Algebraic number theory". (EDIT: In my original version, I translated beweis to example. I shouldn't trust my poor knowledge of German... )

• And to think, the number of times we tell students 'an example is not a proof'... Oct 3, 2011 at 23:49

In "Functional Analysis" by Kosaku Yosida he denotes the closure of a set $M$ by $M^a$. He explains that it is a shortcut from German abgeschlossene Hulle.

"schlichtartig" refers to a surface on which every simple closed curve which separates locally, also separates globally. Hence it means roughly "planar". This is used in the conformal mapping theory of Riemann surfaces. Introduction to Riemann Surfaces, Springer, p. 91. I know only a little German but it seems to translate something like "simply behaved"?

Anzahl-theorems is one I have recently read in Wan's book on classical groups.

Kegelspitzen. There are directed complete orders equipped with a convex structure such that all relevant operations are (Scott) continuous. Introduced by Klaus Keimel and Gordon Plotkin in https://arxiv.org/abs/1612.01005 It literally means "tip of a cone"; the motivation is that you would obtain a Kegelspitze by considering a cone and cutting of its tip. I guess that because the English description would be three words instead of a single one, the authors chose for the German translation, probably also because one of the authors was German.

Gleichverteilungssatz, which refers to both H. Weyl's result in complex analysis and ergodic theory ([1] §5, pp.18-19) and to several theorems in statistical physics (Boltzmann's,... etc.).

[1] Heins, Maurice (1962), Selected Topics in the Theory of Functions of a Complex Variable, New York: Holt, Rinehart and Winston, Athena Series. Selected Topics in Mathematics, xi+160, MR0162913, Zbl 1226.30001.

In Swedish, a field is called a 'kropp', a body. This of course from the German word Körper.

• That's is really stretching things; isn't it just as likely from the French "corps"? Jun 5, 2011 at 4:50
• Fields (in this algebraic sense) are called bodies in most languages. Mar 13, 2012 at 22:01
• Some early English papers use the term "corpus" for field. Jun 6, 2012 at 14:40

It's early in the morning, so maybe I missed it in the answers above, but, if we're including symbols, then the obvious example is $\mathbb{Z}$, the integers, or zahlen!

Ooops! It is early in the morning... I see that Roland noted that the symbol for the integers (which I also can't seem to get to process properly) just a few comments above.

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

• 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. Apr 19, 2011 at 11:50
• Even because Leibnitz wrote in Latin. Apr 19, 2011 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  Apr 22, 2011 at 7:07