A follow-up question to the use of "sei" in mathematics.

Compare the two following sentences, which are both common:

  • Es sei f eine stetige Funktion.
  • Sei f eine stetige Funktion.

Would you regard one version more correct than the other, at least historically?

  • I am more interested in the grammatical correctness than in the frequency. I do agree that "Sei" is much more frequent, but I had a heated dispute with one of the non-mathematical lectors of one of my math papers on the grammatical correctness.
    – Phira
    Aug 10 '11 at 11:23
  • Any German lector who rejects "Sei ..." in a mathematical text should get a chance to look at a few high-quality mathematical publications in German. If he/she then doesn't admit they were wrong, they should probably be fired for incompetence. But really the first step shouldn't be necessary.
    – user2183
    Oct 20 '15 at 22:34
  • 1
    In general language, sentences using typical "Sei ..." structure would usually not be correct. However, in mathematics it's the norm nowadays and the longer form "Es sei ..." just sounds antiquated and slightly weird nowadays if it is overused.
    – user2183
    Oct 20 '15 at 22:40
  • 3
    Gerne würde man auch sagen: "f sei eine stetige Funktion", aber dann prallen zwei starke Konventionswelten aufeinander, die Mathematik,die will, dass man f klein schreibt, und die dt. Grammatik, die will, dass man am Satzanfang F groß schreibt. Einen von beiden müsste man verletzen - so kommt vielleicht das 'Sei' oder 'Es sei' an den Satzanfang. Oct 21 '15 at 1:51

I've studied mathematics for a long time and in my experience the second form (i.e. without the "es" at the beginning) is used more often or almost exclusively. So I would use the second version although the first one seems perfectly correct and understandable.

EDIT: Since there seems to be some disagreement here, I picked 4 random books about mathematics and computer science from my bookshelf:

  1. "Sei G eine endliche Gruppe von Automorphismen eines Körpers L, und sei K ihr Fixkörper." (M. Artin, "Algebra", Birkhäuser Verlag, 1998, S. 633)
  2. "Sei v ein Multiplikatorsystem vom Gewicht r/2 bezüglich einer Kongruenzgruppe \Theta" (E. Freitag / R. Busam, "Funktionentheorie", Springer, 2000, S. 363)
  3. "Sei G ein Graph mit den Cliquen C_1, ... , C_m" (C. Beierle, "Methoden Wissensbasierter Systeme", Vieweg, 2003, S. 341)
  4. "Es sei K ein Körper, p(x) ein separables Polynom in K und E ein Zerfällungskörper für p(x)." (E. Artin, "Galoissche Theorie", Verlag Harry Deutsch, 1988, S. 67)

So, only in one out of four books the "es" at the beginning was used. Although this selection is by no means representative, it suggests a certain trend, which I also observed in all the mathematics classes i had to take in the past.

  • 3
    As a German who had some math lectures - both are correct. I assume that most people skip the "Es" just because it's shorter. Mathematicians are lazy. Jul 12 '11 at 8:43
  • As a German who had Math in school (a while back) I would tend to use the first version with "Sei" because it feels more natural to me. However, by no means can I remember what version was used way back when I was in school. Looking at the publishing dates of the books mentioned here and the one mentioned by @GeorgesElencwajg, the books using "Es sei" are older then the "Sei" books. So may be there is a trend from one to the other. May 14 '14 at 18:21
  • 1
    The date you give for Artin is misleading. That book was translated to German in 1959, which explains the antiquated language to some extent.
    – user2183
    Oct 20 '15 at 22:48
  • Nice selection of books! And I agree you can use both versions without problems; the statistics might be more difficult to ascertain. Oct 21 '15 at 7:31

I consider both of them equally correct and I don't feel that one of them is used more frequently than the other.


Both are correct, but "Es sei ..." is the old-fashioned way of saying it and "Sei ..." is the modern one. Fortunately, the Google Books corpus includes mathematics books, and fortunately it treats the simplest formulas as words. Since the n-gram viewer is capitalisation-sensitive, we can get a diagram comparing the relative frequencies of "Es sei ..." and "Sei ..." for the typical variable or function names x, y, f, g.

See here.

You can see that Sei ... has always been around, but that ever since the 1970s it has been more common than Es sei ... and that more recently it is in fact a lot more common.

I think that if one were to look closer one would also see that Sei ... and Es sei ... are used differently. Consider the following four equivalent ways of postulating continuity of a function f:

  1. Sei f stetig.
  2. Sei f eine stetige Funktion.
  3. Es sei f eine stetige Funktion.
  4. Es sei f stetig.

1 and 2 are standard nowadays. 3 sounds old-fashioned. 4 just sounds wrong. I think this is primarily because it combines modern colloquial brevity with the antiquated use of an unnecessary extra word. I guess that this combination, even if it should have occurred occasionally, can never have been very popular.

  • Why do you think that the construction is specific to mathematics? Do you see a difference between sei f stetig and f sei stetig? Do you think it is different from möge der bessere gewinnen?
    – Carsten S
    Oct 21 '15 at 12:29
  • Good point about Möge der Bessere gewinnen. I removed the half-sentence with that claim. I had looked for non-mathematical examples and couldn't think of any. But even here, Es möge der Bessere gewinnen doesn't really sound right any more.
    – user2183
    Oct 21 '15 at 12:55
  • Sei f stetig and f sei stetig differ only in stress. German usually puts the stressed part of speech first. Sei f stetig is the default word order, which stresses the verb. The word order of f sei stetig stresses the subject, as in f sei stetig und g sei differenzierbar.
    – user2183
    Oct 21 '15 at 12:59
  • Ich kann mir vorstellen, dass die Ähnlichkeit zum Imperativ die Stellung des Verbs am Anfang ermöglicht hat.
    – Carsten S
    Oct 21 '15 at 13:00
  • Verben konnten meines Wissens schon im Protogermanischen ganz vorne stehen, und ich glaube nicht, dass es dazwischen eine Zeit gab, wo das nicht ging. Aktuell ist Deutsch ja V2, was bedeutet, dass das Verb an der zweiten Stelle nach dem betonten Satzteil steht, es sei denn, das Verb ist selbst der betonte Teil und steht deshalb selbst ganz vorne. Das trifft regelmäßig nicht nur bei Befehlen zu, sondern auch bei Fragen ohne Fragewort.
    – user2183
    Oct 21 '15 at 21:25

Im ersten deutschen Buch dass ich aufschlage, Leutbechers Zahlentheorie, wird ausschliesslich "es sei" benutzt. Willkürliches Beispiel Seite 222: "Es sei L/K eine galoissche Zahlkörpererweiterung...".
Im Plural scheint mir, dass "Es seien" häufiger vorkommt als "seien": Leutbecher schreibt (Seite 233) "Es seien K_1,K_2 zwei endliche Erweiterungen des Zahlkörpers...".
Ich habe aber keine Statistik (hard numbers!) um diesen Eindruck zu bestätigen.


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