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Consider that the question does not concern the origin of the ideas of equivalence relation and equivalence class. It exactly concerns the origin of the terms "equivalence relation" and "equivalence class". It seems that the terms weren't in use at least until 1903 where Russell writes:

Peano has defined a process which he calls definition by abstraction, of which, as he shows, frequent use is made in Mathematics. This process is as follows: when there is any relation which is transitive, symmetrical and (within its field) reflexive, then, if this relation holds between u and v, we define a new entity Ø (u), which is to be identical with Ø (v).

Relations which possess these properties are an important kind, and it is worthwhile to note that similarity is one of this kind of relations.

UPDATE: Thanks to the suggestions given in the comments and a very informative answer of Francois Ziegler, suddenly the following piece from Russell's Introduction to Mathematical Philosophy came into a new light:

The question “what is a number?” is one which has been often asked, but has only been correctly answered in our own time. The answer was given by Frege in 1884, in his Grundlagen der Arithmetik. Although his book is quite short, not difficult, and of the very highest importance, it attracted almost no attention, and the definition of number which it contains remained practically unknown until it was rediscovered by the present author in 1901.

So, perhaps that was Frege himself who used the terms! Although, I couldn't find anything in Frege's writings yet. But now, considering Eugen Netto's paper (see Francois' update below) a very more important question now is: Is he indeed Russell who should be credited with rediscovering Frege in 1901?!

PS. I am well aware that asking a new question inside another question is not a good idea. However, up until this post I had the feeling that apart from the origin of the terms I know "everything" about the history of the notions of equivalence relation and equivalence class (part of which has been published here). But, this new information caught me by surprise, and I could not help myself to add the new question. You may just ignore it.

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My guess (without any checking): Bourbaki I –  Peter Michor Jun 30 '13 at 17:37
@PeterMichor He (!) indeed defined "equivalence relation" and "equivalence class" in "Theory of sets". But, there is a big time gap between what I quoted from Russell and the Bourbaki's book. It is very strange that such an important notion remains unnamed for such a long time. –  Amir Asghari Jun 30 '13 at 18:27
It certainly predates Bourbaki. Birkhoff was using the term "equivalence relation" in 1934 (abstract number 323 in ams.org/journals/bull/1934-40-11/S0002-9904-1934-05986-X/…), and in a 1935 paper he mentions the 1927 book Höhere Algebra by Hasse (see the footnote on page 446 of math.hawaii.edu/~ralph/Classes/619/birkhoff1935.pdf). Hasse uses the term Äquivalenzrelation also in a 1927 paper (Crelle, volume 157, page 120). I assume this goes further back, but I don't know how far. –  Henry Cohn Jun 30 '13 at 18:33
@HenryCohn That's great. The time gap is now only 24 years. –  Amir Asghari Jun 30 '13 at 18:50
@FrancoisZiegler I am so sorry to misspelled your name. I wrote it in a piece of paper and then typed it. Lesson: copy paste works better :) –  Amir Asghari Jul 6 '13 at 15:26
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3 Answers

up vote 15 down vote accepted

Von Neumann uses "equivalence class" in Zur Prüferschen Theorie der idealen Zahlen, Acta Sci. Math. (Szeged) 2 (1926) 193-227, p. 197 (viewable after free registration):

Wir nennen $R$ und $S$ äquivalent, in Zeichen: $R\sim S$, wenn (...)

Satz 2. Es ist stets $R\sim R$. Aus $R\sim S$ folgt $S\sim R$. Aus $R\sim S$ und $S\sim T$ folgt $R\sim T$.


Infolge des Satzes 2. zerfällt die Menge der Folgen realer Zahlen in paarweise elementefremde Klassen untereinander äquivalenter Folgen.


Definition 4. Eine Aequivalenzklasse, die lauter Fundamentalfolgen enthält, nennen wir eine ideale Zahl.

This seems like an early example, in that he sees fit to add here the footnote: "We define the ideal number as the corresponding set of fundamental sequences, itself; naturally one could also regard it as an ideal element attached to this set."

I note also that Hasse's book Höhere Algebra (where Henry Cohn found the earliest occurrence of "equivalence relation" so far) appears to have a 1926 edition too.

Update 1: One might also quote a paper by Eugen Netto, Über die arithmetisch-algebraischen Tendenzen Leopold Kronecker's, in: Mathematical Papers read at the International Mathematical Congress (Chicago, 1893), Macmillan 1896, pp. 243-252, who writes:

"Jede wissenschaftliche Forschung geht darauf aus, Aequivalenzen festzustellen und deren Invarianten zu ermitteln (...)."

Jede Abstraction, z. B. die von gewissen Verschiedenheiten, welche eine Anzahl von Objecten darbietet, statuirt eine Aequivalenz; alle Objecte, die einander bis auf jene Verschiedenheiten gleichen, gehören zu einer Aequivalenzclasse, sind unter einander aequivalent, und der aus der Abstraction hervorgehende Begriff bildet die "Invariante der Aequivalenz."

Update 2: Finally(?) one should probably also quote Vol. 2 of Weber's Lehrbuch der Algebra (1896). Literally he uses neither "equivalence relation" nor "equivalence class", but see how close he gets in §152 "Aequivalenz":

  1. Zwei ganze oder gebrochene Functionale $\varphi$, $\psi$ im Körper $\Omega$ heissen äquivalent, wenn (...)

  2. Zwei Functionale, die mit einem dritten äquivalent sind, sind auch unter einander äquivalent.

Theilt man hiernach alle Functionale des Körpers $\Omega$ in Classen ein, indem man zwei Functionale in dieselbe oder in verschiedene Classen wirft, je nachdem sie äquivalent sind oder nicht, so ergiebt sich (...)

Same for Dedekind's Ueber die Theorie der algebraischen Zahlen (1879), §175:

Wir wollen nun zwei Ideale $\mathfrak a$, $\mathfrak a'$ äquivalent nennen, wenn (...)

Zugleich ergiebt sich hieraus, dass (...) je zwei Ideale $\mathfrak a'$, $\mathfrak a''$, die mit einem dritten Ideal $\mathfrak a$ äquivalent sind, stets auch miteinander äquivalent sein müssen. Auf diesem Satze beruht die Möglichkeit, alle Ideale in Idealclassen einzutheilen; (...) der Inbegriff $A$ aller mit $\mathfrak a$ äquivalenten Ideale $\mathfrak a$, $\mathfrak a'$, $\mathfrak a''$ (...) nennen wir eine Idealclasse oder kürzer eine Classe

(Of course, calling class a family of objects related by some equivalence is a custom that can be traced to much older work — cf. Dirichlet [1863, p. 172] or Eisenstein [1847, p. 118] or Gauss [1801, §223].)

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How did you find that? Once upon a time, before the age of MO, I did a two years search and I failed. Shame on me :) –  Amir Asghari Jun 30 '13 at 21:01
google.com/search?tbm=bks&q=aequivalenzklasse -- Shame on me ;) –  Francois Ziegler Jun 30 '13 at 21:18
I wait a few days to see if there is an earlier date. If not, I "accept" your answer. Considering that you reduced the time suggested by Henry Cohn in a few minutes, I am sure a few days would be fair to history. –  Amir Asghari Jun 30 '13 at 21:23
J. Hurwitz and J. König use "Äquivalenzbeziehung" in papers from 1902 and 1905 (google.com/…) but I'm not certain the meaning is quite the same as you want. –  Francois Ziegler Jun 30 '13 at 22:20
@Amir The term is "Äquivalenzbeziehung" and used in the 1905 paper "Zum Kontinuum-Problem" in the Mathematische Annalen. There are no equivalence classes and he only uses the term for "equal in cardinality". –  Michael Greinecker Jul 1 '13 at 20:00
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Re: the two questions in your UPDATE, I'd like to recommend the 2005 paper Frege's Natural Numbers: Motivations and Modifications by Erich Reck:

  • Regarding other people's awareness of Frege's definition of number, Reck involves — as if by coincidence! — two of our previous protagonists, p. 276:

    the Frege-Russell conception of numbers (...) seems to have been in the air already before Russell's writings. For example, the mathematician Heinrich Weber proposes essentially the same conception, independently of both Frege and Russell, in an 1888 letter to Richard Dedekind (...)

  • Regarding Frege's awareness of "equivalence relations", he raises (but leaves open) precisely that question in endnotes 5 and 41:

    It would be interesting to know to what degree, and in what form exactly, Frege was also aware of equivalence class constructions in the algebra of his time (...) I am not aware of any historical account of the use of such methods in nineteenth-century mathematics, or even earlier; thus I am not sure how safe it is, in the end, to assume that Frege knew about them.

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Your new answer encouraged me to add a follow up as an answer. MO, I am sorry, it wasn't fit as a comment :) –  Amir Asghari Jul 3 '13 at 12:51
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This is to add something to Francois' answer to the UPDATE. This is from "Frege : philosophy of mathematics" by Dummett (1991), p. 50:

One of the mental operations most frequently credited with creative powers was that of abstracting from particular features of some object or system of objects, that is, ceasing to take any account of them. It was virtually an orthodoxy, subscribed to by many philosophers and mathematicians…, that the mind could, by this means, create an object or system of objects lacking the features abstracted from, but not possessing any others in their place. It was to this operation that Dedekind appealed in order to explain what the natural numbers are. …Frege devoted a lengthy section of Grundlagen, §§29-44, to a detailed and conclusive critique of this misbegotten theory; it was a bitter disappointment to him that it had not the slightest effect.

I had never a chance to read Frege's Grundlagen. Worse, at the moment, I don't know how Reck, Dummett and Russell views about Grundlagen may come together!

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I think the critique/controversy here is between those (like Frege, Russell, or Weber in his letter) who would take the newly defined objects to be the equivalence classes themselves, and those (like Kummer, Peano, or allegedly Dedekind) who'd rather have them be "corresponding abstract entities", somehow created by the mind out of thin air. But my quote of Dedekind 1879, or his 1888 answer to Weber, show that he could also do without such extra entities (just like von Neumann in his 1926 footnote). –  Francois Ziegler Jul 3 '13 at 18:20
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