Sets and Classes

The history of the mathematical use of the English word "set" is complicated because papers and books written in the early 20C which were unquestionably about what English mathematicians and logicians now call "set theory", were not written in English. In translations since at least the 1960's, the word "set" (for the German Menge or Inbegriff) or "set theory" (Mengenlehre) is used. For example, the title of Zermelo's "Untersuchungen Uber die Grundlagen der Mengenlehre I" is translated as "Investigations in the foundations of set theory" by Heijenoort, in 1967.

But earlier translators used different words. For example, in P.E.B. Jourdain's translation of Cantor's "Contributions to the Founding of the Theory of Transfinite Numbers," published in 1915, he uses "set" to translate "Menge", but only on the first page of the preface. Thereafter he clearly prefers "aggregate". For example:

This volume contains a translation of the two very important memoirs of Georg Cantor on transfinite numbers which appeared in the Mathematische Annalen for 1895 and 1897* under the title: "Beitrage zur Begrundung der transfiniten Mengenlehre." It seems to me that, since these memoirs are chiefly occupied with the investigation of the various transfinite cardinal and ordinal numbers and not with investigations belonging to what is usually described as "the theory of aggregates" or "the theory of sets" (Mengenlehre, theorie des ensembles), - the elements of the sets being real or complex numbers which are imagined as geometrical "points" in space of one or more dimensions, - the title given to them in this translation is more suitable. Preface, p.v.

Let us now briefly consider again the meaning of the word "Mannichfaltigkeitslehre" which is usually translated as "theory of aggregates." In a note to the Grundlagen, Cantor remarked that he meant by this word "a doctrine embracing very much, which hitherto I have attempted to develop only in the special form of an arithmetical or geometrical theory of aggregates (Mengenlehre). By a manifold or aggregate I understand generally any multiplicity which can be thought of as one (jedes Viele, welches sich als Eines denken lasst), that is to say, any totality of definite elements which can be bound up into a whole by means of a law." Introduction, p.54.

By an "aggregate" (Menge) we are to understand any collection into a whole (Zusammenfassung zu einem Ganzen) M of definite and separate objects m of our intuition or our thought. These objects are called the "elements" of M. (First sentence of the translation, p.85 (Cantor's p.481).

The section titles of Principia Mathematica (first published 1910) nowhere use the word "set". For example, "Classes and Relations," "Products and Sums of Classes", "Unit Classes and Couples," "Sub-Classes, Sub-Relations, and Relative Types," &c. (However, see sec. 37 "… generally, given any set of objects such that, if we suppose the set to have a total, it will contain members which presuppose this total, then such as set cannot have a total. By saying that the set has ‘no total’, we mean, primarily, that no significantstatement can be made about ‘all its members'" In a "The Continuum and other types of serial order " by E.V. Huntington (1917), uses the word "class" as Russell uses it, but also the word "set", when particular numbers are in question. For example, let K be the class of all the natural numbers, but let K consist of two sets of integers (i.e. chosen from that class).

Wittgenstein apparently talks about "set theory" in the 1920's, but one forgets, because he is a philosopher who taught in England, and whose books are published in English, that he wrote only in German, and that the books are translations made after his death in the 1950's

Presumably somebody at some point made the decision to translate "Mengenlehre" as "set theory" and not "class theory". Nineteenth-century texts, such as by Mill and Boole, only use the word "class" when talking of what we now call sets, and this practice continued into early discussion of set theory in English. In Principles of Mathematics, written in the period 1900-1902, which was (according to Monk's biography) largely influenced by Cantor, Russell used the word "class" for, probably, whatever Cantor means by "Menge".

The word "class" also has a complicated history. It came to English in the 17C. Peirce says it acquired its logical signification at around the same time. Pierre Gassendi in (Institutio logica, 1658) apparently introduces the idea of a "class", however it is not clear what Latin word he used. Hobbes does not use the word "class" at all. There are five instances of the word in Book III of Locke's Essay on Human Understanding (1690). He makes it plain that a class is really nothing different from what the medieval notion of a genus or species or universal. These correspond to the English"kind" and "sort". He uses "class" to make it clear that being in a class is not essential to the objects that make up the class. A ("kind" of thing, by contrast, suggests some essential feature that belongs to each thing of that kind, and which makes it of that kind).

100 years later, Reid says, in his essay on the Intellectual Powers of Man, that we can divide infinitely things into "a limited number of classes (my emph.), which are called kinds and sorts; and, in the scholastic language, genera and species. Observing many individuals to agree in certain attributes, we refer them all to one class, and give a name to the class." (Essay V, "Of General Conceptions").

60 years later (1847) in The Mathematical Analysis of Logic, Boole says "that all reasoning ultimately depends on an application of the dictum of Aristotle, de omni et nullo. 'Whatever is predicated universally of a *class* of things, may be predicated in like manner of any thing comprehended in that class'". For the late traditional logicians like Mill and Whately, it seems interchangeable with "genus" or "species".

This is evidence that, whatever early modern writers meant by "class", they meant something similar to what they thought the medieval writers meant by a "kind" (genus) or "sort" (species) of thing. That, of course, doesn't imply they understood the medieval writers properly!

But none of the medieval Latin texts seem to use the word "classis" in this sense. "Class" is derived from Latin. The French root for "class" is "classe", which derives from "classis" in Latin. The original Indo-European root is "kel", which means "shout", from which the English word "call" derives. In ancient Greek there are two verbs "keleuo" which means to "order, urge, or entreat" and "kaleo", which means to "call"; these come from the same root. The Romans called forth their citizens into divisions for service in the legion, in a way presumably more forceful than gentle. Thus it came to mean a division or section of the people.

But it is not used by scholastic logicians in the sense in which it is used by logicians from the seventeenth century onwards (i.e. as a translation of the Latin words genus and species) Thus it is unlike other non-Latin words (like singular, definition, reductio) whose technical sense is acquired from the technical sense of Latin words.

John Aldrich (in this site maintained by Jeff Miller) says

CLASS (in set theory). In the early English literature the word class was used where set would be used today, e.g. in Bertrand Russell's Principles of Mathematics (1903). Russell was following Peano. Classe appears in his “Dizionario di Matematica” Revue de mathematiques, 7, (1900-1), 160-172 and is described as "idea primitiva." In the von Neumann set theory classes and sets are distinguished, so that every set is a class, but not conversely. The von Neumann publications begin with "Eine Axiomatisiereung der Mengenlehre" (1925) (translated in van Heijenoort (1967)) with later work in English by Bernays and Godel.

Bernays writes: "According to the leading idea of the von Neumann set theory we have to deal with two kinds of individuals, which we may distinguish as sets and classes." from "A System of Axiomatic Set Theory," 2, (1937) p. 66.

This is partly correct. Russell certainly uses '"class" where mathematicians would now use "set", also in the later Prinicipia. And Peano also uses "class" though of course (for some eccentric reason) he wrote in Latin, so used the word classis. But of course when the scholastic philosophers wrote about classes, they didn't use the Latin word classis at all. The mention of Von Neumann's paper supports the idea that as soon as the distinction between "set" and "proper class" was made in German or other languages, English had to follow.

The New Zealand logician Arthur Prior, writing in 1949 on the history of logic, refers only to classes, never to "sets". Even Quine, in the first edition of *Methods of Logic* (1950) refers only to "classes", and "class theory" . Here he is definitely talking about sets, referring to "class theories in the Zermelo line". Oddly though he mentions a paper of his own called "set-theoretic foundations for logic" (1936

"Menge", "Inbegriff"

Bolzano and Cantor used the words Menge and Inbegriff for what we now call a set. The word "Inbegriff" means, according to the Deutsches Worterbuch by Jacob and Wilhelm Grimm, vol. 10, col. 2103-4:

INBEGRIFF, m. das was innerhalb seiner etwas anderes begriffen, umschlossen hält, summa, complexio, comprehensio; die umschlossene sache steht im genitiv oder mit der praep. von vermittelt. das wort wird zuerst von STEINBACH verzeichnet (seine hauptquelle HEDERICH hat es nicht): inn begrieff, compendium, ein inn begrieff aller tugent, conspectus omnium virtutum. 1, 641; dann von FRISCH 1, 373a mit wesentlich weiterer bedeutung: inbegriff, area ambitu adstricta, locus in finibus suis contentus; item, quae in libro aliquo continentur; in letzterer beziehung braucht KANT der inbegriff eines buches. 8, 17, der sonst das wort im heute gewöhnlichen sinne öfters anwendet: der weite inbegriff einer vorzüglichen erkenntnis. 8, 10; die sphäre der ausgebildeten natur ist nur ein kleiner theil desjenigen inbegriffs, der den samen zukünftiger welten in sich hat. 324; die dauerhaftigkeit, die bei den anstalten der schöpfung an den groszen gliedern ihres inbegriffs angetroffen wird. 9, 3; bei andern: mit dem geliebten heimgehen ist der inbegriff aller seligkeit. BETTINE briefe 2, 107.

Jose Ferreiros translates it always by "collection". He says (per litt.) that "Inbegriff suggests two things. In normal language it is used for collecting, for reunions of things, e.g. an Inbegriff of straw (this comes from the verb begreiffen). This is the main reason why I translate collection; "epitome", "ideal" or "embodiment" seem to me very poor translations. However, I believe that for cultivated Germans in the 19th century it also suggests a connection with Begriff, which means "concept." Bolzano's Mengen are particular types of Inbegriffe, collections, but in general all collections are specified through a concept. The word Inbegriff was also used by Cantor and Dedekind (sometimes) and I think in most cases they have in mind something like the principle of comprehension".

The word "Menge" means "mass" or "crowd" or "aggregate" of things. Gabriela Muller writes

In its early usage die Menge describes a crowd of people (Menschenansammlung, Volkshaufen, Schar). This meaning remains the same in cases in which the emphasis is on the fact that the Menge in question is made up of individuals. In this sense of mass the word is the opposite of individual or little (wenig). At the same time the societal or state connections of a crowd come to the fore in some of the word's usages, dorfmenge (village crowd), being a case in point.

Die Menge has acquired a negative connotation only in new German and this valuation stems, according to the Grimms, "from the perception that such a closely connected social bond destroys independence and individual judgement." They liken this negative connotation to that of the word Haufe(n) (literally pile) and further compare it to Pöbel (rabble). The examples they cite here stem from Herder and Goethe among others, which suggests that die Menge began to assume negatively valences only in the eighteenth century. They write that die Menge was described from this point on as large, colorful, confused, incapable of judgement, betrayed, gaping, restless, and so on. In this sense die Menge is opposed to leaders, those higher up, and rulers. Nevertheless they still find the word used, simultaneously, in a less disparaging way.

The Dictum de Omni

This was a well-known medieval maxim (in full "Dictum de Omni et Nullo"), supposed to derive from Aristotle, An. Prior I.i.8. "We say that an attribute is predicated of All a subject, when there is no one of the parts of the subject, of which the attribute is not predicated; and similarly in the case, in which it is predicated of None". Roughly, whatever is affirmed (or denied) of any subject is affirmed (or denied) of everything that is "a part of" that subject. It is mentioned by as late a writer as John of St Thomas (1589-1644) " Quidquid universaliter dicitur de aliquo subjecto, dicitur de omni quod sub tali subjecto continetur: quidquid negatur de aliquo subjecto, negatur et de omni contento sub tali subjecto. (Logica. Pars I., Lib.3, c.x). (See also Aquinas An.Post. I., lect 9. Scotus, Sup. Lib. I Priorum, Q.7.)

Ockham and Buridan say "Dici de omni". Lukaziewicz quotes it as "quidquid de omnibus valet, valet etiam de quibusdam et de singulis" and adds testily that "this obscure principle" is not found in Aristotle, since Aristotle had no theory of singular propositions (the "Socrates is mortal syllogism is not his). The An. Prior I passage (he says) "is only an explanation of the words 'to be predicated of all/none'"

What does it mean? The difficulty is understanding what "a part of" the subject means. It is important that the "subject", for example "all men" is NOT regarded as a collection or aggregate of individuals. Then the syllogism reduces to a tautology: "all men are mortal" simply says of Socrates, Aristotle, Plato, that they are men, and so directly implies "Socrates is mortal" without the intervening premiss "Socrates is a man". (In fact, there's a dispute that was famous in the nineteenth century, now probably forgotten, as to whether the universal syllogism does involves a petitio principii see Mill, System of Logic, II, II §2).

So you could say the medieval logicians did have the concept of a class, and that the dictum de omni expresses this. The subject of a universal proposition is the species or genus or "class" as such, the logical whole, it is not just a collection or aggregate of things, and the proposition is not a mere enumerative judgement. Hence you could argue that the idea of a set, as a class or a species was very familiar to them, indeed was something essential to the whole of medieval logic and metaphysics.