J:
"I think it's only philosophers, and a few mathematicians who would say there are classes or sets with only one member."
Well, there's nothing in the word 'set' (as per word) that implicates, implies, entails ... plurality. Surely a hen can set (is that the word) just ONE egg.
"The table is set for ONE".
I wouldn't know why the English think of 'set' (which is such a bit of an emtpy word) to refer to what the Germans call 'menge', as I think they do, as per 'set theory'. The French may be even more charismatic.
"class" may be a more pedantic Latinate choice. I think Carroll used it, and I can be pedantic at times.
But after studying the 'classics', I found out that Cicero and his ilk were never sure as to why the 'classics' had to be 'classics'. Apparently, it's from 'class'. But then implicated to mean 'FIRST' class.
I rather feel nominalistic ANYday!
---
Or not.
Subscribe to:
Post Comments (Atom)
Posts
In set theory one now speaks of both sets and classes which have different meanings.
ReplyDeleteThe early development of set theory, from Cantor through to Von Neumann seems to be dominated by German publications, in which the term "mengenlehre" is used.
English translations of these works use the term "Set Theory" for mengenlehre.
The distinction between set and class originates with a distinction made by Cantor between consistent and inconsistent multiplicities, but does not seem to appear in axiomatic set theory until Von Neuman in 1925 published an axiomatic set theory in which a similar distinction appears.
Ontologically this is a strange set theory, since the entities involved are functions rather than sets. Nevertheless the paper, in german, again uses the term mengenlehre in the title.
It may however be the source of the migration of mathematical logicians from talk of classes (which is the term used by Russell) to sets (with classes as collections which do not qualify as sets).
Von Neumann's introduction into axiomatic set theory of Cantor's distinction between consistent and inconsistent multiplicities leads eventually to the NBG (Neumann, Bernays, Goedel) set theory in which the ontology involves two kinds of multiplicity called classes and sets. The sets are the ones which correspond to the multiplicities in theories which do not make the distinction (they lack the distinction because their ontologies lack the entities which are known in NBG as classes).
It is possibly because of the need to make this distinction that we use two terms in English for the German menge, and the (possibly arbitrary) choice to use "class" for the "inconsistent" multiplicities (which are rendered consistent in the formal theories in which both kinds occur), seems to have forced a migration of terminology in the English speaking world from talk of classes to talk of sets (bearing in mind that the set theories in which the distinction appears, and in which there are classes in this new sense, have largely disappeared from active use, so far as I am aware). The term class, once use as menge, was appropriated for use specifically for a kind of collection which we don't really need, and mengenlehre, principally the study of the ones we really do need, became set theory.
How this played out in German I have no idea.
Presumably German set theorists do now have to make the same distinction, at least when they speak of systems like NBG, but how they do it I have no idea.
Possibly by talking of sets and classes!
This seems to me likely a case where prior usage is of marginal influence. If one had considered prior usage of the words "set" and "class" one might come up with grounds for choosing which to use for (let us say) consistent or inconsistent multiplicities (or grounds perhaps connected with the fact that classes are larger than sets) but it is doubtful that any such exercise was ever undertaken or that mathematicians ever go into such matters when choosing their terms.
(another example one might speculate about is the choice of the term "category theory", which actually seems to me quite appropriate).
Apparently the word set used as a noun for some kind of collection goes back to the 15C (and possibly came from the French "sette").
Roger Jones
Beautiful!
ReplyDeleteCantor--that's the real culprit. Wittgenstein-- not my fave guru, but a somewhat bright mensch-- considered Cantor's collection of infinities fairly ridiculous. Quine for one mostly agrees, and consistently objected to the shall we say the reification of set theory, not to say the platonic aspects.
ReplyDeleteThanks to J and Jones for the commentary. I would agree with J that the blame is on Cantor -- but then I can blame MORE than one!
ReplyDeleteJones: "This ['class' vs. 'set' vs. neutral 'Menge'] seems to me likely a case where prior usage is of marginal influence."
Indeed. But I enjoyed Jones's suggestion that a class is larger than a set, and that we don't really them (classes).
Etc.