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11 [‘Collection’ in The Century Dictionary] [Whitney 1889; Ms 1597; Peirce 1888(?)–1914(?); Whitney 1909] The original definition of ‘collection’ in The Century Dictionary was written, not by Peirce, but by an anonymous contributor; it is reproduced here for reference. Peirce’s critical notes on that definition, written in his own copy of the dictionary , date from some time around the turn of the century; they are followed here by his definitions of two additional senses of ‘collection’ for the 1909 Supplement to the dictionary. The notes open with a definition: A collection (or plural) is an individual object whose existence consists in the existence of whatever individuals there may exist having one character, these being called the members of the collection. This is a dry run for, but differs importantly from, the definition of the second new sense Peirce contributed to the Supplement: A plural object; an individual object whose existence consists in the existence of whatever individuals may have been mentally connected and regarded as parts of it. The latter is more mentalistic than the former; the approach Peirce takes in the notes, which makes collections ontologically dependent on their defining characters, is more typical of the selections that follow this one. Since the Supplement postdates those, the second new sense may betoken a major doctrinal change, or else may just be a “sop to Cerberus” forced upon Peirce by the space constraints of a dictionary entry. In the notes Peirce gives extensional identity criteria for collections, thereby differentiating them from such intensional entities as characters. (Note that for Peirce a set is an ordered collection, by marked contrast with what has since become standard usage.) In his notes Peirce recognizes the empty collection as a collection; he also distinguishes a collection with one member from that member, a collection from its defining attribute, and a collection of collections from the union of its members. All of these ideas are now basic to the conceptual equipment of set theory. Beings of Reason.book Page 85 Wednesday, June 2, 2010 6:06 PM 86 | Philosophy of Mathematics A collection in the first new sense is its members regarded as an individual object, whereas in the second sense a collection is an individual object. The first new sense is very close to one of Cantor’s definitions of set (“[a] multiplicity which can be thought of as one” (Cantor 1883, 916)); Peirce may have separated the senses in order to distinguish his view from Cantor’s. According to both the annotations and the entry, a collection’s existence consists in that of its members. Yet though sameness of members is both necessary and sufficient for sameness of collections, Peirce adds in both the notes and the entry that every collection has a defining character (an essence), which is intensional. In the dictionary entry Peirce tries to sort out the relationship between existence and essence for collections.1 Logicians think of their collections as comprising distinct individuals, whose properties may change without changing the identity of the collections. Mathematicians , on the other hand, deal with collections of hypothetical objects whose identities are more sensitive to changes of property. As a result, essence and existence are not as distinct for a mathematician’s collections as they are for a logician’s. The selections that follow take up these ontological problems at greater length; it is left to the reader to determine whether this entry, which as just noted is later than them all, shows that Peirce’s metaphysics of collections had reached a reflective equilibrium or was still on the way to one. 1. ‘COLLECTION’ FROM THE CENTURY DICTIONARY 2. An assemblage or gathering of objects; a number of things collected, gathered, or brought together; a number of objects considered as constituting one whole of which the single objects are parts: as, a collection of pictures; a collection of essays; a collection of minerals. 2. PEIRCE’S CRITICAL ANNOTATIONS collection 2. Not a very successful definition. A collection (or plural) is an individual object whose existence consists in the existence of whatever individuals there may exist having one character, these being called the members of the collection. It is commonly limited to the case in which more than one individual exist[s] having that character. It is, however, desirable to consider a collection which happens to contain one individual as different from that individual itself. Moreover, nothing is that sole collection of which Beings of Reason.book Page 86 Wednesday, June 2...

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