Page:The World and the Individual, First Series (1899).djvu/529

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SUPPLEMENTARY ESSAY

end at any given point, and by then saying, “A series taken thus as without end, may be called infinite.” We ourselves, so far in this discussion, have defined our infinite processes on the whole in a negative way. But the new definition of the infinity of our system uses positive rather than negative terms. The conception of a representation or of an imaging of one object by another, is wholly positive. This conception, if applied to the elements of a system A, with the proviso that A’, the image or the representation of A, shall form a constituent portion of A itself, remains still positive. But the system A, if defined as capable of this particular type of self-representation, proves, when examined, to contain, if it exists at all, an infinite number of elements. Whatever the metaphysical fate of the ideal object thus defined, the method of definition has a decided advantage over the older ones.^[1] It may be well at once to quote Dedekind’s original statement and illustration of the conception in question, in the passage cited in the note: —

“A System S is called ‘infinite’ when it is similar^[2] to a

↑ More or less vaguely this positive property of infinite multitudes was observed as a paradox whenever the necessity of conceiving “one infinite as greater than another,” or as containing another as a part of itself, was recognized. The paradox was in this sense felt already by Aristotle in the third Book of the Physics, ch. 5 (cf. Spinoza’s Ethics, Part I, Prop. XV. Scholium, where the well-known solution is that the true infinite is essentially indivisible, having no parts and no multitude). Explicitly the property of infinite multitudes here in question was insisted upon by Bolzano in his Paradoxien des Unendlichen (1851). Cantor, and, in America, Mr. Charles Peirce, have since made this aspect of the infinite multitudes prominent. Most explicitly, however, Dedekind has built up his entire theory of the number concept upon defining the infinite multitude or system simply in these positive terms, without previous definition of any numbers at all. See his op. cit., §5, 64, p. 17.
↑ In previous definitions, in Dedekind’s text, two systems have been defined as similar (ähnlich), when one of them can be made to correspond, element for element, with the other, any two different elements having different representations. And a proper part (echter Theil), or constituent portion, of a system, has been defined as one produced by leaving out some elements of the whole.

[1] More or less vaguely this positive property of infinite multitudes was observed as a paradox whenever the necessity of conceiving “one infinite as greater than another,” or as containing another as a part of itself, was recognized. The paradox was in this sense felt already by Aristotle in the third Book of the Physics, ch. 5 (cf. Spinoza’s Ethics, Part I, Prop. XV. Scholium, where the well-known solution is that the true infinite is essentially indivisible, having no parts and no multitude). Explicitly the property of infinite multitudes here in question was insisted upon by Bolzano in his Paradoxien des Unendlichen (1851). Cantor, and, in America, Mr. Charles Peirce, have since made this aspect of the infinite multitudes prominent. Most explicitly, however, Dedekind has built up his entire theory of the number concept upon defining the infinite multitude or system simply in these positive terms, without previous definition of any numbers at all. See his op. cit., §5, 64, p. 17.

[2] In previous definitions, in Dedekind’s text, two systems have been defined as similar (ähnlich), when one of them can be made to correspond, element for element, with the other, any two different elements having different representations. And a proper part (echter Theil), or constituent portion, of a system, has been defined as one produced by leaving out some elements of the whole.

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