Multitude. We shall here see how there are cases, — and
cases, too, of the most fundamental importance for the Theory
of Being, where a single purpose, definable as One, demands
for its realization a multitude of particulars which could not
be a limited multitude without involving the direct defeat of
the purpose itself. We shall in vain endeavor to escape from
the consequences of this discovery by denouncing the purposes
of the type in question as self-contradictory, or the Infinite in
question as das Schlecht-Unendliche. On the contrary, we
shall find these purposes to be the only ones in terms of which
we can define any of the fundamental interests of man in the
universe, and the only ones whose expression enables us to
explain how unity and diversity are harmonized at all, or how
Being gets its individuality and finality, or how anything
whatever exists. Having made this clear, we shall endeavor
to show, positively, that the concept of infinite variety in
unity, to which these cases lead us, is consistent in itself, and
is able to give our Theory of Being true definition.
Section III. Theory of the Sources and Consequences of any Recurrent Operation of Thought. The Nature of Self-Representative Systems
I shall begin the present section with illustrations. I shall make no preliminary assumption as to how our illustrations are related to the ultimate nature of things. For all that we at first know, we may be dealing, each time, with deceptive Appearance. We merely wish to illustrate, however, how a single purpose may be so defined, for thought, as to demand, for its full expression, an infinite multitude of cases, so that the alternative is, “Either this purpose fails to get expression, or the system of idealized facts in which it is expressed contains an infinite variety.” Whether or no the concept of such infinite variety is itself self-contradictory, remains to be considered later.[1]
- ↑ The discussion of the instances and conceptions of Multitude and Infinity, contained in what follows, is largely dependent upon various recent contributions to the literature of the subject. Prominent among the later authors who have dealt with our problem from the mathematical side, is George Cantor. For his now famous theory of the Mächtigkeiten or grades of infinite multitude, and for his discussions of the purely mathematical aspects of his problem, one may consult his earlier papers, as collected in the Acta Mathematica, Vol. II. With this theory of the Mächtigkeiten I shall have no space to deal in this paper, but it is of great importance for forming the conception of the determinate Infinite. Upon the more philosophical aspects of the same researches, Cantor wrote a brief series of difficult and fragmentary, but fascinating discussions in the Zeitschrift für Philosophic und Philosophische Kritik: Bd. 88, p. 224; Bd. 91, p. 81; Bd. 92, p. 240. In recent years (1895-97) Cantor has begun a systematic restatement of his mathematical theories in the Mathematische Annalen: Bd. 48, p. 48; Bd. 49, p. 207. Some of Cantor’s results are now the common property of the later text-books, such as Dini’s Theory of Functions, and Weber’s Algebra. Upon Cantor’s investigations is also based the remarkable and too much neglected posthumous philosophical essay of Benno Kerry: System einer Theorie der Grenzbegriffe (Leipzig, 1890) — a fragment, but full of ingenious observations. The general results of Cantor are summarized in a supplementary note to Couturat’s L’Infini Mathematique (Paris, 1896), on pp. 602-655 of that work. Couturat’s is itself the most important recent general treatment of the philosophical problem of the Infinite; and the Third Book of his Second Part (p. 441, sqq.) ought to be carefully pondered by all who wish fairly to estimate the “contradictions” usually attributed to the concept of the Infinite Multitude. A further exposition of Cantor’s most definite results is given, in a highly attractive form, by Borel, Leçons sur la Théorie des Fonctions, Paris, 1898. Side by side with Cantor, in the analysis of the fundamental problem regarding number, and multitude, stands Dedekind, upon whose now famous essay, Was Sind und Was Sollen die Zahlen? (2te Auflage, Braunschweig, 1893), some of the most important of the recent discussions of the nature of self-representative systems are founded. See also the valuable discussion of the iterative processes of thought by G. F. Lipps, in Wundt’s Studien (Bd. XIV, Hft. 2, for 1898); and the extremely significant remarks of Poincaré on the nature of mathematical reasoning in the Revue de Metaphysique et de Morale for 1894, p. 370. Other references are given later in this discussion.
