Synthetic activity and the continuum : a criticism of Russell's account of extensive magnitude

Abstract

This essay deals with the concept of continuity, as it has been developed both in philosophy and in mathematics during the 19th and 20th Centuries. In particular, the problem of focus is the relation which the continuum has to number. The debate on whether the continuum can be given as a class of atomic individuals is the principle item of consideration in this work. The negation of this claim is argued for. -- The two sides of this debate are presented in terms of the philosophical characterizations of extensive magnitude found in the writings of Bertrand Russell (representing the claim that such a reduction is possible) and of Immanuel Kant (representing the negation of this claim). In particular, the epistemological distinction between the two figures is connected with their relative positions on this debate, discussed mainly in connexion with the issue of synthetic a priori judgments. -- The principal claim argued for in this paper is that the classical analysis of the geometric continuum, and hence Russell's logical reduction of space and time, tacitly presupposes an original undifferentiated continuum among its initial principles. This point is intended to lend support to the more general view of the continuum holding the undifferentiated whole to be utterly prior over its parts. In addition to Kant, one should attach to this view the names of Peirce and Brouwer. In particular, I shall attempt to establish an understanding of the 'spatial point' as an entity which can be individuated only as the result of a synthetic act. Finally, an examination of the relation of intuitionist choice sequences with the classical set of real numbers is presented, concluding with the conjecture that no law-like system can exhaust all possible positions on the line.