Abstract

This book on mathematics, logic, and experience by the doyen of the Vienna Circle is an almost completely revised reprint of the volume originally published in 1957. The book is organized into four chapters, the first of which treats "mathematics and logic" and is one page in length. The other three chapters, "Logic and Experience," "Mathematics and Experience," and "Conventionalism and Its Refutation," share equally the remaining pages of the book. The validity of the principle of logic, Kraft argues, is the indispensable condition for order and knowledge. Consequently, logic cannot be an abstraction from the natural laws and cannot contain the formalized laws of reality. Rather logic has to be--and this is in accordance with Port-Royal and Wundt--a normative science. Kraft reminds us that only propositions which occur in predicate logic or class logic may have a truth value. Kraft therefore characterizes the principle of identity and the principle of contradiction as rules which are prelogical in nature, since they are the basis of order and serve as the norms of correct thinking. The validity of these prelogical rules rests on the fact that they constitute order and must consequently be accepted as necessary. The conception of logic as normative science puts the author in opposition to the view held by the Vienna Circle, namely, that the application of logic does not contain any problem. In Kraft's position, on the contrary, the necessity arises of showing the conditions under which logic is applicable. In respect of the applicability of logic, the nature of the reality serving as material in this application makes a difference; for the logical relation between the universal and the particular finds its fulfillment and its determination with regard to the content provided by reality. The condition which logic requires for its application is that the "manifold" be capable of being ordered and that it thereby make universality possible. In a similar way Kraft attempts to work out the conditions for the applicability of arithmetic and geometry, which like logic are independent of experience in respect to their validity. According to the author, arithmetic can be applied insofar as operative relations between numbers correspond to classes of sets. Geometry is applicable insofar as the coordination of certain empirical phenomena with geometrical elements and relations is possible. The applicability of geometry consequently presupposes the validity of certain natural laws, thereby denying the thesis of conventionalism that these can be arbitrarily chosen. Kraft argues convincingly in the third part of his book for his own thesis by showing the shortcomings of Dingler's procedure of exhaustion.--H. H.