MATHEMATICISM

the effort to employ the formal structure and rigorous method of mathematics as a model for the conduct of philosophy. Mathematicism is manifested in Western philosophy in at least three ways: (1) General mathematical methods of investigation can be used to establish consistency of meaning and completeness of analysis. This is the revolutionary approach introduced in the first half of the 17th century by Ren Descartes. The perfection of this approach led to the Age of Analysis in the first half of the 20th century. (2) Descartes also pioneered the subjection of metaphysical systems, expressing the nature of ultimate reality, to axiomatizationi.e., to a procedure that deduces tenets from a set of basic axioms, on the model of Euclid's axiomatization of geometry. The method was elaborately used later in the 17th century by Benedict de Spinoza. (3) Calculi, or syntactic systems, on the model of mathematical logic, have been developed by several 20th-century analytic philosophers, among them Bertrand Russell, Ludwig Wittgenstein, and Rudolf Carnap, to represent and to explicate philosophical systems, as well as to solve and to dissolve metaphysical problems. Descartes gave four rules of method in philosophy based on mathematical procedure: (1) accept as true only indubitable (self-evident) propositions, (2) divide problems into parts, (3) work in order from simple to complex, and (4) make enumerations and reviews complete and general. When a philosopher approaches metaphysical problems in this way, it may appear to be natural or useful for him to organize his philosophical knowledge in the form of definitions, axioms, rules, and deduced theorems. In this way he can assure consistency of meaning, correctness of inference, and a systematic way to discover and to exhibit relationships.