Chapter 1 - INTRODUCTION

Abstract

This chapter discusses the general characterization of mathematical logic, dialectic of the relation between mathematical and metamathematical aspects, metamathematico-mathematical parallelism and its natural limits, principal mathematical tools of mathematical logic, constructivism in metamathematics, and philosophy related to mathematical logic. Mathematical logic becomes an exact science only when use is made of the method of formalization, or formal codification, of the mathematical language, that is, when an exact procedure for registration and analysis of the formal aspects of the relation of consequence is applied. By this, the characterization of that aspect of the relation of consequence can be understood which is immediately accessible to mathematical analysis. If the verbal mathematical language is replaced by an artificial language of symbolic abbreviations, in some systematic and precise manner, then the relation of consequence is carried over to a strictly combinatorial relation between the symbolized assumptions and the symbolized assertion of a mathematical statement. This relation neither reflects in any sense whatsoever the mathematical and logical meaning of the individual signs nor of the symbols composed of these signs; it is a relation defined only by the rules of composition of the individual signs. The procedure of formalization is inseparably bound up with the converse procedure of interpretation, that is, the materially intuitive exposition of the symbols obtained previously in a strictly formal combinatorial manner. Only the balanced unity of both procedures—formalization and interpretation—may yield useful results.