Philosophy

Abstract

THIS is a very useful introduction to mathematical logic, which does full justice to its title, in so far as it really carries the reader into the principles of mathematics interpreted mainly in terms of logical notions. Most elementary works on this subject make a mountain of a molehill by confining themselves to the exposition of its purely logical prolegomena and dismissing their application to mathematics with the excuse that they are too difficult for lay readers. Dr. Tarski rightly thinks otherwise. After a restricted account of the leading doctrines of symbolic logic (pp. 1-97), he gives a short exposition of the idea of number, of the simple arithmetical operations, followed by a short discussion of the methodology and the axiom-system of arithmetic. This is an illuminating and sufficient introduction to the analysis of the logical foundations of elementary arithmetic. But once the purpose and technique of the method are thoroughly understood, it will be easier for the reader to tackle the more advanced works on the subject. Dr. Tarski helps him in this connexion by suggesting a number of easy examples after each chapter. Controversial questions are carefully avoided, and the technical apparatus is reduced to a minimum, though one may have wished for practice' sake more symbolic formulae and proofs. Yet, as it is, Dr. Tarski's book, with the authority of its author, who is a leading member of the Polish school of mathematical logicians, fulfils a real want and is well worthy of an English translation.

Einführung in die mathematische Logik:

und in die Methodologie der Mathematik. Von Alfred Tarski. Pp. x + 166. (Wien und Berlin: Julius Springer, 1937.) 7.50 gold marks.