This enlightening survey of mathematical concept formation holds a natural appeal to philosophically minded readers, and no formal training in mathematics is necessary to appreciate its clear exposition. Contents include examinations of arithmetic and geometry; the rigorous construction of the theory of integers; the rational numbers and their foundation in arithmetic; and the rigorous construction of elementary arithmetic. Advanced topics encompass the principle of complete induction; the limit and point of accumulation; operating with sequences and differential quotient; remarkable curves; real numbers and ultrareal numbers; and complex and hypercomplex numbers.

Friedrich Waismann was born in Vienna in 1896 and lived there until the time of the Anschluss in 1938. From then until his death in 1959 he lived in England; this, apart from a brief period at Cambridge early on, was almost wholly at Oxford, \,Vhere he held the posts, first, or reader in the philosophy of mathematics and then of reader in the philosophy of science. He was of Jewish descent -his father being Russian, his mother German. He studied mathematics and physics at the University of Vienna and attended the lec­ tures of Hahn. Beginning his career as a teacher of mathematics he soon be­ came an unofficial assistant to Moritz Schlick. It was Schlick's concern to see that the new philosophical ideas developed by Wittgenstein from the time of his return to philosophy in the later 1920s were made public that de­ termined the subsequent shape of Waismann's activities. Until the out­ break of the war in 1939 his main task was the preparation of a book in which Wittgenstein's thought was to be systematically expounded. Be­ tween 1927 and 1935 this project was carried on in close personal conjunc­ tion with Wittgenstein. A first version of the planned book, Logik. Sprache. Philosophie seems to have been completed by 1931. A very differ­ ent version came to England with Waismann in 1938. It finally appeared, in an English translation, as Principles of Linguistic Philosophy.