Mathematics and Logic in History and Contemporary Thought (review)

98 HISTORY OF PHILOSOPHY freedom, moral community, and obligation. The book is valuable for its critical analysis, its historical depth, and its relevance to contemporary social and political philosophy. JOE EDWARDBARNHART University o] Redlands Mathematics and Logic in History and Contemporary Thought. By Ettore Carruccio. Translated by Isabel Quigly. (Chicago: Aldine Publishing Company, 1964. Pp. 398.) Professor Carruccio aims in this book to trace the history of both mathematics and logic in relation to its cultural setting. This is, of course, a very ambitious task, and I fear that the book does not accomplish it--no book of this size could. Rather, what we are given are glimpses of mathematicians, and occasionally logicians, working at different periods. We get a sense of the continuity of mathematics from the Greeks to the Renaissance; the thought of pre-Platonic mathematicians is linked to the rise of philosophy; there are remarks on the infinite in theology and in Renaissance art; but more often the book simply drops names of the famous, coupled with some anecdote about their life. Some major historical problems (Why did geometry dominate mathematics for so long?) are interestingly discussed, others (Why did logic decline in the Renaissance?) never even mentioned. The connection of mathematics with science is elaborated in the Roman period, largely ignored in the last three centuries. After a brief chapter on pre-Greek mathematics, the first third of the book traces the Greco-Roman theories of mathematics and logic from Thales to St. Augustine. There is a survey of pre=Platonic and Platonic philosophical schools that relate to mathematics, a rather unsatisfactory chapter on Aristotelian logic, and a helpful discussion of Euclid's elements, Archimedes and the method of exhaustion, and Apollonius. The following two chapters--on Roman developments---contain interesting sections on Pappus and Diophantus. A short and again unsatisfactory chapter on medieval logic leads to the Reniassance development of algebra, particularly by such Italians as Cardano and Tartaglia, and then Descartes, whose mathematical work is carefully related by Carruccio to his philosophical thinking. A long chapter discusses the rise of the calculus; Italian anticipators of Newton and Leibnltz--Galileo, Salviati, Cavalieri, Torricelli, Mengoli--are given their due, so that Newton and Leibnitz are seen in better perspective. The chapter ends with a disappointingly short two pages (237-238) on criticisms of the foundations of the calculus; Berkeley, for example, is not mentioned, and, though we are told that Cauchy resolved the worries by his concept of limit, anticipated by Mengoli, we are not given the actual definition. There follow a chapter on projective geometry, a long and fascinating chapter on the work of Saccheri , Gauss, Golyai, Lobachevsky, and Riemann in connection with non-Euclidean geometries , and a far too brief chapter on probability theory. The last four chapters are on roughly contemporary developments, and they are the least satisfactory. Cantor's theory of sets is outlined, but the exposition contains several errors (e.g., the definition of infinite cardinal, p. 294; the proof of Cantor's theorem, p. 303; the definition of well-ordering, p. 304). Peano's development of logic and arithmetic is given nine pages of attention, Frege not quite one, and it is not said that Frege (or Pierce, for that matter) anticipated Peano's use of quantifiers by a number of years. Carruccio chooses to present next the propositional calculus in the Hilbert-Ackermann formulation, thus giving the impression that the system was new in 1928--no hints are given as to the development of this material from Boole through Frege, Russell, Post, etc. His account also fails to give any indication of which are the primitive and which the defined symbols of this system. His remarks on the antinomies are marred by the statement "From these paradoxes...it turns out that the usual rules of logic do not apply to all sets and affirmations" (p. 344), whilst, of course, the 'usual' logic is preserved in most modern forms of set theory (apart from brief references to Zermelo, these are nowhere discussed). To Wittgenstein is attributed the founding of the Vienna Circle (p. 359). Finally, Goedel's undecidability theorem, though BOOK REVIEWS 99 outlined, is quite inadequately presented (pp. 361...