*Thinking about Mathematics: The Philosophy of Mathematics*, Stewart Shapiro, 2000, 308pp. $29.95, paper, ISBN 0 – 19 – 289306 – 8, New York; Oxford University Press, http:// www.oup.com .

Many students of the history of mathematics have found themselves pondering the question ‘Just what is mathematics anyway?’ Is it the ability to discern geometric shape, to be numerate, to weave a basket, solve an equation or prove a theorem? Is it an act, a concept or a collection of techniques designed to solve problems? It is interesting, and a bit telling, that mathematics was the first intellectual discipline to be so examined both historically and philosophically. The need for a philosophy of mathematics was realized quite early in human history. Stewart Shapiro, in *Thinking about Mathematics*, introduces the readers to the historical quest of understanding the real essence of mathematics.

This book presents a survey of the major schools of thought and movements that have attempted to define a philosophy of mathematics. Shapiro begins with Plato’s conception of mathematical ideals existing outside of the mind of man. He then considers the Age of Scientific Enlightenment with its philosophical dichotomy of rationalism, mathematics as pure abstraction, versus empiricalism, mathematics arises from observation of the real world. In these eras as well as in the era of the analytic movements of the late nineteenth century, the reader is introduced to the ideas and persons that have shaped philosophies of mathematics. The work of Immanuel Kant (1724 – 1804) and John Stuart Mills (1806 – 1873) in their attempts to mediate positions between rationalism and empiricalism is discussed. We are introduced to the theories of Frege, Russell, Hilbert, Bernays and Brouwer and many other mathematical thinkers. The philosophies of logicism, formalism and intuitionism are discussed in detail. Shapiro himself is an advocate for structuralism, that is, that the true understanding of mathematics resides in knowing its structures and patterns. This survey is thorough and informative. However, it is concept dense and at times the reading is difficult even for the mathematically knowledgeable.

Frank J. Swetz, Professor Emeritus, the Pennsylvania State University