In this illuminating collection, Charles Parsons surveys the contributions of philosophers and mathematicians who shaped the philosophy of mathematics over the course of the past century.

Parsons begins with a discussion of the Kantian legacy in the work of L. E. J. Brouwer, David Hilbert, and Paul Bernays, shedding light on how Bernays revised his philosophy after his collaboration with Hilbert. He considers Hermann Weyl’s idea of a “vicious circle” in the foundations of mathematics, a radical claim that elicited many challenges. Turning to Kurt Gödel, whose incompleteness theorem transformed debate on the foundations of mathematics and brought mathematical logic to maturity, Parsons discusses his essay on Bertrand Russell’s mathematical logic—Gödel’s first mature philosophical statement and an avowal of his Platonistic view.

Philosophy of Mathematics in the Twentieth Century insightfully treats the contributions of figures the author knew personally: W. V. Quine, Hilary Putnam, Hao Wang, and William Tait. Quine’s early work on ontology is explored, as is his nominalistic view of predication and his use of the genetic method of explanation in the late work The Roots of Reference. Parsons attempts to tease out Putnam’s views on existence and ontology, especially in relation to logic and mathematics. Wang’s contributions to subjects ranging from the concept of set, minds, and machines to the interpretation of Gödel are examined, as are Tait’s axiomatic conception of mathematics, his minimalist realism, and his thoughts on historical figures.

The philosopher’s toolkit.

Aristotle, great Greek philosopher, researcher, reasoner, and writer, born at Stagirus in 384 BC, was the son of a physician. He studied under Plato at Athens and taught there (367–347); subsequently he spent three years at the court of a former pupil in Asia Minor. After some time at Mitylene, in 343–342 he was appointed by King Philip of Macedon to be tutor of his teen-aged son Alexander. After Philip’s death in 336, Aristotle became head of his own school (of “Peripatetics”), the Lyceum at Athens. Because of anti-Macedonian feeling there after Alexander’s death in 323, he withdrew to Chalcis in Euboea, where he died in 322.

Nearly all the works Aristotle prepared for publication are lost; the priceless ones extant are lecture-materials, notes, and memoranda (some are spurious). They can be categorized as follows:

I Practical: Nicomachean Ethics; Great Ethics (Magna Moralia); Eudemian Ethics; Politics; Economics (on the good of the family); On Virtues and Vices.
II Logical: Categories; Analytics (Prior and Posterior); Interpretation; Refutations used by Sophists; Topica.
III Physical: Twenty-six works (some suspect) including astronomy, generation and destruction, the senses, memory, sleep, dreams, life, facts about animals, etc.
IV Metaphysics: on being as being.
V Art: Rhetoric and Poetics.
VI Other works including the Constitution of Athens; more works also of doubtful authorship.
VII Fragments of various works such as dialogues on philosophy and literature; and of treatises on rhetoric, politics, and metaphysics.

The Loeb Classical Library edition of Aristotle is in twenty-three volumes.

This book is meant to be a primer, that is, an introduction, to probability logic, a subject that appears to be in its infancy. Probability logic is a subject envisioned by Hans Reichenbach and largely created by Adams. It treats conditionals as bearers of conditional probabilities and discusses an appropriate sense of validity for arguments such conditionals, as well as ordinary statements as premisses.

This is a clear well-written text on the subject of probability logic, suitable for advanced undergraduates or graduates, but also of interest to professional philosophers. There are well-thought-out exercises, and a number of advanced topics treated in appendices, while some are brought up in exercises and some are alluded to only in footnotes. By this means, it is hoped that the reader will at least be made aware of most of the important ramifications of the subject and its tie-ins with current research, and will have some indications concerning recent and relevant literature.

In Pursuit of Truth W. V. Quine gives us his latest word on issues to which he has devoted many years. As he says in the preface: "In these pages I have undertaken to update, sum up, and clarify my variously intersecting views on cognitive meaning, objective reference, and the grounds of knowledge?'The pursuit of truth is a quest that links observation, theory, and the world. Various faulty efforts to forge such links have led to much intellectual confusion. Quine's efforts to get beyond the confusion begin by rejecting the very idea of binding together word and thing, rejecting the focus on the isolated word. For him, observation sentences and theoretical sentences are the alpha and omega ofthe scientific enterprise. Notions like "idea" and "meaning" are vague, but a sentence-now there's something you can sink your teeth into.

Starting thus with sentences, Quine sketches an epistemological setting for the pursuit of truth. He proceeds to show how reification and reference contribute to the elaborate structure that can indeed relate science to its sensory evidence.In this book Quine both summarizes and moves ahead. Rich, lively chapters dissect his major concerns-evidence, reference, meaning, intension, and truth. "Some points;' he writes, "have become clearer in my mind in the eight years since Theories and Things. Some that were already clear in my mind have become clearer on paper. And there are some that have meanwhile undergone substantive change for the better." This is a key book for understanding the effort that a major philosopher has made a large part of his life's work: to naturalize epistemology in the twentieth century. The book is concise and elegantly written, as one would expect, and does not assume the reader's previous acquaintance with Quine's writings. Throughout, it is marked by Quine's wit and economy of style.