Social revolutions--that is critical periods of decisive, qualitative change--are a commonly acknowledged historical fact. But can the idea of revolutionary upheaval be extended to the world of ideas and theoretical debate? The publication of Kuhn's The Structure of Scientific Revolutions in 1962 led to an exciting discussion of revolutions in the natural sciences. A fascinating, but little known, off-shoot of this was a debate which began in the United States in the mid-1970's as to whether the concept of revolution could (...) be applied to mathematics as well as science. Michael Grove declared that revolutions never occur in mathematics, while Joseph Dauben argued that there have been mathematical revolutions and gave some examples. This book is the first comprehensive examination of the question. It reprints the original papers of Grove, Dauben, and Mehrtens, together with additional chapters giving their current views. To this are added new contributions from nine further experts in the history of mathematics, who each discuss an important episode and consider whether it was a revolution. The whole question of mathematical revolutions is thus examined comprehensively and from a variety of perspectives. This thought-provoking volume will interest mathematicians, philosophers, and historians alike. (shrink)

This book argues against the view that mathematical knowledge is a priori,contending that mathematics is an empirical science and develops historically,just as ...

First published in 2000. Routledge is an imprint of Taylor & Francis, an informa company.

" 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 ...

Provability, Computability and Reflection.

Offers a text of mathematical logic on a sophisticated level, presenting the reader with several of the most significant discoveries, including the independence of the continuum hypothesis, the Diophantine nature of enumerable sets and the impossibility of finding an algorithmic solution for certain problems.

§30. Significance of Desargues's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 CHAPTER VI. PASCAL'S THEOREM. §31. ...

In this book, which is both a philosophical and historiographical study, the author investigates the fallibility and the rationality of mathematics by means of rational reconstructions of developments in mathematics. The initial chapters are devoted to a critical discussion of Lakatos' philosophy of mathematics. In the remaining chapters several episodes in the history of mathematics are discussed, such as the appearance of deduction in Greek mathematics and the transition from Eighteenth-Century to Nineteenth-Century analysis. The author aims at developing a notion (...) of mathematical rationality that agrees with the historical facts. A modified version of Lakatos' methodology is proposed. The resulting constructions show that mathematical knowledge is fallible, but that its fallibility is remarkably weak. (shrink)

The various papers and short "discussions" contained in this latest addition to the "Studies in Logic" series were presented at the 1965 International Colloquium in the Philosophy of Science, in London. Of the nine "problems" considered in this symposium, seven have directly to do with philosophy, one is an historical study of the origins of Euclid's axiomatics, and the last is an interesting—if one-sided—discussion of the "new math" controversy in the pre-college curriculum. Happily, this book demonstrates that the important issues (...) in the philosophy of mathematics somewhat outrun the usual ism-mongering debate between representatives of various schools. Unhappily, it also demonstrates that a good deal of undeserving material seems to find its way between the covers of scholarly books. After reading certain of the articles in this volume one is left with the feeling that the authors have not come to grips with the issues which they raise, that a lot of effort has been expended on peripheral matters, and that, in some instances, a sledge hammer has been brought down upon an egg. Abraham Robinson's article, "The Metaphysics of the Calculus," deals with a series of important linguistic and ontological problems concerning the foundations of analysis. Moreover, these are problems which can no longer be ignored, especially in view of the evolution of non-standard analysis. But, unfortunately, the interested reader would do far better to turn to Robinson's recent book on the subject, and pay special attention to the second, third, and last chapters therein. "On a Fregean Dogma" by Fred Sommers gives a rather curious non-quantificational algorithm for the monadic fragment of syllogistic. This Sommers does in the interest of "a conservative effort to restore categorical statements to predicative status." Despite the heat and the occasional interesting remarks which the discussion generated, one is inclined to ask, "So what?" The sequence of three papers by Mostowski, Bernays, and Körner is best considered as a group, since each of these is concerned with the philosophical significance of recent results in set theory and logic. Students of the philosophy of mathematics would do well to read this part of the book very carefully, although one wishes that the authors would elaborate a bit on their specific philosophical views. Finally, there is Georg Kreisel's difficult and amazing paper, "Informal Rigor and Completeness Proofs," in which is presented a mathematical treatment of the problem of selecting the primitive rules and concepts of mathematical systems. Kreisel's ideas are refreshing, unorthodox, dubious, and a genuine contribution to the philosophy of mathematics. In sum, this book is a mixed bag, uneven in the quality of its contents, but worth reading.—H. P. K. (shrink)