Logicism

Abstract: Bertrand Russell is the most important philosopher of mathematics of the twentieth century The author of The Principles of Mathematics and, with Alfred Whitehead, the massive Principia Mathematica , Russell brought together his formidable knowledge of the subject and his skills as a gifted communicator to provide a classic introduction to the philosophy of mathematics Introduction to Mathematical Philosophy sets out in a lucid and non-technical way the main ideas of Principia Mathematica It is as inspiring and useful to the beginner now as it was when it was first published in 1919 This paperback edition of Introduction to Mathematical Philosophy includes a new introduction by John G Slater, University of Toronto

Abstract: The fundamental texts of the great classical period in modern logic, some of them never before available in English translation, are here gathered together for the first time. Modern logic, heralded by Leibniz, may be said to have been initiated by Boole, De Morgan, and Jevons, but it was the publication in 1879 of Gottlob Frege's "Begriffsschrift" that opened a great epoch in the history of logic by presenting, in full-fledged form, the propositional calculus and quantification theory. Frege's book, translated in its entirety, begins the present volume. The emergence of two new fields, set theory and foundations of mathematics, on the borders of logic, mathematics, and philosophy, is depicted by the texts that follow. Peano and Dedekind illustrate the trend that led to "Principia Mathematica." Burali-Forti, Cantor, Russell, Richard, and Konig mark the appearance of the modern paradoxes. Hilbert, Russell, and Zermelo show various ways of overcoming these paradoxes and initiate, respectively, proof theory, the theory of types, and axiomatic set theory. Skolem generalizes Lowenheim's theorem, and heand Fraenkel amend Zermelo's axiomatization of set theory, while von Neumann offers a somewhat different system. The controversy between Hubert and Brouwer during the twenties is presented in papers of theirs and in others by Weyl, Bernays, Ackermann, and Kolmogorov. The volume concludes with papers by Herbrand and by Godel, including the latter's famous incompleteness paper. Of the forty-five contributions here collected all but five are presented "in extenso." Those not originally written in English have been translated with exemplary care and exactness; the translators are themselves mathematical logicians as well as skilled interpreters of sometimes obscure texts. Each paper is introduced by a note that sets it in perspective, explains its importance, and points out difficulties in interpretation. Editorial comments and footnotes are interpolated where needed, and an extensive bibliography is included.

Abstract: No one has figured more prominently in the study of the German philosopher Gottlob Frege than Michael Dummett. His magisterial "Frege: Philosophy of Language" is a sustained, systematic analysis of Frege's thought, omitting only the issues in philosophy of mathematics. In this work Dummett discusses, section by section, Frege's masterpiece" The Foundations of Arithmetic" and Frege's treatment of real numbers in the second volume of "Basic Laws of Arithmetic," establishing what parts of the philosopher's views can be salvaged and employed in new theorizing, and what must be abandoned, either as incorrectly argued or as untenable in the light of technical developments.Gottlob Frege (1848-1925) was a logician, mathematician, and philosopher whose work had enormous impact on Bertrand Russell and later on the young Ludwig Wittgenstein, making Frege one of the central influences on twentieth-century Anglo-American philosophy; he is considered the founder of analytic philosophy. His philosophy of mathematics contains deep insights and remains a useful and necessary point of departure for anyone seriously studying or working in the field.