Wittgenstein's role was vital in establishing mathematics as one of this century's principal areas of philosophic inquiry. In this book, the three phases of Wittgenstein's reflections on mathematics are viewed as a progressive whole, rather than as separate entities. Frascolla builds up a systematic construction of Wittgenstein's representation of the role of arithmetic in the theory of logical operations. He also presents a new interpretation of Wittgenstein's rule-following considerations - the `community view of internal relations'.

Ludwig Wittgenstein is perhaps the greatest philosopher of the twentieth century, and certainly one of the most original in the entire Western tradition. Given the inaccessibility of his work, it is remarkable that he has inspired poems, paintings, films, musical compositions, titles of books -- and even novels. In his splendid biography, Ray Monk has made this very compelling human being come alive in a way that perfectly explains the fascination he has evoked. Wittgenstein's life was one of great moral (...) and spiritual depth. Although his work addresses problems of logic, language, mind, and knowledge, he often described it as having an ethical point, and once said that he "could not help seeing every problem from a religious point of view." In successfully bridging the gap assumed by many to exist between Wittgenstein's life and his work, Monk's biography helps to explain the interest in Wittgenstein's later philosophy on the part of Buddhist scholars. (shrink)

This lucid and comprehensive essay by a distinguished philosopher surveys the views of Plato, Aristotle, Leibniz, and Kant on the nature of mathematics. It examines the propositions and theories of the schools these philosophers inspired, and it concludes by discussing the relationship between mathematical theories, empirical data, and philosophical presuppositions. 1968 edition.

Intellectual diary of a thinker of the school of Logical Positivism showing the day-by-day development of his philosophical ideas.

The question is raised of the source of the extreme verificationist views which Wittgenstein put forward immediately after his return to philosophy in 1929. Since these views appear to be radically different from the ideas put forward in theTractatus some explanation of this dramatic new turn in Wittgenstein''s thought certainly seems to be called for. Wittgenstein''s very low level of interest in philosophy between 1918 and shortly before his return to philosophy is documented. Attention then focuses on the crucial period (...) immediately before Wittgenstein''s return to Cambridge, and it is shown that in this period he encountered only two new philosophical influences. These were the ideas of Brouwer and the ideas of the Vienna Circle. Each of these is examined and neither is found capable of providing an explanation of the source of Wittgenstein''s verificationism. This leads to a reconsideration of the underlying assumption that Wittgenstein''s verificationism does represent the radical departure from the ideas of theTractatus which it appears to. It is argued that the only way we can account for Wittgenstein''s evident approval of the reading of theTractatus which he must have encountered in his meetings with members of the Vienna Circle is by concluding that, far from being incompatible with his earlier ideas, some form of verificationism must always have been implicit in theTractatus. (shrink)

When in May 1930, the Council of Trinity College, Cambridge, had to decide whether to renew Wittgenstein's research grant, it turned to Bertrand Russell for an assessment of the work Wittgenstein had been doing over the past year. His verdict: "The theories contained in this new work . . . are novel, very original and indubitably important. Whether they are true, I do not know. As a logician who likes simplicity, I should like to think that they are not, but (...) from what I have read of them I am quite sure that he ought to have an opportunity to work them out, since, when completed, they may easily prove to constitute a whole new philosophy." "[ Philosophical Remarks ] contains the seeds of Wittgenstein's later philosophy of mind and of mathematics. Principally, he here discusses the role of indispensable in language, criticizing Russell's The Analysis of Mind . He modifies the Tractatus 's picture theory of meaning by stressing that the connection between the proposition and reality is not found in the picture itself. He analyzes generality in and out of mathematics, and the notions of proof and experiment. He formulates a pain/private-language argument and discusses both behaviorism and the verifiability principle. The work is difficult but important, and it belongs in every philosophy collection."--Robert Hoffman, Philosophy "Any serious student of Wittgenstein's work will want to study his Philosophical Remarks as a transitional book between his two great masterpieces. The Remarks is thus indispensible for anyone who seeks a complete understanding of Wittgenstein's philosophy."--Leonard Linsky, American Philosophical Association. (shrink)

En logique mathématique, on doit distinguer entre une conception « axiomatique »de la logique, qui fut celle de Frege, Russell et Hilbert, et une conception plus « pragmatique »en termes d’actes de preuves, que l’on retrouve dans les systèmes de déduction naturelle de Gentzen. Des parallèles sont esquissés entre la conception de l’inférence et de la logique dans le Tractatus Logico-philosophicus de Wittgenstein et celle de Gentzen. Ce cadre permet en outre de jeter un regard neuf sur l’argument de Wittgenstein (...) sur « suivre une règle ». (shrink)

The philosophy of arithmetic of Wittgenstein's Tractatus is outlined and the central role played in it by the general notion of operation is pointed out. Following which, the language, the axioms and the rules of a formal theory of operations, extracted from the Tractatus, are presented and a theorem of interpretability of the equational fragment of Peano's Arithmetic into such a formal theory is proven.

Luitzen Egburtus Jan Brouwer founded a school of thought whose aim was to include mathematics within the framework of intuitionistic philosophy; mathematics was to be regarded as an essentially free development of the human mind. What emerged diverged considerably at some points from tradition, but intuitionism has survived well the struggle between contending schools in the foundations of mathematics and exact philosophy. Originally published in 1981, this monograph contains a series of lectures dealing with most of the fundamental topics such (...) as choice sequences, the continuum, the fan theorem, order and well-order. Brouwer's own powerful style is evident throughout the work. (shrink)

Wittgenstein's role was vital in establishing mathematics as one of this century's principal areas of philosophic inquiry. In this book, the three phases of Wittgenstein's reflections on mathematics are viewed as a progressive whole, rather than as separate entities. Frascolla builds up a systematic construction of Wittgenstein's representation of the role of arithmetic in the theory of logical operations. He also presents a new interpretation of Wittgenstein's rule-following considerations - the `community view of internal relations'.

This massive two-volume reference presents a comprehensive selection of the most important works on the foundations of mathematics. While the volumes include important forerunners like Berkeley, MacLaurin, and D'Alembert, as well as such followers as Hilbert and Bourbaki, their emphasis is on the mathematical and philosophical developments of the nineteenth century. Besides reproducing reliable English translations of classics works by Bolzano, Riemann, Hamilton, Dedekind, and Poincare, William Ewald also includes selections from Gauss, Cantor, Kronecker, and Zermelo, all translated here for (...) the first time. (shrink)

pt. 1. The proposition and its sense.--pt. 2. On logic and mathematics.

Der zweisprachige Band führt in die "Wiener Ausgabe" der Werke Ludwig Wittgensteins (1889-1951) ein, in der erstmals dessen Manuskripte vollständig und in getreuer Wiedergabe erscheinen werden. In jahrzehntelanger Arbeit wurde der Nachlaß mithilfe einer Editionsmethode ediert, die für dieses ungewöhnliche Werk speziell entwickelt wurde. Der Einführungsband erklärt diese Methode wie die editorische Situation des Nachlasses und stellt darüber hinaus Arbeitsunterlagen für das Studium von Wittgensteins Texten zur Verfügung: Eine biographische Skizze zeigt die Verbindung von Leben und Werk, eine schematische Chronologie (...) erleichert die Zuordnung der jeweiligen Manuskriptseiten und Typoskripte. Die Gegenüberstellung von Faksimileabdrucken einzelner Manuskriptseiten und der edierten Seiten läßt die einzelnen Schritte der Textedition transparent werden. (shrink)

Since the first publication of Insight and Illusion in l972, a wealth of Wittgenstein's writings have become accessible. Accordingly, in this edition Professor Hacker has rewritten six of his eleven original chapters and revised the others to incorporate the new abundant material. Insight and Illusion now fully clarifies the historical backgrounds of Wittgenstein's highly different masterpieces, the Tractatus and the Investigations, and traces the evolution of Wittgenstein's thought. Hacker explains all of Wittgenstein's writings in detail, focusing on his critique of (...) metaphysics, his famous "private language argument," and his account of self-consciousness. (shrink)

From Brouwer To Hilbert: The Debate on the Foundations of Mathematics in the 1920s offers the first comprehensive introduction to the most exciting period in the foundation of mathematics in the twentieth century. The 1920s witnessed the seminal foundational work of Hilbert and Bernays in proof theory, Brouwer's refinement of intuitionistic mathematics, and Weyl's predicativist approach to the foundations of analysis. This impressive collection makes available the first English translations of twenty-five central articles by these important contributors and many others. (...) The articles have been translated for the first time from Dutch, French, and German, and the volume is divided into four sections devoted to (1) Brouwer, (2) Weyl, (3) Bernays and Hilbert, and (4) the emergence of intuitionistic logic. Each section opens with an introduction which provides the necessary historical and technical context for understanding the articles. Although most contemporary work in this field takes its start from the groundbreaking contributions of these major figures, a good, scholarly introduction to the area was not available until now. Unique and accessible, From Brouwer To Hilbert will serve as an ideal text for undergraduate and graduate courses in the philosophy of mathematics, and will also be an invaluable resource for philosophers, mathematicians, and interested non-specialists. (shrink)

Wittgenstein's role was vital in establishing mathematics as one of this century's principal areas of philosophic inquiry. In this book, the three phases of Wittgenstein's reflections on mathematics are viewed as a progressive whole, rather than as separate entities. Frascolla builds up a systematic construction of Wittgenstein's representation of the role of arithmetic in the theory of logical operations. He also presents a new interpretation of Wittgenstein's rule-following considerations - the `community view of internal relations'.

This text offers major re-evaluation of Wittgenstein's thinking. It is a collection of essays that presents a significantly different portrait of Wittgenstein. The essays clarify Wittgenstein's modes of philosophical criticism and shed light on the relation between his thought and different philosophical traditions and areas of human concern. With essays by Stanley Cavell, James Conant, Cora Diamond, Peter Winch and Hilary Putnam, we see the emergence of a new way of understanding Wittgenstein's thought. This is a controversial collection, with essays (...) by highly regarded Wittgenstein scholars that may change the way we look at Wittgenstein's body of work. (shrink)