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'. (shrink)

Wittgenstein's philosophy of mathematics has long been notorious. Part of the problem is that it has not been recognized that Wittgenstein, in fact, had two chief post-Tractatus conceptions of mathematics. I have labelled these the calculus conception and the language-game conception. The calculus conception forms a distinct middle period. The goal of my article is to provide a new framework for examining Wittgenstein's philosophies of mathematics and the evolution of his career as a whole. (...) I posit the Hardyian Picture, modelled on the Augustinian Picture, to provide a structure for Wittgenstein's work on the philosophy of mathematics. Wittgenstein's calculus period has not been properly recognized, so I give a detailed account of the tenets of that stage in Wittgenstein's career. Wittgenstein's notorious remarks on contradiction are the test case for my theory of his transition. I show that the bizarreness of those remarks is largely due to the calculus conception, but that Wittgenstein's later language-game account of mathematics keeps the rejection of the Hardyian Picture while correcting the calculus conception's mistakes. (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'. (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'. (shrink)

Hilary Putnam, James Conant; On Wittgenstein's Philosophy of Mathematics, Proceedings of the Aristotelian Society, Volume 97, Issue 1, 1 June 1997, Pages 195–22.

This paper has two main purposes: first, to compare Wittgenstein's views to the more traditional views in the philosophy of mathematics; second, to provide a general outline for a Wittgensteinian reply to these two objections. Two fundamental themes of Wittgenstein's account of mathematics title the following two sections: mathematical propositions are rules and not descriptions and mathematics is employed within a form of life. Under each heading, I examine Wittgenstein's rejection of alternative views. (...) My aim is to make clear the differences and too suggest some similarities. As will become soon clear, Wittgenstein often rejects opposing views for the same or similar reasons. This comparison will provide the necessary background for better understanding Wittgenstein's philosophy of mathematics, for appreciating its many unappreciated advantages and, finally, for defending a conventionalist account of mathematics. (shrink)

For Wittgenstein mathematics is a human activity characterizing ways of seeing conceptual possibilities and empirical situations, proof and logical methods central to its progress. Sentences exhibit differing 'aspects', or dimensions of meaning, projecting mathematical 'realities'. Mathematics is an activity of constructing standpoints on equalities and differences of these. Wittgenstein's Later Philosophy of Mathematics grew from his Early and Middle philosophies, a dialectical path reconstructed here partly as a response to the limitative results of Gödel and (...) Turing. (shrink)

This paper has two main purposes: first to compare Wittgenstein's views to the more traditional views in the philosophy of mathematics; second, to provide a general outline for a Wittgensteinian reply to two objections against Wittgenstein's account of mathematics: the objectivity objection and the consistency objections, respectively. Two fundamental thesmes of Wittgenstein's account of mathematics title the first two sections: mathematical propositions are rules and not descritpions and mathematics is employed within a (...) form of life. Under each heading, I examine Wittgenstein's rejection of alternative views. My aim is to make clear the differences and to suggest some similarities. As will become clear, Wittgenstein often rejects opposing views for the same or similar reasons. This comparison will provide the necessary background for better understanding Wittgenstein's philosophy of mathematics, for appreciating its many unappreciated advantages and, finally, for defending a conventionalist account of mathematics. (shrink)

An opinionated survey of the main topics in later Wittgenstein's philosophy of mathematics.

Wittgenstein's remarks on mathematics have not received the recogni tion they deserve; they have for the most part been either ignored, or dismissed as unworthy of the author of the Tractatus and the I nvestiga tions. This is unfortunate, I believe, and not at all fair, for these remarks are not only enjoyable reading, as even the harshest critics have con ceded, but also a rich and genuine source of insight into the nature of mathematics. It is (...) perhaps the fact that they are more suggestive than systematic which has put so many people off; there is nothing here of formal derivation and very little attempt even at sustained and organized argumentation. The remarks are fragmentary and often obscure, if one does not recognize the point at which they are directed. Nevertheless, there is much here that is good, and even a fairly system atic and coherent account of mathematics. What I have tried to do in the following pages is to reconstruct the system behind the often rather disconnected commentary, and to show that when the theory emerges, most of the harsh criticism which has been directed against these re marks is seen to be without foundation. This is meant to be a sym pathetic account of Wittgenstein's views on mathematics, and I hope that it will at least contribute to a further reading and reassessment of his contributions to the philosophy of mathematics. (shrink)

Sebastian Grève interviews Felix Mühlhölzer on his work on the philosophy of mathematics.

This chapter contains section titled: Some Personal Prefatory Remarks Introduction: Wittgenstein's Chief Contribution? The Reception of Wittgenstein's Philosophy of Mathematics in His Own Lifetime The Post 1956 Reaction.

Notes taken by these last four are the basis for the thirty-one lectures in this book.

Drawing mainly from the Tractatus Logico-Philosophicus and his middle period writings, strategic issues and problems arising from Wittgenstein’s philosophy of mathematics are discussed. Topics have been so chosen as to assist mediation between the perspective of philosophers and that of mathematicians on their developing discipline. There is consideration of rules within arithmetic and geometry and Wittgenstein’s distinctive approach to number systems whether elementary or transfinite. Examples are presented to illuminate the relation between the meaning of an arithmetical generalisation (...) or theorem and its proof. An attempt is made to meet directly some of Wittgenstein’s critical comments on the mathematical treatment of infinity and irrational numbers. (shrink)

During the course of about ten years, Wittgenstein revised some of his most basic views in philosophy of mathematics, for example that a mathematical theorem can have only one proof. This essay argues that these changes are rooted in his growing belief that mathematical theorems are ‘internally’ connected to their canonical applications, i.e. , that mathematical theorems are ‘hardened’ empirical regularities, upon which the former are supervenient. The central role Wittgenstein increasingly assigns to empirical regularities had profound implications (...) for all of his later philosophy; some of these implications (particularly to rule following) are addressed in the essay. (shrink)

Bazzocchi disposes the text of the Tractatus in a user-friendly manner, exactly as Wittgenstein's decimals advise. This discloses the logical form of the book by distinct reading units, linked into a fashioned hierarchical tree. The text becomes much clearer and every reader can enjoy, finally, its formal and literary qualities.

From his return to Cambridge in 1929 to his death in 1951, Wittgenstein influenced philosophy almost exclusively through teaching and discussion. These lecture notes indicate what he considered to be salient features of his thinking in this period of his life.

Philosopher Ludwig Wittgenstein had an enormous influence on twentieth-century philosophy even though only one of his works, the famous Tractatus Logico-Philosophicus, was published in his lifetime. Beyond this publication the impact of his thought was mainly conveyed to a small circle of students through his lectures at Cambridge University. Fortunately, many of his ideas have survived in both the dictations that were subsequently published, and the notes taken by his students, among them Alice Ambrose and the late Margaret Macdonald, (...) from 1932 to 1935. These notes, now edited by Professor Ambrose, are here published, and they shed much light on Wittgenstein's philosophical development. Among the topics considered are the meaning of a word and its relation to common usage, rules of grammar and their relation to fact, the grammar of first person statements, language games, and the nature of philosophy. This volume is indispensable to any serious discussion of Wittgenstein's work. (shrink)

This dissertation provides a careful reading of the later Wittgenstein’s philosophy of mathematics centered around three major themes: reality, determination, and infinity. The reading offered gives pride of place to Wittgenstein’s therapeutic conception of philosophy. This conception views questions often taken as fundamental in the philosophy of mathematics with suspicion and attempts to diagnose the confusions which lead to them. In the first essay, I explain Wittgenstein’s approach to perennial issues regarding the alleged reality to (...) which mathematical truths or propositions correspond. Wittgenstein diagnoses exotic pictures of mathematical reality as stemming from misleading analogies formed across empirical and mathematical propositions. The second essay explains Wittgenstein’s treatment of perplexity regarding the ability of a mathematical rule to determine its applications in advance. This too is found to depend on a misleading analogy, in this case across behavioral and mathematical senses of ‘determine’. The third and final essay discusses Wittgenstein’s general critical approach to “the infinite”. Philosophical perplexity about infinity is shown to depend on the verbal imagery regularly associated with proofs and their power to take hold of one’s imagination. Wittgenstein dampens the imaginative excesses often associated with “the infinite” by offering a sober redescription of Cantor’s famous method of diagonalization. (shrink)

Philosopher Ludwig Wittgenstein had an enormous influence on twentieth-century philosophy even though only one of his works, the famous Tractatus Logico-Philosophicus, was published in his lifetime. Beyond this publication the impact of his thought was mainly conveyed to a small circle of students through his lectures at Cambridge University. Fortunately, many of his ideas have survived in both the dictations that were subsequently published, and the notes taken by his students, among them Alice Ambrose and the late Margaret Macdonald, (...) from 1932 to 1935. These notes, now edited by Professor Ambrose, are here published, and they shed much light on Wittgenstein's philosophical development. Among the topics considered are the meaning of a word and its relation to common usage, rules of grammar and their relation to fact, the grammar of first person statements, language games, and the nature of philosophy. This volume is indispensable to any serious discussion of Wittgenstein's work. (shrink)

According to Mark Steiner, Wittgenstein’s intense work in the philosophy of mathematics during the early 1930s brought about a distinct turning point in his philosophy. The crux of this transition, Steiner contends, is that Wittgenstein came to see mathematical truths as originating in empirical regularities that in the course of time have been hardened into rules. This interpretation, which construes Wittgenstein’s later philosophy of mathematics as more realist than his earlier philosophy, challenges another influential (...) interpretation which reads Wittgenstein as moving in the opposite direction, from a more realist toward a less realist position. Both of these readings, I argue here, tend to overlook the crucial role of conventionalism in shaping Wittgenstein’s later philosophy. I show that during the transition period, Wittgenstein was first strongly attracted to conventionalism, but then, upon discovery of the rule following paradox, came to realize its weaknesses. Did this realization mark the end of the liaison with conventionalism and a wholehearted acceptance of the empirical regularities account? Illustrating Wittgenstein’s ongoing ambivalence toward conventionalism and pointing to his novel understanding of this position, I answer this question in the negative. (shrink)

(No abstract is available for this citation).

This five-volume German-English edition presents, for the first time, new translations of all of Wittgenstein's mature 1937-1944 writings on mathematics and logic. The first (1956) and third (1978) editions of Wittgenstein's Remarks on the Foundations of Mathematics omitted, unsystematically, more than half of Wittgenstein's later writings on mathematics; for that reason, the reader will here read some entire manuscripts for the first time, and other manuscripts for the first time as unabridged, sustained pieces of (...) writing. Philosophers and other interested readers will gain fresh insight into Wittgenstein's perspectives on a wide range of topics, from Gödelian propositions and the Cantorian conception of real numbers to the nature of mathematical propositions and the diversity of proof techniques. Other subjects covered include: mathematical sense; axioms and self-evidence; prudish proofs; the functionality of extra-mathematical application; undecided mathematical conjectures; rule-governed unwinding; and G. H. Hardy's conceptions and claim. (shrink)

Introduction 1. Perspectives on Wittgenstein: An Intermittently Opinionated Survey: Hans-Johann Glock. 2. Wittgenstein's Method: Ridding People of Philosophical Prejudices: Katherine Morris. 3. Gordon Baker's Late Interpretation of Wittgenstein: P. M. S. Hacker. 4. The Interpretation of the Philosophical Investigations: Style, Therapy, Nachlass: Alois Pichler. 5. Ways of Reading Wittgenstein: Observations on Certain Uses of the Word 'Metaphysics': Joachim Schulte. 6. Metaphysical/Everyday Use: A Note on a Late Paper by Gordon Baker: Hilary Putnam. 7. Wittgenstein and Transcendental Idealism: A. W. (...) Moore. 8. Simples and the Idea of Analysis in the Tractatus: Marie McGinn. 9. Words, Waxing and Waning: Ethics in/and/of the Tractatus Logico-Philosophicus: Stephen Mulhall. 10. The Uses of Wittgenstein's Beetle: Philosophical Investigations §293 and Its Interpreters: David G. Stern. 11. Bourgeois, Bolshevist or Anarchist?: The Reception of Wittgenstein's Philosophy of Mathematics: Ray Monk. 12. Wittgenstein and Ethical Naturalism: Alice Crary. (shrink)

Wittgenstein's work remains, undeniably, now, that off one of those few philosophers who will be read by all future generations.

Th e con fusion a nd b arren ness o f psycho logy is no t to be e xplain ed b y calling it a “yo ung science”; its state is not comparable with that of physics, for instance, in its beginnings. (Rather with that of certain branches of mathematics. Set theory.) For in psychology there are experimental methods and conceptual confusion. (As in the oth er case, con cep tual co nfusion and m ethod s of pro (...) of.) The existence of the experimental method makes us think we have the means of solving the problems that trouble us; though problem and method pass one another by. (PI p. 232). (shrink)

Since the publication of the Remarks on the Foundations of Mathematics, Wittgenstein’s interpreters have endeavored to reconcile his general constructivist/anti-realist attitude towards mathematics with his confessed anti-revisionary philosophy. In this article, the author revisits the issue and presents a solution. The basic idea consists in exploring the fact that the so-called “non-constructive results” could be interpreted so that they do not appear non-constructive at all. The author substantiates this solution by showing how the translation of mathematical results, (...) given by the possibility of translation between logics, can be seen as a tool for partially implementing the solution Wittgenstein had in mind. (shrink)