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This book offers a detailed account and discussion of Ludwig Wittgenstein’s philosophy of mathematics. In Part I, the stage is set with a brief presentation of Frege’s logicist attempt to provide arithmetic with a foundation and Wittgenstein’s criticisms of it, followed by sketches of Wittgenstein’s early views of mathematics, in the Tractatus and in the early 1930s. Then (in Part II), Wittgenstein’s mature philosophy of mathematics (1937-44) is carefully presented and examined. Schroeder explains that it is based on two key ideas: the calculus view and the grammar view. On the one hand, mathematics is seen as a human activity — calculation — rather than a theory. On the other hand, the results of mathematical calculations serve as grammatical norms. The following chapters (on mathematics as grammar; rule-following; conventionalism; the empirical basis of mathematics; the role of proof) explore the tension between those two key ideas and suggest a way in which it can be resolved. Finally, there are chapters analysing and defending Wittgenstein’s provocative views on Hilbert’s Formalism and the quest for consistency proofs and on Gödel’s incompleteness theorems.
Mathieu Marion offers a careful, historically informed study of Wittgenstein's philosophy of mathematics. This area of his work has frequently been undervalued by Wittgenstein specialists and by philosophers of mathematics alike; but the surprising fact that he wrote more on this subject than on any other indicates its centrality in his thought. Marion traces the development of Wittgenstein's thinking in the context of the mathematical and philosophical work of the times, to make coherent sense of ideas that have too often been misunderstood because they have been presented in a disjointed and incomplete way. In particular, he illuminates the work of the neglected 'transitional period' between the Tractatus and the Investigations. Marion shows that study of Wittgenstein's writings on mathematics is essential to a proper understanding of his philosophy; and he also demonstrates that it has much to contribute to current debates about the foundations of mathematics.
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 (1934–1951) grew from his Early (1912–1921) and Middle (1929–33) philosophies, a dialectical path reconstructed here partly as a response to the limitative results of Gödel and Turing.
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'.
First published in 2005. Routledge is an imprint of Taylor & Francis, an informa company.
This monograph examines the private annotations that Ludwig Wittgenstein made to his copy of G.H. Hardy’s classic textbook, A Course of Pure Mathematics. Complete with actual images of the annotations, it gives readers a more complete picture of Wittgenstein’s remarks on irrational numbers, which have only been published in an excerpted form and, as a result, have often been unjustly criticized. The authors first establish the context behind the annotations and discuss the historical role of Hardy’s textbook. They then go on to outline Wittgenstein’s non-extensionalist point of view on real numbers, assessing his manuscripts and published remarks and discussing attitudes in play in the philosophy of mathematics since Dedekind. Next, coverage focuses on the annotations themselves. The discussion encompasses irrational numbers, the law of excluded middle in mathematics and the notion of an “improper picture," the continuum of real numbers, and Wittgenstein’s attitude toward functions and limits.
This book offers a lucid and highly readable account of Wittgenstein's philosophy, framed against the background of his extraordinary life and character. Woven together with a biographical narrative, the chapters explain the key ideas of Wittgenstein's work, from his first book, the Tractatus Logico-Philosophicus, to his mature masterpiece, the Philosophical Investigations. Severin Schroeder shows that at the core of Wittgenstein's later work lies a startlingly original and subversive conception of the nature of philosophy. In accordance with this conception, Wittgenstein offers no new philosophical doctrines to replace his earlier ones, but seeks to demonstrate how all philosophical theorizing is the result of conceptual misunderstanding. He first diagnoses such misunderstanding at the core of his own earlier philosophy of language and then subjects philosophical views and problems about various mental phenomena understanding, sensations, the will to a similar therapeutic analysis. Schroeder provides a clear and careful account of the main arguments offered by Wittgenstein. He concludes by considering some critical responses to Wittgenstein's work, assessing its legacy for contemporary philosophy. Wittgenstein is ideal for students seeking a clear and concise introduction to the work of this seminal twentieth-century philosopher.
Since the middle of the 20th century Ludwig Wittgenstein has been an exceptionally influential and controversial figure wherever philosophy is studied. This is a comprehensive volume on Wittgenstein where 35 scholars explore the whole range of his thought, offering critical engagement and original interpretation.