Wittgenstein, Finitism, and the Foundations of Mathematics

Wittgenstein, Finitism, and the Foundations of Mathematics

Wittgenstein, Finitism, and the Foundations of Mathematics

Wittgenstein, Finitism, and the Foundations of Mathematics

Synopsis

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.

Excerpt

During his life, Wittgenstein published only one book, the Tractatus Logico-Philosophicus (1921), one paper, 'Some Remarks on Logical Form' (1929), one page-long book review (1913), and one letter to the editor (1930). But he left behind an extensive Nachlass. From the amount of secondary literature devoted to it, it is surprising to discover that Wittgenstein wrote more on philosophy of mathematics than on any other subject. That Wittgenstein's remarks on mathematics did not receive their fair share of attention is not to be explained only by the fact that it is a comparatively unpopular topic. When Wittgenstein Remarks on the Foundations of Mathematics (RFM) were first published in 1956, reviewers' assessments were negative. For example, the logician Georg Kreisel, a close friend who had frequent discussions with Wittgenstein in the early 1940s, ended his review of the book with these much-quoted words: 'it seems to me to be a surprisingly insignificant product of a sparkling mind' (Kreisel 1958a: 158). Since specialists were in agreement in their negative assessment, followers and commentators of Wittgenstein simply hived off issues in philosophy of mathematics from those concerning language and psychology, more or less assuming that, although Wittgenstein may have erred when tackling issues in mathematical logic and foundations of mathematics, this was of no consequence to the rest of his philosophy: they could then continue with the business of interpreting these other parts in isolation. To my mind, this is ultimately as unacceptable as it would be for someone to interpret Frege's or Russell's philosophy in ignorance of their work in mathematical logic and the foundations of mathematics.

A variety of factors explain the negative reception of Wittgenstein's remarks on the philosophy of mathematics in the 1950s. First, the book comprised remarks selected by the editors (G. E. M. Anscombe, R. Rhees, and G. H. von Wright) from various manuscripts dating from 1937 to 1944 and it is clear that editorial choices, which produced a truncated version of the original text, have hindered rather than helped us to understand Wittgenstein's thoughts. He was an undoubtedly difficult writer, but when whole paragraphs are cut out in between two published remarks without any indications to that . . .

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