Wittgenstein and Philosophy of Mathematics

This page is about works that discuss the influence of Wittgenstein on contemporary philosophy of mathematics.

Wright, Crispin, Wittgenstein on the Foundations of Mathematics , Duckworth/Harvard, 1981/1980.

Book information
Hardcover: 500 pages
Publisher: Harvard Univ Pr (April 1980)
Language: English
ISBN-10: 0674953851
ISBN-13: 978-0674953857

Paperback: 500 pages
Publisher: Duckworth (1981)

Book description
'Wright’s most significant contribution has been his development of a distinctive neo-Fregean philosophy of mathematics. Wright’s first book, Wittgenstein on the Foundations of Mathematics, was dense, disorganized, and exhibited a prose-style whose complexity and ungainliness drove many reviewers to despair. But it’s a pioneering work, and introduced several themes which, with hindsight, have come to seem crucial to the philosophies of mind and language, as well as to that of mathematics. Of particular significance is its groundbreaking treatment of Wittgenstein’s discussion of rule-following, which predates Saul Kripke’s hugely influential Wittgenstein on Rules and Private Language. Kripke, notoriously, attributes to Wittgenstein the sceptical thesis that there are no facts about linguistic understanding and other intentional states, or, consequently, about linguistic meaning. Wright, in contrast, does not read Wittgenstein as sceptical of the reality of intentional and semantic phenomena, but rather of a pervasive realist or ‘objectivist’ conception of them. The nub of this conception – as far as linguistic meaning is concerned - is the idea that the meaning of an expression at a given time is constituted independently of speakers’ linguistic behaviour and judgements about it at (or after) that time. Speakers are, as it were, contractually answerable to independently-constituted standards of correct use, and understanding involves cognition of such standards. In Wittgenstein on the Foundations of Mathematics and various subsequent papers (many of which are collected in his Rails to Infinity) Wright develops Wittgenstein-inspired arguments against this objectivist conception, and defends a ‘constructivist’ alternative. In earlier work, this constructivism involved the view that the facts about meanings, such as there are, are (perpetually) determined by speakers’ behaviour and judgements about them. In later papers Wright has tended to focus on parallels between understanding and other intentional states, and to argue that principal determinants of the contents of such states in general are subjects’ first-person judgements about them. Wright’s interpretation of Wittgenstein and his constructivist proposals are vigorously opposed by most of his contemporary interlocutors (John McDowell’s energetic denunciations deserve special mention) but his arguments against the objectivist position have not been conclusively refuted.' Dr. Darragh Byrne's appraisal of the work in his 'Wright, Crispin', in Brown, S. (ed.), The Dictionary Of Twentieth-Century British Philosophers, Thoemmes Press (2005); the full text of the entry can be found in: http://www.philosophy.bham.ac.uk/staff/byrne_Crispin_Wright.pdf

Fann, K.T, ed., Ludwig Wittgenstein The Man and His Philosophy, New York: Delta (Dell Publishing Co. Inc.) 1963.

Book Information:
Paperback: 412 pages

Brief Discussion:
An interesting early collection of essays notable for its high proportion of personal recollections, and for its inclusion of a number of essays relating to mathematics and logic. All the pieces except for the contribution by Wolf Mayes previously published elsewhere, Contains the following essays: G.H.Von Wright: A Biographical Sketch; 4 Memoirs of Wittgenstein individually by Bertrand Russell, Rudolph Carnap, G.E. Moore and John Wisdom; D.A. Gasking & A.C. Jackson: Wittgenstein as a Teacher; Karl Britton: Portrait of a Philosopher; A Symposium: Assessments of the Man and the Philosopher individually by Erich Heller, M.C.O.Drury, Norman Malcolm & Rush Rhees; Jose Ferrater Mora: Wittgenstein, A Symbol of Troubled Times; Gilbert Ryle: Ludwig Wittgenstein; George A. Paul: Ludwig Wittgenstein; Morris Lazerowitz: Wittgenstein on The Nature of Philosophy; O.K. Bouswma: The Blue Book; Leonard Linsky: Wittgenstein on Language & Some Problems of Philosophy; Norman Malcolm: Wittgensteinʼs Philosophical Investigations ; Paul Feyerabend: Wittgensteinʼs Philosophical Investigations ; Rush Rhees: Wittgensteinʼs Builders; Alice Ambrose: Wittgenstein on Some Questions in Foundations of Mathematics; Joseph L.Cowan: Wittgensteinʼs Philosophy of Logic; Arnold Levison: Wittgenstein and Logical Laws; George Pitcher: Wittgenstein, Nonsense and Lewis Carrol; Alice Ambrose: Wittgenstein on Universals; John Wisdom: A feature of Wittgensteinʼs Technique; Albert W. Levi: Wittgenstein as Dialectitian; Dennis Oʼ Brien: The Unity of Wittgensteinʼs Thought.

de Bruin, Boudewijn, "Wittgenstein on Circularity in the Frege-Russell Definition of Cardinal Number." Philosophia Mathematica 16.3 (2008): 354-373.

Several scholars have argued that Wittgenstein held the view that the notion of number is presupposed by the notion of one-one correlation, and that therefore Hume's principle is not a sound basis for a definition of number. I offer a new interpretation of the relevant fragments from Wittgenstein's Nachlass, showing that if different uses of "presupposition" are understood in terms of de re and de dicto knowledge, Wittgenstein's argument against the Frege-Russell definition of number turns out to be valid on its own terms, even though it depends on two epistemological principles the logicist may find too "constructivist."

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