Journal ArticleDOI
Wittgenstein and finitism
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In this paper, elementary but hitherto overlooked connections are established between Wittgenstein's remarks on mathematics, written during his transitional period, and free-variable finitism.Abstract:
In this paper, elementary but hitherto overlooked connections are established between Wittgenstein's remarks on mathematics, written during his transitional period, and free-variable finitism. After giving a brief description of theTractatus Logico-Philosophicus on quantifiers and generality, I present in the first section Wittgenstein's rejection of quantification theory and his account of general arithmetical propositions, to use modern jargon, as claims (as opposed to statements). As in Skolem's primitive recursive arithmetic and Goodstein's equational calculus, Wittgenstein represented generality by the use of free variables. This has the effect that negation of unbounded universal and existential propositions cannot be expressed. This is claimed in the second section to be the basis for Wittgenstein's criticism of the universal validity of the law of excluded middle. In the last section, there is a brief discussion of Wittgenstein's remarks on real numbers. These show a preference, in line with finitism, for a recursive version of the continuum.read more
Citations
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Journal ArticleDOI
Remarks on the Foundations of Mathematics. By L. Wittgenstein. Edited by G. H. Von Wright, R. Rhees and G. E. M. Anscombe. Translated by G. E. M. Anscombe. Pp. 400. 37s. 6d. 1956. (Basil Blackwell, Oxford)
TL;DR: Wittgenstein's work remains, undeniably, now, that off one of those few few philosophers who will be read by all future generations as mentioned in this paper, regardless of whether it is true or not.
Journal ArticleDOI
Insight and illusion.
TL;DR: In this article, the authors failed to account for costs which are shifted away from the hospital in-patient service but which are borne by other agencies, including the costs incurred by the hospital outpatient service, primary care services, the criminal-justice system, community agencies and informal carers.
Journal ArticleDOI
Recursive Number Theory. By R. L. Goodstein. Pp. ix, 190. 36s. 1957. Studies in logic and the foundations of mathematics. (North-Holland, Amsterdam)
Book ChapterDOI
On Saying What You Really Want to Say: Wittgenstein, Gödel, and the Trisection of the Angle
TL;DR: Wittgenstein's remarks on the first incompleteness theorem have often been denounced, and mostly dismissed as mentioned in this paper, despite indirect historical evidence to the contrary, despite indirect evidence that Wittgenstein rejected Godel's proof because he did not, or even could not, understand it.
References
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Book
Tractatus Logico-Philosophicus
TL;DR: The Tractatus Logico-Philosophicus as discussed by the authors was the only philosophical work that Ludwig Wittgenstein (1889-1951) published during his lifetime, and it immediately convinced many of its readers and captured the imagination of all.
Book
Introduction to Mathematical Philosophy
TL;DR: Bertrand Russell is the most important philosopher of mathematics of the twentieth century The author of The Principles of Mathematics and, with Alfred Whitehead, the massive Principia Mathematica, Russell brought together his formidable knowledge of the subject and his skills as a gifted communicator to provide a classic introduction to the philosophy of mathematics as discussed by the authors.
Book
Culture and Value
TL;DR: Wittgenstein's "Vermischte Bemerkungen" as mentioned in this paper contains a number of passages from various periods of Wittgenstein life, over half were written during his later years, and this edition adds material probably written in 1944.
Book
From Frege to Gödel: a source book in mathematical logic, 1879-1931
TL;DR: The fundamental texts of the great classical period in modern logic, some of them never before available in English translation, are here gathered together for the first time as discussed by the authors, and they are presented with a note that sets it in perspective, explains its importance, and points out difficulties in interpretation.
Journal ArticleDOI
Remarks on the Foundations of Mathematics. By L. Wittgenstein. Edited by G. H. Von Wright, R. Rhees and G. E. M. Anscombe. Translated by G. E. M. Anscombe. Pp. 400. 37s. 6d. 1956. (Basil Blackwell, Oxford)
TL;DR: Wittgenstein's work remains, undeniably, now, that off one of those few few philosophers who will be read by all future generations as mentioned in this paper, regardless of whether it is true or not.