Synthese 105 (2):141 - 176 (1995)
AbstractIn 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.
Similar books and articles
The Gödel Paradox and Wittgenstein's Reasons.Francesco Berto - 2009 - Philosophia Mathematica 17 (2):208-219.
The Later Wittgenstein and the Philosophy of Religion.Stig Børsen Hansen - 2010 - Philosophy Compass 5 (11):1013–22.
Wittgenstein, Finitism, and the Foundations of Mathematics.Mathieu Marion - 1998 - Oxford, England: Oxford University Press.
Added to PP
Historical graph of downloads
Citations of this work
Wittgenstein on Irrationals and Algorithmic Decidability.Victor Rodych - 1999 - Synthese 118 (2):279-304.
Wittgenstein, Ramsey and British Pragmatism.Mathieu Marion - 2012 - European Journal of Pragmatism and American Philosophy 4 (2).
The Hidden Set-Theoretical Paradox of the Tractatus.Jing Li - 2018 - Philosophia 46 (1):159-164.
Misunderstanding Gödel: New Arguments About Wittgenstein and New Remarks by Wittgenstein.Victor Rodych - 2003 - Dialectica 57 (3):279–313.
Misunderstanding Gödel: New Arguments About Wittgenstein and New Remarks by Wittgenstein.Victor Rodych - 2003 - Dialectica 57 (3):279-313.
References found in this work
Tractatus Logico-Philosophicus.Ludwig Wittgenstein, G. C. M. Colombo & Bertrand Russell - 1922 - Fratelli Bocca.
The Foundations of Mathematics and Other Logical Essays.Frank Plumpton Ramsey - 1925 - London, England: Routledge & Kegan Paul.
Introduction to Mathematical Philosophy.Bertrand Arthur William Russell - 1919 - London, England: Dover Publications.