David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
History and Philosophy of Logic 31 (3):219-245 (2011)
When it comes to Wittgenstein's philosophy of mathematics, even sympathetic admirers are cowed into submission by the many criticisms of influential authors in that field. They say something to the effect that Wittgenstein does not know enough about or have enough respect for mathematics, to take him as a serious philosopher of mathematics. They claim to catch Wittgenstein pooh-poohing the modern set-theoretic extensional conception of a real number. This article, however, will show that Wittgenstein's criticism is well grounded. A real number, as an 'extension', is a homeless fiction; 'homeless' in that it neither is supported by anything nor supports anything. The picture of a real number as an 'extension' is not supported by actual practice in calculus; calculus has nothing to do with 'extensions'. The extensional, set-theoretic conception of a real number does not give a foundation for real analysis, either. The so-called complete theory of real numbers, which is essentially an extensional approach, does not define (in any sense of the word) the set of real numbers so as to justify their completeness, despite the common belief to the contrary. The only correct foundation of real analysis consists in its being 'existential axiomatics'. And in real analysis, as existential axiomatics, a point on the real line need not be an 'extension'
|Keywords||No keywords specified (fix it)|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
Alexander George (2002). Philosophies of Mathematics. Blackwell Publishers.
Timm Lampert (2008). Wittgenstein on the Infinity of Primes. History and Philosophy of Logic 29 (1):63-81.
Victor Rodych (1999). Wittgenstein on Irrationals and Algorithmic Decidability. Synthese 118 (2):279-304.
S. G. Shanker (1987). Wittgenstein and the Turning Point in the Philosophy of Mathematics. State University of New York Press.
Ludwig Wittgenstein (1979). Notebooks, 1914-1916. University of Chicago Press.
Citations of this work BETA
No citations found.
Similar books and articles
Claude Laflamme & Marion Scheepers (1999). Combinatorial Properties of Filters and Open Covers for Sets of Real Numbers. Journal of Symbolic Logic 64 (3):1243-1260.
Kenneth L. Manders (1986). What Numbers Are Real? PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1986:253 - 269.
W. Balzer & M. Reiter (1989). Completeness for Systems Including Real Numbers. Studia Logica 48 (1):67 - 75.
Stewart Shapiro (2000). Frege Meets Dedekind: A Neologicist Treatment of Real Analysis. Notre Dame Journal of Formal Logic 41 (4):335--364.
Mathieu Marion (1995). Wittgenstein and Finitism. Synthese 105 (2):141 - 176.
António M. Fernandes & Fernando Ferreira (2002). Groundwork for Weak Analysis. Journal of Symbolic Logic 67 (2):557-578.
Michael Dummett (2000). Is Time a Continuum of Instants? Philosophy 75 (4):497-515.
William J. Collins & Paul Young (1983). Discontinuities of Provably Correct Operators on the Provably Recursive Real Numbers. Journal of Symbolic Logic 48 (4):913-920.
Anne Newstead & Franklin James (2008). On the Reality of the Continuum. Philosophy 83 (01):117-28.
Added to index2010-08-11
Total downloads49 ( #34,092 of 1,102,850 )
Recent downloads (6 months)7 ( #36,679 of 1,102,850 )
How can I increase my downloads?