David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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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'.
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References found in this work BETA
Frank Plumpton Ramsey (1960). The Foundations of Mathematics and Other Logical Essays. Paterson, N.J.,Littlefield, Adams.
Ludwig Wittgenstein (1979). Notebooks, 1914-1916. University of Chicago Press.
Ludwig Wittgenstein (1975/1980). Philosophical Remarks. University of Chicago Press.
Citations of this work BETA
Ryan Dawson (2015). Wittgenstein on Set Theory and the Enormously Big. Philosophical Investigations 38 (4).
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