Article ID: CBB001210976

Wittgenstein and the Real Numbers (2010)

unapi

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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Authors & Contributors
Loner, David
John Erik Hmiel
Chambris, Christine
Ahmet Süner
Nakano, Anderson
Visnovska, Jana
Concepts
Philosophy
Philosophy of mathematics
Number theory; number concept
Mathematics
Logic
Epistemology
Time Periods
20th century
Ancient
21st century
20th century, early
19th century
17th century
Places
United States
France
Great Britain
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