David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Philosophy 83 (01):117-28 (2008)
In a recent article (‘The Continuum: Russell’s Moment of Candour’), Christopher Ormell argues against the traditional mathematical view that the real numbers form an uncountably inﬁnite set. He rejects the conclusion of Cantor’s diagonal argument for the higher, non-denumerable inﬁnity of the real numbers. He does so on the basis that the classical conception of a real number is mys- terious, ineffable, and epistemically suspect. Instead, he urges that mathematics should admit only ‘well-deﬁned’ real numbers as proper objects of study. In practice, this means excluding as inadmissible all those real numbers whose decimal expansions cannot be calculated in as much detail as one would like by some rule. We argue against Ormell that the classical realist account of the continuum has explanatory power in mathematics and should be accepted, much in the same way that "dark matter" is posited by physicists to explain observations in cosmology. In effect, the indefinable real numbers are like the "dark matter" of real analysis.
|Keywords||continuum Cantor's diagonal argument realism philosophy of mathematics Bertrand Russell real numbers infinity|
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