On the Reality of the Continuum Discussion Note: A Reply to Ormell, ‘Russell's Moment of Candour’, Philosophy
Philosophy 83 (1):117-127 (2008)
Abstract
In a recent article, 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 analysisAuthor's Profile
DOI
10.1017/s003181910800034x
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Citations of this work
An Aristotelian Realist Philosophy of Mathematics: Mathematics as the science of quantity and structure.James Franklin - 2014 - London and New York: Palgrave MacMillan.
References found in this work
Achievements and fallacies in Hume's account of infinite divisibility.James Franklin - 1994 - Hume Studies 20 (1):85-101.
Wittgenstein, Finitism, and the Foundations of Mathematics.Mathieu Marion - 1998 - Studia Logica 66 (3):432-434.