David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Noûs 47 (3):467-481 (2013)
It is often alleged that, unlike typical axioms of mathematics, the Continuum Hypothesis (CH) is indeterminate. This position is normally defended on the ground that the CH is undecidable in a way that typical axioms are not. Call this kind of undecidability “absolute undecidability”. In this paper, I seek to understand what absolute undecidability could be such that one might hope to establish that (a) CH is absolutely undecidable, (b) typical axioms are not absolutely undecidable, and (c) if a mathematical hypothesis is absolutely undecidable, then it is indeterminate. I shall argue that on no understanding of absolute undecidability could one hope to establish all of (a)–(c). However, I will identify one understanding of absolute undecidability on which one might hope to establish both (a) and (c) to the exclusion of (b). This suggests that a new style of mathematical antirealism deserves attention—one that does not depend on familiar epistemological or ontological concerns. The key idea behind this view is that typical mathematical hypotheses are indeterminate because they are relevantly similar to CH
|Keywords||Continuum Hypothesis Undecidability indeterminacy Disagreement|
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References found in this work BETA
Mark Balaguer (1998). Platonism and Anti-Platonism in Mathematics. Oxford University Press.
George Boolos (1998). Must We Believe in Set Theory? In Richard Jeffrey (ed.), Logic, Logic, and Logic. Harvard University Press. 120-132.
Michael A. E. Dummett (1991). The Logical Basis of Metaphysics. Harvard University Press.
Solomon Feferman, Harvey M. Friedman, Penelope Maddy & John R. Steel (2000). Does Mathematics Need New Axioms? Bulletin of Symbolic Logic 6 (4):401-446.
Hartry Field (1989). Realism, Mathematics & Modality. Basil Blackwell.
Citations of this work BETA
Justin Clarke-Doane (2014). Moral Epistemology: The Mathematics Analogy. Noûs 48 (2):238-255.
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