Noûs 47 (3):467-481 (2013)
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Abstract |
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.
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Keywords | Continuum Hypothesis Undecidability indeterminacy Disagreement |
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DOI | 10.1111/j.1468-0068.2012.00861.x |
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References found in this work BETA
Science Without Numbers: A Defence of Nominalism.Hartry H. Field - 1980 - Princeton, NJ, USA: Princeton University Press.
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Citations of this work BETA
When Expert Disagreement Supports the Consensus.Finnur Dellsén - 2018 - Australasian Journal of Philosophy 96 (1):142-156.
A Metasemantic Challenge for Mathematical Determinacy.Jared Warren & Daniel Waxman - 2020 - Synthese 197 (2):477-495.
In Defense of Countabilism.David Builes & Jessica M. Wilson - forthcoming - Philosophical Studies:1-38.
Set-Theoretic Justification and the Theoretical Virtues.John Heron - 2020 - Synthese 199 (1-2):1245-1267.
View all 10 citations / Add more citations
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