Against the Judgment-Dependence of Mathematics and Logic
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Although the case for the judgment-dependence of many other domains has been pored over, surprisingly little attention has been paid to mathematics and logic. This paper presents two dilemmas for a judgment-dependent account of these areas. First, the extensionality-substantiality dilemma: in each case, either the judgment-dependent account is extensionally inadequate or it cannot meet the substantiality condition (roughly: non-vacuous specification). Second, the extensionality-extremality dilemma: in each case, either the judgment-dependent account is extensionally inadequate or it cannot meet the extremality condition (roughly: absence of independent explanation). The paper concludes with a moral concerning the judgment-dependence of a posteriori areas of discourse that emerges from consideration of these two a priori cases.
KeywordsBasic Equation Logical Consequence Turing Machine Logical Truth Continuum Hypothesis
This paper was written in 2005 under the welcoming roof of the New York Public Library. I am grateful to audiences at the 2005 Bristol Logic Colloquium, the Oxford Philosophy of Mathematics Seminar and the Oxford Philosophical Society for subsequent discussion, and to Mark Sprevak and Ralph Wedgwood for comments.
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