Breaking the Tie: Benacerraf’s Identification Argument Revisited

Philosophia Mathematica 31 (1):81-103 (2023)
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Abstract

Most philosophers take Benacerraf’s argument in ‘What numbers could not be’ to rebut successfully the reductionist view that numbers are sets. This philosophical consensus jars with mathematical practice, in which reductionism continues to thrive. In this note, we develop a new challenge to Benacerraf’s argument by contesting a central premise which is almost unanimously accepted in the literature. Namely, we argue that — contra orthodoxy — there are metaphysically relevant reasons to prefer von Neumann ordinals over other set-theoretic reductions of arithmetic. In doing so, we provide set-theoretical facts which, we believe, are crucial for informed assessment of reductionism.

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References found in this work

What numbers could not be.Paul Benacerraf - 1965 - Philosophical Review 74 (1):47-73.
Frege’s Conception of Numbers as Objects.Crispin Wright - 1983 - Critical Philosophy 1 (1):97.
Finitism.W. W. Tait - 1981 - Journal of Philosophy 78 (9):524-546.

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