Canadian Journal of Philosophy 48 (2):200-222 (2018)

Sharon Berry
Oakland University
Since Benacerraf’s ‘What Numbers Could Not Be, ’ there has been a growing interest in mathematical structuralism. An influential form of mathematical structuralism, modal structuralism, uses logical possibility and second order logic to provide paraphrases of mathematical statements which don’t quantify over mathematical objects. These modal structuralist paraphrases are a useful tool for nominalists and realists alike. But their use of second order logic and quantification into the logical possibility operator raises concerns. In this paper, I show that the work of both these elements can be done by a single natural generalization of the logical possibility operator.
Keywords nominalism  Hellman  philosophy of math  logical possibility  potentialism  second order logic  modal structuralism
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Reprint years 2018
DOI 10.1080/00455091.2017.1344502
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References found in this work BETA

On the Plurality of Worlds.David Lewis - 1986 - Wiley-Blackwell.
Modal Logic as Metaphysics.Timothy Williamson - 2013 - Oxford University Press.
Saving Truth From Paradox.Hartry Field - 2008 - Oxford University Press.
On the Plurality of Worlds.David Lewis - 1986 - Revue Philosophique de la France Et de l'Etranger 178 (3):388-390.
Realism, Mathematics and Modality.Hartry Field - 1988 - Philosophical Topics 16 (1):57-107.

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Can All Things Be Counted?Chris Scambler - forthcoming - Journal of Philosophical Logic:1-28.

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