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Modal structuralism simplified

Published online by Cambridge University Press:  01 January 2020

Sharon Berry*
Van Leer Jerusalem Institute, Jerusalem, Israel.


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.

Copyright © Canadian Journal of Philosophy 2017

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Benacerraf, P. 1965. “What Numbers could not be.” The Philosophical Review 74(1): 4773.Google Scholar
Etchemendy, J. 1990. The Concept of Logical Consequence. Cambridge, MA: Harvard University Press.Google Scholar
Field, H. 1989. Realism, Mathematics & Modality. Oxford: Basil Blackwell.Google Scholar
Field, H. H. 2008. Saving Truth from Paradox. Oxford: Oxford University Press.CrossRefGoogle Scholar
Hellman, G. 1994. Mathematics without Numbers. Oxford: Oxford University Press.Google Scholar
Hellman, G. 1996. “Structuralism without Structures.” Philosophia Mathematica 4(2): 100123.Google Scholar
Lewis, D. K. 1986. On the Plurality of Worlds. Oxford: B. Blackwel.Google Scholar
Linnebo, Ø. 2010. “Pluralities and Sets.” Journal of Philosophy 107(3): 144164.Google Scholar
McGee, V. 1997. “How we learn mathematical language.” Philosophical Review 106(1): 3568.CrossRefGoogle Scholar
Parsons, C. 1977. What is the Iterative Conception of Set?, 335367. Dordrecht: Springer.Google Scholar
Parsons, C. 2007. Mathematical Thought and its Objects. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Putnam, H. 1967. “Mathematics without Foundations.” Journal of Philosophy 64(1): 522.CrossRefGoogle Scholar
Rayo, A. 2013. The Construction of Logical Space. Oxford: Oxford University Press.CrossRefGoogle Scholar
Shapiro, S., and Wright, C.. 2006. “All Things Indefinitely Extensible.” In Absolute Generality, edited byRayo, Agust\’{\i}n and Uzquiano, Gabriel, 255304. Oxford University Press.Google Scholar
Weisstein, Eric W.. “Peano’s Axioms.” From MathWorld-A Wolfram Web Resource. Scholar
Williamson, T. 2013. Modal Logic as Metaphysics. Oxford University Press: Oxford.CrossRefGoogle Scholar