Platonism and aristotelianism in mathematics

Philosophia Mathematica 16 (3):310-332 (2007)
Philosophers of mathematics agree that the only interpretation of arithmetic that takes that discourse at 'face value' is one on which the expressions 'N', '0', '1', '+', and 'x' are treated as proper names. I argue that the interpretation on which these expressions are treated as akin to free variables has an equal claim to be the default interpretation of arithmetic. I show that no purely syntactic test can distinguish proper names from free variables, and I observe that any semantic test that can must beg the question. I draw the same conclusion concerning areas of mathematics beyond arithmetic. This paper is a greatly extended version of my response to Stewart Shapiro's paper in the conference 'Structuralism in physics and mathematics' held in Bristol on 2–3 December, 2006.
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DOI 10.1093/philmat/nkm035
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References found in this work BETA
Fraser MacBride (2003). Speaking with Shadows: A Study of Neo-Logicism. British Journal for the Philosophy of Science 54 (1):103-163.

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Francesca Boccuni (2013). Plural Logicism. Erkenntnis 78 (5):1051-1067.

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