Structuralism's unpaid epistemological debts

Philosophia Mathematica 4 (2):124--47 (1996)
One kind of structuralism holds that mathematics is about structures, conceived as a type of abstract entity. Another denies that it is about any distinctively mathematical entities at all—even abstract structures; rather it gives purely general information about what holds of any collection of entities conforming to the axioms of the theory. Of these, pure structuralism is most plausibly taken to enjoy significant advantages over platonism. But in what appears to be its most plausible—modalised—version, even restricted to elementary arithmetic, it requires defence of a very strong possibility claim: that there could be a completed w-sequence of concrete objects. There are very serious epistemological difficulties in the way of providing the requisite defence.
Keywords structuralism   logicism   mathematics   WAYS
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DOI 10.1093/philmat/4.2.124
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Charles Parsons (2004). Structuralism and Metaphysics. Philosophical Quarterly 54 (214):56--77.
Robin Jeshion (2014). Intuiting the Infinite. Philosophical Studies 171 (2):327-349.

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