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Fregean One-to-one Correspondence and Numbers as Object Properties

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Author(s): Boris Grozdanoff

Journal: Principia : an International Journal of Epistemology
ISSN 1414-4247

Volume: 13;
Issue: 3;
Start page: 327;
Date: 2009;
Original page

Keywords: Numbers | properties | one-to-one correspondence | Frege

ABSTRACT
The paper critically examines an unpopular line of Frege’s view on numbers in the Foundations of Arithmetic. According to this view, which analyzes numbers in terms of properties and not in terms of extensions, numbers are properties of concepts vs. properties of objects. The latter view is held by Mill and is famously criticized in the Foundations. I argue that on the property account numbers cannot only be properties of concepts but they also have to be properties of objects. My main argument rests on purely metaphysical grounds. It stems from the motivation that were numbers only properties of concepts we would not have been able to explain mathematical truths about the physical world or those truths would have been miraculous. On pains that we do have mathematical truths about the physical world that are not miraculous we cannot agree with Frege’s property line about the metaphysical nature of numbers.
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