In this essay, I examine three interrelated issues in the philosophy of mathematics.
The first is an issue in ontology: do mathematical objects exist and, if so, what kind of objects are they? The second is an issue in epistemology: how do we have mathematical knowledge? The third is an issue about the application of mathematics: how can mathematics be applied to the physical world? These issues are, of course, interconnected, and I explore the relations between them by examining different philosophical conceptions about mathematics.
I start with four versions of Platonism (Fregean Platonism, Godelian Platonism, Quinean Platonism, and structuralist Platonism), and I focus on the different epistemological strategies that have been developed to explain the possibility of mathematical knowledge. Given that, on these views, mathematical objects exist and are not physically extended, accounts of how we can have knowledge of these objects are in order. I then consider three alternative nominalist approaches (mathematical fictionalism, modal structuralism, and deflationary nominalism), which deny the existence of mathematical objects. With this denial, nominalists owe us, in particular, an explanation of the application of mathematics. I then examine their strategies to make sense of this issue, despite the non-existence of mathematical entities. I conclude the essay by bringing together the various issues and the philosophical conceptions.1.
Source:
Allhoff F.. Philosophies of the Sciences: A Guide. N.-Y.: Wiley-Blackwell,2010. — 386 p.. 2010
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