Title: How are Mathematical Objects Constituted? A Structuralist Answer
Abstract: In my view, structuralism as presented by Shapiro (1991, 1997), Resnik(1991), and elsewhere offers the most plausible philosophy of mathematics:Mathematics is about structures, indeed it is the science of pure structures.Structures have no mysterious ontological status, and hence mathematicsis not ontologically mysterious, either. Again, it is no mystery how we canacquire knowledge about structures and thus mathematical knowledge. Wefind structures everywhere. Hence, if mathematics is about structures, wecan apply mathematics everywhere. In this way, structuralism promises tooffer straightforward answers to the most pressing problems in the philo-sophy of mathematics.However, there are not only structures, there are also mathematical ob-jects, numbers, pairs, triangles, sets, etc. Concerning their nature, struc-turalism tends to metaphorics, the most preferred metaphor being thatmathematical objects are places in mathematical structures. Maybe it isnot really necessary to say more, since it is only the structures that reallymatter. Still, I think one should be explicit and precise about mathema-tical objects, and this is what this paper is intended to achieve. I tend tothink that the amendment it adds to structuralism is both trivial and ob-ligatory. Maybe, though, it is contested and hence of substantial interest.In order to say what mathematical objects are one needs to have aconception of general ontology. I shall present such a conception very sket-chily in section 1, just as much as to able to explain my preferred versionof Leibniz’ principle in section 2, which will become important when I amgoing to present and defend in section 3 how I think that mathematicalobjects are constituted.
Publication Year: 2006
Publication Date: 2006-01-01
Language: en
Type: article
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Cited By Count: 2
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