How are Mathematical Objects Constituted? A Structuralist Answer

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 mathematics is not ontologically mysterious, either. Again, it is no mystery how we can acquire knowledge about structures and thus mathematical knowledge. We find structures everywhere. Hence, if mathematics is about structures, we can apply mathematics everywhere. In this way, structuralism promises to offer straightforward answers to the most pressing problems in the philosophy of mathematics. However, there are not only structures, there are also mathematical objects, numbers, pairs, triangles, sets, etc. Concerning their nature, structuralism tends to metaphorics, the most preferred metaphor being that mathematical objects are places in mathematical structures. Maybe it is not really necessary to say more, since it is only the structures that really matter. Still, I think one should be explicit and precise about mathematical objects, and this is what this paper is intended to achieve. I tend to think that the amendment it adds to structuralism is both trivial and obligatory. Maybe, though, it is contested and hence of substantial interest. In order to say what mathematical objects are one needs to have a conception of general ontology. I shall present such a conception very sketchily in section 1, just as much as to able to explain my preferred version of Leibniz’ principle in section 2, which will become important when I am going to present and defend in section 3 how I think that mathematical objects are constituted.