Ontological free snack
This Tuesday Øystein Linnebo gave a talk at the department on what he called „meta-ontological minimalism” (see the abstract here). His aim was to defend realism about mathematical entities (aka „Platonism„) by arguing that such an ontological addition is rather „cheap”, and he proposed an interpretation of the Fregean, abstractionist approach to ontological commitment that would generate a sufficient condition for the existence of mathematical objects (I won’t go into details now; on his website there are available a couple of papers that elaborate on this approach). The talk was very nice, in fact, one of the nicest we’ve had at CEU since I got here, and it complements nicely the talk given by Bob Hale in November (abstract here).
Crucial to his view is the distinction between „thick” and „thin” objects. Thick objects are those whose existence imposes costly requirements on the world; for instance, physical objects require that some physical laws be satisfied, which means that some physical properties must be instantiated and quite a few objects of the same and lawfully correlated kinds must exist as well. By contrast, thin objects have very lax requirements of the world; numbers (conceived as abstract objects) do not require the existence of physical objects or some law of nature. All this, however, is only the intuitive way of drawing the distinction. As Kati pointed out during discussion, Platonic (ante rem) universals are difficult to place on one or the other side of the divide; they come out as „real” semantic values for nominalizations of predicates, just as mathematical objects do, and do not occupy space-time regions. Yet they seem to somehow determine how the world is in a rather substantial way. Similar worries may arise for possibilia, and perhaps intrinsic values. (In the case of ante rem universals, Øystein replied that such a theory must conceal some kind of inconsistency.) The issue gets complicated by the fact that he wants to retain an unambiguous notion of existence (the handout contained the phrase „more attenuated existence”, which seems to imply a form of gradualism, but he made clear in discussion that he didn’t intend it in this sense).
One possible solution that he didn’t take into consideration is to employ a kind of pragmatic or epistemic criterion for delineating physical (thick) and mathematical (thin) objects from other putative (here’s a nice label: „kooky”) objects, for instance, that our current best scientific theories (physical and mathematical alike) make reference to both thick and thin objects, but not to kooky ones. I am not sure whether or not this line of reasoning would lead to a kind of indispensability argument, and, if so, whether he would accept it (some people who endorse the indispensability argument, such as Quine or Resnik, are non-Fregean about ontological commitment–that is, they take it to be cashed out by means of quantification rather than reference). In any case, he sometimes seemed to suggest that all wannabe Platonists are in the same boat, so the topic and his own solution surely is worth pursuing for those craving for this kind of ontological free snack.