Explanation, Extrapolation, and Existence

Abstract

Colyvan [2010] raises two problems for "easy road" nominalism about mathematical objects. The first is that a theory's mathematical commitments may run too deep to permit the extraction of nominalistic content. Taking the math out is, or could be, like taking the hobbits out of Lord of the Rings. I agree with the "could be," but not (or not yet) the "is." A notion of logical subtraction is developed that supports the possibility, questioned by Colyvan, of bracketing a theory's mathematical aspects to obtain, as remainder, what it says "mathematics aside." The other problem concerns explanation. Several grades of mathematical involvement in physical explanation are distinguished, by analogy with Quine's three grades of modal involvement. The first two grades plausibly obtain, but they do not require mathematical objects. The third grade is likelier to require mathematical objects. But it is not clear from Colyvan's example that the third grade really obtains.

Putnam and contemporary fictionalism

“…In other words, it does not seem that Hellman's argument would convey a real problem for nominalist or fictionalist programs in case they succeeded to provide an account of the applicability of mathematics understood in the narrow sense mentioned above. After all, they are happy to accept that mathematics has an indispensable descriptive and structural role (Yablo 2012(Yablo , 1020, or even that mathematics is indispensable in order to prove what follows from his synthetic version of Newtonian Physics (Field 1989, 241).…”

Nominalism, Trivialism, Logicism

“…Another way to see that extricability claims can be problematic is to consider the question of what would be left if one subtracted someone is thirsty from I'm thirsty (Yablo 2012); or the question of what would be left if one subtracted the tomato is red from the tomato is scarlet (Searle & Körner 1959, Woods 1967, Kraemer 1986, Yablo 2012)? It's not clear that there are well-defined answers to be given-unless, of course, one is prepared to say 'nothing'.…”

“…φ is true as far as its non-mathematical subject-matter is concerned, where a claim's non-mathematical subject-matter is defined as the set of worlds which agree in all non-mathematical respects with a world at which the claim is literally true (Yablo 2012).…”

Infinitesimal idealization, easy road nominalism, and fractional quantum statistics

“…Nevertheless, the case study that I consider, the relevance of Norton's distinction between approximation and idealization, and the particular problems posed by infinite/infinitesimal idealizations for issues in philosophy of mathematics have been largely ignored to my best knowledge. 2 E.g., Colyvan (2010Colyvan ( , 2012, Azzouni (2012), Bueno (2012), Leng (2012), Liggins (2012), and Yablo (2012). Also see even more recent essays in the special issue Molinini et al (2016).…”

“…In what follows, I first introduce Leng's (2005Leng's ( , 2010Leng's ( , 2012 approach to easy road nominalism in Section 2 and show how it depends on the idea that a physical system can "approximately instantiate" mathematical structure where a mathematical explanation is at hand. Since Leng (2005Leng ( , 2010Leng ( , 2012 does not does explicate what is meant by "approximate instantiation," I critically consider a possible way of expounding on her account via John Norton's (2012) well--received distinction between approximations and idealizations in Section 3: physical structure can approximately instantiate 8 In hope to thwart an objection early, note that I'm only partially defending Colyvan (2010) since there are other approaches to easy road nominalism, e.g., Azzouni (2012), Bueno (2012), Liggins (2012), andYablo (2012), which purport to make sense of mathematical explanation. Also recall that the overarching goal of this paper is to make interesting connections between infinite and infinitesimal idealization in science and the easy road nominalism debate-it is not to mount a comprehensive defense of Colyvan's (2010) thesis.…”