This book not only outlines the indispensability argument in considerable detail but also defends it against various challenges.

This book does three main things. First, it defends mathematical platonism against the main objections to that view (most notably, the epistemological objection and the multiple-reductions objection). Second, it defends anti-platonism (in particular, fictionalism) against the main objections to that view (most notably, the Quine-Putnam indispensability objection and the objection from objectivity). Third, it argues that there is no fact of the matter whether abstract mathematical objects exist and, hence, no fact of the matter whether platonism or anti-platonism is true.

The subject with which Mark Colyvan's book deals is timely indeed. While discussions of mathematical ontology have been a mainstay in philosophy of mathematics for the last century (at least), for the last thirty years or so this discussion has begun with (and often not left) the Quine/Putnam indispensability argument. Though the argument is widely cited, to my knowledge this is the first book-length project exclusively dedicated to articulating and defending the Quine/Putnam indispensability argument for mathematical platonism. In the first (...) section below, I shall run through a summary of the book (following its chapter layout) with a few critical remarks. In the second section I shall raise a few critical issues organized topically. (shrink)

This paper sets out a new framework for discussing a long-standing problem in the philosophy of mathematics, namely the connection between the physical world and a mathematical domain when the mathematics is applied in science. I argue that considering counterfactual situations raises some interesting challenges for some approaches to applications, and consider an approach that avoids these challenges.

We examine, from the partial structures perspective, two forms of applicability of mathematics: at the “bottom” level, the applicability of theoretical structures to the “appearances”, and at the “top” level, the applicability of mathematical to physical theories. We argue that, to accommodate these two forms of applicability, the partial structures approach needs to be extended to include a notion of “partial homomorphism”. As a case study, we present London's analysis of the superfluid behavior of liquid helium in terms of Bose‐Einstein (...) statistics. This involved both the introduction of group theory at the top level, and some modeling at the “phenomenological” level, and thus provides a nice example of the relationships we are interested in. We conclude with a discussion of the “autonomy” of London's model. (shrink)