Unification and mathematical explanation

This paper provides a sorely-needed evaluation of the view that mathematical explanations in science explain by unifying. Illustrating with some novel examples, I argue that the view is misguided. For believers in mathematical explanations in science, my discussion rules out one way of spelling out how they work, bringing us one step closer to the right way. For non-believers, it contributes to a divide-and-conquer strategy for showing that there are no such explanations in science. My discussion also undermines the appeal to unifying power in support of the enhanced indispensability argument.

Platonic Relations and Mathematical Explanations

Some scientific explanations appear to turn on pure mathematical claims. The enhanced indispensability argument appeals to these ‘mathematical explanations’ in support of mathematical platonism. I argue that the success of this argument rests on the claim that mathematical explanations locate pure mathematical facts on which their physical explananda depend, and that any account of mathematical explanation that supports this claim fails to provide an adequate understanding of mathematical explanation.

Quine and the Incoherence of the Indispensability Argument

It is an under-appreciated fact that Quine’s rejection of the analytic/synthetic distinctionwhen coupled with some other plausible and related viewsimplies that there are serious difficulties in demarcating empirical theories from pure mathematical theories within the Quinean framework. This is a serious problem because there seems to be a principled difference between the two disciplines that cannot apparently be captured in the orthodox Quienan framework. For the purpose of simplicity let us call this Quine’s problem of demarcation. In this paper this problem will be articulated and it will be shown that the typical sorts of responses to this problem are all unworkable within the Quinean framework. It will then be shown that the lack of resources to solve this problem within the Quinean framework implies that Quine’s version of the indispensability argument cannot get off the ground, for it presupposes the possibility of making such a distinction.

Joseph Melia’s nominalism and the indexing theory of numbers

According to the Quine-Putnam indispensability argument, we are committed to all the entities that are indispensable to our best scientific theory. John Melia argues contra Quine-Putnam by claiming that even though such entities as numbers are indispensable to our best science, there is reason to deny their existence. In order to defend Melia?s theory from criticism put forth by Mark Colyvan, who demands that Melia provide a nominalistically acceptable paraphrase of our best scientific theory, supporters of this view have argued for the stronger claim that numbers are not indispensable. They all claim that numbers have an indexing role in the scientific explanation. In this article, I will consider some of the arguments for the indexing theory and point out its inadequacies.

Response to Van Inwagen and Welty

In response to my critics, I argue that Peter van Inwagen, despite his protestations, is an advocate of an indispensability argument for Platonism. What remains to be shown by van Inwagen is that his version of the argument overcomes his own presumption against Platonism and survives defeat by besting every anti-Platonist alternative. While acknowledging Greg Welty’s helpful responses to my worries about divine conceptualism as a realist alternative to Platonism, I express ongoing reservations about some of those responses.

Explaining with Mathematics? From Cicadas to Symmetry

The putative explanatory role of mathematics is further pursued in this chapter in the context of the so-called indispensability argument. Our conclusion here is that the possibility of mathematical entities acquiring some explanatory role is not well motivated, even within the framework of an account of explanation that might be sympathetic to such a role. We also consider the claim that certain scientific features have a hybrid mathematico-physical nature, again in the context of a specific example, namely that of spin, but we argue that the assertion of hybridity also lacks strong motivation and comes with associated metaphysical costs. Furthermore, such claims fail to fully grasp the details of the interrelationships between mathematical and physical structures in general and the distinction between the mathematical formalism and its interpretation in particular.