Here is a recent draft of a paper I have been working on throughout my year at the Pittsburgh Center for the Philosophy of Science. It corresponds roughly to chapters III and IV of my book project where I go into more detail with examples and the significance for confirmation. I hope to post a more comprehensive overview of the project soon, but for now this may interest those working on both modeling and indispensability arguments.
Abstract: Many philosophers would concede that mathematics contributes to the abstractness of some of our most successful scientific representations. Still, it is hard to know what this abstractness really comes to or how to make a link between abstractness and success. I start by explaining how mathematics can increase the abstractness of our representations by distinguishing two kinds of abstractness. First, there is an abstract representation that eschews causal content. Second, there are families of representations with a common mathematical core that is variously interpreted. The second part of the paper makes a connection between both kinds of abstractness and success by emphasizing confirmation. That is, I will argue that the mathematics contributes to the confirmation of these abstract scientific representations. This can happen in two ways which I label "direct" and "indirect". The contribution is direct when the mathematics facilitates the confirmation of an accurate representation, while the contribution is indirect when it helps the process of disconfirming an inaccurate representation. Establishing this conclusion helps to explain why mathematics is prevalent in some of our successful scientific theories, but I should emphasize that this is just one piece of a fairly daunting puzzle.
Update (July 23, 2009): I have now linked to a new version of the paper.
Update (Sept. 30, 2010): This paper has been removed.