Mathematics and Scientific Representation, Claim 2: Individuate Contributions via Content

With the move to Missouri complete and my book safely in press, there is now an opportunity to continue blogging through the key claims of the book. The first key claim is that mathematics contributes many things to the success of science. Next comes

2. These contributions can be individuated in terms of the contents of mathematical scienti c representations.

In principle, we might divide up the contributions which mathematics makes in any number of ways. These include historically, by scientific discipline, by the most successful and by mathematical technique. But it seemed most productive to consider different sorts of scientific representations, where the 'sorts' were isolated by the content of that representation. The hope is then that representations are available which are more or less pure instances of this or that kind of representation. And, for the successful representations, we can then look at the mathematics and see how it is contributing to the success of that sort of representation.

Approached in this way, there is a crucial difference between two ways in which some parts of mathematics might relate to a given representation. Some mathematics will be intimately part of the mathematics, in the following sense: the content of the representation is that some mathematical structure bears some structural relation to a target system, under some interpretation. So, the mathematics related directly to that structure is what I call intrinsic to the representation. This has immediate consequences for grasping the content of the representation, which I think of as given by the accuracy conditions for that representation. To understand the representation, one must believe that the claims of the intrinsic mathematics are true.

On the other hand, there may be additional mathematical claims which are extrinsic to the representation, but which are still relevant to the success of that representation. As a central example, consider the case where some stronger mathematical theory is deployed to solve the equations given in some weaker mathematical theory. It is wrong to say that an understanding of the representation requires a belief in the stronger mathematical claims, but these claims are still relevant to a consideration of the ultimate success of the representation. More generally, the mathematical relationships between representations may go beyond the intrinsic mathematics of the representations, but these relationships can still be central to an account of the success of the representations.