3. A list of these contributions should include at least the following fi ve: concrete causal, abstract acausal, abstract varying, scaling and constitutive.The contrast between concrete causal and abstract acausal is perhaps the most intuitive. Many successful scientific representations purport to accurately represent the causal relationships which obtain in a target system. Different accounts of causation present different views on what this special causal content comes to. For example, one may require the representation of a certain sort of process or mechanism. Alternatively, approaches like Woodward's insist only on the representation of what would happen under a certain kind of intervention or manipulation of the system. If this is how we understand causal representations, then it is clear that many representations are acausal. They may abstract away from the constituents and their causal interactions. This can happen in several ways. In this chapter of the book I consider how mathematics helps both with causal representations and with acausal representations.

There is a second sort of abstraction: abstraction by varying. In this case, we have a family of representations with a different physical interpretation, but with a core overlap in their mathematics. In such cases, the mathematical links between the representations take center stage. It may be the case that all members of the family are causal representations, or some may abstract away from causes.

Aspects of many successful representations turn on considerations of scale. The scale of a feature can be thought of as a comparison between that feature and some given parameters. So, for example, we may consider the relative time scales of two processes and use this comparison to adjust our representation of some target system. More generally, procedures for understanding the relative scale of this or that magnitude are central to simplification and idealization. Unsurprisingly, there is a central place for mathematics in determining which manipulations are acceptable and what the best interpretation of the resulting representations should be.

Finally, I distinguish between constitutive and derivative representations. As neutrally as possible, we can think of a derivative representation as one which is successful only if some related 'constitutive' representations are successful. Carnap, Kuhn and Michael Friedman have all tried to motivate this distinction, and for each philosopher mathematical claims play a central role in the story. I side largely with Friedman on the need for constitutive representations, but try to pin down exactly what sort of success is at issue and how these two sorts of representations are related.