Book excerpt: This essay was written following discussions with friends and colleagues from different disciplines about the use of mathematics in the social sciences. It sets out some of the advantages of mathematical modeling (MM) in the social sciences, and defends it against some of the common criticisms that I have encountered. The main argument is that theories of the social world necessarily involve abstractions and trade-offs between the detail of the analysis and its tractability. Mathematics is a way to make these trade-offs in the clearest possible way, and this clarity makes discussion and empirical testing easier. Not all MM is based on such Popperian arguments. Often, models have a more artistic aspiration, of providing a metaphor or caricature of some feature of reality in order to understand it better. Finally, like any research methodology, mathematical theorizing does have its limits and associated problems, which I will also briefly discuss.