Chapters 7–9 offer a general account of the growth of mathematics. Introduce the notion of a mathematical practice, a multidimensional entity consisting of a language, accepted statements, accepted questions, accepted means of inference, and methodological maxims. Mathematics grows by modifying one or more components in response to the problems posed by others. So new language, language that is not initially well understood, may be introduced in order to answer questions taken to be important but resisting solution by available methods; in consequence, there may be a new task of clarifying the language or taming the methods that the extended language allows. The chapters attempt to show how such processes have occurred in the history of mathematics, and how they link the rich state of present mathematics to the crude beginnings of the subject. In Chapter 7, in particular, Kitcher compares mathematical change with scientific change, attempting to show that the growth of mathematical knowledge is far more similar to the growth of scientific knowledge than is usually appreciated.
Keywords: mathematical knowledge, observation, scientific knowledge, theory, theory change