In this Introduction, I distinguish between two a posteriori views with respect to mathematical knowledge. According to the epistemological a posteriori position, mathematical knowledge is acquired empirically. According to the methodological a posteriori approach, empirical research is important for understanding what mathematical knowledge is like. I emphasise the need for the latter in the epistemology of arithmetic, while also accepting the importance of a priori methodology. However, empirical researchers and philosophers of mathematics do not share a common conceptual framework, which makes successful interdisciplinary research difficult. After pointing out some of the key problems, I provide a coherent conceptual framework and consistent terminology.

In Chapter 8, I deal with the threat that the present account strips arithmetical knowledge of all the important characteristics traditionally associated with it: apriority, objectivity, necessity and universality. I argue that apriority can be saved in the strong sense of arithmetical knowledge being contextually a priori in the context set by our cognitive and physical capacities. Objectivity can be saved in the sense of maximal inter-subjectivity, while necessity can be saved in the sense of arithmetical theorems being true in all possible worlds where cognitive agents with proto-arithmetical abilities have developed. Finally, universality of arithmetical truths is saved through arithmetic being universally applicable and shared by all members of cultures that develop arithmetic based on proto-arithmetical abilities.