Abstract: This chapter consists in development of some of the key ideas of the previous chapter. It begins by showing how an arithmetical epistemology along the lines suggested in the previous chapter sits well with a structuralist conception of arithmetic and arithmetical concepts (although it is also compatible with other views). It cautions against a simplistic understanding of the envisaged grounding relationship between concepts and sensory input, arguing that the proposed account allows us to say that our arithmetical beliefs count as knowledge by the lights of Chapter 3. It shows that the account is consistent with realism as characterized in Chapter 1. The chapter spends some time discussing the crucial notion of unconceptualized sensory input, and also offers some comments on what is called here ‘ungrounded’ and ‘unfitting’ concepts.

Keywords: structuralism, simplistic, unconceptualized, sensory input, non-conceptual content, ungrounded, unfitting, concepts,