In Chapter 9, I move the focus from epistemology to ontology. I ask what kind of objects natural numbers are and show why different types of realist answers are problematic. I then endorse a constructivist account, according to which natural numbers exist as social constructs through our shared number concepts. This account is compatible with the social constructivist views of Cole and Feferman. However, I see my account as an improvement on their work because it provides a strong explanation as to why numbers are widely applicable social constructs. This explanation is based on the important role that proto-arithmetical abilities have both for the way we experience our environment and the development of arithmetic. Finally, I discuss the question of whether natural numbers as social constructs exist primarily as ordinals or cardinals, showing that, although more empirical research is needed, there is no reason to currently believe that natural numbers must be fundamentally ordinal.

This chapter contrasts Parsonian potentialist set theory with the Putnamian potentialism and raises some questions about recent developments of the former by Linnebo and Studd.