Aviv Keren

Ordinality and cardinality, in the finite domain, are ordinarily considered as mere aspects of the very same objects, the natural numbers. Yet Steiner (Mathematics – application and applicability. In: Shapiro S (ed) The Oxford handbook of philosophy of mathematics and logic. Oxford University Press, 2005) draws attention to the intricate interplay between them, which is made implicit by this conception of them. In this chapter, I present a fitting cognitive framework and use it to account for how this situation comes to be. The framework concerns basic principles of object representation in the cognitive system, and particularly when and how different “aspects” of objects come, developmentally, to be integrated into a deeper, combined representation. Our coming to master the natural numbers is then presented as the process of such integration of ordinals and cardinals, based on their well-behaved interaction in the finite domain. This regular connection in the finite domain between ordinality and cardinality, however, is grasped automatically by the cognitive system as a statistical pattern rather than being stated explicitly, mathematically. Philosophically, this case study thus serves as a warning against ignoring cognition’s inner workings. The upshot is a novel, cognitively motivated account of the natural numbers.