Abstract

According to various conceptions of arithmetical reality, numbers are sui generis entities, whose definition does not fundamentally depend on counting as an activity of the mind. Such views raise a challenge, however, which is to determine how numbers are represented in computers and other counting systems. In this paper, we take a counting-first perspective, examining how numbers could be defined from the internalist perspective of a counting system. First, we highlight the basic modules needed to perform counting in both human and artificial systems. Our main premise is that counting is a stateful activity involving memory. To analyze the basic structure of counting, we use stacks, namely memory structures that are in one sense simpler and in another sense richer than the structure given by the standard model of the Dedekind-Peano axioms. We go on to propose a definition of natural numbers as equivalence classes of memory states of a counting system. We argue that, contrary to appearances, this definition does not imply a narrow form of anti-realism. We use it to develop an argument for the epistemological priority of ordinal numbers over cardinal numbers.