Tansu Alpcan

According to Frege's influential conception of arithmetic reality, numbers are not adequately conceived as ordinal structures, and their existence is independent of the activity of counting. In this paper, we challenge this view by taking a counting-first perspective, investigating how numbers might be defined from the internal perspective of a counting system. First, we spell out 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 counting, we use stacks, namely memory structures that are in one sense richer and in another sense more general 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.