Since the logicist turn in the philosophy of mathematics, it is customary to think of the structure of natural numbers as ontologically prior to any account of counting. In this paper, we take a systems-first perspective, in which numbers are defined from counting systems. First, we highlight the basic modules needed to perform counting in both human and artificial systems. Our central premise is that counting is a stateful activity involving memory. To expose the basic structure of counting, we…
Read moreSince the logicist turn in the philosophy of mathematics, it is customary to think of the structure of natural numbers as ontologically prior to any account of counting. In this paper, we take a systems-first perspective, in which numbers are defined from counting systems. First, we highlight the basic modules needed to perform counting in both human and artificial systems. Our central premise is that counting is a stateful activity involving memory. To expose 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. Based on that, we present a counting-first approach to the definition of natural numbers as equivalence classes of states of a counting system. Finally, we present an argument for the epistemological priority of ordinal numbers over cardinal numbers.
