The counting process and category theory: The psychogenesis of the natural numbers

The laws of arithmetic and basic theorems of elementary number theory were established long before Peano axioms were formulated. Counting was usually supposed to be the source of this knowledge. This idea has been suppressed in philosophical circles since the advent of set theory and mathematical logic, especially after Frege\u27s attacks against psychologism. However, as Willard Van Orman Quine says, those set theoretic definitions of number are notoriously irrelevant to psychogenesis ( Roots of Reference , 116). In our psychogenetic account of numbers, we revive the old idea that our knowledge of arithmetic and number theory is based upon counting. Since counting is a process, and category theory is the mathematical theory of morphisms, which naturally represent processes, we use category theory as a tool in modelling the counting process. We argue that by attending to some basic features of the counting process, we can discover the facts about natural numbers; in fact, making use of a theorem of Lawvere\u27s, we derive the Peano axioms from the basic features of counting. In doing this we try to have our account satisfy the two Benacerrafian constraints: that the semantics should agree with the standard semantics of scientific discourse, and that an account of our mathematical knowledge should reveal the epistemic connection with the source of our beliefs about numbers. We argue that the surface semantics of number statements is justified by recourse to the counting techniques and the deep semantics of counting processes. As an important aside we obtain an explanation of how we come up with the standard notion of natural numbers. The counting process goes all the way with the standard natural numbers, and not a step beyond them, thus it provides us with the standard notion of natural numbers without provoking any nonstandard notion of natural numbers. Finally, we discuss why our account escapes Frege\u27s criticism from psychologism and we defend the possibility of communication about numbers against the Wittgensteinian admonitions about rule following