ON THE BASIC QUALITATIVE ASPECTS OF ARITHMETIC
Jorge Carrera-Bolaños · 2012
In a world of computing, the question of the limits of computability takes grear importance and there have been a lot of attention to it. But no attention has been given to the theoretical origins of calculating. In this paper it is shown that there can be an Arithmetic (even if very restricted) without a calculating structure (the semigroup structure given by addition). This is done showing that there is a semantically closed proper subset of Peano’s axiom system that does not include addition but even so gives rise to what can be called a pre-arithmetical structure in the set of natural numbers. That means that the set of axioms defining the successor function builds a semantic entity independent of the axioms for addition; this concept is defined. Even if this is not a historical analysis, this situation implies that “counting” can be defined independent of “adding”. This discussion allows some interesting speculations concerning the possibility that human beings began “counting” as a qualitative process.