Recursion Only on the Actual

Our purpose here is to focus attention on a development in computer science that has remarkable implications for the philosophy of mathematics. Bellantoni and Cook [BC] gave a purely syntactical characterization of polynomial-time computation. They indicate that their work was strongly influenced by previous work of Leivant, who subsequently [L] gave a more elegant form to the characterization, and it is appropriate to call the result the BCL theorem. This theorem is in strong contrast to the situation for general algorithms, where the familiar diagonalization argument precludes a syntactical characterization. (Enumerate the algorithmic descriptions of functions that satisfy the putative syntactical characterization, ordering them first by length and then lexicographically. Let fn be the nth function in the list, and let g(n) = fn(n) + 1. This is an algorithmic description of a function not in the list.) The key idea of the BCL theorem is to distinguish between actual numbers (called normal in [BC] and tier 1 in [L]) and potential numbers (called safe in [BC] and tier 0 in [L]). Potential numbers are numbers in the process of being constructed. In the BCL formalism the schema of primitive recursion is modified so that recursions occur only on actual numbers. The distinction between potential and actual was fundamental to Aristotle, but it has had scant influence on the static Platonic ethos of mathematics. The BCL theorem, of which this paper is an appreciation, appears to be the first mathematical work in which the distinction plays a major role.