Right. Unfortunately…! But what is counting, anyway? Surely, the word and the mental operations associated with it are not exclusively concerned with those maddening “problems” of leaking taps and the exact quantity of water they are liable to spill in a given amount of time; or with the number of miles it will take for a certain train to catch up with another one, taking into account their respective speed.

As a matter of fact, counting, for our distant forebears, did not apply exclusively to items belonging to the physical world. To put it differently, calculus was not necessarily understood as an abstract but useful tool for the study of nature, devoted to the task of reckoning, distributing, or classifying the elements of reality, as we know it. The agency of numbers was in fact extended to questions pertaining to the immaterial layers of being as well, inasmuch as numerical entities were perceived as governing central concepts of cosmology, such as harmony, measure, balance or proportion, which could also apply outside the limits of physical reality proper.

The question of knowing whether numbers actually exist separately on a plane of their own (mathematical “realism”), or whether they are no more than abstractions generated by the working of the human mind (“idealism”), is still being debated today among professional scientists and philosophers. Both stances, which the historical names of Plato (or Pythagoras) and Aristotle (fifth-fourth cent. BCE) have respectively come to stand for, can be traced back to classical Antiquity. From that period onwards, until well into the seventeenth century at least, numbers have often been closely related to the qualitative aspects of both cosmic and human nature.

This particular cultural attitude refers to a type of understanding of mathematics which imbues numbers with the capacity of signifying more than just the quantity they materially refer to. In this perspective, numbers are construed as manifesting effectively the invisible energies at work behind the veil of exterior reality, thus structuring and harmonising the physical world from inside (or from “above”). This view actually presupposes the idea (“neoplatonic,” in a very general way) of a multi-layered universe, in which numerical entities occupy a higher level of reality than the one corresponding to objects belonging to the ordinary physical plane.