Conceptual and mathematical modelling - the platonic dichotomy and the Euclidean anthyphairetic method

摘要

This article focuses on the concept of modelling and more precisely on the relation between conceptual and mathematical modelling. A simple example of an exact numerical mathematical model is the ratio of two numbers, which arises from the geometrical and philosophical concept of analogy. The ratio of two numbers expresses the relationship between two magnitudes A and B. The discovery of irrational ratios and the need to suggest ways to approximate them has imposed the philosophical and mathematical thought of that time to go a little bit further. In the field of philosophy, this was accomplished by Plato and his dichotomous division method, and in the field of mathematics, it was expressed later by Euclid and his method of anthyphairesis, which introduces the fundamental notions of creating sequences approximating irrational numbers. Such developments introduce the notion of approximation, which is central in defining models of dynamical processes in the modern context of applications.