The Origin of Mathematics 13: The Role of Definitions

We have seen how Euclid formulates a definition of the sphere that goes back to the actual process of making one. Plato, in this case more advanced than Euclid, seems to waver between the artisan’s and the mathematician’s definition when he adds the condition of equidistance from a center. The example shows how a particular mathematical concept changes from a practical and realistic origin toward a more abstract notion that fits better into a developed theory. Usually in the Elements we find this mathematically advanced kind of definition. Many of them are explicit descriptions of geometrical objects like “equilateral triangle” or “gnomon,” which simply determine certain notions for later use. However, not everything can be defined. Every mathematical theory starts with some undefined concepts. Today these may be in principle no more than “set” and “function” or one of these two, but one cannot start from nothing. In the Elements a typical example of this sort is “to measure.” We are never told what “measuring” is, in spite of its fundamental importance in Book V for magnitudes and in Book VII for numbers. Somehow one knows what is meant, and it works. We have strong connotations of an everyday procedure that help us to make sense of the concept.