Mathematical Vectors and Physical Vectors

Abstract

From a metaphysical point of view, it is important clearly to see the ontological difference between what is studied in mathematics and mathematical physics, respectively. In this respect, the paper is concerned with the vectors of classical physics. Vectors have both a scalar magnitude and a direction, and it is argued that neither conventionalism nor wholesale anti-conventionalism holds true of either of these components of classical physical vectors. A quantification of a physical dimension requires the discovery of ontological order relations among all the determinate properties of this dimension, as well as a conventional definition that connects the number one and (in case of vector quantities) mathematical unit vectors to determinate spatiotemporal physical entities. One might say that mathematics deals with numbers and vectors, but mathematical physics with scalar quantities and vector quantities, respectively. The International System of Units (SI) distinguishes between basic and derived scalar quantities; if a similar distinction should be introduced for the vector quantities of classical physics, then duration in directed time ought to be chosen as the basic vector quantity. The metaphysics of physical vectors is intimately connected with the metaphysics of time. From a philosophical-historical point of view, the paper revives W. E. Johnson's distinction 'determinates-determinables' and Hans Reichenbach's notion of 'coordinative definition'.