The Relation of Mathematics to the Other Sciences

Abstract

Within Western civilization mathematics has always been deeply related to the idea (or the ideal) of science as such, and this for several reasons. The first is that the concept of knowledge, in its fullest meaning and significance, was quickly identified by early Greek philosophers with something more demanding than simple truth. While Parmenides had distinguished truth (alétheia) from opinion (dóxa), Plato noted that we certainly have “true opinions”, but they do not constitute knowledge in a full sense, that form of knowledge that he calls science (epistéme). According to this view there is a weak form of knowledge (namely opinion, which may be true, but is contingent and unstable), and a strong form of knowledge, which is science, and is characterized as being demonstrative and, in such a way, endowed with necessity and stability¹. It is not difficult to recognize that such a requirement was imposed upon the ideal of science by the historical fact that mathematics had already attained in Greek culture the status of a demonstrative discipline. Indeed, several particular “mathematical truths” had been found by Egyptian and Mesopotamian scholars, but they consisted in the discovery of single instantiations of certain geometrical or numerical properties, while early Greek mathematicians were able to demonstrate general theorems, under which the said particular instantiations appeared to be contained, along with a potentially infinite amount of similar examples.