The Priority of Object over Method in Early Greek Mathematics

Abstract

In modern mathematics, whose origins date back to the 17th c., method takes priority over the object, as the generality of method (which relies on a set of axioms and the laws of formal logic, applicable to any kind of objects whatsoever) implies the generality of the studied object. This epistemological attitude was alien to Greek mathematics, in which the generality of method did not necessarily mean the general character of its object. This is manifest, e.g., in Euclid's Elements, where the axiomatic-deductive method does not eradicate the particularity of the studied objects: numbers, lengths, plane figures, and solids. The aim of this paper consists in showing that in the early stages of Greek mathematics, when axiomatic-deductive way of argumentation was still unknown, object took priority over method in the sense that an investigation was not accomplished according to a definite method, fixed in advance, but rather relied upon methods that were suggested, in each particular case, by the structure of the appropriate objects. I will examine such a state of affairs by proposing reconstructions of archaic demonstrations of theorems taken from early geometry, arithmetic, and the theory of ratios and proportions.