Evolution of the Modes of Systematization of Mathematical Knowledge

Abstract

There is a widespread opinion that prior to the beginning of theoretical mathematics there was only a conglomeration of heterogeneous empirical results dealing with the measurement of areas and volumes and working with the simplest diagrams. In many cases these results had an approximate character and were not related to one another. On this view, empirical material simply accumulated in the mathematics of ancient Egypt and Mesopotamia, while in ancient Greece, mathematical knowledge became a system of interrelated propositions. B.L. van der Waerden’s hypothesis about the existence of theoretical mathematical knowledge in certain nations of Western Europe in the middle of the third millenium B.C. is at first glance quite at odds with this interpretation (van der Waerden 1983). However, his hypothesis also opposes theoretically systematized knowledge to the empirical, unsystematized knowledge of earlier ages. It is considered self-evident that the only way to organize mathematical knowledge, to unite mathematical material, is through the logical derivation of one statement from others. Such an approach to the origin of mathematical knowledge may be defined as a transition “from empirical aggregate to theoretical system.” This approach also appears in a subtle way in N. Bourbaki’s assessment: “…but Babylonian algebra, by virtue of the elegance and reliability of its methods, could not be conceived as a mere collection of problems resolved by empirical groping” (Bourbaki 1960, 9). Bourbaki contrasts the supposedly highly-systematized Babylonian algebra with the conglomerate of fragmentary, unsystematized problems.