Newton AB Omni Naevo Vindicatus (1)

Abstract

Two theories, Euclid’s geometry and Newton’s mechanics, have persistently influenced the history of the exact sciences for centuries. Common to both is that their historical and systematic impact has been due not only to their outstanding positive achievements, but also to their specific deficiencies. As is well known, Euclid’s geometry, which is oriented upon the Aristotelian Ideal of theory, begins with a series of definitions which are a) insufficient from the modern logical standpoint or the standpoint of the theory of definition; (and which b) play a role neither in the proof of the theories, nor in the propositional content of the theory. Gauss was not the first to find (2) with the absence of (in modern terms) axioms of order - thus, essentially, of rules for the use of the word ‘between’ - in Euclid. It can be shown historically that the logical weakness of these definitions, as well as their systematic insignificance for geometry resulting from these weaknesses, have led to a twothousand year history of attempted repairs and, finally, to the modern, formalistic conception of geometry with D. Hilbert. According to this conception, the defining of fundamental terms was banned from mathematics by programmatic decree. (3) Many mathematicians, as well as philosophers of science have, somewhat too quickly, coupled this conception together with the notion that the basic concepts of geometry are not definable because they have not yet been defined.