Abstract

The lack of specific arithmetical axioms in Book VII has puzzled historians of mathematics. It is hardly possible in our view to ascribe to the Greeks a conscious undertaking to axiomatize arithmetic. The view that associates the beginnings of the axiomatization of arithmetic with the works of Grassman [1861], Dedekind [1888] and Peano [1889] seems to be more plausible. In this connection a number of interesting historical problems have been raised, for instance, why arithmetic was axiomatized so late. This question was first posed and quite conclusively answered by Yanovskaja [1956]. Her major and quite conclusive argument was that “algorithms in arithmetic have absolute character, while in geometry we have to do with reducibility algorithms”. In this paper we are going to draw attention on certain peculiarities of the construction of the arithmetical Books of Euclid’s "Elements" and show that Euclidean arithmetic is constructed by effective procedures. In spite of the use of inference by reductio ad absurdum and methods equivalent to mathematical induction, Euclidean arithmetic retains its finitary character. It is not full arithmetic, but a finitary fragment of classical arithmetic, and thereby there was not any internal reason for its axiomatization.