Part 1. Euclid’s Way of Building Mathematical Theories 3

Abstract. The persisting gap between the formal and the informal mathematics is due to an inadequate notion of mathematical theory behind the current formalization tech-niques. I mean the (informal) notion of axiomatic theory according to which a mathemat-ical theory consists of a set of axioms and further theorems deduced from these axioms according to certain rules of logical inference. Thus the usual notion of axiomatic metho