Introduction

Abstract

It is commonly believed that mathematics is a pattern and an ideal of certainty and precision, and that within mathematics itself the role of an ideal is played by geometry. The sources of such opinions could be seen in the fact that just geometry has been developed (and taught) in the axiomatic-deductive form almost since its beginning. The origins of the axiomatic method should be detected in ancient Greece — it seems that Plato was the first who invented it and considered it to be the proper method of mathematics. This was connected with his assertion that objects of mathematics belong to the world of ideas and that mathematics which discovers interrelations between those objects and describes them is just a science about ideas. This implies in particular that the mathematical knowledge should be based on reason and that the proper method of mathematics is the axiomatic method consisting of accepting certain theorems without proof (axioms and postulates) and next deriving from them all other theorems. As an ideal example (and pattern) of applying this method to mathematics were considered Elements by Euclid (about 300 B.C.). This work played an enormous role in the development of the methodology of mathematics. In fact the paradigm established by Elements was the working paradigm till the beginning of the twentieth century.