Za horizont euklidovského geometrického názoru

Abstract

In the article we have mentioned questions, which are connected with the notion of “parallel straight lines” and with Euclid’s postulate about them. These problems arise in ancient geometrical world, which contains only those objects, which are placed in front of geometer’s horizon. Vopěnka’s Alternative set theory (or namely its application to the geometrical world, introduced in Calculus infinitesimalis pars prima1) is suitable for modeling of this ancient geometrical approach. There the properties “to be finite” or “to be finitely far” are mathematically exactly defined. Many attempts to solve the problem of parallel straight lines had been failing since Euclid until Gauss. This history is briefly mentioned in the second section. The solution of the problem of parallel straight lines came with Riemann’s lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen, where the context, showing the existence of non-Euclidean geometry in an obvious way, was introduced. This context consists in the notion of the multiply extended manifoldness, which contains not only the Euclidean but also the curved one. And the curved one is similarly obvious like the geometry on the sphere. Finally, thanks to Vopěnka’s conception of “horizon”, in the last section here is also shown, which relation may have the Riemann’s curved space and the classical euclidean one.