Vector Spaces

Abstract

There are at least two methods of defining the basic notions of geometry. The one which appears more natural at first sight, and which is in many ways more satisfactory from the logical point of view, is the so called synthetic approach. This begins by postulating objects such as points, lines and planes and builds up the whole system of geometry from certain axioms relating these objects. In order to progress beyond a few trivial theorems, however, there must be sufficient axioms. Unfortunately, it is difficult to foresee the kind of axioms which are required in order to be able to prove what one regards as “fundamental” theorems (such as Pappus’ Theorem and Desargues’ Theorem). This method is very difficult as an introduction to the subject.