ABSTRACT
What I shall call “elementary synthetic (plane) geometry” is essentially the practice of elementary school geometry. There we learned to draw two-dimensional figures on a sheet of paper, using certain fixed means such as a sufficiently plane surface to draw upon, a straight rule with two marked points on it (representing the chosen unit length), and a circle. And we “proved” geometrical statements by describing geometrical constructions represented by such drawings, together with some comments on possible movements of surfaces or on possible mirror-mappings. It is this special concept of “proof” that I propose to call “demonstration.” In this way we prove, as it well known, theorems like those of Thales or Pythagoras in elementary synthetic geometry. Hence, elementary synthetic geometry is not an axiomatized systemdespite the fact that Hilbert’s famous book Foundations of Geometry tried to reconstruct an axiomatic theory of geometry just as Peano had done for arithmetic, and Zermelo and others for set theory. In a fully axiomatized system a “proof” is just a series of deductions, that is, a series of applications of the rules of first-order predicate calculus, where one starts with a set of formulas called “axioms.” Hilbert’s geometry is no such theory for the following reasons: neither his Archimedean axiom nor his versions of the completeness axiom are formulated in a first-order language; and his proofs are not all first-order deductions, but are often semantic proofs.
