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Essay on Projective Geometry
Geometry distinguishes itself from other natural sciences because it obtains only very few concepts and laws directly from experience, and aims at obtaining from them the laws of more complex phenomena by purely deductive means. The empirical foundation of geometry was encapsulated by Pasch in a core of basic concepts and basic statements or axioms. The basic concepts refer to the shape and size of bodies and their positions relative to one another. They are not defined, for no definition could replace the "exhibition of appropriate natural objects," which is the only road to understanding such simple, irreducible notions (ibid., p. 16). All other geometric concepts must be ultimately defined in terms of the basic ones. The basic concepts are connected to one another by the axioms, which "state what has been observed in certain very simple diagrams" (p. 43). All other geometric statements must be proved from the axioms by the strictest deductive methods.
Everything that is needed to prove them must be recorded, without exception, in the axioms. These must therefore embody the whole empirical material elaborated by geometry, so that "after they are established it is no longer necessary to resort to sense perceptions" (p. 17). "Every conclusion which occurs in a proof must find its confirmation in the diagram, but it is not justified by the diagram, but by a definite earlier statement (or definition)" (p. 43). Pasch understood clearly the implications of his method.
If geometry is to be truly deductive, the process of inference must be independent in all its parts from the meaning of the geometric concepts, just as it must be independent from the diagrams. All that need be considered are the relations between the geometric concepts, recorded in the statements and definitions. In the course of deduction it is both permitted and useful to bear..................
