Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

purity of method in hilbert’s grundlagen 237called by Hilbert non-Pythagorean geometry. Moreover, since OA < OA ′ ,thesquare over OA will fit inside the square over OA ′ with a non-zero area leftover, so we have a square which is of equal area-content to a proper part ofitself, violating one of the central pillars of the Euclidean/Hilbertian theory oftriangle area-content.⁴⁰It is important to see that these are not just technical results, for theygive us a great deal of information about what axioms are necessary for thereconstruction of classical Euclidean geometry. Hilbert sums it up in his 1905lectures as follows:The result is particularly interesting again because of the way continuity isinvolved. In short:1.) The theorem about the isoceles triangle and with it the Congruence Axiomin the broader sense is not provable from the Congruence Axiom in the narrowersense when taken with the other plane axioms I–IV [thus excluding continuityassumptions].2.) Nevertheless, it becomes provable when one adds continuity assumptions,in particular the Archimedean Axiom.From this we see therefore that in the Euclidean system properly construed, andwhich allows us to dispense with continuity assumptions, the broader CongruenceAxiom is a necessary component. The investigations which I have here set intrain throw new light on the connections between the theorem on the isocelestriangle and many other propositions of elementary geometry, and give rise tomany interesting observations [Bemerkungen]. Only the axiomatic method couldlead to such things. (Hilbert, ∗ 1905, pp. 86–87)This last remark is amplified by Hilbert in the direct continuation of this passage,which also ties the nature of the investigation directly to the view of geometrywhich we have seen emerging, namely emancipation from interpretation andintuition without losing contact with what underlies the axioms:When one enquires as to the status within the whole system of an old familiartheorem like that of the equality of the base angles in a triangle, then naturally onemust liberate oneself completely from intuition and the origin of the theorem,and apply only logically arrived at conclusions from the axioms being assumed.In order to be certain of this, the proposal has often been made to avoid theusual names for things, because they can lead one astray through the numerousassociations with the facts of intuition. Thus, it was proposed to introduce intothe axiom system new names for point, straight line and plane etc., names whichwill recall only what has been set down in the axioms. It has even been proposed⁴⁰ Hilbert states the conclusion as follows: ‘The theory of surface content depends essentially on the theoremconcerning the base angles of an isoceles triangle; it is thus not a consequence of the theory of proportions on its own’(Hilbert ( ∗ 1902, 125a), or Hallett and Majer (2004, 597)).