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

purity of method in hilbert’s grundlagen 227elliptical ‘lens’ such that in the interior of the ellipse they describe arcs ofcircles, and the new ‘straight lines’ are these straight line/circle arc/straight linecomposites. (See Fig. 8.3 and then Hilbert’s Grundlagen, pp. 52–54, in Hallettand Majer (2004, pp. 488–491).) Once again the idea has connections withperceptual experience; once more, we see a case where the result issues frominterplay between geometrical intuition and abstract, non-intuitive calculation.But again note that it is this calculation which shows that the model works,for once again, precision is the key. We also have a case which is similarto the Isoceles Triangle Theorem result in Section 8.4.2, for what is shownhere is that spatial assumptions can be avoided if (here) we adopt the full setof axioms governing plane congruence. It is also worth bearing in mind thathighly refined mathematical/numerical models like these are what show forHilbert that certain ‘intuitive’ geometries ‘exist’.One of the results of Hilbert’s investigation is that one need not look anyfurther for a purely plane proof (from I 1–2, II) of the Planar Desargues’sTheorem; one can, as Hilbert puts it, ‘sich die Mühe sparen’. But this is byno means the end of the matter. As Hilbert states, our ‘drive’ for mathematicalknowledge is only satisfied when one can establish why matters cannot goin the initially expected way. (See the passage from the Grundlagen cited onp. 201.) Since Desargues’s Theorem can be proved from the axiom groups Iand II, which among other things put conditions on the ‘orderly’ incidence oflines and planes, one might say that the theorem is a necesssary condition forsuch incidence. Hilbert now conjectures the following:Is the Desargues Theorem also a sufficient condition for this? i.e. can a system ofthings (planes) be added in such a way that all Axioms I, II are satisfied, and thesystem before can be interpreted as a sub-system of the whole system? Then theDesargues Theorem would be the very condition which guarantees that the planeitself is distinguished in space, and we could say that everything which is provablein space is already provable in the plane from Desargues. (Hilbert ( ∗ 1898/1899,p. 33), p. 240 in Hallett and Majer (2004))Hilbert shows that this conjecture is indeed corrrect, and the result is achievedby profound investigations of the relationship between the geometrical situationand the analytic one, the overall result being a re-education of our geometricalintuition, for what it reveals is that the Planar Desargues’s Theorem in effectactually has spatial content. This provides an explanation of why it is (in theabsence of congruence and the Parallel Axiom) that the Planar Desargues’sTheorem cannot be proved without the use of spatial assumptions, and itprovides a beautiful example of Hilbert establishing in the fullest way possiblewhy ‘impure’ elements are required in the proof of Desargues’s Theorem,the grounds for the ‘impossibility of success’ in trying to prove Desargues’s