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

purity of method in hilbert’s grundlagen 235rotated into it. Parallel transport does not alter length, but crucially rotationsometimes does.³⁸Hilbert now considers the point (1, 0) on the x-axis, and rotates thispositively through the infinitesimal angle t. This will give a segment OA,where O is the origin and A has coordinates (x ′ , y ′ ) determined byx ′ + iy ′ = e (1+i)t · (x + iy) = e (1+i)t · x = e (1+i)tsince y = 0andx = 1. This is then:e t+it = e t · e it = e t · (cos t + isint) = e t cos t + ie t sin tgiving the coordinatesx ′ = e t cos t,y ′ = e t sin tAccording to the method of determining length, segment OA necessarily haslength 1.Now we construct the reflection of A in the x-axis, giving A ′ withcoordinates (x ′ , −y ′ ), and A and A ′ are then connected by a line perpendicularto the x-axis; denote by B the point where this line cuts the axis (Fig. 8.4).qA (x′,y′)OtB(1,0)(e 2t , 0)qA′ (x′, −y′)Fig. 8.4. The triangle showing the failure of the Isoceles Triangle Theorem.³⁸ A key part of the proof is showing, of course, that the Congruence Axioms hold. Hilbert’s workis not explicit about this. In his new version of Appendix II to the Seventh Edition of the Grundlagen,Schmidt uses one ‘congruence mapping’ in place of the distinct notions of parallel transport androtation. This is essentially the rotation function described above, with ‘transport’ parameters added,i.e. x ′ + iy ′ = e iθ+(1+i)τ · (x + iy) + λ + iµ, whereλ, µ are taken to be elements of T. Schmidt thenproceeds to give careful verifications of the Congruence Axioms.