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

206 michael hallettSimilar points hold about the Parallel Axiom. In his 1891 lectures onprojective geometry, Hilbert remarks in a note that:This Parallel Axiom is furnished by intuition. Whether this latter is innate or nurtured,whether the axiom corresponds to the truth, whetheritmust be confirmed by experience,or whether this is not necessary, none of this concerns us here. We treat of intuition,and this demands the axiom. (Hilbert ( ∗ 1891, p.18), p. 27 in Hallett and Majer(2004). See also the 1894 lectures, pp. 88–89, pp. 120–121 of (Hallett and Majer,2004).In the Ausarbeitung of the 1898/1899 lecture notes, Hilbert is somewhatmore circumspect:... the question as to whether our intuition of space has an apriorior empiricalorigin remains unelucidated. (Hilbert, ∗ 1899, p.2) p.303 in Hallett and Majer(2004).But the proof of the independence of the Axiom shows again that a distinctjustification has to be given:Even the philosophical value of the investigation should not be underestimated. Ifwe wish to apply geometry to reality [Wirklichkeit], then intuition and observationmust first be called on. It emerges as advantageous to take certain small bodiesas points, very long things with a small cross-section, like for instance taughtthreads etc., as straight lines, and so on. Then one makes the observation that astraight line is determined by two points, and in this way one observes [as correct]the other facts expressed in the Axioms I–III of the schema of concepts. Non-Euclidean geometry, i.e. the axiomatic investigation of the Parallel Proposition,states then that to know that the angle sum in a triangle is 2 right angles a newobservation is necessary, that this is no way follows from the earlier observations(respectively from their idealised and more precise contents). (Hilbert ( ∗ 1905,pp. 97–98))For Hilbert, this shows that Euclid’s instincts had been quite right, that theaxiom is required as a new assumption to prove certain central, intuitivelycorrect ‘facts’ such as the angle-sum theorem or the existence of a rectangle:Even if Euclid did not state these axioms [incidence, order, and congruence]completely explicitly, nevertheless they correspond to what was intended by himand his successors down to recent times. However, when Euclid wanted to provefurther propositions immediately furnished by intuition, propositions such as thepresence of a quadrilateral with four right-angles, so he recognised that theseaxioms do not suffice, and therefore erected his famous Parallel Axiom. ...Thebrilliance it demanded to adopt this proposition as an axiom can best be seen inthe short historical sketch: Stäckel–Engel Parallellinien, Teubner, 1895 (Stäckeland Engel, 1895. (Hilbert’s lecture notes for 1898/1899, p.70; p.261 in Hallettand Majer (2004))