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

purity of method in hilbert’s grundlagen 241same breath with another, namely that, given a straight line and a circle inthe same plane, if the line has both a point in the interior of the circle anda point outside it, then it must intersect it, and in exactly two points.⁴¹ Callthis the line-circle property. Similar to this is what can be called the circle-circleproperty, namely a circle with points both inside and outside another circle hasexactly two points of intersection with it.⁴² The connection to the TriangleInequality Theorem is not surprising, for the circle-circle property is preciselywhat Euclid implicitly relies on in the proof of I, 22.It is important to see Hilbert’s question as one concerning ‘purity of method’.It is well known that similar questions arise with the very first proposition ofEuclid’s Elements (I, 1) which shows how to construct an equilateral triangleon a given base AB. Euclid’s construction takes the two circles whose centresare the endpoints of the base and whose radii are equal to the base; either ofthe two points of intersection of the circles can be taken as the third apex, C.But do the circles actually intersect? A standard objection is that there is noguarantee of this. Heath notes:It is a commonplace that Euclid has no right to assume, without premissing somepostulate, that the two circles will meet in a point C. To supply what is wantedwe must invoke the Principle of Continuity. (Heath, 1925, Volume 1, p.242)And by a ‘Principle of Continuity’, Heath means something like Dedekindcontinuity (op. cit., pp. 237–238). Heath also cites one of Hilbert’s contemporaries,Killing, as invoking continuity to show the line-circle property (Killing,1898, p.43). And many commentators on Euclid (for example, Simson in the1700s) raised this point with respect to the proof of the Triangle InequalityProperty, while at the same time stating that it is ‘obvious’ that the circlesintersect, and that Euclid was right not to make any explicit assumption. WhatHilbert investigates is what formal property of space corresponds to the implicitunderlying existence/construction assumption.Hilbert constructs a model of his geometry (i.e. Axiom Groups I–III)in which the existence assumption fails, and where the Triangle InequalityTheorem (Euclid’s I, 22) also fails. Thus, the necessary conditions for Hilbert’sproof of the Three Chord Theorem are not present in a geometry based solelyon I–III. Moreover, since the Euclidean proof of I, 22 is based on a simplestraightedge and compass construction, Hilbert’s result is tantamount to sayingthat his axiom system does not have enough existential ‘weight’ to match thisparticular construction. This result is interesting, because, beginning with an⁴¹ Note that this has to be assumed for the power of point to be defined for all points and all circles.⁴² For similar properties, see pp. 62, 65 of Hilbert’s lecture notes, and pp. 63–64 of the Ausarbeitung,i.e. Hallett and Majer (2004, 335–337).