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

246 michael hallettradii using the compass as, say, a pair of dividers. What this means is thatthe constructions licensed by Hilbert’s geometrical system of the 1890s canbecarried out (i.e. the existence of the points constructed justified) in any analyticgeometry whose coordinates form a Pythagorean field, even the minimalPythagorean field built over the rationals. (These are Theorems 40 and 41 ofthe first edition of the Grundlagen, pp. 79–81.)⁴⁸ Hilbert then remarks that:From this Theorem [41], we can see immediately that not every problem solvableby use of a pair of compasses can be solved by means of a ruler and segmentmover [Streckenübertrager] only. (Hilbert, Grundlagen, p.81, i.e. p. 516 in Hallettand Majer (2004))In other words, the unrestricted use of the pair of compasses in constructions isnot justified in his system.To show this, Hilbert first gives an example of a real number which cannot bein the minimal Pythagorean field built over the rationals, namely √ 2| √ 2| − 2,despite the fact 1 and | √ 2| − 1 are both in the field. (The example is thusslightly different from that given in the lectures.) It follows that we cannotconstruct by means of ‘Lineal und Streckenübertrager’ a right-angled trianglewith sides of length 1 (hypothenuse), | √ 2| − 1and √ 2| √ 2| − 2, since thelatter length cannot correspond to an element in the minimal Pythagoreanfield; hence the construction problem is not soluble in Hilbert’s geometry.But, as Hilbert remarks (Grundlagen,p.82), the problem is immediately solubleby a compass construction; the number Hilbert specifies ( √ 2| √ 2| − 2) is, ofcourse, in any Euclidean ordered field built over the rationals.The central point is now this: If K is the set of all real numbers obtainedfrom the rationals by the operations of addition, multiplication, subtraction, anddivision, and such that K contains square roots for all of its positive elements,then K is Euclidean and is the smallest field over which straightedge andcompass constructions can be carried out. (See Hartshorne, 2000, p.147.) Thesalient point is even clearer in Hilbert’s Theorem 44 (p. 86), which deals withthe problem of characterizing which straightedge and compass constructionscan be carried out in his geometry (i.e. reduced to constructions by straightedgeand Streckenübertrager). In the statement of the condition (see Grundlagen,Theorem 44, p.86), Hilbert quite clearly expresses the fact that the algebraiccondition corresponding to the compass construction is that each number in⁴⁸ Kürschák showed that the Streckenübertrager can be dispensed with in favour of an Eichmaß, i.e. adevice which measures off a single fixed segment. (See Kürschák 1902.) Hilbert makes a correspondingadjustment, with acknowledgement to Kürschák’s work, in the Second Edition of the Grundlagen(Hilbert, 1903, 74, 77).