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

244 michael hallettordered Pythagorean field, one can always construct a triangle from threesides satisfying the triangle inequality if and only if the underlying coordinatefield is also Euclidean. Indeed, for an analytic geometry based on an orderedfield, the Euclidean field property is equivalent to the line-circle property,and this is in turn equivalent to the circle-circle property, the propertydirectly relevant to Hilbert’s proof of the Three Chord Theorem (Hartshorne,2000, pp. 144–146). Given this, the connection between the Euclidean fieldproperty and the formation of a triangle from any three lines satisfying thetriangle inequality is obvious.To repeat, the result is a thoroughly abstract one, a result about fields. Theinspiration is again intuitive, but this time the major fruit is a new theorem inabstract mathematics. (The result can be found in the Grundlagen der Geometrie,though the background work concerns solely the algebraic equivalents toelementary constructions.)This algebraic result is strongly hinted at in the Ferienkurs Hilbert gave in1898. (See Hallett and Majer, 2004, Chapter4; see pp. 22–23 of Hilbert’scourse.) Hilbert poses the question of whether, given a segment product c · d,there is a segment x such that x 2 = c · d, i.e. a square root for c · d. Hiscomment suggests that he thinks this is not always the case, and this is preciselywhat the counterexample outlined above confirms. For example, in the seconddiagram given, consider the horizontal and vertical products formed by the foursegments arising from the intersection of the horizontal and vertical chords.The horizontal segment product (our c · d) is(1 + π/4) · (1 − π/4), whichequals the vertical segment product √ (1 − (π/4) 2 ) · √(1− (π/4) 2 ); thus, thex sought is √ (1 − (π/4) 2 ), which does not exist in the model, as we haveseen. Thus, a question from the 1898 Ferienkurs is answered.Hilbert notes that the problem exhibited here does not just arise because ofthe involvement of the transcendental number π; he gives an example of anelementary number which will be in any Euclidean field over the rationals, butwhich is not in the minimal Pythagorean field, namely √ 1 + √ 2. (See Hilbert’sown lecture notes, p. 67, andalsop.67 of the Ausarbeitung. Hartshorne (2000,p. 147) gives further details of the counterexample.) Another example is givenin the Grundlagen itself, to which we will come in a moment.What is now interesting is how this abstract result is used to yield moreinformation at the intuitive level, at the level of synthetic, Euclidean geometryrooted in elementary constructions. In the 1898/1899 lectures, Hilbert himselfseems to suggest that the problem might have to do with a continuityassumption. On p. 64 of the Ausarbeitung, he says when assuming either theline-circle or circle-circle properties, one is actually assuming that ‘the circle