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

purity of method in hilbert’s grundlagen 247the field of coordinates has a square root in the field, i.e. is Euclidean.⁴⁹ Thus,the problem is not to do with a failure of continuity, as Sommer suggests, verypossibly leaning on Hilbert’s original remark, but rather with the failure of avery much weaker field property, and it seems that Hilbert would certainlyhave been fully aware of this, by the time of the composition of the Grundlagenif not earlier. However, having said this, it is not entirely clear what principleshould be added to the axiom system to guarantee Euclideanness; adding thecircle-circle property itself as an axiom might appear somewhat ad hoc.Finally, to come back to the constructions involved in Euclid’s proofs ofI, 1 and I, 22, although both apparently involve ruler and circle constructions,an adequate construction for the first case can be given using Pythagoreanoperations alone, thus, uses the compass only in the ‘restricted’ sense. Anequilateral triangle can be constructed by Pythagorean operations just incase √ 3 is in the underlying coordinate field. But √ 3 = √ 1 + ( √ 2) 2 ,and√2 =√1 + 12 . Hilbert shows this: see the Ausarbeitung, p.173. Hence,theequilateral triangle can be constructed in Hilbert’s axiom system. (The actualconstruction is given in the 1898 Ferienkurs, p.15: see Hallett and Majer(2004, p.169).) Thus, Euclid’s construction here does not in the least assumecontinuity, or even ‘Euclideanness’.In sum, what we have here is another investigation which begins with a‘purity of method’ question, which then employs higher mathematics in itspursuit, achieves an abstract result, and also uses the knowledge gained toinform us and instruct us about elementary geometry, the geometry closest tointuition.8.5 ConclusionLet us draw some general conclusions from these examples.The concern with ‘purity of method’ usually focuses on some generalconsideration of ‘appropriateness’; this at least is the way that Hilbert casts⁴⁹ In the 1902 lectures, for the purposes of showing the independence of the Vollständigkeitsaxiom,Hilbert contructs a (countable) model based on a minimal Pythagorean field. He adds that this geometry(i.e. model) is particularly interesting, for... it contains only points and lines which can be found solely by measuring off segments andangles.As we have shown in our Grundlagen, page81ff., not every segment can √ be constructed bymeans of measuring off segments alone. Take as an example the segment 2 √ 2 − 2. (Hilbert( ∗ 1902, 89–90); Hallett and Majer (2004, 581)).