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

purity of method in hilbert’s grundlagen 24311 −π420 π4Fig. 8.7. Model of the Failure of the Triangle Inequality Property, continued.In his review of Hilbert’s Grundlagen, Sommer remarks that one cannot provethe line-circle property in Hilbert’s system (Sommer, 1899/1900,p.291). WhatSommer observes already follows from what Hilbert shows in his lectures, andwhat he says there explicitly (e.g. pp. 64–65). Sommer, of course, certainlyknew that Hilbert had shown this. For one thing, this is the same JuliusSommer, the ‘friend’, who, along with Minkowski, is thanked by Hilbert atthe end of the first edition of the Grundlagen for help with proofreading. Foranother, Sommer refers to Hilbert’s lectures course for 1898/1899 directly,in a different context (loc. cit., p.292). Furthermore, the mathematical coreof the result is in fact present in the Grundlagen, if only implicitly and in amore abstract form. Let us turn to this now, and then return to Sommer’sremark.The model described above really presents a result about the abstractconceptual structures which mature axiom systems represent, for what Hilberthas in effect shown, in modern algebraic terms, is that not every Pythagoreanfield is Euclidean. An ordered field is said to be Pythagorean when thePythagorean operation holds, that is, when √ x 2 + y 2 is in the field wheneverx and y are. (Note that, for each real r, there is a minimal, and countable,Pythagorean sub-field of the reals containing the rationals and r, afactwhichHilbert frequently employs in his independence proofs.) An ordered field Kis said to be Euclidean when for any non-negative element x ∈ K, wealsohave √ x ∈ K. It is obvious that every ordered Euclidean field is Pythagorean,but Hilbert shows here that the converse fails, for his model is formedπ4from a field which is Pythagorean; is an element, given that π is, andso are ( π 4 )2 and 1 − ( π 4 )2 ;but,aswehaveseen, √ 1 − (π/4) 2 is not, andso the field cannot be Euclidean.⁴⁴ The key point is summed up in thefollowing result: In an analytic geometry whose coordinates are given by an⁴⁴ That Hilbert had in fact shown this was first pointed out to me by Helmut Karzel.