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

208 michael halletthe calls the Pascal Theorem (usually called Pappus’s Theorem) using just theplane part of the elementary axioms together with congruence, but withoutany involvement of continuity, a result new to Hilbert. More generally, thewhole of ‘school geometry’ can be developed without continuity assumptions,showing that Euclid was right not to include explicit continuity principlesamong his axioms.Of course, there are ways in which Hilbert’s treatment is not ‘purely’or historically Euclidean, for example in developing the core of projectivegeometry on the basis of the order and incidence axioms alone (i.e. beforethe Congruence and Parallel Axioms have been introduced), and in its furtherconcentration on the Desargues and Pascal theorems. An underlying concernhere is the relationship between Euclidean geometry and the coordinatestructure of analytic geometry, for part of what Hilbert investigates are thehidden field properties in segment structure showing that the basic magnitudeprinciples (represented by the core of the ordered field axioms) are true oflinear segments once addition and multiplication operations have been definedfor them in a reasonable way. Part of the point is surely to defend Euclid, inthat Hilbert shows that the ‘theory of magnitudes’ arises intrinsically, and doesnot have to be imposed from without by some extra assumptions, but anothermotive is clearly to show that the central guiding assumption of analyticgeometry, coordinatization by real numbers, is not ad hoc, a central concern ofHilbert’s since at least the 1893/1894 lectures. The guiding insight is clearly that,since analytic geometry was, at the time Hilbert was writing, the pre-eminentway to pursue Euclidean geometry, careful analysis of any suitable syntheticreplacement should reveal some central conceptual parallels with analyticstructures.¹⁰ Indeed, despite the desire to keep synthetic Euclidean geometry asfar as possible independent of analytic geometry, Hilbert did impose a strongadequacy condition, namely ‘completeness’ with respect to analytic geometry,i.e. the demand that a satisfactory synthetic axiomatization should be able toprove all the geometrical results that analytic geometry could.¹¹Nevertheless, despite these modern flourishes, it is clear (above all fromthe 1898/1899 lectures) that Hilbert’s own investigations were profoundlyinfluenced by Euclid’s. In the Introduction to his 1891 lectures on projectivegeometry, Hilbert gives a short but highly illustrative survey of geometry. He(2004, pp. 177, 284, 392 respectively). The problem reappears as Problem 3 in Hilbert’s famous list ofmathematical problems set out in 1900; see Hilbert (1900c, 266–267).¹⁰ Desargues’s Theorem is essential to this: see Section 8.4.1.¹¹ For a discussion of what Hilbert meant by completeness in 1899 when calling for a ‘complete’axiomatization of geometry, see Section 5 of my Introduction to Chapter 5 in Hallett and Majer (2004,426–435).