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

100 kenneth mandersattributes (as we can learn from the ubiquity of the notion of generic point inalgebraic geometry earlier in this century). With respect to co-exact attributes,however, no single diagram need be generic. For non-trivial configurations,the effect naively envisaged as atypicality avoidance must instead be achievedthrough redrawing practice together with disjunctive case examination.¹⁶4.3.2 Constructions and diagram controlConstructions contribute to diagram appearance control by their determinateness:they allow us to form a particular one of the many diagrams with curveswith the desired connectivity.Recent commentators have extensively debated whether Euclid’s constructionpostulates and propositions have existential import, but virtually withoutmention of their uniqueness import or determinateness. Mueller (1981) deniesthat Euclid shows ‘any concern with the question whether a constructed objectis unique’—in connection with I.30 (p. 19), and a straight line connectingtwo points (p. 32); not, though, because Euclid would doubt uniqueness, butbecause it is clear what uniqueness results from the basic constructions, andthis need not be explicitly addressed.¹⁷It seems plain that geometric constructions purport to give unique outcomes,or outcomes unique up to finitely many variants in any given case. For example,the line connecting two given points is plainly understood to be unique (seealsoProclusonanobjectiontoI.4). The equilateral triangle constructionof I.1 is plainly understood to give two possible triangles, congruent butperhaps differently located with respect to the data, if (as Proclus 225.16appears to regard as relevant) space allows. Occasional Euclidean propositions,such as I.7, could, moreover, be read as addressing uniqueness questions forconstructions. We propose that traditional practice had, in principle if tovarying extent depending on the case at hand, access to two quite distinctresources in regards uniqueness of the outcome of constructions subject todiagram discipline.¹⁶ There are other levels of function of diagrams in traditional geometry, on which we have nottouched. Besides providing ‘inferential’ licenses, diagrams serve to organize and motivate geometricalinquiry. These considerations have not yet found their proper philosophical casting; and it goes beyondour present purpose to attempt that task here. In the context of such non-justificational functions,diagrams license a wider range of ‘attribution’, including what we have called exact conditions. Thereis a corresponding need for stricter diagram discipline, as well as different criteria for case distinctions inatypicality control. In particular, diagram discipline in this sense can be expected to extend to suitablyselected inequalities.¹⁷ The index to Knorr (1986), for example, contains a series of headings under ‘existence’; notably,‘problems as existence proof ’ gives 10 entries, but there is nothing on ‘uniqueness’. Mueller mentionsno work on the topic since Heath.