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

the euclidean diagram (1995) 101The primary resource is a non-propositional acquaintance with control inexecuting constructions in a given diagram. The practice requires one to graspthat the attempt to connect two points by a straight line, for example, has nolegitimately competing alternate outcomes. In this attenuated and inarticulatesense, uniqueness is a part of what must be undertaken and subscribed to inPostulate 1. Little does it matter that Euclid might have asserted uniqueness,making it a propositional content of Postulate 1, by using some appropriateverbal form such as a definite article (as Mueller objects, p. 32), but did notdo so. Just so, one is required to grasp that the control one exercises inattempting to make the circles in the construction of I.1 is such that we shouldtake precisely two intersection points to arise as seen from the diagrams weproduce. At this level, there is no need of I.7 or III.10 (as Heath envisages,p. 261) to establish the uniqueness of the triangle in I.1. Such a requirementof non-propositional grasp is of course compatible with the lack of explicitlyarticulated concern with uniqueness which Mueller observes in Euclid.¹⁸In spite of this non-propositional grasp, challenges arise to what ‘justhappens’ in making diagrams. Perhaps the realm of non-propositional graspis a fool’s paradise? As constructions pile up, adequate grasp of what controlmay be exerted through proper diagram discipline may be unattainable forparticipants. For example, the two possibilities in I.1 could each lead to multipleand dissimilar possibilities in further constructions. What possibilities arise inthis way will typically depend on the initial data. It helps that, as one can nowestablish by metatheoretical means, a determinate construction has at mostfinitely many outcomes; each corresponding to an in principle independentlyintelligible geometrical condition on initial data.But also, the conditions on initial data under which various possibilities ariseneed not be open to grasp in carrying out the construction in a particulardiagram: in the all-triangles-are-isoceles argument, what clearly happens forclearly non-isoceles triangles might inscrutably cease to occur for certainclose-to-isoceles triangles; perhaps with some unimagined threshold. Wemay therefore need to survey the possible outcomes at each stage. Undersuch circumstances, a secondary resource in the practice comes to the fore:propositionally explicit claims and demonstrations concerning the outcome of¹⁸ The expectation of control in constructions is illustrated, for example, by a remark of Newton:he compares the traditional geometrical neusis construction of the conchoid with a regulation Cartesianconstruction of solutions to its cubic locus equation (pointwise by intersecting a varying circle andfixed hyperbola), and comments ‘... this [Cartesian] Solution is too compounded to serve for anyparticular Uses. It is a bare Speculation, and geometrical Speculations have just as much Elegancyas Simplicity, and deserve just so much praise as they can promise Use.’ (Universal Arithmetic 1728pp. 229–230.)