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

92 kenneth mandersand therefore could not allow them without dissolving into a disarray ofirresolvably conflicting judgements.The distinction, stated loosely, is this. Consider all (imperfect) diagramswhich might reasonably be drawn from given specifications; some of whichmay be regarded as related by deformation or variation. Co-exact attributesare those conditions which are unaffected by some range of every continuousvariation of a specified diagram; paradigmatically, that one region includesanother (which is unaffected no matter how the boundaries are to someextent shifted and deformed), or the existence of intersection points such asthose required in Euclid I.1 (which is unaffected no matter how the circlesare to some extent deformed). Exact attributes are those which, for at leastsome continuous variation of the diagram, obtain only in isolated cases;paradigmatically, straightness of lines or equality of angles (neither of whichsurvive any except exceptional types of deformation, no matter how small).The distinction is perfectly apparent to anyone who has made enough diagramsto sense what one can and cannot control in a reasonably simple diagram.¹¹Many attributions have both co-exact and exact components. A triangle,for example, is a non-empty region bounded by three visible curves (all that isco-exact) which are in fact straight line segments (exact). Once realized, thisdoes not give any difficulty.Exact attributes can, at least since since Descartes’ time, be expressed byalgebraic equations. In traditional geometry, many were expressed or definedby equalities or proportionalities. Prominent examples include equality oflines (segments), angles, or other magnitudes, congruence of triangles or otherfigures, proportionality of lines; that an angle is right (not that it is an angle), thatfour points are con-cyclic (lie on a common circle); the geometric characterof lines or curves, or the regions they delimit: a circle, an ellipse; thatlinesarestraight (derivatively also, rectilinear angles, triangles, and so on) or parallel;that three lines or curves, while not initially stipulated to do so, intersect ina common point (rather than pairwise in three distinct points); that a line istangent to a curve (rather than intersecting it in two or more distinct pointsclose together).¹²¹¹ It is a matter of mere patience to provide an explicit modern topological account of diagrams, (interms of smooth embeddings of linkages of lines and circles into the Real plane) and their variation (interms of homotopic families of such embeddings), which does not suppose them made with perfectaccuracy (because a smooth embedding of a straight line need not be straight).¹² ‘Expressing’ here carries no philosophical weight, as their equational expressability is not whatmakes attributes exact. But it is perhaps prudent to note that this ‘expressing’ involves considerablecomplications. Notably, the notion of straight line in the Euclidean diagram sense is presupposed: itis an unexplained primitive in Descartes’ geometry, and must be independently available in order torelate coordinate geometry to Euclidean diagrams. The algebraic properties of the coordinate equation