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

104 kenneth mandersequal-angles position (Euclidean parallel postulate); (ii) there is a tiny intervalof intersectionless positions, symmetric around the equal-angles intermediateposition (hyperbolic geometry); (iii) at the equal-angles position, there isan intersection in each direction, as there is in any other position (ellipticgeometry); and (iv) as in every position, there is a unique intersection inthe intermediate position, and it lies simultaneously in both directions (Realprojective geometry). The history of the subject, however, brings out wellthat such stipulative resolutions raise questions: does the practice control theappearance independently of the stipulation, in some as yet unseen way, so asto render the stipulation either superfluous or incompatible?4.4 Case distinctionsTraditional geometrical reasoning distinguishes many cases. Poncelet describesthe situation perceptively, if with discontent:The diagram is drawn, and never lost from view. One always reasons aboutreal magnitudes; every conclusion must be pictured ... One stops as soon as [invarying the diagram] objects cease to have definite, absolute, physical existence.Rigor is even taken to the point of not accepting conclusions of reasoning onone general arrangement of a diagram, for another equally general and perfectlyanalogous arrangement. This restrained geometry forces one to go through thewhole sequence of elementary reasoning all over, as soon as a line or a point haspassed from the right to the left of another, etc. (Poncelet, 1822, pp. xii–xiii)For example, De Sectione Ratione of Apollonius Apollonius concerns theproblem:Given two lines, a point on each (Q, Q ′ ), and a point P not on either line: tolocate a line through P that (with the given points) cuts off from the given linessegments QR, Q ′ R ′ in a given ratio.Starting with the major division as to whether the two given lines intersect,Apollonius treats some eighty-seven (!) cases, with occasional further subdivisions.Although such complete treatments appear to be rare (Euclid, forexample, tends to leave many cases to the reader), Apollonius’ careful treatmentappears to reflect the highest standard of geometrical reasoning as far as rigor(disarray avoidance) is concerned. The standards of traditional diagram-basedgeometrical reasoning force case distinctions: ‘diagrams individuate claims andproofs’! This we must see. We recognize a ‘case distinction’ when capacityfor uniform proceeding (in artifact and response) in one aspect or stage within