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

3Diagram-Based GeometricPracticeKENNETH MANDERSDemonstrations in Euclid’s Elements, from Proposition I.1 on, use their diagramessentially to introduce items such as that notorious intersection point of thetwo circles, for which Euclidean demonstration has no alternative justificationalresources.In the 19th century this style of reasoning received critical attention frommathematicians needing to articulate various alternative geometries and theirinterrelationships. Twentieth century philosophers of mathematics have tendedto dismiss it altogether as a means of justification, often citing modern logic assetting not only a more appropriate standard for mathematical justification, butthe only acceptable one. The article in this volume by Giaquinto cites somesources on these matters.One student, apparently the beneficiary of enthusiastic instruction in thevirtues of the modern logical account, recently compared the study of Euclideandemonstration as mathematical justifications to the study of the Flat Earth.While myself not in a position to judge the contributions of the flat earthtradition to modern plate tectonics, I believe this expresses at least two complaintsthat many professional philosophers of mathematics would endorse: thatdiagram-based reasoning in Euclid is unreliable and justificationally inadequate,and that the study of a tradition of argument so obsolete cannot benefit thephilosophy of mathematics. Let me take these in turn.Assessments based on diagrams are held to be unreliable on several grounds.Drawn diagrams are imperfect in that, say, lines are not perfectly straight;regardless, human assessments of straightness or equality of line segmentswould be imperfect. Moreover, geometrical figures are individual, or at leastatypical compared to the generality of geometrical conclusions. Next, there aredifferent forms of geometry, which differ in their conclusions, and so a single