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

the euclidean diagram (1995) 109Thus, practice with diagrams, and not the geometrical text, controls theindividuation of claim and proof in traditional geometry; in contrast with Cartesianand projective geometrical practices, which exploit different representationin order to individuate geometrical claims and proofs less finely.4.5 Diagrams in reductio proof: torturingthe diagram?Although traditional diagram-based geometrical practice has built-in qualitycontrolstandards, both on the production/reading of diagrams and throughthe case/objection proposal mechanism (discussed below), it lacks an externalvantage point from which to impose a standard (truth?) by which diagrambasedreasoning can, in a unified and independent way, be evaluated forstepwise cogency. The verdict of a diagram can be objected to; and if so,localized responses can be made; but by and large, these in turn must acceptthe verdict of diagrams.It is therefore strange to encounter arguments (in Euclid) in which thediagram is simply impossible, or clearly violates some condition invoked in theargument; for it is unclear what force the verdict of such diagrams could have.²⁰The matter is worth careful examination, for after all, Euclidean practice showsno signs of having broken down in disarray; and we can hardly dismiss its useof such diagrams. We find them also used to refute objections; that will requireseparate analysis.4.5.1 Example: Euclid I.6In the Elements, we first encounter reductio in I.6. GivenatriangleABC withthe angle ABC equal to the angle ACB, it is to be shown that the sides AB andAC are equal. For reductio, we deny this conclusion in the presence of thepremises. If AB and AC were not equal, one would exceed the other, say ABexceeds AC (by symmetry, one case alone suffices). The inequality is convertedinto the combination of (i) an equality between AC and BD constructed onAB (the equality is registered in the discursive text, perhaps manifestly violatedin the diagram) and (ii) the topologically explicit diagrammatic presentationof the inclusion of BD in AB. Draw the diagram, starting from an isoscelestriangle.²⁰ I thank David Israel for pressing me on this point.