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

the euclidean diagram (1995) 91diagram, rather than inferred from prior entries in the text, require not justequality but instead its most explicit possible form: coincidence that arises inan extremely direct way from the constraints under which the (imperfect)diagram is produced; such as the common side in the triangles which arisein the proof of I.6. This restriction on diagram-based attribution is not sosurprising, as equality among less directly related diagram elements cannotbe controlled by the standards under which diagrams are produced; but therestriction applies just so to inequality (which appears to pass our ‘co-exactness’stability test below).A striking example is Euclid I.20, ‘any two sides of the triangle takentogether are greater than the third.’ The sometimes maligned requirementthat this diagrammatically evident fact be proved no doubt reflects a relativelyadvanced theoretical foundational development in Greek geometry; but it isthat tradition so developed which we are to characterize. Geometrical reasoningfrequently obtains inequalities directly from the diagram, but (corresponding tothe restriction for equality) only when an object in the diagram is a proper partof another, rather than from any kind of indirect comparison. Equalities andinequalities (greater, less) among objects distributed throughout the diagramare of great importance, and inference principles for both are available withinthe discursive text. In order to allow these notions to function also in diagrambasedgeometrical argument, the Elements makes it its first order of businessto secure their transfer around the diagram so that direct comparisons may bemade (notably, I.2, I.3, I.4): both (i) so that inequalities already available inthe discursive text may be realized as proper inclusions in the diagram (I.6),and (ii) so that inequalities in the diagram may be related to explicit inclusionsthere which then may be read off (I.16).4.2.2 Exact and co-exact attributionGeometric attribution standards may be understood further by distinguishingwhat will be called exact and co-exact attributes. Typical alleged ‘fallacies ofdiagram use’ rest on taking it for granted that an—to the eye apparentlyrealized but false—exact condition would be read off from a diagram; butthe practice never allows an exact condition to be read off from the diagram.Typical ‘gaps in Euclid’ involve reading off some explicit co-exact feature froma diagram; and this is permissable. It will not be hard to discover a rationalefor this: the distinction of exact/co-exact attributes is directly related to whatresources the practice has to control the production of diagrams and provide forresolution of disagreement among judgements based on their appearances. Thepractice has resources to limit the risk of disagreement on (explicit) co-exactattributions from a diagram; but it lacks such resources for exact attributions,