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

the euclidean diagram (1995) 125by some metrically relaxed standard, and then providing further reductio adabsurdam arguments showing which variants are not viable as outcomes of thedeterminate constructions (Dubnov, p. 24). But it might also take the form ofa direct argument showing that a particular variant is the appropriate diagram(Maxwell, pp. 24–25).Again, however, we seem to have gained little by our sharper distinctionbetween case and objection.While probing is directly evident in the commentary tradition, it is of greatinterest to consider what in Euclid might best be understood as already aresponse to probing, even if only conjecturally. III.2 is a striking case. III.2argues that a chord of a circle lies inside it, a matter read off from the diagramin the proof of III.1. From a logical point of view, this order of presentationis puzzling. Mueller (p. 179) notes that the theory of circles in Book III makesmany ‘implicit assumptions’, ‘facts of spatial intuition’, sometimes taking themfor granted and sometimes, as in III.2, arguing for them. Though this stilloffends logical sensitivity, its status as a refutation of an objection to III.1 couldaccount for its following rather than preceding that result.In view of other unprobed diagram-based steps in Book III, it is perhapssomewhat puzzling why this point of appearance control would be probed.Unless we worry about situations in which the end-points of the chord onthe circle lie so close together that the separation of chord midpoint and circlebecomes questionable, there seems no particular metric sensitivity or risk ofdisarray addressed by demonstrating III.2—but the same may be said for manypropositions in Book III, which might readily be read off directly from thediagram.Alternatively, III.2 might be a probing of a matter needed in the demonstrationof its only application III.13, that two circles cannot have two distinctpoints of outer tangency. The diagram in our text shows two circles intersectingtwice, in accord with the theory of diagrams for reductio given here:the diagram is subjected to all but the tangency conditions, which are exactconditions from the hypothesis for reductio. From this diagram, one couldread off directly that there are common interior points, contradicting externaltangency without invoking III.2. Given the opposite curvatures of the circles,the diagram presents no particular appearance control sensitivity unless the twopoints of tangency are particularly close together, so it is not completely clearwhy this observation would be an urgent candidate for probing. The diagramcould, however, be made differently, still in accord with our expectationsfor diagrams for reductio: with either or both circles distorted to keep themdisjoint with two points of tangency. In that case, an argument would beneeded; and not only the statement of III.2 but a reductio argument with