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

diagram-based geometric practice 71specialists suggest this is the basic tool of ancient practice, with reductio argumentfor the exclusion of putative alternatives as backup. Harari (2003) discussesthe 20th-century alternative view that constructions serve as existence proofs.Some treatments (Miller (2001); Manders (2007), arguably) take eachdifferent-appearance diagram after an additional entry to require separatecase treatment. Using Euclid I.2 as example, Mumma (2006) shows that limitingcase branching to when a different-appearance feature is actually attributed,while sufficient for generality, vastly reduces the number of separate cases toabout what we find in Euclid and Proclus.5. Sensitivity of co-exact conditions.As long as the ‘circles’ in I.1 are continuous closed curves, no amountof distortion can eliminate intersection points. Most co-exact conditions,however, will be affected by sufficiently large distortions violating exactconditions: a ‘straight’ line segment, say, that loops out far enough, willspuriously intersect any other line in the diagram. Normal drawing practicesobviously avoid such egregious spurious co-exact conditions; nonetheless,this raises a challenge to the justificatory ability of diagram-based attribution:might we not encounter ‘sensitive’ diagram situations, where normal drawingpractices would not suffice to avoid a spurious co-exact occurrence, which ademonstration would then be entitled to attribute?Traditional practice aims to avoid such sensitive situations. There seem tobe two mechanisms.(a) By breaking up a development into a sequence of separate propositions,Euclid keeps each diagram simple: omitting construction lines from priorlemmas avoids point- and region-forming interactions with those absent lines.(b) No matter how simple a diagram, if it is too small we cannot drawit accurately enough to rely on its looks. Even a good-sized diagram mighthave too much going on in too tight a corner—in a seminar paper, MatthewWeiner trenchantly called this a ‘smudge’. I propose that a diagram in ademonstration is required to provide a clear case for its co-exact attributions,on pain of rejection. For example, to show what happens when an exactcondition fails (say in I.6, in a reductio proof of the exact condition), one mustuseadiagraminwhichitfailsinanexaggeratedway.Inactualgeometricalreasoning, providing demonstrative grounds for a given co-exact attributionmight require re-drawing the diagram. We will, however, need to consider theconsequences of this proposal for the epistemology of generality in geometricaldemonstration.In the face of such complications, keep in mind that ancient diagram-basedreasoning did work! The complications are only in our understanding of whyit works.