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

the euclidean diagram (1995) 121of the intellectual strategy by which the practice, in the long run, achievesunanimity on ‘general’ claims from the manipulation of a small number ofphysical diagrams. The generality so attained is neither full uniformity oftreatment (distinct cases are argued with varying degrees of analogy), noruniversality in the modern model-theoretic sense with respect to specifiedcollections of ‘individuals’; it is that acceptable diagram variants not coveredby the argument don’t come up any more.Case-proposing criticism works in traditional geometry because the openendednessof case-branching control is in fact modest; for the scope ofcase-criticism is nontrivially constrained. In a way that is not less real for beingdifficult to formalize, the diagram of an argument gives access to a ‘space’ tobe probed for variants; for variants may generally be located by distortion ofmetric features of a diagram. The tradition thanks its stability in part to the quitelimited repertoire of variants requiring separate treatment that come up in thisway. In today’s Real Algebraic Geometry, this might now be made precise. Itfollows from this work that, even by traditional geometrical standards, only afinite repertoire of alternative diagrams need be considered for any argumentin ‘elementary’ geometry. As we will see later, moreover, traditional geometryhad resources to keep diagrams relatively simple; and in this way to controlthe numbers of cases, so they were, by and large, manageable rather thanmerely finite.According to Proclus, ‘... an ‘‘objection’’ (enstasis) prevents an argumentfrom proceeding on its way by opposing either the construction or thedemonstration’ (212). Proclus speaks so often of objection that the Teubneredition does not undertake to index all occurrences. Among objections that herecords: the triangle constructed in I.1 need not be equilateral if two distinctlines can coincide in a segment and then branch apart; the diagram of I.7(triangles with the same base, and the same sides upstanding at the same endsof the base, have the same vertex) admits a variant not covered in Euclid.Proclus takes making objection to be a responsive role distinct from proposinga case: ‘A case and an objection are not identical; the case proves the samething in another [context], but an objection is adduced to show an absurdity(eis atopian epagei to enistamenon) in the proof objected to.’ (289) Helays responsibility for dealing with case and objection on different shoulders:‘... unlike the proposer of a case, who has to show that the proposition is trueof it, he who makes an objection does not need to prove anything; rather itis necessary [for his opponent] to refute the objection and show that he whouses it is in error ...’ (212).It is puzzling, though, that the distinction should go very deep in thisway. For if a case is viable, in the sense that its initial diagram satisfies all