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

the euclidean diagram (1995) 111A proof is unaffected by any diagrammatic incompatibility between thepremises and the denial of the conclusion (diagrammatic incompatibility: nodiagram indicates all the co-exact conditions and is subject to all exact ones)in so far as responsability for move licensing is divided between diagram anddiscursive text so as to leave it indifferent whether the diagram is subjectedto all exact conditions in force. This allows one to use a perfectly ordinaryEuclidean diagram, in co-exact conditions if not in every respect metricallyappropriate to the discursive text.Whether or not the diagram was subjected to specific equalities indicatedin the discursive text—indeed, any metric property of the diagram—couldaffect which topological conclusions it would indicate; notably, if the proofemployed constructions more metrically sensitive than our straight-line joiningof DC in I.6. In so far as a diagram is not subject to all exact conditionsin the discursive text, there would seem to be no general assurance for thestanding of co-exact conclusions which emerge after constructions; althoughthe case/objection proposal mechanism would in any case provide means forcriticism of such conclusions.Because the exact conditions to which the diagram is subjected are a subsetof those indicated in the discursive text, it is easier than one would expectfrom those conditions taken together (rather than harder because of theirultimate incompatibility) to produce variant diagrams subjected to the sameexact conditions but indicating different co-exact ones. One would then seekcontradiction for those cases in turn, employing all exact conditions availablein the discursive text together with such co-exact conditions as might beindicated by the diagram. (To keep variants from proliferating in this process,one would subject the diagram to as many at a time as appears possible of theexact conditions indicated in the discursive text.)There is thus no special reason to doubt the cogency of diagram-basedreasoning in reductio proof: it does not presuppose that everything asserted inthe discursive text be ‘true’ of the diagram, not even in the attenuated senseprovided by the notion of diagram discipline.4.5.2 The hypothesis for reductioSo far, we have taken something else for granted: for straightforward proceedingaccording to the account given above, the data of a reductio context must forma (conjunctive) system of conditions (unsuccessfully) put forward concerning asingle diagram. It would of course also be possible to consider several variantdiagrams disjunctively, though this does not seem to have been the preference.In any case, we can give sufficient conditions for this success in launching areductio argument.