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

122 kenneth mandersstipulations in force, but no proof of the claim is readily available, it surelybecomes material for an objection. Moreover, an omitted but treatable casein a reductio argument would look to Proclus as an objection; this happens,for example, in I.7 (262–263). Unsurprisingly, then, Proclus continues aftersetting out the distinction, ‘By not discriminating between these, commentatorshave introduced them all together and have not made it clear whetherthey are asking us to diagram (graphein) cases or objections’ (289). It wasevidently not a commonplace for two roles to be distinguished at all. Perhapsparticipants have standing indifferently to propose cases or objectionswhich they discern. An uncovered case too is potentially a defect of argument.That responsibility for a straightforward case, once recognized, remainwith the proposer, could be justified by pedagogy alone: often enough, generationand treatment of variant diagrams is just routine. Thus, whateverother serious reasons there might be to distinguish case and objection, differencesin allocation of the task of responding need not have any theoreticalsignificance.Our evidence on cases as well as objections is skewed because of the way inwhich ancient geometrical texts come to us through a long process of revisionand recasting.²¹ Even in the commentary literature, a successful case-proposingwould tend to be recorded only if there was some special interest in thevariant diagram or proof. An objection which in fact succeeded in blocking anargument might result in the deletion or modification of the argument objectedto. Notably, if a variant diagram were discovered in which a construction couldnot be carried out, the result would most likely be revision of the text toexclude the variant by an additional hypothesis in the construction theorem.Proclus (330) regards the condition of I.22, that the two sides taken togetherexceed the third, as having such an objection-refuting role. On the other hand,successful response to an objection might be incorporated into the text as asubsidiary argument. As the order of propositions, though evidently not alldetails of their formulation, appears to have been fixed in Euclid’s time, thatparticular mode of inquiry must predate him.4.6.2 ProbingIn addition to the role of protagonist, we propose to recognize one single role astheoretically central in traditional geometric practice, that of providing criticalscrutiny,oraswewillcallit,probing. Participants may, of course, recognizevarious distinctive styles of critical scrutiny, including those of proposing case²¹ cf.W.Knorr(1989).