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

120 kenneth mandersprotagonist’s role so as to render it incompatible both with contemporaryagentless conceptions of mathematics, and with our initial suggestion thatgeometrical practice requires ‘unqualified assent’.4.6.1 Proposing a case or an objectionAccording to Proclus, ‘the proposer of a case ... has to show that the propositionis true of it...’ (212). The proposer of a case exercises standing to respond tothe protagonist: ‘you have not dealt with a diagram like this .... Here is how itgoes ...’ A ‘case’ shows how a claim made by the protagonist applies to a variantdiagram topologically inequivalent to those considered by the protagonist, beit the initial diagram of a claim or one arising from a construction (all illustratedin Proclus’ case analysis of I.3, pp. 228–232): ‘a case proves the same thingin another [diagram] (alloos)’ (289). It is perhaps preferable that the proposedproof be analogous to the proof being responded to: Proclus praises Euclidfor a construction which can be modified to fit a great variety of cases (222).There may be some room for debate as to whether a proposed case meetingthese standards is to the point.It may be unclear why the role of proposer of a case merits seriousphilosophical consideration. Case arguments are often either similar to theones given, or concern degenerate situations in which a simpler argumentis available. Shouldn’t their treatment be governed by strictly expository orpedagogical considerations? To write out every case would be boring andunnecessarily long; better to let students exercise by working out the variants.So it’s sensible to leave the burden of proof on the proposer of a case.It would be wrong to dismiss case-proposing in that way. As we argued,traditional geometrical practice has inadequate facilities for case-branchingcontrol; so that case distinction management remains in principle open-ended.The critical attitude toward the protagonist’s diagram choices required in traditionalpractice is not supported by any clear-cut or complete procedure, andtherefore leaves geometric inference open-ended in a way which we moderns,spoiled by complete systems of iron-clad inference licenses, hardly expect.In the 20th century, Tarski (refining Pasch and Hilbert) was able todeploy complete first-order inferential machinery with which one can, if notlocate cases, at least conclusively establish that given enumerations of casesare complete. But Tarski uses representational (artifact) resources that Greekgeometric practice lacked: formal quantifier-predicate logic with completeproof systems and associated foundational methodology.In a geometrical practice without such means at its disposal, the criticalattitude toward the protagonist’s diagram choices displayed by the proposer ofa case has an intrinsically theoretical, even inferential, function; it is an element