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

116 kenneth mandersnot just to sanction refraining from attributing the spurious angles, but indeed,to block (or disarm) the obvious objection: ‘But EBGDF is a pentangle, not atriangle!’This goes to the heart of reductio demonstration. For according to ouraccount, reductio employs an altogether ordinary Euclidean diagram, in anonly slightly unusual way: one refrains from subjecting the diagram to oneor more exact conditions in force in the discursive text. In the case at hand,diagram discipline is clearly suspended with respect to keeping EG and FGstraight. But according to our account, the diagram functions in the ordinaryfashion; in particular, we must endorse such licenses to attribute co-exactconditions based on the appearance of the diagram as allow the argument toproceed to contradiction. The mere incompatibility of some such conditionwith some attribution in force in the discursive text cannot suffice to block it,or diagram-based contributions to reductio arguments would essentially all beblocked.In our initial example I.6, theElements makes the diagram-based attribution(d-vi) that DBC is a triangle, and applies the side-angle-side criterion I.4 toconclude that DBC equals ACB. Only then does it conclude from the diagramthat DBC is a (proper) part of ACB; if this attribution were blocked, nocontradiction would be available.There are several resources available here to deal with this issue. First, onecould accept the pentangle objection to this particular diagramming strategy;as an alternative would seem available. Even accepting the decision to subjectthe diagram to all exact conditions except that EG and FG be straight, we canconnect BG and DG by producing EB and FD by circle segments tangent atB and D respectively. Then no spurious angles can be attributed. Similarly forthe ‘line’ DCF in Proclus’ converse of I.15 (above).On the other hand, one can imagine that a suitably selective rule on defeasibilityof attribution licenses would allow the present argument to go forwardwithout undercutting the very possibility of diagram-based attribution inreductio argument; a natural candidate emerges. Genuinely diagram-utilizingarguments by reductio require a diagram; hence (if need be, after disjunctivecase analysis) some construal of the hypothesis for reductio as a conjunctivecollection of exact and co-exact conditions D 1 &... &D m . According to observation(2) just above, there is a lower priority for subjecting the diagram toexact conditions among these, compared to exact conditions coming into forceat an earlier stage. It would thus be natural to single out—as having theirattribution licenses undercut—co-exact conditions which taken by themselvesdirectly contradict exact conditions from among the Ds towhichthediagramis not being subjected.