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

70 kenneth manderspermitting, see ‘sensitivity avoidance’ below). Egregiously un-straight or uncircularentries can be immediately rejected; but on these exact criteria, entriesdo not appear to admit of immediate decisive acceptance. What matters hereis whether inaccuracies can cause spurious co-exact diagram consequences;accounts of demonstrative justification could invoke sensitivity avoidanceoptions, but also deferred rejection in the form of challenges to subsequentco-exact attribution (a logical novelty).It is striking that diagram-entries, here typically one-dimensional, controlzero-, and two-dimensional diagram entries (distinguished points, regions).This mutual relatedness of diagram elements appears to underlie the facilityfor shifts in emphasis and groupings of diagram elements that Macbeth (2007)thematizes in diagrammatic reasoning.4. Case-branching. How additional entries in the diagram change itsappearance (attributable co-exact character) may depend on previously nonattributable(metric) features. Example: every triangle, taken by itself, has thesame appearance; but whether a perpendicular dropped from the vertex pointto the base falls within the base or outside it (co-exact) depends on the shape ofthe triangle. Thus the same construction, applied to same-appearance diagrams(in our technical sense) may give different-appearance results.A demonstration may attribute a feature to its diagram (say, that theperpendicular from the vertex falls within the base) that would not arise hadwe made the prescribed entries in a different initial diagram for the proposition.Generality then requires case-branching: separate diagram and continuation ofdemonstration for each combination of attributions that could so arise froman instance of the initial configuration (see below). How Euclidean practiceensures this is thus critical to its ability to justify general claims.There are two theories on how a demonstration is to determine, giventhe inaccuracy of drawn diagrams, which diagram appearances may actuallyarise when it adds an entry: enumeration/exclusion or diagram control. Miller’s(2001, 2007) work exemplifies enumeration/exclusion theory: for any diagramentry, he enumerates all conceivable topological arrangements; the demonstrationmust explicitly argue to exclude unrealizable arrangements before itdemonstrates its conclusion for the remaining ones. This gives vastly morecases than we encounter in the texts, limiting its utility as an explication ofancient geometrical method. Mumma’s version (2006) implements a stricterconceivability, giving significantly fewer cases.Diagram control theory (see Manders (2007)) invokes our ability, usinggeometrical constructions, to produce reasonably accurate physical diagrams,and so limit the diagram appearance outcomes to be considered by physicaldiagram production rather than discursive argument. Conversations with