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

the euclidean diagram (1995) 105a practice falls short of such capacity elsewhere, in a finite-to-one fashion: aswhen a geometrical assertion or argument admits of finitely many cases.The capacity for uniformity in diagram-based reasoning is limited bythe dependence of geometrical demonstration on what we have called itsappearance or topology—inclusions and contiguities of regions, segments, andpoints in the diagram. A geometrical proof typically responds to the appearanceof its diagram. If, in varying a diagram, ‘a line or a point has passed from the rightto the left of another’, as Poncelet puts it, we would need to reconsider whethersome region or inclusion invoked in the proof might have disappeared. Theall-triangles-are-isoceles argument shows that such considerations as whethera point lies on one or the other side of a given line are crucial; if differentialresponsiveness to such conditions were allowed to relax, disarray is a virtualcertainty.Geometric text is typically more flexible, giving the appearance of greateruniformity of presentation than can be attained in a diagram-based demonstration.In the case of Apollonius, the resources of geometric text allowuniformity of formulation ultimately 87-fold beyond what geometric argumentrequires.As one would expect from a practice which engages its artifacts in thatway, topologically distinct diagrams are treated in separate argument; whereas—maximizinguniformity in artifact use—separate argument is inappropriatefor diagrams with the same appearance, however otherwise dissimilar. We willrefer to this as the ‘one appearance–one diagram’ principle. In this way, theartifact strategy of the practice, combining diagram and discursive text, attainsa ‘local optimum’ in making more manageable the diversity of spatial forms:by reducing it while retaining an important inferential grip.Euclidean geometry often prefers to avoid making case distinctions at theoutset of a proof: the resources of geometric text allow one instead to substitutea formulation in which the appearance of diagrams is more fully described.Thus, a claim (p → q), where diagrammatically, p is one of p 1 or p 2 , wouldtend to be replaced by separate discursive claims (p 1 → q) and (p 2 → q).Indeed, under some circumstances, separate claims are required: constructionproblems often have preconditions of possibility (diorismoi), which must bestated as assumptions of a claim; these preconditions are often different fordifferent diagram topologies. The claim must then be made separately for suchdifferent diagram types.In other circumstances, case distinctions arise only within geometrical demonstrationand so are not readily avoidable. Arguments by reductio often require acase distinction because of trichotomy: when two quantities can be compared,the first is either greater, equal, or less than the second. If, for reductio,