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

diagram-based geometric practice 75one’s diagram lies. Mumma (2007) is currently expanding his analysis of casedistinctions in Euclidean geometry by developing this.6. Lest relaxing the currently favored representation-enforced-unresponsivenessmeans to generality, by giving a role to response prohibition, seemradical: contrast both with a third mathematically crucial strategy (though not intraditional geometry) for justifying a general claim: reduction to a special case.A striking geometrical example is Poncelet’s use of ‘maximally-Euclidean’configurations to prove projectively invariant claims. Faced with an ellipse andsome lines, he specializes to the case of a circle with some parallel lines; allowinghim to efficiently exploit various kinds of equalities and congruent triangles toget the desired result, in the form it takes in this special situation (Poncelet,1822) [Figs.57–67]. This argument is supplemented by the reduction: ananalysis that identifies the projectively invariant form of the claim, togetherwith a demonstration that any instance of its projectively invariant hypothesismay be connected by projections to the special configuration considered.Reduction to a special case is distinctive in that its treatment of the specialcase expressly relies on special features not shared with typical instances of thegeneral claim under consideration (otherwise why bother with reduction), andfails for precisely for this reason to justify it. Instead, further argument, thereduction, justifies the general claim based on the special case. In their relaxingof representational unresponsiveness, such generality strategies are thus moreradical than what we find in Euclid.3.3 Tasks for the futureThe analysis of diagram use in ED is specific to plane geometry in the style ofEuclid I, III. Focus on one definite, cogent reasoning practice is its strength:it leads to substantive, sustained inquiry that invites us to go beyond ouringrained ideas. The philosopher’s task is to articulate, using such a specialcase, notions, or principles of general interest, a distinctive usage for discussingrigorous diagram-based reasoning practices.This conceptual framework then puts us in a position to articulate differencesthat matter between the base case and the role of diagrams in any othercomparable area of geometry. Doing so would also test the broader import ofour theoretical notions: what range of related practices do they allow us tounderstand?1. Natural candidates include: roles of diagrams in Euclid Book II, or inApollonius’ De ratione sectionis, or Euclid’s solid geometry; in ancient work