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

the euclidean diagram (1995) 103and to how the configuration to which the lemma applies is embeddedin a diagram enriched by constructions in the course of demonstration ofthe lemma.In view of the long record of entrenchment of the sense that geometricproperty attribution behaves like truth, as well as the central role of context-freeapplication relationships among the chain of propositions, it does not seem thatthis particular type of deep conflict has been widely felt to be in force for anynotable interval in the documented history of the practice. This may motivatewhat seems to be the standard of traditional geometrical practice for dealingwith the possibility of multiple outcomes of specified constructions on onegiven initial diagram: it suffices to treat one outcome (provided its standing assuch is not in question), leaving it to the process of probing (see later) whethersomething funny comes up when a different outcome diagram is considered.Perhaps this is best seen as a gamble, which serves to reduce quasi-uniquenessto effective uniqueness; together with an implicit acceptance of vulnerabilityto the underlying risk.The determinateness import of constructions is plainly not exhausted bytheir (quasi-) uniqueness (and existence) properties: the information that acertain construction, if properly carried out, will give a unique result neednot help in establishing what the metric properties or even appearance of theresult-diagram will be. Although extreme-sensitivity cases such as arbitrarilyclose-to-isoceles triangles may press constructional determinateness past itslimits, most diagram appearance issues are adequately dealt with by employingconstructions, subject to diagram discipline.4.3.3 Stipulative diagram controlA practice can resolve a limited number of artifact control problems bystipulation. Perhaps the Parallel Postulate should be understood to arise in thatway. As we vary the angle between two intersecting lines, we get a transitionof behavior across the range of angles in which the intersection lies too faraway to draw. The intersection now lies to one side, ever farther out; butif we approach the same range of angles (for which the intersection lies toofar away to draw) from the other side, the intersection lies to the other side,ever farther out. Because of this transition of behavior, something must besaid about the appearance of diagrams in this range; but precisely when theintersection lies too far away to draw, we have lost our diagramming artifactsupport: appearance control by diagram has been lost.With the benefit of hindsight, we can list some stipulations which couldbe made in this situation, compatibly with enough of traditional geometricalpractice to render the proposals competitors: (i) there is a single intersectionless,