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

96 kenneth manders(i)A(ii)ARQBDOCBRDCQFig. 4.4.Orespectively AC. Either display is explicit and co-exact. Then one threatens toconclude (i) AB = AR + RB = AQ + QC = AC or (ii) AB = AR − RB =AQ − QC = AC, so the triangle is isoceles!The resources of traditional geometry to deal with arguments of thisnature consist in control of the diagram; backed up by probing of diagrambehavior. For triangles that are not close-to-isoceles, all but the initially givenappearance would be eliminable by refinement of the diagram subject to‘diagram discipline’ discussed below; so that one could block attribution ofthe other appearances that lead to the undesired conclusion. But the ability ofgeometric argument to exploit this is restricted by the unlimited sensitivity, forclose-to-isoceles triangles, of the appearance of the diagram constructed.4.3 Controlling diagram appearanceDiagram appearance, then, in the technical sense—the inclusions and contiguitiesof regions, segments, and points—plays a decisive role in geometricaldemonstration. The basic general resource of traditional geometrical practicein controlling the appearance of the diagram is diagram discipline, standards forthe proper production and refinement of diagrams.These standards are implemented by exploiting the considerable (thoughnonetheless limited) practical determinateness of geometrical constructions,and our (also nonetheless limited) ability to detect certain types of defects incandidate diagrams. The control that may be obtained in this way is furtherlimited by the (sometimes extreme) sensitivity of diagram appearance to metric