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

visualizing in mathematics 25unreliable, if a process of thinking through an argument contains some nonsuperfluousvisual thinking, that process lacks the epistemic security to bea case of thinking through a proof. This view, claim (a) in particular, isthreatened by cases in which the reliability of visual thinking is demonstratednon-visually. The clearest kind of example would be provided by a formalsystem which has diagrams in place of formulas among its syntactic objects, andtypes of inter-diagram transition for inference rules. Suppose you take in sucha formal system and an interpretation of it, and then think through a proof ofthe system’s soundness with respect to that interpretation; suppose you theninspect a sequence of diagrams, checking along the way that it constitutes aderivation in the system; suppose finally that you recover the interpretationto reach a conclusion. That entire process would constitute thinking througha proof of the conclusion; and the visual thinking involved would not besuperfluous. Such a case has in fact been realized. A formal diagrammaticsystem of Euclidean geometry called ‘FG’ has been set out and shown to besound by Nathaniel Miller (2001). Figure 1.1 presents Miller’s derivation in FGof Euclid’s first theorem that on any given finite line segment an equilateraltriangle can be constructed.Fig. 1.1.