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

106 kenneth mandersone of these possibilities is denied, the other two must be considered, whichtypically requires a case distinction. (Reductio argument is considered morefully below.)Far greater difficulties arise because—quite aside from ambiguities as to theappearance of the diagram formed under proper discipline—there is anotherreason that the appearance of a diagram at one stage in its construction does notdetermine its appearance after further constructions are applied in the courseof an argument, even if those further constructions are fully constrained. Thetopological data of initial appearance simply does not determine what topologyarises from construction of additional elements. Only more detailed metricaldata—such as might now be given by algebraic inequalities—can determinethis. For example, whether a point in the diagram will lie inside a newly enteredcircle or not would depend on whether its distance from the center exceedsthe radius. Such metric data is unavailable (not attributable) in a Euclideandiagram.As a consequence, topologically similar initial diagrams may become dissimilarthrough constructions in the course of a proof. When this occurs, a proofmust, from that point on, deal separately with the dissimilar diagrams whichhave arisen. We might call this case branching. It occurs, here as elsewhere,because capacity for uniform proceeding after the construction falls short (ina finite way) of such capacity before it. (It seems a merely expository quirkthat written proof typically gives variant diagrams from the outset; the formatmakes it inconvenient to display the stages in which a diagram is built up andcase distinctions arise.)If the practice is to avoid disarray in applying propositions, proofs must beapplicable to any diagram which satisfies the conditions stated in the text ofa proposition. In light of ‘one appearance–one diagram’, this requires thatprecisely those appearances which might possibly arise in an argument bepresented in the argument, one diagram each.¹⁹Any diagram arises from arbitrarily chosen, approximately metrically determinate,initial data (say, the ratios of distances of the vertices of a triangle)by constructions—be it the initial diagram of a proposition, or a subsequentelaboration by auxiliary constructions. We have encountered the limitations ofdiagram control in obtaining the appearance of a single diagram, constructedusing a single system of arbitrary choices. Here, an additional and independentlimitation on case branching control comes to the fore: even if there were¹⁹ This is perhaps too strong: once it is clear how an argument will go, one could in principledispense with differences in appearance that were plainly immaterial to it. Authors anyhow seem todiffer in the completeness of their case enumerations.