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

the euclidean diagram (1995) 117We here touch upon a general problem inherent in any use of (quasi-)semantic methods for reductio along the lines suggested, not just diagrams, butalso most other forms of ‘semantic representation’ and ‘models’. Suppose theartifact d (here, a diagram) is not subjected to a constraint e (here, exact). Onthe one hand, if e fails in any d satisfying the appropriate constraints, then itwould be appropriate to consider this to conclude the reductio, in that ourcommitment to e (in the text) is incompatible with the failure of e in anyd satisfying the constraints we impose on it. On the other hand, it is partof what we expect from (quasi-)semantic artifact use that e may fail in oneartifact satisfying the constraints we impose but not in another; that is, thatour production and reading of d-type artifacts proceeds less uniformly that oursetting up of conditions (e-type artifacts).In view of the former possibility, we would want to be able to read off ¬efrom d to conclude the reductio; but in view of the latter possibility, we cannotafford to do so on pain of disarray: our text inferences would then proceedmore uniformly than corresponds (via production and reading procedure) tod-type artifacts. What is a practice to do in such a bind?In the example of I.6, reading off of ¬e is simply blocked, in that it is aninequality, and hence not explicit in our technical sense. Most negations of exactconditions are in fact not diagram conditions, and this blocks their attribution;evidently, geometrical reductio typically just has to do without the ability toclose off by attributing ¬e even if a d-type object satisfying the constraintswe impose on it cannot be subjected to e. Undercutting attribution licensesshould not cause worry of unfounded conclusions or disarray in diagram-basedreductio argument: it only weakens the argumentative resources at hand.This, however, has nothing to do with the source of the problem withquasi-semantic artifact use in reductio proof, which we already identified:‘quasi-semantic’ artifact use provides less support for uniform proceeding thanordinary text. The appropriate solution then, as always when capacity foruniform proceeding in one aspect or stage within a practice falls short ofsuch capacity elsewhere, in a finite-to-one or otherwise surveyable fashion,is case distinction. In reductio, then, we should not aim to simply block theattribution ¬e outright. Rather, if there are different diagrams with potential toaffect the argument, we should consider them as cases, and if we fail to do so,an objection is in order, presenting some diagram of a type not yet considered.There is nothing special about reductio, or geometry, in this.If this general policy is in force, however, then the present difficulty cannotbe avoided simply by diagramming the lines smoothly to avoid ascribing apentangle; for the pentangle diagram could still be brought as an objection.There must, therefore, be a way of dealing with that diagram, either by