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

86 kenneth mandersthe text is literally true, then treating diagrams actually drawn in geometricaldemonstrations as approximations to perfect ones; finally deriving from all thisan understanding of the bearing of the imperfect diagram on inferences in thetext. But no detour through ontology and semantics which treats of truth in adiagram in a sense which entails joint compatibility of all claims in force inthe reductio context can speak to the difficulty with the role of diagrams inreductio arguments, which are pervasive in Euclid.³Thus one is forced back to a direct attack on the way diagrams are usedin reductio argument; the problem of the relationship between diagram andgeometric inference here turns out to be one of standards of inference notreducible in a straightforward way to an interplay of ontology, truth, andapproximate representation. But once this is admitted, there seems to be noreason why direct inferential analysis of diagram-based geometrical reasoningshould not be the approach of choice to characterizing geometrical reasoningoverall, with or without reductio. If this order of analysis proves fruitful,ontological and semantic considerations will seem decidedly less central to thephilosophical project of appreciating geometry as a means of understanding. Forin their then remaining role of making the standards of geometrical reasoningseem appropriate, ontological-and-semantic pictures will have to competewith other types of considerations which we will find have potential to shapea reasoning practice. The failure, since Plato’s time, of diagram ontologystrategies to sponsor a convincing account of Greek inferential practice withdiagrams, perhaps does not bode well for their prospects.I therefore take it that traditional geometric demonstration has a verbalpart, which for contrast I will call the discursive text; and a graphical part, thediagram. The discursive text consists of a reason-giving ordered progression ofassertions, each with the surface form of an ascription of a feature to the diagram(attributions). A lettering scheme facilitates cross-references to the diagram. Astep in this progression is licensed by attributions either already in force in thediscursive text or made directly based on the diagram as part of the step, or³ Jerry Seligman has suggested that one might avoid the argument via some kind of ‘compositionalsemantics’ of diagrams. To avoid the argument, however, geometrically incompatible sets of sentenceswould have to be ‘compositionally true’ in the same diagram. That is to say (and the point hasnothing to do with diagrams or even geometry), a ‘semantics’ that fills the bill here thereby lacksminimal soundness, the minimum requirement for notions of (weak) truth, that a set of sentences all‘true’ in the same situation, cannot trivially entail a contradiction. Because of its failure of mininalsoundness, such a ‘semantical’ conception of diagrams could not account for a role of diagrams ingeometric inference (the original point of the detour); and it would deprive ontology of its traditionalphilosophical point, taking truth really seriously. A corollary: the problem of reductio argument forcessemantics/truth considerations apart from semantics/meaning considerations, which might indeedeffectively be approached by compositional semantics, presenting separate situations in which thecomponents of a contradiction are true.