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

the euclidean diagram (1995) 99perhaps occasionally threes and fours (proportionalities, con-cyclicity). In eachcase, quality control operations (both the detection of defects which requirequestioning the suitability of the diagram and corrective control when defectsarise) involve relationships between diagram elements specifically identified inthe given stipulation.When diagram discipline is in force with respect to an exact claim (wewill see that in reductio arguments this does not always take place), we saythat ‘the diagram is subject to’ the exact attribution; say, that a line AB isstraight. The term ‘subject to’ marks off that although the diagram underthese circumstances need not (and in general will not) satisfy the condition inunequivocally readable fashion, it is to be held to certain standards in relation toits function in argument, standards which might require diagram replacementat any point. The explicit focus of the standards is indicated by the attributionin the ‘to’ position of ‘subject to’. Any explicit co-exact condition whichis properly attributable in a diagram, i.e. not eliminable by refinement withrespect to the exact conditions to which the diagram is subject, will be saidto be indicated in that diagram. On the other hand, at least the justificationalburdens which moderns think to target by quality control standards on thediagram as a whole or on unspecified subsets of diagram elements, notablyrequiring the diagram to avoid special properties (atypicality), would have beencarried by other features of ancient practice:(i) Because exact attributions may anyhow not be read off from the diagram,it has no inferential import if some special relations of that type might seem(spuriously) realized in a diagram. This is just as well, as there are no limitsto the equalities which might be judged to emerge in the construction ofa complex diagram; and except in the very simplest diagrams it is thereforeimplausible that participants could effectively control their work so as to satisfysuch an atypicality requirement, or agree whether they had succeeded indoing so.(ii) Diagrams may be atypical, relative to what is called for, in their co-exactproperties as well. If such atypical properties are explicit in the diagram (seeabove), traditional practice cannot afford to simply block us from readingthem off, as the discursive text has inadequate resources to acquire by othermeans the co-exact attributions it needs. Something else is therefore requiredto control the effects of atypicality at the global diagram topology level; thisfunction is fulfilled by practices of case and objection proposal (see below).Here too, the modern idea of atypicality avoidance would not work: it has thewrong structure. For the atypicality avoidance standard presumes that there issuch a thing as a ‘generic’ diagram, and as long as you re-draw until you haveone, you will then be OK. This is fine for genericity with respect to exact