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

98 kenneth mandersdiagram of a proposition; for arbitrarily complex patterns of connectivity mightbe required; and certain patterns of connectivity would be incompatible withexact attributes such as that the lines be straight. Only certain co-exact requirements,moreover, can be directly controlled: only lines can be drawn directly;planar regions (as well as certain distinguished points and segments on them)just pop up as a result of this.Whether a region is, for example, three- or four-sided, may depend on theinitial data of the construction in a manner sufficiently inscrutable to thwartcontrol. One purpose of Euclid’s treating numerous construction problemsin Book I may therefore be to underwrite the availablity of diagrams whichpropositions might require us to consider. It is probably prudent to regard theavailability of diagrams satisfying complicated co-exact conditions as potentiallya sore point in traditional practice; and potentially contributing to the pressureto get inferential weight away from such diagrams. Such pressure, however, isonly exerted to the extent that one has the expressive means, and intellectualpriorities, to bring complex diagram specifications in play in the first place;one should not simply extrapolate the urgency of such problems back from the19th century to ancient mathematics.On the other hand, exact stipulations concerning diagram elements, suchas that lines be straight or circles perfect, cannot be fully attained even forthe simplest of configurations. The needs of Euclidean practice are higherhere than one might imagine; rather than unequivocal readability of exactattributes (which is superfluous in so far as the discursive text records thestipulations), what will turn out to be needed is metric accuracy sufficientto render unequivocally readable co-exact properties associated with furtherconstructions which might need be applied to the diagram: both the co-exactpre-conditions of applicability of those constructions, and the appearance ofthe diagram resulting by their application.I take it that presentation of exact stipulations in the diagram is thereforesubject to quality control: defects are recognizable, and when they appear severe,or pertinent to co-exact attributions made from the diagram, complaints arein order. In some situations a diagram must be re-drawn, or rejected asinadequate. (i) If a line said to be straight is so crooked in your diagram thatI can noticeably improve it, and it looks like switching to the improved linemight affect the topology of regions, you are required to re-draw; (ii) equallyso if I can point to outrageous features of the circle or circle segment in yourdiagram, or (iii) clear failures of alleged equalities or (iv) of parallelism, whichappear of some moment for what you are doing.These quality control standards for exact attributes concern diagram elementsindividually (straight lines), or in pairs (equalities, parallelism of lines) and