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

130 kenneth manderswe sought to attribute rules to the practice which would account for this.Having recognized several rings of defense against error and indecisiveness,we find that remaining threats to geometry fail to motivate further discernablemeasures; indeed, fail to motivate the major actions of the practice.This embarrassment results from too severe a restriction of philosophicalfocus, by which we have deprived ourselves of an explanatory perspective ongeometrical diagram and proof. We have looked at inferential roles, demonstration,and probing alike, exclusively as defense against error, unreliability, ordisagreement. This leaves us scarcely able to motivate why Euclid proves whathis diagram shows!Drawing on our broader perspective on intellectual practices as means ofcontrol, however, we can overcome this restriction of focus. For justificationby no means exhausts the control intellectual practices aspire to. To admit thatsomething just happens when we construct a chord in a circle, for example,that a particular region just pops up inside another one, is to admit defeat alonga broad front. In their disdain for ‘empirical geometry’ or ‘mechanical curves’,participants in the geometrical tradition express revulsion for such impotence.This sensation that confronts us when something ‘just happens’ in a diagrammay be contrasted with (a) the diverse measures that if taken may enhanceour control; (b) the diverse questions, to which the ability to answer is part ofbeing in control; and also (c) the diverse virtues we recognize when we aremore in control.Starting with measures: guided by the diagram, we can (as in III.2 as aresponse to the chord location inside a circle) probe variant diagrams by wayof case and objection, and seek claims and arguments to clarify the originaldiagram.For another example, we might have gone on our way to I.20, startingfrom circling two segments around the two endpoints of another segmentand noticing a triangle pop up when we draw lines from the intersection tothe two endpoints. This brute occurrence could then be probed by trying toget it to fail. If we stretch the base segment but not the others, we can seewhat happens to the intersection point; this could lead to the diorism of I.20.We could, however, apply the challenge to the construction of a triangle inthis way in the demonstration of I.1, reported by Proclus: the straight linesfrom the circle intersection to the ends of the base might have a commonsegment, and so not give a triangle with the given segments we circled aroundas upstanding sides. Or we might give the diagram of I.7 and object to theimplicit assumption of diagram control in the uniqueness of the outcome of theconstruction. Plainly, the brute diagram occurrence we started with does notcall forth a unique challenge; and although the initial diagram (and, perhaps,