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

90 kenneth mandersFig. 4.1.Traditional geometric practice is surprisingly literal in what it recognizesin the diagram beyond items and relationships attributed stipulatively in thediscursive text. Proclus points out that ‘... the triangle as such never has anexterior angle’ (In Euclidem, 305). One must explicitly say, ‘if one of the sides isproduced’, and so indicate in the diagram, before referring to an exterior anglein a demonstration. Even so, Proclus counts a proof (Euclid I.32) asinferior,with ‘the middle term used here only a sign,’ instead of a ‘demonstration basedon the cause,’ because it involves extending one of the sides to form an exteriorangle (206.16–207.3).Another example: Proclus accepts a four-sided triangle (!) (Fig. 4.1), becauseit ‘... was the practice of Greek geometers not to recognize as an ‘‘angle’’ anyangle not less than two right angles’ (Morrow).Evidently, the outer angle does not count here; presumably because anglesare regions delimited in a certain way, and two lines at two right angles ormore do not do so.¹⁰The primary recognitions sanctioned by traditional practice include suitablybounded regions or extended objects (line segments, triangles, angles), associatedcontainment relationships (part-whole, point-in-region, line-in-region), andintersections of two (but not more) curves. For example, when, as in the proofof Euclid I.6, we draw a line from one vertex of a triangle to (the interior of)the opposing side, we are to recognize that the triangle has been partitionedinto two triangles, parts of the original, sharing a common side; that the angleat the vertex in question is divided into a sum of two angles, and that the lineforms a pair of angles where it meets the opposing side.Standards for treatment of equality and inequality (greater and less) helpfurther bring out the explicitness requirement in diagram-based attribution.Many things are said to be equal in geometry; and this is a key to the powerof geometrical inference. But judgements of equality, if to be made from a¹⁰ Morrow, fn. to Proclus 329.7; see also Heath Euclid Ip.264. At301–302, Proclus introducesa three-grade distinction which can plausibly be aligned with explicitness: between the familiar (i)knowledge about already existing things (that the base angles of the isoceles are equal), and (iii) makingsomething (bisecting an angle), he suggests (ii) a ‘finding of things’, ‘to bring what is sought into viewand exhibit it before the eyes’ (to find the center of a given circle, or the greatest common measurebetween two given commensurable magnitudes).