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

the euclidean diagram (1995) 93Exact attributes (indeed, by definition) are unstable under perturbation of adiagram. This renders them well beyond our control in drawing diagrams andjudging them by sight. Thus, diagramming practice by itself provides inadequatefacilities to resolve discrepancies among responses by participants as they drawand judge diagrams. It is therefore unsurprising that exact attribution is licensedonly by prior entries in the discursive text; and may never be ‘read off’ fromthe diagram. We observe this throughout ancient geometrical texts. That thecurves introduced in the diagram in the course of the proof of Euclid I.1are circles, for example, is licensed by Postulate 3 in the discursive text, and isrecorded in the discursive text to license subsequent exact attributions, such asequality of radii (by Definition 15, again in the discursive text).Exact attributions license a variety of extremely powerful inferences, whichare central and pervasive in traditional geometrical discourse. Besides substitutioninferencesa ≃ b,(b) ⇒ (a)which notably include transitivity inferences, we find equalities licensed bydefinitions (circle has equal radii), and congruence attributions, and licensingcongruence attributions. We frequently find proportionality attributionslicensed by the various rules for manipulating proportionalities, similarity offigures, and 4-point con-cyclicity (by Euclid III 35–37). Without such inferences(whether in this or some other form), spatial reasoning is handicappedbeyond recognition.¹³Co-exact attributes express recognition of regions (and their lower-dimensionalcounterparts, segments, and points) and their inclusions and contiguities in thediagram. We might say they express the topology of the diagram. Prominentexamples include ascription of non-empty delimited planar regions: triangles,squares, ... (but not that sides are straight); circles (but not that they are circularrather than elliptical or just irregular); angles (must be less than two rightangles, to delimit a region). In lower dimension, non-empty segments (but notthe character of the curve of which they are segments); points as two-place(but not three-place or tangent) intersections of curves; non-parallel lines; nontangencies.Further, contiguity and inclusion relations among these: point lieswithin region or segment; side opposed to vertex; line divides region into twoparts; one segment, angle, or triangle is part of another; alternate angles, a pointlying between two others, and so on.of the straight line no doubt does constrain its own applications; but the equation still captures only acertain inference potential of the Euclidean diagram notion of straight line, not that notion itself.¹³ As in Fuzzy Logic, compare Hjelmslev (1923).