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

diagram-based geometric practice 69such as a circle with center A and radius AB; only a later demonstrationstep admits it. That a diagram for the entire demonstration may alreadybe present detracts from this no more than that subsequent claims of thedemonstration text are already on the page detracts from the fact that at agiven step in the proof, only claims up to that point are available as premises.The linear structure of text allows one to track how far the process oftaking responsibility has progressed; including how much of the diagram iscurrent.The general form of a single demonstration step license is thus:claims in prior text, attributions to current diagramnew claims in text, new elements in diagramNot all elements need be present.2. Diagram-based attribution: exact vs. co-exact claims.Recent critics of traditional geometrical demonstration see potential forerror, precluding justification, when we claim based solely on what is in thediagram, as with the intersection point of circles in I.1; the logical traditionthen claims a gap in a traditional proof. This, however, ignores how the ancienttexts limit diagram-based attribution.The only claims based on diagram appearance in a demonstration recognizeconditions that are insensitive to the effects of a range of variation in diagramentries: lines and circles that are not perfectly straight or circular, and cannot betaken to be without thickness. As we distort the ‘circles’ in I.1, their intersectionpoint C may shift but it does not disappear. Such conditions I call co-exact.They include: part–whole relations of regions, segments bounding regions,and lower-dimensional counterparts. Call the totality of these conditions inthe current diagram its appearance.In contrast, many (most?) conditions considered in Euclidean geometrywould fail immediately upon almost any diagram variation: notably, equalitiesof non-identicals and proportionalities. Such conditions I call exact; they arenever claimed based on what the diagram looks like. (Some naive diagramunreliabilitycriticisms presuppose that they are.)3. Diagram entries in elementary geometry are line segments and circles, byPostulates I.1–3 directly or via prior constructions. Whenever a diagram entryis made, the text records the exact character (straight, circular) of the elemententered. There is thus no need to later judge this from the diagram.Diagram entries must be adequate both with respect to their co-exact andexact character. They are (at a minimum) continuous non–self-intersectingcurves, a line segment connecting its endpoints, a circle closed. Participantscan apply these co-exact criteria immediately and decisively (diagram size