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

the euclidean diagram (1995) 89a diagram, some may be inferred only from prior entries in the discursivetext. More careful consideration reveals that the inferential contribution ofthe diagram depends also on standards of diagram production, in perhaps notentirely expected ways.Directly attributable features are ones with (a) certain explicit perceptualcues that are (b) fairly stable across a range of variation of diagram (‘co-exact’rather than ‘exact’) and (c) not readily eliminable by diagram discipline, properexercise of skill in producing (still imperfect) diagrams required by the practice.For later reference, we define the appearance or topology of a diagram tocomprise the inclusions and contiguities of regions, segments, and points inthe diagram (strictly speaking, only in so far as not eliminable by diagramdiscipline). These will turn out to determine its attributable features.Already in antiquity, diagram-based moves were open to challenges, as wesee from objections recorded by Proclus and others, discussed below; but themoves were defended and retained their license. By the middle and late 19thcentury, one does indeed want to consider alternative geometries such as thosewith restricted coordinate domains characterized by algebraic conditions; onedoes indeed want to consider the properties of ‘curves’ such as space-fillingor continuous nowhere-differentiable ones. Traditional diagram usage doesindeed provide inadequate representational support for the reasoning requiredin these cases.⁸ But such challenges did not arise within traditional geometry;and the unquestionable need to shift to other representational means to meetthem should not (as it has for most of this century) cloud the question of howtraditional geometry met its challenges in such exemplarily stable fashion.4.2.1 Explicit perceptual cuesGreek geometry requires us to respond to diagrams as made up primarily ofregions, not the curves which bound them. Euclid defines, ‘A figure is thatwhich is contained by any boundary or boundaries.’ The Greek word forfigure here, schema, refers to a shaped region; in contrast, we usually attend tothe delimiting curve and treat as incidental such regions as appear. In analyticgeometry, a circle is a curve satisfying an equation; in Euclid, it is ‘a planefigure contained by one line such that ...’⁹⁸ This, and not the outright invalidity of diagram-based reasoning (‘spatial intuition’) would be theproper conclusion from similar considerations adduced by Hans Hahn (1980).⁹ Euclid I definitions 14–15; see further Heath Euclid I, 182–3. Proclus brings out this sense ofschema very clearly: ‘the circle and the ellipse ... are not only lines, but also productive of figures. ... ifthey are thought of as producing the sorts of figures mentioned, ...’ (103). He attributes to Posidoniusa contrary view: ‘it seems he is looking at the outer enclosing boundary, while Euclid is looking at thewhole of the object.’ (143.14)