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

diagram-based geometric practice 77ancient diagram-handling practice, eventually perhaps to the point where wecan settle the original question about PP.3. The role of diagram use in ancient geometric analysis is highly non-trivial,and 20th-century discussions have shed little light on this. Compared to whatwe find in the Elements, ancient analytic practice (Pappus’ Collectio IV, VII)seems much less affected by case distinctions; probably due to matters of logicalorder (Behboud (1994)) as well as to the structuring of analytic reasoningby the collections of case-free what-determines-what lemmas such as Euclid’sData. Indeed, one as yet undescribed factor in geometrical analysis is thesense of what-determines-what that one only obtains by actually constructingdiagrams. (Study of geometrical analysis must include the rich Islamic tradition,cf. Bergren and van Brummelen (2000).)4. The nature of geometricality. Descartes’ method in geometry is guided byalgebraic form rather than by a geometrical diagram, and its algebraic manipulationsare not intelligibly related to the original problem diagram—perhapsbecause a problem has the same equation as infinitely many others, whose diagramsneed not be at all similar to the one intended. This strikes many authors,notably Leibniz, as a lack of geometricality; motivating two centuries of ultimatelyfruitful work to provide an ‘analysis situs’, some complementary theoryof spatial arrangements that would respect geometrical form independent ofthe merely metric properties to which Descartes’ method reduces geometry.For example, Poncelet (1822) aims to regain diagrammatic intelligibility ofgeometrical reasoning while selectively maintaining uniform treatment of variantdiagrams achieved by algebraic methods. Via a system of reinterpretationsguided by algebra and continuous variation of diagrams, invoking complex(‘imaginary’) and ideal objects (‘at infinity’), he grounds a novel diagram readingpractice and an accompanying algebra of generalized line segments that togethereliminate case-branching even more than Descartes. (This is a clear caseof enhanced uniformity of conception and argument that exploits standards ofreasoning rather than choice of representation to suppress differential responseto certain different-looking diagrams.) At the same time, Poncelet’s diagrammaintains his grasp on a similarity class of Euclidean-individuated diagramsmuch narrower than those sharing a common Cartesian problem equation.ED and most other recent discussions cited address, in the spirit of logicalfoundations, primarily justificational roles of the geometrical diagram. Butdiagrams in Poncelet have lost some cogency roles that ED attributes themin Euclidean argument. Because of this, further contrasting Descartes’ ungeometricalgeometry with Poncelet’s (and Steiner’s) diagram use may helpbring out diagram contributions to geometricality that we have so far missedin looking at Euclid.