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

68 kenneth mandersdemonstration might be mere grunt-work and not teach us anything newphilosophically. Here too he would be wrong.Euclidean diagram use forces us to confront mathematical demonstrativepractice, in a much richer form than is implicit in the notions of mathematicaltheory and formal proof on which so much recent work in philosophy ofmathematics is based; and to confront rigorous demonstrative use of non-propositionalrepresentation. The philosophical opportunities are extrordinary.In the remainder of this introductory chapter, we outline a theory of diagrambasedinference in Euclidean plane geometry based on the exact/co-exactdistinction; and then survey further issues about diagram use in geometricaldemonstration: how particular diagrams justify general conclusions, and therole of our theory in understanding diagram use in geometry beyond Euclideanplane geometry. For a view of what can be understood about diagram use fromcareful analysis of ancient texts, see Netz (1999).Our topic parallels that of Giaquinto in this volume; but the two contributionsaddress very different issues. Justificatory diagram use requires stronglyshared standards of inference; Giaquinto looks at the benefits of visualizationwhere such agreement is unavailable. The difference is stark. Geometricaldemonstration got the ‘famously agonistic Greeks’ (Lloyd, 1996, pp. 21–23),and the later Arabic, Latin, and modern worlds, to agree on a body of detailedmathematical claims and arguments. In contrast, every time a mathematicallyinformed audience responds to a repertoire of mathematical picture-proofs,they disagree on what the pictures show; even though all agree that the picturesshow something important.3.1 Basic rules of Euclidean diagram-based planegeometryWe give only the briefest sketch, anticipating ‘The Euclidean Diagram’.1. Demonstration step. Euclidean demonstrations read as discrete sequencesof claims, each licensed by prior claims and the ‘current’ diagram. Beyondtraditional text–text licensing, demonstration steps may draw from the diagram(diagram-based premises), and they may take responsability for additionaldiagram elements (diagram-based conclusions). The current diagram consistsof those items previously so introduced, together with the regions, segments,and distinguished points arising by their interaction.When in I.1, for example, we have said ‘let AB be the given straightsegment’, we have not thereby admitted further features or diagram elements