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

the euclidean diagram (1995) 834.1.1 Diagram and controlAs does control itself, diagnosis and remedy of failures of control in geometrycan take many forms. The grip on living a practice can give arises in a delicateinterplay between the way its artifacts lie in the game, and the endowmentsand limitations humans bring to it. Two examples (a–b) of control throughattaining uniformity, in quite different guises, may illustrate this.(a) We take participants to be responsible for attaining uniformity of responsein reading off features in diagrams. But the production and reading of diagramscannot humanly be controlled so as to obtain a uniform response to diagramsdirectly on questions such as equality of lines. If a practice licenses a type ofresponse to artifacts, then in order to attain uniformity of response, it mustprovide means by which participants can resolve their differences in responses tothose physical artifacts they might reasonably produce. In traditional geometry,the division of responsibility between diagram and text, in which the text tracksequality information, is a way to meet this challenge. Subsequent geometricalrepresentations provide alternatives.If participants are to be able to respond in a stable and stably shared fashion,a practice must limit the repertoire of basic responsive roles which it requires.Geometers need to recognize a triangle in a diagram; and about thirty othersuch items. This repertoire of responses is drastically less diverse than are thephysical objects which would qualify as diagrams (even more so, if we allowdiagram imaginings).(b) Under the banner of ‘generality’, practices exploit repertoire restrictionalso in artifact production. Why make distinct artifacts to secure the sameresponse, after all? Why not support uniformity of response through uniformityof artifact? Geometrical diagrams are drastically less diverse than are thecircumstances in life with which geometrical practice may let us cope; in thatits generality (in Aristotle’s sense?) consists. And in any particular geometricaldemonstration, the diagrams displayed are drastically less diverse than arethe circumstances in life with which that argument can help us cope; inthat its generality (in Frege’s sense) consists. We want to get the control orcoping ability we need, through argument or otherwise, with the scantestartifact-and-response repertoire that will do.The advances with respect to Euclidean geometry in Descartes’ geometry,and later in projective geometries, are practice modifications to exploit moreuniform artifact strategies, which implausibly somehow retain argumentativegrip. The uniformity is achieved by suitably reducing the diversity either ofartifacts or of responses to them. On the other hand, unsuitable limitationon the diversity of artifact-and-response repertoire leaves a practice blind and