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

76 kenneth manderson conics (Apollonius), spherical geometry (Menelaos), and mathematicalastronomy. Attractive targets also include later non-analytic work in geometry(Steiner!) and its foundations (Leibniz, Legendre); and early projective authorswho aim to clarify Euclidean geometry rather than play a different game(Desargues, Poncelet). Diagrams in early modern physics (Galileo, Huygens,Newton) provide the additional ability to study the geometry of tangency toas yet unknown curves, hence to tackle problems now handled by first-orderdifferential equations. What non-diagrammatic means (sign conventions) makeit possible for Lagrange’s Méchanique Analytique to dispense with diagramsaltogether in theoretical mechanics?I myself have used ED to contrast with Descartes’ analytic geometry andlater versions. One striking difference: analytic geometries have expressiveresources allowing certain cogent inferences based solely on exact geometricinformation (using polynomial equations) separately from co-exact information(polynomial inequalities); Euclidean diagram-based reasoning requires both intandem. These expressive means include invariants of a determinate geometricproblem, the coefficients of its (irreducible) polynomial equation, which, up toselection from a finite set using inequalities, suffice for its solution. Moreover,algebraic manipulation, perhaps together with initial geometric constructions,suffices to derive the equation of a problem.2. A special challenge is the role of diagram use in investigations of the parallelpostulate (PP). It licenses inferences that would be based on the behavior of thediagram ‘infinitely far’ out, or, as is easily seen equivalent by Book I argumentsnot using PP, in the ‘infinitely small’: precisely circumstances under which ourability to control or attribute features in physical diagrams must give out.Neither the mere consistency of alternatives to PP (hyperbolic geometry)with the remaining tenets of Euclidean geometry, nor even the ‘Euclideanmodels’ of such alternatives (hyperbolic geometry as the restriction of theEuclidean plane to the interior of an ellipse) fully settle the original question:whether Euclidean diagram-use has resources to obtain the results for which ourEuclid texts invoke PP. For at least in their modern forms, consistency resultsare based only on propositionally articulated aspects of Euclidean practice, andthe ‘Euclidean models’ explicitly modify the way diagram elements (even linesegments) are handled. Both are obvious distortions of ancient practice.The methods of ED in contrast, may be applied to earlier, still diagrambasedinvestigations of PP (Saccheri, Lambert). Notably, Saccheri (1733) triesto exclude alternatives to PP by reductio analysis; but he clearly treats evenline segments differently from what one would encounter in a non-reductioEuclidean demonstration. Is his treatment within the limits for a properEuclidean reductio? Such questions will force us to refine our conception of