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

diagram-based geometric practice 67historical interest, characterize an ancient reasoning practice warts and all, thestudent would not contest. But there are many reasons for a philosopher todayto analyze diagram-based geometrical reasoning, starting with the challenge ofexplaining its epistemic success in the face of so many apparent challenges.Though some philosophers haughtily deny there could be mathematics,properly speaking, before modern logic, mathematicians generally, includingones at the forefront of radical 20th-century reconceptualizations of geometry,recognize Euclid as rigorous mathematical reasoning to nontrivial correctconclusions (witness recent books of Hartshorne (2000) and Artmann (1999)).Maybe they just like having a long history?No, modern mathematics subsumes Euclid’s geometrical conclusions in realanalytic geometry, real analysis, and functional analysis. In research profile, onlya corner of modern geometry; but in terms of the footprint of mathematics inthe modern world, that is most of modern mathematics.Euclid (see Vitrac (2004)), and Apollonius and Archimedes, are virtuallywithout error: their every result has a counterpart in modern mathematics,even if subsumed in patterns of claims and proofs recognized much later. Butthere was ample scope for error: some claims even of the early books of Euclidare subtle, and especially the many equality claims are versatile in indefinitelyextendable combinations of great power. Think of the equalities of angles,sum-of-angles, and Pythagorean theorems in Book I, and in Book III theequality of angles on a chord from the points on a circle and the equality ofrectangles (products) of secants through a point to a circle.Moreover, Euclid’s Elements already sets patterns that are extrapolated bymore abstract 20th-century mathematics. The Pythagorean theorem lives on inEuclidean and even Riemannian metric spaces; even Euclidean results ‘everydog knows’ set the framework of more abstract modern mathematical thought,as with the triangle equality I.20 in the definition of metric spaces.Surely one of the most basic intellectual responsibilities of justificationcenteredphilosophy of mathematics is to account for the justificatory successof diagram-based geometry. Nor should we fool ourselves that modern logicalreconstructions by Hilbert (1899/1902) and Tarski (1959), however greattheir interest in other respects, give such an account: ancient geometersachieved their lasting, subtle and powerful results precisely by the meansthat philosophers dismiss so high-handedly today, without the benefit ofmodern logic and Hilbert’s refined control of coordinate domains. (Mumma(2006) gives a logical reconstruction that is more sensitive to diagram use indemonstration.)Still, our student might protest, even if it discharges a pressing intellectualresponsibility, giving an account of the success of ancient mathematical