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

4The Euclidean Diagram (1995)KENNETH MANDERS[Geometrical] figures must also be regarded as characters, for the circle describedon paper is not a true circle and need not be; it is enough that we take it for acircle.(Loemker, Leibniz: Philosophical Papers and Letters, p. 84.)In Euclidean geometry, a diagram has standing to license inference, just as dorelationships recognized in the text. It is now commonly held that this is a defectof rigor. But the extraordinary career of Euclidean practice justifies a fullerconsideration. It was a stable and fruitful tool of investigation across diversecultural contexts for over two thousand years. During that time, it generallystruck thoughtful and knowledgeable people as the most rigorous of humanways of knowing, even in the face of centuries of internal criticism in antiquity.We here undertake to reconstruct the role of diagrams in the inferentialstandards of Euclidean practice, seeking a more accurate view of the strengthsand weaknesses of traditional geometrical argument. It is interesting to reclaimour philosophical grip on traditional geometrical reasoning especially becausesuch reasoning coordinates two means of expression with very differentcharacteristics: diagrams and ordinary text-based argument. Philosophers of ourage seem to have ignored such representational contrasts, which nonethelessappear to be of great interest.If it is to give a non-trivial grip on life, in its particular way, an intellectualpractice must give us—all too finite and human beings—a game we can play;and play well, together, and to our profit. To succeed in this, intellectualpractices harness our abilities to engage their artifacts, as I will call it: toproduce, preserve, and respond to artifacts in controlled ways—to play games,Research supported by NSF Grants DIR-90-23955 and SBER-94-12895, and a Howard FoundationFellowship for 1994/95.