, Volume 186, Issue 1, pp 55-102

The twofold role of diagrams in Euclid’s plane geometry

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Abstract

Proposition I.1 is, by far, the most popular example used to justify the thesis that many of Euclid’s geometric arguments are diagram-based. Many scholars have recently articulated this thesis in different ways and argued for it. My purpose is to reformulate it in a quite general way, by describing what I take to be the twofold role that diagrams play in Euclid’s plane geometry (EPG). Euclid’s arguments are object-dependent. They are about geometric objects. Hence, they cannot be diagram-based unless diagrams are supposed to have an appropriate relation with these objects. I take this relation to be a quite peculiar sort of representation. Its peculiarity depends on the two following claims that I shall argue for: (i) The identity conditions of EPG objects are provided by the identity conditions of the diagrams that represent them; (ii) EPG objects inherit some properties and relations from these diagrams.

Some views expounded in the present paper have been previously presented in Panza (2002), whose first version was written in 1996, during a visiting professorship at the Universidad Nacional Autónoma de México. I thank all the people who supported me during my stay there. Several preliminary versions of the present paper have circulated in different forms and two of them are available online at http://hal.archives-ouvertes.fr/hal-00192165. This has allowed me to benefit from many comments, suggestions and criticisms and to change some of my views. I thank in particular, for their comments, suggestions and criticisms: Carlos Alvarez, Andrew Arana, Jeremy Avigad, Jessica Carter, Karine Chemla, Annalisa Coliva, Davide Crippa, Paolo d’Allessandro, Enzo Fano, Michael Friedman, Massimo Galuzzi, Giovanna Giardina, Bruce Glymour, Pierluigi Graziani, Jan Lacki, Abel Lassalle Casanave, Danielle Macbeth, Paolo Mancosu, Sébastien Maronne, John Mumma, Michael Hallett, Ken Manders, Michael Otte, Mircea Radu, Ferruccio Repellini, Giuseppina Ronziti, Ken Saito, Wagner Sanz, and Bernard Vitrac.