Diagrams in mathematics: history and philosophy

Synthese, May 2012

John Mumma, Marco Panza

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Diagrams in mathematics: history and philosophy

John Mumma 0 Marco Panza 0 0 J. Mumma Division of Logic , Methodology, and Philosophy of Science at the Suppes Center for History and Philosophy of Science, Stanford University , Palo Alto, CA, USA 1 The date of publication of Manders' paper is misleading. Though it was not published until 2008, it was written in 1995 and widely circulated, and became highly influential as a manuscript. - of the CNRS and the University of Paris 7). The first workshop was held at Stanford in the fall of 2007, the second in Paris in the fall of 2008. The present special issue is the direct result of these workshops. Because of the central position of Euclids geometry, in both the history and philosophy of mathematics, the diagrams of Euclids geometry have been a major topic in the recent research on diagrams in mathematics. This is reflected in the make-up of the issue. Part I, which contains over half of the contributions to the issue, concerns Euclidean diagrams, addressing either their place and nature in the manuscript tradition, their role in Euclids geometry, or their relevance to early modern philosophy and mathematics. The contributions of part II examine mathematical diagrams in more modern and/or advanced settings. In the remainder of this introduction, we briefly describe the content of these papers. The two articles included in section I.a are devoted to geometric diagrams in the manuscript tradition of ancient Greek texts. As Saito points out in the beginning of his article, the diagrams we find in modern versions of ancient Greek mathematical texts are far from representative of those appearing in the manuscripts from preceding centuries. These manuscripts result from various scribal traditions that exhibit various practices in the production of diagrams. The general aim of both Saitos article and Gregg De Youngs is to illuminate significant features of such practices. Saitos concern is with how the diagrams of the manuscript tradition relate to the geometrical content of the arguments they accompany. They often satisfy conditions not stipulated to hold (e.g. by instantiating a parallelogram as a square) or alternatively misrepresent conditions stipulated or proven to hold (e.g. by depicting angles stipulated as equal as clearly unequal). There are, further, notable features with the diagrams representing reductio ad absurdam arguments, and the diagrams representing arguments that range over many cases. To shed light on the different manuscript traditions in these respects, Saito presents a case study of the manuscript diagrams that accompany two propositions in book III of Euclids Elements. De Young focuses on a specific manuscript tradition. He explores the basic architecture (i.e. the relation of diagrams with white space and text) of medieval manuscripts and early printed editions of Euclidean geometry in the Arabic transmission. Many features of Euclidean diagrams remain constant through a long transmission history, although differences in scribal abilities exist. Yet there also seems to be a degree of freedom to adapt diagrams to the architectural context. Adaptation to print brings subtle changes, such as title pages, page numbers, and fully pointed text. The demands of a long-held calligraphic aesthetic ideal and the necessity to compete against traditional manuscripts within the educational marketplace combined to favor the use of lithography over typography in Arabic printed geometry. The articles included in section I.b all share the general goal of deepening our philosophical understanding of the diagrams of Euclids geometry. In his paper, Marco Panza advances an account where geometric diagrams are fundamental to Euclids plane geometry in fixing what the objects of the theory are. Specifically for Panza geometric diagrams perform two indispensable theoretical roles. First, they provide the identity conditions for the objects referred to in Euclids propositions; second, they provide the basis via their concrete properties and relations for the attribution of certain properties and relations to these objects. After explicating this conception in general terms, Panza demonstrates what it amounts to vis--vis the Elements with a detailed and thorough analysis of the definitions and the first 12 propositions of book I. A consequence of Panzas account is that Euclids geometry is in a sense a constructive theory. It does not concern a pre-existing domain of objects, but instead objects brought into existence by the constructions of geometers. The issue of John Mummas paper is the extent to which such an interpretation of Euclid can be given in formal terms. At the center of his discussion is a proof system he developed where Euclids diagrams are formalized as part of the systems syntax. Mumma argues that his formalization is superior to others in accounting for the spatial character of the theorys constructions. It is not immediate, however, that the resulting picture of the theory qualifies as a (...truncated)


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John Mumma, Marco Panza. Diagrams in mathematics: history and philosophy, Synthese, 2012, pp. 1-5, Volume 186, Issue 1, DOI: 10.1007/s11229-011-9988-3