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.
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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)