Mathematizing Space: The Objects of Geometry from Antiquity to the Early Modern Age

320 Pages·2015·5.63 MB·English

by  Vincenzo De Risi (eds.)

Trends in the History of Science
Vincenzo De Risi
Editor
Mathematizing
Space
The Objects of Geometry
from Antiquity to
the Early Modern Age

Trends in the History of Science
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Vincenzo De Risi
Editor
Mathematizing Space
The Objects of Geometry from
Antiquity to the Early Modern Age
Editor
Vincenzo DeRisi
Historyof Science
Max PlanckInstitute
Berlin
Germany
ISSN 2297-2951 ISSN 2297-296X (electronic)
Trends intheHistoryofScience
ISBN 978-3-319-12101-7 ISBN 978-3-319-12102-4 (eBook)
DOI 10.1007/978-3-319-12102-4
LibraryofCongressControlNumber:2014957136
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Foreword
The present book collects the essays presented at the conference on Space,
Geometry and the Imagination from Antiquity to the Early Modern Age, jointly
organized by the Max Planck Institute for the History of Science (Berlin) and the
CentroMatematico‘EnnioDeGiorgi’oftheScuolaNormaleSuperiore(Pisa),and
hostedbytheMaxPlanckInstituteinAugust27–29,2012.Theconferencewaspart
of the activities of the MPI Research Group on Modern Geometry and Space,
directed by Vincenzo De Risi. The Group’s principal aim is to study the inter-
connections between the history of geometry and the philosophy of space in the
pre-Modern and Early Modern Age. In particular, several investigations of the
fellows of the Research Group were directed toward understanding the complex
epistemological revolution that transformed the classical geometry of figures into
themoderngeometryofspace.Theabove-mentionedconferencerepresentedoneof
the high points of this research, and we are grateful to Birkhäuser for accepting to
publish the proceedings of the meeting.
We acknowledge the generous support of the two institutions, the Max Planck
Institute and the Scuola Normale, for organizing the conference. We would par-
ticularlyliketothanktheDirectoroftheCentro‘EnnioDeGiorgi’inPisa,Mariano
Giaquinta, whose early involvement in the organization was crucial for the reali-
zation of the project.
We are especially grateful to the distinguished scholars who took part in the
conferenceandacceptedtodiscusstheirviewswithusonspaceandgeometry,and
for their willingness to contribute to the present volume. Their intellectual gener-
osity made it possible for a real scientific exchange to take place that enlightened
the workshop in Berlin and still shines in their written essays. We also thank all
other participants in the conference, fellows of the Max Planck Institute, students,
andvisitingscholars,whosepointsofviewandobjectionsplayedanimportantpart
in enriching the discussion.
Wefinallythank Chiara Fabbrizi andFredSengmuellerfortheirhelp inediting
the present volume.
v
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Vincenzo De Risi
What’s Location Got to Do with It? Place, Space, and the Infinite
in Classical Greek Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Henry Mendell
A Note on Lines and Planes in Euclid’s Geometry . . . . . . . . . . . . . . . 65
Jeremy Gray
Theon of Smyrna and Ptolemy on Celestial Modelling in Two
and Three Dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Alexander Jones
Proclus’ Conception of Geometric Space and Its Actuality. . . . . . . . . . 105
David Rabouin
Subject, Space, Object: The Birth of Modernity . . . . . . . . . . . . . . . . . 143
Franco Farinelli
On Natural Geometry and Seeing Distance Directly in Descartes. . . . . 157
Gary Hatfield
Hobbes’s Theory of Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Douglas Jesseph
Mathematics and Infinity in Descartes and Newton. . . . . . . . . . . . . . . 209
Andrew Janiak
vii
viii Contents
Leibniz’s Transcendental Aesthetic. . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Daniel Garber
Hume’s Skepticism and Inductivism Concerning Space
and Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
Graciela De Pierris
Kant on Geometry and Experience. . . . . . . . . . . . . . . . . . . . . . . . . . . 275
Michael Friedman
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
Contributors
Graciela De Pierris Department of Philosophy, Stanford University, Stanford,
CA, USA
VincenzoDeRisiMaxPlanckInstitutefortheHistoryofScience,Berlin,Germany
Franco Farinelli Dipartimento di Filosofia e Comunicazione, Università di
Bologna, Bologna, Italy
MichaelFriedmanDepartmentofPhilosophy,StanfordUniversity,Stanford,CA,
USA
Daniel Garber Department of Philosophy, Princeton University, Princeton, USA
Jeremy Gray Department of Mathematics and Statistics, The Open University,
Buckinghamshire, UK
GaryHatfieldDepartmentofPhilosophy,UniversityofPennsylvania,Philadelphia,
PA,USA
Andrew Janiak Department of Philosophy, Duke University, Durham, NC, USA
Douglas Jesseph Department of Philosophy, University of South Florida, Tampa,
FL, USA
Alexander Jones Institute for the Study of the Ancient World, New York
University, New York, USA
Henry Mendell Department of Philosophy, California State University, Los
Angeles, CA, USA
David Rabouin Université Paris 7—CNRS Laboratoire SPHERE UMR 7219,
Paris Cedex 13, France
ix
Introduction
Vincenzo De Risi
Today the definition of geometry as the science of space is generally accepted by
both epistemologists and mathematicians. The history of modern geometry is
entirely builtaround themathematical notion of space, and different approaches to
this science, from Gauss’ studies of intrinsic curvature to the Erlangen Program,
from the discovery of General Relativity to the most recent developments in
topology (take, for instance, Thurston’s geometrization conjecture and its proof)
rely on a general understanding of mathematical space that remains constant
through different perspectives and offers a common ground for regarding all these
developments as parts of a single enterprise. Modern geometry is simply incon-
ceivable without the notion of space.
Nonetheless, the definition of geometry as the science of space, however stan-
dard, is properly speaking modern. Should we go through the thirteen books of
Euclid’s Elements, or in fact the entire corpus of ancient mathematics, we would
find almost no occurrence of spatial concepts or terms. Were we to follow the
millennial development of Classical geometry in the Middle Ages or the Renais-
sance,wewouldstillnotfindanyreferencetospace.Thefirst(andquiterhetorical)
mention of spatium in a geometrical essay does not predate the last decades of the
sixteenth century. To see spatial notions effectively employed in geometrical rea-
soning,wehavetowaitforanotheronehundredyears.Leibniz’workongeometry
(the analysis situs) is probably the first attempt in this direction, and in any case it
ushered in a general discussion about the object of geometrical investigation. The
eighteenth century debated whether geometry had to be regarded as a science of
space, and this new idea initially attracted more opponents than supporters; by the
end of the century the spatial backers won their battle, and the nineteenth century
V.DeRisi(&)
MaxPlanckInstitutefortheHistoryofScience,Boltzmannstraße22,14195Berlin,Germany
e-mail:[email protected]
©SpringerInternationalPublishingSwitzerland2015 1
V.DeRisi(ed.),MathematizingSpace,TrendsintheHistoryofScience,
DOI10.1007/978-3-319-12102-4_1