Table Of ContentMinnesota Studies in the Philosophy of Science
Minnesota Studies in the Philosophy of Science
editorial board
Roy T. Cook
(Philosophy, University of Minnesota)
Helen E. Longino
(Philosophy, Stanford University)
Ruth G. Shaw
(Ecol ogy, Evolution, and Be hav ior,
University of Minnesota)
Jos Uffi nk
(Philosophy, University of Minnesota)
C. Kenneth W aters
(Philosophy, University of Calgary)
also in this series
Scientific Pluralism
Stephen H. Kellert, Helen E. Longino, and C. Kenneth W aters, Editors
volume 19
Logical Empiricism in North Ameri ca
Gary L. Hardcastle and Alan W. Richardson, Editors
volume 18
Quantum Meas ure ment: Beyond Paradox
Richard A. Healey and Geoff rey Hellman, Editors
volume 17
Origins of Logical Empiricism
Ronald N. Giere and Alan W. Richardson, Editors
volume 16
Cognitive Models of Science
Ronald N. Giere, Editor
volume 15
the language of nature
Reassessing the Mathematization of Natu ral
Philosophy in the Seventeenth Century
geoffrey gorham, benjamin hill,
edward slowik, and c. kenneth w aters,
editors
Minnesota Studies in the Philosophy of Science 20
University of Minnesota Press
Minneapolis
London
Copyright 2016 by the Regents of the University of Minnesota
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23 22 21 20 19 18 17 16 10 9 8 7 6 5 4 3 2 1
Library of Congress Cataloging-i n-P ublication Data
Names: Gorham, Geoffrey, editor.
Title: The language of nature : reassessing the mathematization of natu ral philosophy
in the 17th century / Geoffrey Gorham [and three others], editors.
Description: Minneapolis : University of Minnesota Press, [2016] | Series: Minnesota
studies in the philosophy of science ; volume 20 | Includes bibliographical
references and index.
Identifi ers: LCCN 2015036898 | ISBN 978-0-8166-9950-6 (hc) |
ISBN 978-0-8166-9989-6 (pb)
Subjects: LCSH Physics— Philosophy— History—17th c entury. | Physics—P hilosophy.
Mathe matics— Philosophy—H istory—17th century. | Mathe matics— Philosophy.
Classifi cation: LCC QC7 .L224 2016 | DDC 530.15— dc23
LC reco rd available at http://lccn. loc . gov2015036898
contents
Introduction
geoffrey gorham, benjamin hill, and edward slowik 1
1.Reading the Book of Nature: Th e Ontological and Epistemological
Under pinnings of Galileo’s Mathematical Realism
carla rita palmerino 29
2. “Th e Marriage of Physics with Mathe matics”: Francis Bacon
on Mea sure ment, Mathe matics, and the Construction of a
Mathematical Physics
dana jalobeanu 51
3.On the Mathematization of Free Fall: Galileo, Descartes,
and a History of Misconstrual
richard t. w. arthur 81
4. Th e Mathematization of Nature in Descartes and the
First Cartesians
roger ariew 112
5.Laws of Nature and the Mathe matics of Motion
daniel garber 134
6.Ratios, Quotients, and the Language of Nature
douglas jesseph 160
7.Color by Numbers: Th e Harmonious Palette in Early
Modern Painting
eileen reeves 178
8. Th e Role of Mathematical Prac ti tion ers and Mathematical Practice
in Developing Mathe matics as the Language of Nature
lesley b. cormack 205
9.Leibniz on Order, Harmony, and the Notion of Substance:
Mathematizing the Sciences of Metaphysics and Physics
kurt smith 229
10.Leibniz’s Harlequinade: Nature, Infi nity, and the Limits
of Mathematization
justin e. h. smith 250
11. Th e Geometrical Method as a New Standard of Truth,
Based on the Mathematization of Nature
ursula goldenbaum 274
12.Philosophical Geometers and Geometrical Phi los o phers
christopher smeenk 308
Contributors 339
Index 343
introduction
geoffrey gorham, benjamin hill,
and edward slowik
conceptual background
No other episode in the history of Western science has been as consequential
as the rise of the mathematical approach to the natu ral world, both in terms
of its impact on the development of science during the scientifi c revolution
but also in regard to the debates that it has generated among scholars who
have striven to understand the history and nature of science. In his recent
summary of this “mathematization thesis,” Michael Mahoney recounts the
stunningly quick ascendancy of the mathematization of nature, a mere two-
hundred- year span that witnessed the overthrow of the Aristotle- inspired
Scholastic approach to the relationship between mathe matics and natu ral
philosophy that had held sway up through the fi rst half of the Re nais sance:
“For although astronomy had always been deemed a mathematical science,
few in the early sixteenth century would have envisioned a reduction of
physics— that is, of nature as motion and change—to mathe matics” (1998,
702). Yet, by the end of the seventeenth century this radical change in approach
had become dominant. In this introduction, we fi rst summarize and explore
some of the main conceptual issues crucial to the mathematization of nature
during the scientifi c revolution. Th e mathematization thesis signifi es above all
the transformation of scientifi c concepts and methods, especially those con-
cerning the nature of matter, space, and time, through the introduction of
mathematical (or geometrical) techniques and ideas (Yoder 1989). We next
analyze the prominence of mathematization as a historiographical framework
within scholarship of the scientifi c revolution, especially in the twentieth
century. Fi nally, we explain how the contributions to this volume explore,
challenge, and reshape these conceptual and historiographical perspectives.
1
2 Introduction
Th e ideal of mathematization has ancient roots (Bochner 1966). Indeed,
as we will see in the next section, modern historiography has emphasized
the revival of Platonism in the seventeenth century’s drive to mathematize.
Th e remnants of Plato’s own Pythagoreanism are evident in the Republic,
where he advocates an a priori astronomy insofar as the vis i ble motions in
the sky “fall short of the true ones— motions that are really fast or slow as
mea sured in true numbers, that trace out true geometrical fi gures, that are
all in relation to one another” (529d1-5; 1997, 1145–46). So Socrates urges:
“let’s study astronomy by means of prob lems, as we do geometry, and leave
the things in the sky alone” (530b6- c1; 1146). And in the Timaeus Plato de-
velops an elaborate geometrical cosmology and matter theory, guided by
the conviction that the creator, in order to produce the best and most in-
telligible world, would produce a “symphony of proportion” (32c2; 1237).
Aft er Plato, Archimedes’s program of mathematization in the sciences of
hydrostatics and mechanics provided a model for Galileo and others
(Clagett 1964).
Controversy about the value and limits of mathematization also goes
back to the beginnings of philosophy. In Aristotle’s view, Pythagoras and
Plato excessively confl ated the abstract realm of mathe matics with the con-
crete realm of nature: “the minute accuracy of mathe matics is not to be
demanded in all cases, but only in the case of things which have no matter.
Hence its method is not that of natu ral science” (995a15–18; 1984, 2:1572).
So Aristotle concludes that the student of nature should not simply assume
that matter and motion will conform to mathematical princi ples. Neverthe-
less, in his own Physics, he acknowledges the importance of “the more phys-
ical of the branches of mathe matics, such as optics, harmonics, and
astronomy” (194a8; 1984, 1:311). And in the methodological treatise Poste-
rior Analytics he indicates that such sciences are subject to geometrical (e.g.,
mechanics and optics) or arithmetical (e.g., harmonics) demonstration
(76a1, 21–25; 1984, 1:123) even though their subject matter is empirical: “it
is for the empirical scientist to know the fact, and for the mathematicians to
know the reason why” (78b32-3; 1984, 1:128). Aristotle assumed that the
theorems of such sciences must be “subordinate” to the theorems of their
corresponding mathematical sciences, since he prohibited demonstrations
that crossed subject- genera (75b3-20; 1984, 1:122). Th is way of conceiving the
“mixed sciences,” as they came to be known, gained additional infl uence
through the pseudo- Aristotelian treatise on mechanics, whose prob lems in-
volving wheels, pulleys, and levers were routinely treated geometrically by
Introduction 3
phi los o phers through the sixteenth century, including Galileo (Bertoloni
Meli 2006). Indeed, arithmetic, geometry, astronomy, and music— already
identifi ed as peculiarly mathematical by Plato (Republic Bk 7; 525a–31d;
1997, 1141–47)— were formally and pedagogically grouped together in the
classical “quadrivium.” Consequently, the idea that mathe matics could be
used to directly represent physical phenomena remained an open and
contested question through the ancient and medieval periods. In the sev-
enteenth century, the main foci of the ongoing debate can be grouped under
three broad conceptual categories: instrumentalism versus realism, types of
mathematization, and social context.
Instrumentalism versus Realism
Two impor tant sources of skepticism about mathematization can be
traced to the Aristotelian strictures mentioned previously, one metaphysi-
cal and one methodological. First, it was claimed that matter did not con-
form to the exactness of mathe matics, and second, that the deductive
structure of mathematical demonstration was inadequate to capture the
causal relationships among natu ral bodies. Hence, outside of the classical
“mixed sciences” of optics, mechanics, and astronomy, the utility of mathe-
matics for understanding nature was severely limited. Based on these con-
cerns, an instrumentalist tradition arose that provided a negative answer to
the question, do mathematical objects and their relationships correspond to
natu ral objects and their relationships? Instrumentalism regards the math-
ematical component of physical theories, for example, the epicycle- deferent
system of Ptolemaic astronomy, as a mere calculating device for predicting
phenomena (Machamer 1976). And this outlook remained infl uential
through the beginning of the early modern period. It is expressed in Osian-
der’s preface to Copernicus’s De Revolutionibus (1543), which stipulates
that since the astronomer “cannot in any way attain to the true causes, he
will adopt what ever suppositions enable the motions to be computed cor-
rectly from the princi ples of geometry . . . these hypotheses need not be
true nor even probable” (1978, xvi).
Yet, during the sixteenth and seventeenth centuries the mathematical
constructions employed in the new Copernican theory of astronomy began
to be accepted by many as providing knowledge of the actual relationships
among celestial bodies. Th us, Kepler and Galileo urged that the aim of as-
tronomy was physical truth, not merely to “save the phenomena” via math-
ematical models (Jardine 1979). And the same realist attitude was extended
