Table Of ContentADAM DE WODEHAM:
TRACT ATUS DE INDIVISIBILIBUS
SYNTHESE HISTORICAL LIBRARY
TEXTS AND STUDIES IN THE HISTORY OF
LOGIC AND PHILOSOPHY
Editors:
N. KRETZMANN, Cornell University
G. NUCHELMANS, University ofL eyden
Editorial Board:
J. BERG, Munich Institute of Technology
L. M. DE Rul<', University of Leyden
D. P. HENRY, University of Manchester
J. HINTIKKA, Florida State University, Tallahassee
B. MATES, University of California, Berkeley
J. E. MURDOCH, Harvard University
G. PATZIG, UniversityofGottingen
VOLUME 31
ADAM DE WODEHAM:
TRACTATUS
DE
INDIVISIBILIBUS
A Critical Edition with Introduction, Translation,
and Textual Notes
by
REGA WOOD
Franciscan Institute, St. Bonaventure University, New York
KLUWER ACADEMIC PUBLISHERS
DORDRECHT I BOSTON I LONDON
Library of Congress Cataloging in Publication Data
Wodeham, Adam, d. 1358.
[Tractatus de indivisibilibus. English & Latini
Adam de Wodeham, Tractatus de indivisibilibus: a critical edition
/ with introduction, translation, and textual notes by Rega Wood.
p. cm. - (Synthese historical library ; v. 31)
English and Latin.
Includes bibliographies and indexes.
ISBN-13: 978-94-010-7138-3 e-ISBN-13: 978-94-009-1425-4
DOl: 10.1007/978-94-009-1425-4
1. Indivisibles (Philosophy) Early works to 1800. I. Wood, Rega.
II. Title. III. Title: Tractatus de indivisibilibus. IV. Series.
8105.153063131987 87-18800
511 - dc19 CIP
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TABLE OF CONTENTS
PART 1 / INTRODUCTION
1. The Indivisibilist Controversy 3
Indivisibilists: 1.1. Henry of Harclay 4
1.2. Robert Grosseteste 5
1.3. Walter Chatton 7
1.4. William Crathorn 8
Divisibilists: 1.5. William of Ockham 10
1.6. John Duns Scotus 12
1.7. William of Alnwick 12
1.8. Walter Burley 13
2. Tractatus de indivisibilibus 13
2.1. The Author 13
2.2. The Contents 14
2.3. Date Written 15
2.4. Quaestio de divisione et compositione continui 16
2.5. The Date of the Quaestio 17
3. Editorial Principles 19
4. Notes to the Introduction 21
PART II / TEXT AND TRANSLATION
Question 1: Whether forms or extended continua are composed
of indivisibles 33
Article 1: Twelve arguments proving that continua are not
composed of indivisibles 35
Article 2: Twelve arguments by Harclay and Chatton pur
porting to show that continua are composed of
indivisibles 93
Article 3: Refutation of Harclay's and Chatton's arguments 103
VI TABLE OF CONTENTS
Question 2: Whether among continua there is any indivisible
extensive quantum, or among augmentable forms
any intensive indivisible 123
Article 1: A proof that neither points nor lines nor surfaces
exist 123
Article 2: Arguments purporting to show that there must be
indivisibles in continua 139
Article 3: Replies to the arguments in article 2 143
Question 3: Doubts about whether a continuum is infinitely
divisible or composed of indivisibles 165
Doubts based on Zeno's arguments 165
Doubt 1 165
Doubt 2 171
Doubt 3 175
Doubt 4 181
Doubt 5: Whether there are actually infinitely many parts in a
continuum 183
Doubt 6: Whether there is a natural minimum or least indivisi-
ble part 191
Doubt 7: Whether a continuum can be divided into infinitely
many parts of the same size 201
Arguments purporting to show that if it is composed of
infinitely many such parts, then it is actually infinite 201
Proof that a continuum may be divided into infinitely many
completely distinct parts 205
Reply to the first arguments 209
Question 4: Whether the infinite divisibility of a continuum can
be reduced to actuality 213
Article 1: A continuum cannot be divided into its parts 213
Article 2: The parts of a continuum can be divided 225
Doubts 227
Question 5: Whether there are more parts of the same propor-
tion in a larger continuum than in a smaller 237
Article 1: Proof that there are more parts of the same propor-
tion in a larger continuum 239
Article 2: Objections against this proof 247
Article 3: Replies to the objections 261
TABLE OF CONTENTS VII
PART III/NOTES TO THE TEXT
Notes to Question 1 277
Notes to Question 2 294
Notes to Question 3 298
Notes to Question 4 300
Notes to Question 5 302
PART IV / APPENDIX
William Crathorn, "De continuo" 309
Indices 321
PART I
INTRODUCTION
INTRODUCTION
The English Franciscan philosopher and theologian, Adam of Wodeham
(d. 1358), was a disciple and friend of William of Ockham; he was also
a student of Walther Chatton. Nevertheless, he was an independent
thinker who did not hesitate to criticize his former teachers - Ockham
sporadically and benevolently, Chatton, frequently and aggressively.
Since W odeham developed his own doctrinal position by a thorough
critical examination of current opinions, the first part of this introduc
tion briefly outlines the positions of the chief figures in the English
controversy over indivisibles. The second part of the introduction pre
sents a summary of Wodeham's views in the Tractatus de indivisibilibus,
lists the contents of the treatise, and considers the question of its date
and its chronological position in the context of Wodeham's other
works. In the third part, the editorial procedures used here are set
forth.
1. THE INDIVISIBILIST CONTROVERSY
In the literature of the 13th and 14th centuries, the term 'indivisible'
refers to a simple, un extended entity. Consequently, these indivisibles
are not physical atoms but either mathematical points, temporal instants
or indivisibles of motion, usually called mutata esse.
I
THOMAS BRADWARDINE (d. 1349), roughly contemporary with
Wodeham, classified the positions it was possible to take regarding
indivisibles. He described his own view as the common view, that of
"Aristotle, A verroes, and most of the moderns," according to which a
"continuum was not composed of atoms (athomis) but of parts divisible
without end." 2
Bradwardine's opponents, the English medieval indivisibilists,3 all
posited the existen<-e of indivisibles to explain the composition of
continua, such as lines, temporal intervals, or motion. The first sort of
indivisibilism holds that a continuum is composed of a finite number of
indivisibles; Bradwardine ascribes it to Walter Chatton and to the
3
4 PART I
ancient authors, Pythagoras and Plato. The second is the view that a
continuum is composed of an infinite number of indivisibles imme
diately conjoined to one another; it is ascribed to Henry of Harclay.
According to the third and last view, not altogether correctly ascribed
to Robert Grosseteste, a continuum is composed of an infinite number
of indivisibles mediately conjoined. In what follows, I will adopt
Bradwardine's classificatory scheme, departing from its order.
INDIVISIBILISTS
1.1. HENRY OF HARCLAY or HARKE LEY (d. 1317), Chancellor
of Oxford from 1312 until his death, was the first medieval straight
forwardly to advocate indivisibilism. Harclay treated the subject of the
composition of the continuum in his question: Utrum mundus poteril
durare in aeternum a parte post, the two articles of this question deal
respectively with whether one infinite can be greater than another and
whether there are potentially more parts in a greater continuous
quantity than in a smaller one. In the second article Harclay attempts
to refute the view that a continuum could not be composed of
indivisibles.4 Given the title of the question, it would appear that
indivisibilism was first advocated in the context of the medieval debate
on the eternity of the world.
As Bradwardine's classificatory scheme indicates. Harclay holds that
indivisibles are immediately conjoined - that is, they touch one
another. His response to the question how points added to points can
result in an increase in magnitude is to claim that indivisibles touch in
respect to distinct places, immediately next to one another (in distincto
situ immediate se habente).
Harclay also maintains three other characteristic theses regarding
indivisibles. First, following Robert Grosseteste, he holds that infinities
can be unequal. Second, he argues that a more general principle than
Euclid's axiom "every whole must be greater than its parts" is required
to cover the case of infinite quantities. Third, he claims that in the case
of infinite numbers, the whole must be understood to be "that which
contains another and something else beside or beyond it." 5
According to Harclay's most influential argument, God actually sees
the first, incipient point of any line and every other possible point
which could be designated in that line. Either, therefore, God sees that
between the incipient point of the line and every other point on the
