On Maxima and Minima: Chapter 5 of Rules for Solving Sophismata, with an anonymous fourteenth-century discussion

202 Pages·1984·6.33 MB·English

by  William Heytesbury (auth.)

ON MAXIMA AND MINIMA
SYNTHESE HISTORICAL LIBRARY
TEXTS AND STUDIES IN THE HISTORY OF
LOGIC AND PHILOSOPHY
Editors:
N. KRETZMANN, Cornell University
G. NUCHELMANS, University of Leyden
L. M. DE RIJK, University of Leyden
Editorial Board:
J. BERG, Munich Institute of Technology
F. DEL PUNTA, Linacre College, Oxford
D. P. HENRY, University ofM anchester
J. HINTIKKA, Florida State University, Tallahassee
B. MATES, University of California, Berkeley
J. E. MURDOCH, Harvard University
G. PATZIG, University ofGOttingen
VOLUME 26
WILLIAM HEYTESBURY
ON MAXIMA AND MINIMA
Chapter 5 of Rules for Solving Sophismata,
with an anonymous fourteenth-century discussion
Translated, with an Introduction and Study, by
JOHN LONGEWA Y
University of Wisconsin at Parkside, Kenosha, Wisconsin, U.S.A.
D. REIDEL PUBLISHING COMPANY
A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP
DORDRECHT/BOSTON/LANCASTER
library of Congress Cataloging in Publication Data
Heytesbury, William, fl. 1340.
On maxima and minima.
(Synthese historical library ; v. 26)
Bibliography: p.
Includes indexes.
Partial contents; Rules for solving sophisms, chapter 5 / by William
Heytesbury - Treatise concerning maxima and minima / anonymous - Tractatus
de maximo et minimo / anonymous.
1. Logic-early works to 1800. I. Longeway, John. II. Treatise
concerning maxima and minima. 1984. III. Tractatus de maximo et minimo.
1984. IV. Title.
BC60.H3813 1984 160 84-18094
ISBN-\3: 978-94-009-6498-3 e-ISBN-\3: 978-94-009-6496-9
DOl: 10.1007/978-94-009-6496-9
Published by D. Reidel Publishing Company,
P. O. Box 17, 3300 AA Dordrech t, Holland.
Sold and distributed in the U.S.A. and Canada
by Kluwer Academic Publishers,
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In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
P.O. Box 322, 3300 AH Dordrecht, Holland.
All Rights Reserved
© 1984 by D. Reidel Publishing Company, Dordrecht, Holland
Softcover reprint of the hardcover 1s t edition 1984
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner
Table of Contents
List of Figures vii
Preface ix
Introduction • • 1
Notes ••••• 4
Rules for Solving Sophisms, Chapter 5: On Maxima
and Minima, by William Heytesbury 7
Par~: Introduction and basic notions 9
Part 2: Conditions under which limits exist 9
Part 3: Rules for the choice of limit in each case • 11
Part 4: Objections and replies concerning
the general program • • • • • • • 14
Part 5: Objections and replies concerning the
conditions under which limits exist • 16
Part 6: Objections and replies concerning
the Choice of limits • 28
Notes • 34
Treatise Concerning Maxima and Minima, Anonymous • • • • 67
Part 1: The four-fold distinction • • 69
Part 2: Exposition of the members of
the distinction • • • • • • • • • • • • 70
Part 3: Requirements for correct application
of the division ••••••• • • 74
Part 4: Rules for choosing the correct part of
the division. • • • • • • • ••• 77
Part 5: Doubts concerning what has been said • • • • 85
Notes 90
Tractatus de Maximo et Minimo, Anonymous • • • • 99
Study • .-. • • • .-. • • • • • • • • • • • • 133
I. The nature of Heytesbury's "De maximo et minimo"
and his theory • • • 135
2. The tradition behind the theory 137
3. The fundamentals of the theory. 141
3.I.That capacities of a single sort can be
measured on a linear continuum of
coordinate capacities ••••• 141
3.2.The possible limits for a capacity 148
3.3.The extension of the theory for physical
capacities to all cases of limits
for a bipartition on a continuum 151
3.4.The problem of the second limit ••••• 152
4. Conditions for the existence of a limit 158
5. The choice of limit 166
6. Conclusion 172
vi TABLE OF CONTENTS
Notes •••• • • • • 174
Bibliography • • • • • • 187
Indices
Index of names and topics 191
Index of sophismata 197
Scholars cited. 201
List of Figures
Figure 1: Heat A • 39
Figure 2: Magnitude A ••••••• 49
Figure 3: The Difform Resistance 50
Figure 4: Difform Heats • • • • • • • 51
Figure 5: The Black and White Object 54
Figure 6: The Case of Difform Quantities 57
Figure 7: The Descent of Object A • • • • 58
Figure 8: Uniformly Difform Surface defg 72
Figure 9: Uniformly Difform Heat A • • 73
Figure 10: Object A, Half Pure White, Half Not 76
Figure 11: The Representation of Variable Capacities on
a Number Line . . • • . . . • . • • 146
Figure 12: Some Possible and Impossible Limits for
Capacities. • • • • • • • • • • • •• 147
vii
Preface
This book began with my edition of the anonymous treatise.
A translation and notes seemed essential if the material of the
treatise was to be understood. It then seemed that Chapter 5
of Heytesbury's Rules for Solving Sophismata, on which the
treatise was based, should also be included. My translation of
the Heytesbury treatise is based on a fifteenth-century
edition, supplemented by readings from a few of the better
manuscripts. (A critical edition from all the manuscripts, of
which Chapter 5 will be mine, is now in progress under the
supervision of Paul Spade, but only a few insignificant changes
in the translation should be necessitated by the completed
edition.) An examination of related materials seemed
reasonable, and these included Heytesbury's commentator
Gaetano, as well as a chapter from a treatise by Johannes
Venator (in an edition in progress provided by Francesco del
Punta). It seemed unnecessary to publish Gaetano's and
Venator's related works in this volume, but all their
departures from Heytesbury and the anonymous treatise are noted
here. I have not examined other works in the tradition in any
detail.
I owe a great deal to my teacher, Norman Kretzmann, not
only as regards the edition and translations, but also as
regards the notes, study and introduction. The referees of the
typescript (to me unknown) made unusually thorough criticisms
and suggestions to which I have paid close attention. The book
is far better for my having done so. Thanks are due Eleonore
Stump and Richard Boyd for helpful remarks, and Francesco del
Punta for making his edition of Venator available to me. I
also thank Timothy Fossum for helping me with the mathematics
in some earlier drafts. The many errors that no doubt remain
are, of course, entirely my own responsibility. To these
colleagues, and to my friends, especially Judy, Peter and
Erica, I owe much for their support and encouragement. I owe
Norman Kretzmann in particular more than can be said in this
connection. Without his unfailing faith in my virtues and
abilities, and his very material assistance at some critical
junctures, I would certainly not be a scholar today. My debt
to my wife Kathy is even greater, given the real sacrifices she
has made for my career, and surely something is owed Maureen
for her patience with Daddy when he works all day at the
"puter." I want to thank the Parkside Library computer center,
and Media Services at Parkside, for assistance in preparing the
final typescript. Thanks are due my colleagues at the Oswego
ix
x PREFACE
branch of the State University of New York, and to my
colleagues at the University of Wisconsin -- Parkside. I am
grateful both to Oswego and Parkside for their recognition of
the usefulness of a young scholar with wide interests and
commitment to teaching, even when he is also committed to a
technical and unpopular area of research.
John Longeway
University of Wisconsin at Parkside
February, 1984.
INTRODUCTION
Our earliest record of William Heytesbury makes him a
fellow at Merton College in 1330, so he must have been born by
1313, probably in Wiltshire, Salisbury Diocese. In 1338-39
Heytesbury served as first bursar of the college. He was named
a foundation fellow of Queen's College in 1340, but soon
returned to Merton, where he must already have completed his
regency in Arts. He had become a Doctor of Theology by July,
1348. Heytesbury may have been Chancellor of the University in
1353-54, if the records do not reflect a tenure merely pro
tempore. In any case, he was certainly Chancellor in ---1
1371-72, and died soon after, within a few months of 1372.
Heytesbury's works, all in logic, were written in M2rton
during his regency in Arts, roughly the years 1331-1339.
The most important and influential is the Regulae solv endi
sophismata, "Rules for Solving Sophismata," written in
1335. The book was written to help beginning students respond
properly in formal disputations. These disputations often
began with the posing of a "sophisma" (plural: sophismata).
This was not a sophistical argument, but rather a statement,
the truth of which, given certain specified conditions, was at
issue. The respondent was to take a position on the truth of
the statement, and attempt to answer the questions of his
opponent without evasion, and without being driven into
absurdity or contradiction. Usually, of course, there was some
logical difficulty in determining the truth value of the
sophisma under the given conditions, ~ften enough connected
with determining its precise meaning. Here are some
examples: "Socrates begins to be whiter than Plato begins to be
white," given that Socrates and Plato are equally white, and
that they are increasing in whiteness at the same rate; "A
horse is a donkey," given that there are no horses or donkeys;
"Every proposition is false," which provides a problem, of
course, even with no presuppositions. Here are a couple of
modern examples, for comparison: "The King of France is Bald,"
given that there is no king of France; "Necessarily the number
of the planets is greater than sevef'" given that there are
indeed nine planets, and not seven.
The Rules for Solving Sophismata is divided into six
chapters, the fifth of which is translated here, as follows:
(1) "On insoluble sentences," that is, cgncerning sophismata
that involve self-referential paradoxes, (2) "On knowing