Plato's mathematical imagination: the mathematical passages in the dialogues and their interpretation

308 Pages·1954·14.131 MB·English

by  Robert S. Brumbaugh

Plato's
Matlumatical Imagination
by ROBERT s. BRUMBAUGH
The Mathematical Passages
in the Dialogues
and Their Interpretation
INDIANA UNIVERSITY PRESS
Bloomington
Preface
PROFESSORS Richard P. McKeon, Raymond Klibansky, Newtoll
P. Stallknecht, and Guido Calogero have read this study in
manuscript, and I am most grateful to them for their sugges
tions and comments. I hope that readers acquainted with Pro
fessor McKeon's approach to Greek philosophy will find in this
study by one of his students and admirers a spirit and method
with which they are familiar. Over a period of years, Professor
Stallknecht has been unfailingly generous of his time and as
sistance. His insights into the aesthetic dimension of my prob.
lems, and his distinctive analysis of the nature and history of
imagination, have added immeasurably to my own apprecia
tion of the subject treated here. Professor Klibansky has con
tributed a number of incisive and illuminating comments on
the Pythagorean and Platonic traditions and on some of the
textual problems crucial in mathematical passages in Plato.
Professor Calogero's general evaluation and specific comments
were most helpful. The responsibility for any shortcomings of
thetextremainsmyown; but because this studydeparts at many
points from any previous work in the Platonic tradition, it is a
responsibility I should have hesitated to assume without the
encouragement and help of these scholars.
I wish to thank Professor Edward Seeber, of Indiana Univer
sity, for his help in the preliminary editing of copy. I am par
ticularly grateful to Mr. Walter Albee, of the Indiana Univer
sity Press, for copy-editing the manuscript and seeing it through
the various stages of design and production and for suggesting
many improvements in the design of the figures and in the co
herence of the text of this study. Mr. Robert Williams and the
Yale Cartographic Laboratory have been most helpful in pre
paring drawings for the figures. Professor Ronald B. Levinson,
vii
viii Preface
of the University of Maine, has helped me to clarify and im
prove Section 6 of Chapter III; I am sorry that his In Defense
of Plato appeared too late to be cited where it is clearly rele
vant. The possibility ofconstructinga mirror which would give
a non-reversed image as the one in Figure 88 does, a possibility
that has been seriously doubted, was proven by Dr. Harrison
Pemberton and Mr. George Starr, of Yale University. Mr.
Thomas B. Rosenmeyer's criticism of my views on the Critias
has also been helpful, not only in Chapter II, where his article
is explicitlycited, but in the wholestudy.
My wife has been a model of patience, help, and industry;
she has copied figures, turned thickets of words into English
sentences, and listened intelligently to sporadic speeches on
Platonic mathematics. I also want to thank Meda Z. Steele for
her help in copying figtlres, and my son Robert, starting his
editorial career at the age of five, for help with various opera
tions and problems.
Parts of this study which have already appeared as notes and
papers are included by permission of publishers and journals.
These are "Note on the Numbers in Plato's Critias," Classical
Philology, XLIII (copyright 1948, Univ. of Chicago), 40-42;
"Note on Plato Republic ix. 587D," ibid.) XLIV (copyright
1949, Univ. of Chicago), 197-99; "Colors ofthe Hemispheres in
Plato's Myth of Er (Republic 616E)," ibid.) XLVI (copyright
1951, Univ. of Chicago), 173-76; "Plato Republic 616E: The
Final 'Law of Nines,' " ibid.) XLIX (copyright 1954, Univ. of
Chicago), 33-34; "TeachingPlato'sRepublicVIIIandIX," The
Classical Journal) 46 (1951), 343-48; "Early Greek Theories of
Sex Determination," Journal of Heredity) XL (1949), 49-50;
HPlato's Divided Line," The Review of Metaphysics~V (1952),
529-34.
Preparation of typescript copies of this manuscript was as
sisted by a Research Grant from Indiana University, and Yale
University has provided funds and facilities for preparation of
the figllre drawings.
Acknowledgments
ACKNOWLEDGMENT of permission to include notes originally
published in journals is made on page viii. Permission to quote
from the following works has also been granted. The abbrevi
ated title used in the text and notes is given first in each case.
Adam, Republic. James Adam, edt The Republic of Plato. 2
vols. Cambridge, 1902. Quoted by permission of Cambridge
University Press.
Bury, Timaeus} Critias} etc. R. G. Bury, trans. Timaeus Critias}
J
Cleitophon} Menexenus} and Epistolae (Loeb Classical Li
brary). New York and London, 1929. Quoted by permission
of Harvard University Press.
Chambray. Platon Oeuvres Completes} t. vii, pts. 1-3, La Re
J
publique} texte etab. par E. Chambray. Paris, 1934. Quoted
by permission of the Societe d'Edition, "Les Belles Lettres."
Cornford, Cosmology. Francis M. Cornford. Plato's Cosmology.
New York, 1937. Quoted by permission of Kegan Paul,
Trench, Trubner & Co., Ltd.
Cornford, Republic. Francis M. Cornford. The RejJublic of
Plato. New York and London, 1945. Quoted by permission of
Oxford University Press.
Greene, Scholia. William C. Greene. Scholia Platonica. Haver
ford, 1938. Schemata from scholia reproduced by permission
of the American Philological Association.
Heath, Euclid. Sir Thomas Heath. The Thirteen Books of Eu
clid's Elements} 2nd ed., 3vols. Oxford, 1928. Quoted by per
rrlission of The Clarendon Press, Oxford.
Heath, History. Sir Thomas Heath. History of Greek Mathe
matics} 2 vols. Oxford, 1921. Quoted by permission of The
Clarendon Press, Oxford.
ix
x Acknowledgments
Jowett, Dialogues. Benjamin Jowett, trans. The Dialogues of
Plato, 3rd ed., 5 vols. New York and London, 1892. Quoted
by permission of The Clarendon Press, Oxford.
Shorey, Republic. Paul Shorey, trans. The Republic (Loeb
Classical Library), 2 vols. New York and London, 1930.
Quoted by permission of Harvard University Press.
Taylor, Commentary. A. E. Taylor. A Commentary on Plato's
Timaeus. Oxford, 1928. Quoted by permission of The Clar
endon Press, Oxford.
Wilson. J. Cook Wilson. "Plato, Republic, 616E." Classical Re
view, XV (1902), 292-93. Quoted by permission of the Clas
sical Review.
Other Works Cited by Abbreviated Title
Dupuis' Summary. J. Dupuis. Le nombre geometrique de Pla
ton: interpretation definitive (in Theon, Exposition des Con
naissances mathematique ... , edt with French trans. by J.
Dupuis). Paris, 1892.
Hermann, Scholia. C. Fr. Hermann, ed. Plato 15: Appendix
Platonica. Leipzig, 1875.
Taylor, Plato. A. E. Taylor. Plato: the Man and His Work, new
edt New York, 1936.
Theon. Theon Smyrnaeus. Expositio rerum mathematicarum
ad legendum Platonem utilum, E. Hiller, edt Leipzig, 1878.
Timaeus (text). Platon, Oeuvres completes, t. x, Timee, Critias,
texte etab. parA. Rivaud. Paris, 1925.
Todhunter, Euclid. The Elements of Euclid, edt by Isaac Tod
hunter (Everyman's Library). New York, 1933.
Contents
PAGE
Introduction
Plato's "Mathematical" Passages 3
Types of Mathematical Metaphor 7
Orientation and Limits of the Present Study 8
Final Comment on Tactics 10
Part One
MATHEMATICAL IMAGES RELATIVELY INDEPENDENT OF THEIR
DIALECTICAL CONTEXTS
CHAPTER I Examples from Pure Mathematics of Methods
and Class-Relations
Introductory Comment: Some Simple Illustrations 15
I. Definition ofOdd and Even Number 17
II. Proof of the Recollection Theory of Knowledge 19
III. The Geometer's Method of Hypothesis 32
IV. Definition of Roots and Surds 38
Part Two
MATHEMATICAL IMAGES CLOSELY DEPENDENT ON THEIR
DIALECTICAL CONTEXTS
CHAPTER II "Social Statistics": Arithmetic Detail
I. Atlantis and Its Institutions 47
II. TheSocial Institutions of the Laws 59
6. Mathematics and the Law 59
xi
xii Contents
PAGE
b. Administrative Logistic 61
c. Mathematics in Education for Citizenship 62
d. Statutory Mathematics: Fines and Sumptuary Laws 66
III. Arithmetic Details in Myth and Chronology 68
CHAPTER III Geometric Metaphor
VERBAL MATRICES: Introductory Remarks 72
I. Construction of a Matrix 83
IMAGES OF HARMONYAND CYCLE: The Republic 85
II. Mathematical Imagery in the Republic; the State and
the Musical Scale 85
III. The Cycle of Social Progress 87
IV. The Divided Line 91
V. Mathematics in Higher Education 104
VI. The Nuptial Number 107
a. Introductory Remarks 107
b. Problems of Translation and Pa.raphrase 109
c. The PhilosophyofHistory 112
d. The Rulers' Problem 113
e. Why the Muses Speak Playfully 114
f. Detailed Interpretation 115
g. Interpretation Established from Text of the Passage
Alone 131
h. Historical Comments: Other Interpretations 143
VII. The Tyrant's Number 151
VIII. The Myth of Er-Astronomy 161
Q. The Unity ofRepublic x 161
b. The Allegorical Intention of the Myth 167
c. Detailed Interpretation 171
d. Textof the Passage: Theon's Version 196
IX. The Myth of Er-Transmigration 203
CHAPTER IV Algebraic Metaphor
Introductory Comment 209
I. Constructionofthe World-Soul 221
II. The Theory of Vision 230
III. The TheoryofGeometrical Elements 238
xvi Figures
PAGE
26. Multiplication Table 75
27. Pythagorean Table of Contraries 76
28. Hippocratic Genetic Theory: A Combination Table 77
29. Standard Spatial Orientation of a Platonic Matrix 78
30. Political Matrix from Laws 78
31-35. Some Typical Uses of Matrices in Scholia
31. Gorgias465C 78
32. Gorgias 477A 79
33. Republie 435B 79
34. Republie 440E 79
35. Correction ofRepublic Matrices 79
36. Schematization and Reduction of Three Times Three
Plane Matrix 80
37. Multiplication of Two Similar Matrices 81
38. Schematized Multiplication of Two Similar Matrices 82
39. Sophist Matrix 84
40. Changes ofTopic and Locations of Mathematical Images
in Rep'ublic 87
41. Growth of State: Circular Impulsion 90
42. Growth of State: Circle under Construction 90
43. Divided Line 101
44. Alternative Construction of Divided Line, Carrying Out
Inequality of Segments 103
45-48. Figures from Scholia
45. Republic 5l0B 105
46. Republic 508A 105
47. Republic 510D 105
48. Republic 534A 105
49. Principles of Mathematical Sciences 106
50. Scholion from Proclus 137
51-61. Nuptial Number
51. Life Cycle 138
52. Acme of Human Life Cycle 138
53. The 3-4-5 Triangle 139
Figures xvii
PAGE
54. Shape of Final Figure 140
55. Lines of Genetic Triangle Recombined into Squares 140
56. Identity of "Diagonals" 141
57·}s
quares on Diagonals 142
58.
59. Phaedrus Matrix ofLives 142
60. List Form of Matrix Used in Phaedrus 142
61. Matrix Underlying Republic IX 143
62. Scholion Figure to Illustrate Tyrant's Number 158
63A. Parallels among Phaedrus 248, Republic VIII-IX, Re-
public III-VII 158
63B. Tyrant's Number: First Step of Calculation 159
64. Tyrant's Number: Second Step of Calculation (Squaring) 159
65. Tyrant's Number: Final Calculation 160
66. Law of Nines: Sizes 198
67. Law of Nines: Colors 199
68. Law of Nines: Colors (Alternative Version) 199
69. Size and Color (Volume and Density) 200
70. Size and Color (Ordinal): Another Law of Nines 200
71. Law ofNines: Balance o~Velocities 201
72. ProportionalityofDistance and Forward Velocity 201
73. Final Law of Nines: Exact Balance of Momentum in
System 202
74. Lack of Balance in Proclus· Alternative Text of List of
Sizes 202
75. Necessity's Spindle in Cross Section 203
76. Matrix of Character with Four Levels of Knowledge Dis-
tinguished 207
77. MatrixofCharacterApplied toOlympian Gods Who Are
Patrons of Types of Human Life 208
78. Scholion: Three Kinds of Ratio 218
79. Scholion: Ten Kinds of Motion (Laws 894B) 219
80-86. Construction of World Soul
80. Initial Terms of Scale 227
81. Arithmetic Means Inserted 228
82. Harmonic Means Inserted 228
83. Both Means Inserted 228
xviii Figures
PAGE
84. Matrix Form of Computation of Series 228
85. Traditional Computation 229
86. Traditional Computation in Matrix Form 229
87-93. Theory of Vision: Distorting Effects of Mirrors
87. Reversal in Plane Mirror 234
88. Rectification of Mirror to Prevent Reversal 234
89. Effect ofReversed Intellectual Vision 235
90. Inversionin Mirror 235
91. Effect of Inverted Intellectual Vision 236
92. Effect of Mirror at Oblique Angle 236
93. Effect of Oblique Intellectual Vision 237
94. The Atomic Triangles 244
95. Synthesis of Equilateral Molecular Triangles 245
96. Synthesis ofSquare Molecular Planes 245
97. Decomposition of Polygons into Triangles 246
98. Homoeomereity of First Atomic Triangle 247
99. Homoeomereity of Second Atomic Triangle 247
100. Transmutation Ratios of Elementary Solids 248
101. Ratios of Numbers of Bounding Planes of Elementary
Solids 248
102. "Diagonals and Their Diagonals" 258
103. Locomotion of Animals 259
104. Lack of Balance in Poseidon's Engineering 261
105. Properties of the Image of Concentric Circles in Three
Types of Context 262
106. Later Diagrams of Class-Inclusion: Euler, Leibniz 264
107. Symbolism: Figures Involving Incommensurability in
Plato's Mathematical Imagery 266
A. Meno (1) G. Republic (2)
B. Meno (2) H. Timaeus (1)
C. Theaetetus I. Timaeus (2)
J.
D. Critias Timaeus (3)
E. Laws K. Republic (3)
F. Republic (1) L. Statesman
M. Republic (4)
108. Construction of Divided Line by Divisions in Mean and
Extreme Ratio 270