Pythagorean Plato and the Golden Section Sacred Geometry (dissertation)

226 Pages·1983·8.42 MB·English

by  Scott Anthony

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THE PYTHAGOREAN PLATO AND THE GOLDEN SECTION;
A STUDY IN ABDUCTIVE INFERENCE
BY
SCOTT ANTHONY OLSEN
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1985
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Copyright 1983
By
Scott A. Olsen
This dissertation is respectfully dedicated to two
J*: excellent teachers, the late Henry Mehlberg, and Dan Pedo,
Henry Mehlberg set an impeccable example in the quest for
knowledge. And Dan Pedoe instilled in me a love for the
ancient geometry.
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ACKNOWLEDGMENTS
I would Like to thank my Committee members Dr. Ellen
Haring, Dr. Thomas W. Simon, Dr. Robert D'Amico, and Dr.
Philip Callahan, and outside reader Dr. Joe Rosenshein for
their support and attendance at my defense. I would also
like to thank Karin Esser and Jean Pileggi for their help
in this endeavour.
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TABLE OF CONTENTS
ACKNOWLEDGMENTS iv
LIST OF FIGURES vi
ABSTRACT .vii
CHAPTER I INTRODUCTION 1
Notes 7
CHAPTER II ABDUCTION 8
Peirce • • • • 8
Eratosthenes & Kepler 17
Apagoge 21
Dialectic 32
Meno & Theaetetus 39
Notes '^3
CHAPTER III THE PYTHAGOREAN PLATO 45
The Quadrivium 45
The Academy and Its Members 51
,
On the Good 66
The Pythagorean Influence 73
The Notorious Question of Mathematicals 83
The Divided Line 89
Notes il9
CHAPTER IV THE GOLDEN SECTION 124
Timaeus 124
Proportion 129
Taylor & Thompson on the Epinomis 134
i and the Fibonacci Series 149
The Regular Solids 158
Conclusion 201
Notes 203
BIBLIOGRAPHY 205
•
BIOGRAPHICAL SKETCH 216
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LIST OF FIGURES
Figure # Title Page
1 Plato Chronology 122
2 Divided Line 123
3 Golden Cut & Fibonacci Approximation 147
4 Logarithmic Spiral & Golden Triangle. ,..,,.,..., 153
5 Logarithmic Spiral & Golden Rectangle,..,..,,.,,, 153
6 Five Regular Solids 159
7 1:1:\/T Right-angled Isosceles Triangle. , . . , 163
Right-angled Scalene Triangle 163
8 1:\IT:2 ,
9 Monadic Equilateral Triangle 165
10 Stylometric Datings of Plato's Dialogues 166
11 Pentagon 1^5
12 Pentagons Isosceles Triangle..., 176
13 Pentagon & 10 Scalene Triangles 177
14 Pentagon & 30 Scalene Triangles 178
15 Pentagon & Pentalpha 179
16 Pentagon & Two Pentalphas 179
17 Pentagon & Pentagram 181
18 Pentagonal Bisection > 181
19 Pentagon & Isosceles Triangle 182
20 Two Half-Pentalphas 183
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21 Pentalpha
186
22 Golden Cut
23 Golden Cut & Pentalpha 186
24 Pentalpha Bisection 187
25 Circle & Pentalpha 188
26 Pentagon in Circle 190
27 180 Rotation of Figure # 26, 191
28 Circle, Pentagon, & Half-Pentalphas 192
29 Golden Section in Pentagram 194
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30 Double Square
31 Construction of Golden Rectangle 195
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32 Golden Rectangle
33 Icosahedron with Intersecting Golden Rectangles.. 197
34 Dodecahedron with Intersection Golden Rectangles. 197
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Abstract of Dissertation Presented to the Graduate
Council of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of
Doctor of Philosophy
THE PYTHAGOREAN PLATO AND THE GOLDEN SECTION:
A STUDY IN ABDUCTIVE INFERENCE
By
SCOTT ANTHONY OLSEN
AUGUST 983
t
Chairperson: Dr. Ellen S. Haring
Cochairperson: Dr. Thomas ¥. Simon
Major Department: Philosophy
The thesis of this dissertation is an interweaving
relation of three factors. First is the contention that
Plato employed and taught a method of logical discovery, or
analysis, long before Charles Sanders Peirce rediscovered
the fundamental mechanics of the procedure, the latter
naming it abduction. Second, Plato was in essential
respects a follower of the Pythagorean mathematical
tradition of philosophy. As such, he mirrored the secrecy
of his predecessors by avoiding the use of explicit
doctrinal writings. Rather, his manner was obstetric,
expecting the readers of his dialogues to abduct the proper
solutions to the problems and puzzles presented therein.
Third, as a Pythagorean, he saw number, ratio, and
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proportion as the essential underlying nature of things.
In particular he saw the role of the golden section as
fundamental in the structure and aesthetics of the Cosmos.
Plato was much more strongly influenced by the
Pythagoreans than is generally acknowledged by modern
scholars. The evidence of the mathematical nature of his
unwritten lectures, his disparagement of written doctrine,
the mathematical nature of the work in the Academy, the
mathematical hints embedded in the "divided line" and the
Timaeus, and Aristotle's references to a doctrine of
mathematicais intermediate between the Forms and sensible
things, tend to bear this out. In his method of analysis,
Plato would reason backwards to a hypothesis which would
explain an anomalous phenomenon or theoretical dilemma. In
many ways Plato penetrated deeper into the mystery of
numbers than anyone since his time. This dissertation is
intended to direct attention to Plato's unwritten
doctrines, which centered around the use of analysis to
divine the mathematical nature of the Cosmos.
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CHAPTER
I
INTRODUCTION
The thesis of this dissertation is an interweaving
relation of three factors. First is the contention that
Plato employed and taught a method of logical discovery-
long before Cnarles Sanders Peirce rediscovered the
fundamental mechanics of this procedure, the latter naming
it abduction. Second, Plato was in essential respects a
follower of the Pythagorean mathematical tradition of
philosophy. As such he mirrored the secrecy of his
predecessors by avoiding the use of explicit doctrinal
writings. Rather, his manner was obstetric, expecting the
readers of his dialogues to abduct the proper solutions to
the problems he presented. Third, as a Pythagorean he saw
number, ratio, and proportion as the essential underlying
nature of things. Both epistemologicaliy and
ontologically number is the primary feature of his
,
philosophy. Through an understanding of his intermediate
doctrine of mathematicals and the soul, it will be argued
that Plato saw number, ratio, and proportion literally
infused into the world. The knowledge of man and an
appreciation of what elements populate the Cosmos for Plato
depends upon this apprehension of number in things. And in
particular it involves the understanding of a particular
ratio, the golden section (tome), which acted as a
fundamental modular in terms of the construction and
relation of things within the Cosmos.
Several subsidiary issues will emerge as I proceed
through the argument. I will list some of these at the
outset so that the reader may have a better idea of where
my argument is leading. One feature of my positon is that,
though not explicitly exposing his doctrine in the
dialogues, Plato nevertheless retained a consistent view
throughout his life regarding the Forms and their
mathematical nature. The reason there is confusion about
Plato's mathematical doctrine of Number-Ideas and
mathematicals is because commentators have had a hard time
tallying what Aristotle has to say about Plato's doctrine
with what appears on the surface in Plato's dialogues. The
problem is compounded due to the fact that, besides not
explicitly writing on his number doctrine, Plato's emphasis
is on midwifery throughout his works. In the early
so-called Socratic dialogues the reader is left confused
because no essential definitions are fastened upon.
However, the method of cross-examination (elenchus) as an
initial stage of dialectical inquiry is employed to its
fullest. Nevertheless, the middle dialogues quite
literally expose some of the mathematical doctrine for
those who have eyes to see it. But the reader must employ
abduction, reasoning backwards from the puzzles, problems.
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