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The Pythagorean Plato and the golden section

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Title:
The Pythagorean Plato and the golden section a study in abductive inference
Creator:
Olsen, Scott Anthony
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Language:
English
Physical Description:
viii, 216 leaves : ill. ; 28 cm.

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Subjects / Keywords:
Caves ( jstor )
Dialectic ( jstor )
Geometry ( jstor )
Golden mean ( jstor )
Kidnapping ( jstor )
Mathematics ( jstor )
Metaphysics ( jstor )
Pythagoreanism ( jstor )
Soul ( jstor )
Triangles ( jstor )
Abduction (Logic) ( lcsh )
Abduktion (Logik) ( swd )
Golden section ( lcsh )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1983.
Bibliography:
Includes bibliographical references (leaves 205-215).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Scott Anthony Olsen.

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University of Florida
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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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11337408 ( OCLC )
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Full Text












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


1983
































Copyright 1983

By

Scott A. Olsen






















This dissertation is respectfully dedicated to two

excellent teachers, the late Henry Mehlberg, and Dan Pedoe.

Henry Iehlberg set an impeccable example in the quest for

knowledge. And Dan Pedoe instilled in me a love for the

ancient geometry.



















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.













TABLE OF CONTENTS


ACKNOWLEDGMENTS............................................ iv

LIST OF FIGURES ........................... ............ vi

ABSTRACT.................................................... vii


CHAPTER I


CHAPTER II








CHAPTER III









CHAPTER IV


INTRODUCTION................................... 1
Notes ............... ... ** .......* ... ...... 7

ABDUCTION...................................... 8
Peirce........................................ 8
Eratosthenes & Kepler........................... 17
Apagoge........ ............................... 21
Dialectic........................................ 32
Meno & Theaetetus................ ............ 39
Notes......................................... 43

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.......................................... 119

THE GOLDEN SECTION............................. 124
Timaeus........................................124
Proportion.......................... .. ....129
Taylor & Thompson on the Epinomis..............134
$ and the Fibonacci Series......... .......149
The Regular Solids............................158
Conclusion............ ...................201
Notes.....................................203


BIBLIOGRAPHY........ .......... .......*... ........ ...... ...- 205

BIOGRAPHICAL SKETCH.........................................216













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: CRight-angled Isosceles Triangle.,..,....., 163
8 1:r.T2 Right-angled Scalene Triangle............, 163
9 Monadic EquilaterAl Ttiangle.................... 165
10 Stylometric Datings of Plato's Dialogues......... 166
11 Pentagon ....................................... 175
12 Pentagon & 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
21 Pentalpha............... ...................... 184
22 Golden Cut....................................... 186
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
30 Double Square.................................... 195
31 Construction of Golden Rectangle................. 195
32 Golden Rectangle ................................ 195
33 Icosahedron with Intersecting Golden Rectangles.. 197
34 Dodecahedron with Intersection Golden Rectangles. 197














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 1983



Chairperson: Dr. Ellen S. Haring
Cochairperson: Dr. Thomas W. 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


vii









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

mathematical 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.


viii















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 Charles 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 epistemologically and

ontologically, number is the primary feature of his

philosophy. Through an understanding of his intermediate

doctrine of mathematical 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 position 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

mathematical 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,








and hints to a suitable explanatory hypothesis. This

abductive requirement is even more evident in the later

dialogues, especially the Theaetetus, Parmenides, and

Sophist.3 There, if one accepts the arguments on their

surface, it appears that Plato is attacking what he has

suggested earlier regarding knowledge and the Forms. But

this is not the case.

A further perplexing problem for many scholars enters

the picture when one considers what Aristotle has to say

about Plato's unwritten teachings. It has led to the

mistaken view that Plato changed his philosophy radically

in later life. However, my contention is that a careful

reading of what Aristotle has to say upon the matter helps

to unfold the real underlying nature of the dialogues.

Mathematical concepts are present in one way or another

throughout the dialogues. Possibly obscure at the

beginning, they become central in the middle dialogues,

especially the Republic. And the later Philebus and

Epinomis attest to the retention of the doctrine.

Copleston, who I am in agreement with on this matter,

summed up the position as follows:

There is indeed plenty of evidence that Plato
continued to occupy himself throughout his
years of academic and literary activity with
problems arising from the theory of Forms,
but there is no real evidence that he ever
radically changed his doctrine, still less
that he abandoned it altogether. It has
sometimes been asserted that the
mathematisation of the Forms, which is
ascribed to Plato by Aristotle, was a
doctrine of Plato's old age, a relapse into
Pythagorean "mysticism," but Aristotle does








not say that Plato changed his doctrine, and
the only reasonable conclusion to be drawn
from Aristotle's words would appear to be
that Plato held more or less the same
doctrine, at least during the time that
Aristotle worked under him in the Academy.
(Copleston, 1962, p. 188)

Others, like Cherniss, get around the problem by

accusing Aristotle of "misinterpreting and misrepresenting"

Plato (Cherniss, 1945, p. 25 ). This is ludicrous. We need

only recall that Aristotle was in the Academy with Plato

(until the master's death) for 19 (possibly 20) years.

Surely he should know quite well what Plato had to say.

Fortunately we have some record of what Plato had to say in

his unwritten lectures. This helps to fill the gap. But

unfortunately the remnant fragments are sparse, though very

telling. The view of Cherniss' only indicates the extreme

to which some scholars will move in an attempt to overcome

the apparent disparity. As Copleston goes on to say,

though Plato continued to maintain the
doctrine of Ideas, and though he sought to
clarify his meaning and the ontological and
logical implications of his thought, it does
not follow that we can always grasp what he
actually meant. It is greatly to be
regretted that we have no adequate record of
his lectures in the Academy, since this would
doubtless throw great light on the
interpretation of his theories as put forward
in the dialogues, besides conferring on us
the inestimable benefit of knowing what
Plato's "real" opinions were, the opinions
that he transmitted only through oral
teaching and never published. (Copleston,
1962, pp. 188-189)

My own view on the matter is that if we look closely

enough at the extant fragments, in conjunction with the

Pythagorean background of Plato's thought, and using these








as keys, we can unlock some of the underlying features of

Plato's dialogues. But this is premised on the assumption

that Plato is in fact being obstetric in the dialogues. I

will argue that this is the case, and that further,

sufficient clues are available to evolve an adequate

reconstruction of his mathematical-philosophical doctrine.

Thus, a central theme running throughout this

dissertation is that the words of Aristotle will help to

clarify the position of Plato. Rather than disregard

Aristotle's comments, I will emphasize them. In this way I

hope to accurately explicate some of the features of

Plato's mathematical doctrine and the method of discovery

by analysis that he employed. Plato deserves an even

richer foundation in the philosophies of science and logic

than he has heretofore been credited with. His method of

analysis, of the upward path of reasoning backwards from

conclusion to premises (or from facts to hypothesis or

principle), lies at the very roots of scientific discovery.

The mistaken view of a strictly bifurcated Platonic Cosmos,

with utter disdain for the sensible world, has done unjust

damage to the reputation of Plato among those in science.

This is unfortunate and needs to be remedied.

I set for myself the following problem at the outset.

When we arrive at the Timaeus we will see how the elements,

the regular solids, are said to be constructed out of two

kinds of right-angled triangles, one isosceles and the

other scalene. But he goes on to say,










These then we assume to be the original
elements of fire and other bodies, but the
principles which are prior to to these deity
only knows, and he of men who is a friend of
deity. (Timaeus 53d-e)

I contend that this is a cryptic passage designed by

the midwife Plato to evoke in the reader a desire to search

for the underlying Pythagorean doctrine. To some it

conceals the doctrine. However, to others it is intended

to reveal, if only one is willing to reason backwards to

something more primitive. Thus Plato goes on to say,

anyone who can point out a more
beautiful form than ours for the construction
of these bodies shall carry off the palm, not
as an enemy, but as a friend. Now, the one
which we maintain to be the most beautiful of
all the many triangles is that of which
the double forms a third triangle which is
equilateral. The reason of this would be too
long to tell; he who disproves what we are
saying, and shows that we are mistaken, may
claim a friendly victory. (Timaeus 54a-b)

This is the problem: what more beautiful or primitive form

could there be for the construction of these bodies?

My views are undeniably in the vein of the

Neopythagorean and Neoplatonic traditions. But my

contention is that it is to the Pythagorean Neoplatonists

that we must turn if we are to truly understand Plato. I

have found a much greater degree of insight into Plato in

the Neopythagoreans and Neoplatonists than in the detailed

work of the logic choppers and word mongers. As Blavatsky

once said regarding one of the Neoplatonists, Thomas

Taylor, the English Platonist,











the answer given by one of Thomas Taylor's
admirers to those scholars who criticized his
translations of Plato [was]: "Taylor might
have known less Greek than his critics, but
he knew more Plato." (Blavatsky, 1971, vol.
2, p. 172)

As Flew has written, the origins of the Neoplatonic

interpretation

go back to Plato's own lifetime. Its
starting-point was Plato's contrast between
eternal Ideas and the transient objects of
sense, a contrast suggesting two lines of
speculative enquiry. First, what is the
connection, or is there anything to mediate
between intelligibles and sensibles, the
worlds of Being and of Becoming? Second, is
there any principle beyond the Ideas, or are
they the ultimate reality? (Flew, 1979, p.
254)

This dissertation speaks directly to the former question,

although, I will have something to say about the latter as

well.


Notes

Most subsequent Greek words will be transliterated.
Although I will occasionally give the word in the original
Greek. The golden section, tome, was often referred to by
the Greeks as division in mean and extreme ratio.

Cross-examination, or elenchus, is an important stage
in the dialectical ascent to knowledge. It is employed to
purge one of false beliefs. Through interrogation one is
led to the assertion of contradictory beliefs. This
method is decidedly socratic. Plato emphasized a more
cooperative effort with his students in the Academy.

All citations to works of Plato are according to the
convention of dialogue and passage number. All citations
are to H. Cairns and E. Hamilton, eds., 1971, The Collected
Dialogues of Plato, Princeton: Princeton University Press.
The one major exception is that all Republic quotes are
from D. Lee, transl., 1974, The Republic, London: Penquin.















CHAPTER II

ABDUCTION



Peirce

I choose to begin with Charles Sanders Peirce, because

better than anyone else he seems to have grasped the

significance of the logic of backwards reasoning, or

abduction. Once the position of Peirce is set out, with

some explicit examples, I will return to the examination of

Plato's philosophy.

What Peirce termed abduction (or alternatively,

reduction, retroduction, presumption, hypothesis, or novel

reasoning) is essentially a process of reasoning backwards

from an anomalous phenomenon to a hypothesis which would

adequately explain and predict the existence of the

phenomenon in question. It lies at the center of the

creative discovery process. Abduction occurs whenever our

observations lead to perplexity. Abduction is the initial

grasping at explanation. It is a process by which one

normalizes that which was previously anomalous or

surprising. Peirce's basic formula is very simple:

The surprising fact, C, is observed; But if A
were true, C would be a matter of course,
Hence, there is reason to suspect that A is
true. (Peirce 5.!19)1









Abduction follows upon the initiation of a problem or

puzzling occurrence. The great positive feature about

abduction is that it can lead to very rapid solutions. On

this view discovery takes place through a series of leaps,

rather than a gradual series of developments. As Peirce

said, gradual progression

is not the way in which science mainly
progresses. It advances by leaps; and the
impulse for each leap is either some new
observational resource, or some novel way of
reasoning about the observations. Such novel
way of reasoning might, perhaps, be
considered as a new observational means,
since it draws attention to relations between
facts which should previously have been
passed by unperceived. (Buchler, 1955, p. 51)

Whenever present theories cannot adequately explain a

fact, then the door for abduction opens. Most major

scientific discoveries can be correctly viewed as an

abductive response to perplexing, or anomalous, phenomena.

Thus, Einstein was struck by certain perplexing, seemingly

unaccountable features of the world. Sometimes these

anomalous features cluster about a particular problem.

When this occurs, and is perceived by the individual, great

creative abductive solutions become possible. Thus, as

Kuhn points out,

Einstein wrote that before he had any
substitute for classical mechanics, he could
see the interrelation between the known
anomalies of black-body radiation, the
photoelectric effect and specific heats.
(Kuhn, 1970, p. 89)

Thus, on the basis of these perplexing facts, Einstein was








able to reason backwards to a suitable hypothesis that

would reconcile and adapt each of these puzzling features.

Abduction is the process by which surprising facts

invoke an explanatory hypothesis to account for them.

Thus, abduction "consists in studying facts and devising a

theory to explain them" (Peirce 5.145). It "consists in

examining a mass of facts and in allowing these facts to

suggest a theory" (Peirce 8.209). And it is "the logic by

which we get new ideas" (Peirce 7.98).

For example, Maslow was doing abduction when he

surveyed the data and observations, and, reasoning

backwards, inferred the hypothesis of self-actualization.

The fact that he called it "partly deductive," not knowing

the correct label, does not affect the nature of his

abduction.

I have published in another place a survey of
all the evidence that forces us in the
direction of a concept of healthy growth or
of self-actualizing tendencies. This is
partly deductive evidence in the sense of
pointing out that unless we postulate such a
concept, much of human behavior makes no
sense. This is on the same scientific
principle that led to the discovery of a
hitherto unseen planet that had to be there
in order to make sense of a lot of other
observed data. (Maslow, 1962, pp. 146-147)

Abduction is to be clearly distinguished from

deduction and induction. Nevertheless, the three logical

methods are mutually complementary. However, deduction

only follows upon the initial abductive grasping of the new

hypothesis. The deductive consequences or predictions are

then set out. Induction then consists of the experimental








testing and observation to see if in fact the consequences

deduced from the new hypothesis are correct. If not, then

abduction begins again seeking a new or modified hypothesis

which will more adequately explain and predict the nature

of our observations.

Thus, as Peirce points out,

abduction is the process of forming an
explanatory hypothesis. It is the only
logical operation which introduces any new
ideas; for induction does nothing but
determine a value, and deduction merely
evolves the necessary consequences of a pure
hypothesis. (Peirce 5.171)

Abduction does not have the nature of validity that,

for example, deduction possesses. Abduction is actually a

form of the so-called fallacy of affirming the consequent.

The abducted hypothesis cannot in any way be apprehended as

necessary. It must be viewed as a tentative conjecture,

and at best may be viewed as likely. When we say, the

surprising fact C is observed, but if A were true, C would

follow as a matter of course, the very most that we can do

is say that we therefore have reason to suspect A.

Abduction is a logical method of hypothesis selection, and

it is extremely effective, especially when anomalies are

used to guide one toward the best explanation. But

abductions may turn out to have false results.

The function of hypothesis [abduction] is to
substitute for a great series of predicates
forming no unity in themselves, a single one
(or small number) which involves them all,
together (perhaps) with an indefinite number
of others. It is, therefore, also a
reduction of a manifold to a unity. Every
deductive syllogism may be put into the form:








If A, then B; But A: Therefore, B. And as
the minor premiss in this form appears as
antecedent or reason of a hypothetical
proposition, hypothetic inference [abduction]
may be called reasoning from consequent to
antecedent. (Peirce 5.276)

This notion of reasoning backwards from consequent to

hypothesis is central to abduction. One thing it shares in

common with induction is that it is "rowing up the current

of deductive sequence" (Peirce, 1968, p. 133). But

abduction and induction are to be clearly distinguished.

Abduction is the first step of explanatory discovery, the

grasping of the hypothesis or account. Induction is the

testing of the hypothesis that follows the previous

abductive hypothesis selection and the deductive prediction

of consequences. The operation of testing a
hypothesis by experiment, which consists in
remarking that, if it is true, observations
made under certain conditions ought to have
certain results, and noting the results, and
if they are favourable, extending a certian
confidence to the hypothesis, I call
induction. (Buchler, 1955, p. 152)

There is also a sense in which abduction and induction can

be contrasted as opposing methods.

The induction adds nothing. At the very most
it corrects the value of a ratio or slightly
modifies a hypothesis in a way which had
already been contemplated as possible.
Abduction, on the other hand, is merely
preparatory. It is the first step of
scientific reasoning, as induction is the
concluding step. They are the opposite poles
of reason. The method of either is
the very reverse of the other's. Abduction
seeks a theory. Induction seeks the facts.
(Peirce 7.127-7.218)

The important point here is that abduction occurs in

the process of discovery. It is distinct from the later








process of justification of the hypothesis. Hanson made

this point strongly when he stated that

the salient distinction of "The Logic of
Discovery" consisted in separating (1)
Reasons for accepting a hypothesis, H, from
(2) Reasons for suggesting H in the first
place. (Hanson, 1960, p. 183)

Abduction, the logic of discovery, underlies the latter

above. The robust anomaly R provides reasons to suspect

hypothesis H is true. This is the case, simply because if

H were true, then R would follow as a matter of course.

Hence we have reasons for suggesting or selecting H.

Abduction is the mark of the great theoretical

scientists. Through contemplation of the observables,

especially the puzzling observables, the theoretician

reasoning backwards fastens upon a hypothesis adequate to

explain and predict the occurrence of the anomalies. This,

in effect, defuses the anomalous nature of the observables,

having the effect of normalizing them.

A crucial feature of the activity of abduction is the

role that is played by the anomaly, R. It has the function

of directing one to the type of hypothesis that is

required. As Hanson pointed out, "to a marked degree [the]

observations locate the type of hypothesis which it will be

reasonable ultimately to propose" (Hanson, 1960, p. 185).

The overall interplay of abduction, deduction, and

induction can be appreciated more fully when considering

the following passage by Peirce:

The Deductions which we base upon the
hypothesis which has resulted from Abduction








produce conditional predictions concerning
our future experience. That is to say, we
infer by Deduction that if the hypothesis be
true, any future phenomena of certain
descriptions must present such and such
characters. We now institute a course of
quasi-experimentation in order to bring these
predictions to the test, and thus to form our
final estimate of the value of the
hypothesis, and this whole proceeding I term
Induction. (Peirce 7.115, fn.27)

Abduction, or the logic of discovery, has

unfortunately been ignored for some time. As Paul Weiss

says, "it is regrettable that the logicians are not yet

ready to follow Peirce into this most promising field

[abduction]" (Bernstein, 1965, p. 125). Only recently has

there been a rebirth of interest.

In the case of abduction, Peirce singles out
as an independent form of inference the
formulation of hypotheses for inductive
testing. All this is well known, but, we
fear, too much ignored outside the
constricted space of Peirce scholarship.
Unfortunately, the notion of abductive
inference, which is peculiarly Peirce's, has
not exerted an influence proportionate to the
significance of its insight. (Harris &
Hoover, 1980, p. 329)

The only point where Harris and Hoover err in the

prior statement is in attributing abduction as solely

belonging to Peirce. But this is a mistake. Peirce

himself acknowledged his Greek sources of the abductive

logic. I will argue that the roots of abduction lie in

Plato, and his work in the Academy, and his Pythagorean

predecessors. But first we will consider some of the more

recent developments of Peircean abduction, and then some


actual historical examples.








Consider the following example,

I catch the glint of light on metal through
the trees by the drive, remark that I see the
family car is there, and go on to infer my
son is home. It may be said that taken
literally I have misdescribed things. What I
see, it may be said, is a flash of light
through the trees. Strictly I infer, but do
not see, that the car is there. I
reason backward from what I see, the flash of
light on metal, and my seeing it, to a cause
the presence of which I believe to be
sufficient to explain my experience. Knowing
the situation, and knowing the way things
look in circumstances like these, I infer
that the car is in the drive. (Clark, 1982,
pp. 1-2)

Clark goes on to describe the argument form involved.

Let q be the puzzling perceptual occurrence or anomaly. In

this case it was the glint of light passing through the

trees. Let p stand for the car is in the drive. Let B

stand for the belief that if p (the car is in the drive),

and other things being equal, then q (the glint of light)

would occur. We can then reconstruct the argument as

follows.

1. q (puzzling glint of light),
2. But B (belief that paq),
3. Therefore, p (car in drive).

I conclude from my premises, q and B, that
[hypothesis] p. I conclude that the family
car is there, this being the hypothesis I
draw the truth of which I believe is
sufficient to account for that puzzling
perceptual happening, q. (Clark, 1982, p. 2)

But if this is abduction, are we not simply employing the

fallacy of affirming the consequent? Or is it something


more?








This pattern of reasoning is quite common.
And it is after all a sort of reasoning.
There is here a texture of structured
thoughts leading to a conclusion. Moreover,
there's something sensible about it. It is
not just silly. But of course reasoning this
way, I have sinned deductively. My reasoning
is not deductively valid. (q and B might
after all quite well be true and yet
[hypothesis p] false. Perhaps it is not in
fact the car but a visiting neighbor's camper
whose flash of light on metal I catch).
Peirce insisted that all creativity has its
source in sin: reasoning of this general
sort is the only creative form of inference.
It is the only sort that yields as
conclusions new hypotheses not covertly
asserted in the premises; new hypotheses now
to be tested and examined; hypotheses which
may determine whole new lines of inquiry.
This reasoning is, he thought, quite
ubiquitous, present indeed in all perception
but in nearly every area of contingent
inquiry as well. (It is philosophical
commonplace, too. How frequently we reason
backward from an epistemological puzzle to an
ontological posit.) Peirce, in
characterizing this backward, abductive,
reasoning which runs from effects to
hypotheses about causes sufficient to ensure
them, has implicitly answered the title
question. When is a fallacy valid? Answer:
When it is a good abduction. (Clark, 1982, p.
2)

Clark proceeds admirably, struggling with abduction,

attempting to define its formal standards for validity and

soundness. "It is the need to characterize abductive

soundness which forces the nontrivial nature of abduction

on us"(Clark, 1982, p. 3). In the very process of this

attempt, Clark has reasoned abductively. In a very

analagous manner, this dissertation is an exercise in

abduction, reasoning backwards from the Platonic puzzles

(i.e., the dialogues and the extraneous statements








regarding Plato's doctrines), to an explanatory hypothesis

regarding them.

As an especial philosophical application and
final example, it is perhaps worth remarking
that this account of the nature of abduction
is itself an exercise in abduction. We have
reasoned backward from a puzzling fact--the
widespread employment in philosophical
inquiry of arguments which are deductively
fallacious--to an attempt to characterize an
adequate explanation of the phenomenon. We
have tried to sketch minimal formal standards
by which abductions can be evaluated as valid
or sound, and their employment justified. I
wish I could say more about what is important
about abduction and the competition of
sufficient hypotheses. I wish I could
formulate an articulate formal system of
abduction. But even a sketch like this is
something. It seems to me at least to
override an obvious competitor to explaining
our ubiquitous use of these forms of
inference; the view that these are just
logical lapses--irrational applications of
the fallacy of asserting the consequent.
(Clark, 1982, p. 12)



Eratosthenes & Kepler

The great discovery of Eratosthenes, the Librarian at

Alexandria, provides a good example of abduction. He

pondered over the puzzling fact that on the summer solstice

at noonday the sun was at its zenith directly overhead in

Syene, Egypt, and yet 500 miles north at that precise

moment in Alexandria, the sun was not directly at its

zenith. He abducted that this must be due to the curvature

of the earth away from the sun. He went further and

reasoned that he could determine the amount of curvature of

the earth through geometrical calculation by measuring the

length of shadow cast at Alexandria at noon on the summer








solstice. Knowing the distance from Syene to Alexandria,

he was then able to quite accurately (circa 240B.C.)

calculate the diameter and circumference of the earth.

Eratosthenes worked out the answer (in Greek
units), and, as nearly as we can judge, his
figures in our units came out at about 8,000
miles for the diameter and 25,000 miles for
the circumference of the earth. This, as it
happens, is just about right. (Asimov, 1975,
vol. 1, p. 22)

In view of the perplexing difference in the position

of the sun in the two cities on the summer solstice,

Eratosthenes was able to reason backwards to a hypothesis,

i.e., the earth is round and therefore curves away from the

rays of the sun, which would render that anomalous

phenomenon the expected.
2
Kepler is another example of brilliant abductive

inferences. Both Peirce and Hanson revere the work of

Kepler. Hanson asks,

was Kepler's struggle up from Tycho's data to
the proposal of the elliptical orbit
hypothesis really inferential at all? He
wrote De Motibus Stellae Martis in order to
set out his reason for suggesting the
ellipse. These were not deductive reasons;
he was working from explicanda to explicans
[reasoning backwards]. But neither were they
inductive--not, at least., in any form
advocated by the empiricists, statisticians
and probability theorists who have written on
induction. (Hanson, 1972, p. 85)

The scientific process of discovery may at times be

viewed as a series of explanatory approximations to the

observed facts. An abductively conjectured hypothesis will

often approximate to an adequate explanation of the facts.

One continues to attempt to abduct a more complete









hypothesis which more adequately explains the recalcitrant

facts. Hence there will occasionally occur the unfolding

of a series of hypotheses. Each hypothesis presumably

approximates more closely to an adequate explanation of the

observed facts. This was the case with Kepler's work, De

Motibus Stellae Martis. As Peirce points out,

at each stage of his long
investigation, Kepler has a theory which is
approximately true, since it approximately
satisfies the observations and he
proceeds to modify this theory, after the
most careful and judicious reflection, in
such a way as to render it more rational or
closer to the observed fact. (Buchler, 1955,
p. 155)

Although abduction does involve an element of

guess-work, nevertheless, it does not proceed capriciously.

Never modifying his theory capriciously, but
always with a sound and rational motive for
just the modification he makes, it follows
that when he finally reaches a
modification--of most striking simplicity and
rationality--which exactly satisfies the
observations, it stands upon a totally
different logical footing from what it would
if it had been struck out at random.
(Buchler, 1955, p. 155)

Hence, there is method to abduction. Rather than referring

to it as a case of the fallacy of affirming the consequent,

it would be better to term it directed affirmation of the

consequent. The arrived at hypothesis will still be viewed

as tentative. But as Peirce indicated there is a logical
3
form to it.

Abduction, although it is very little
hampered by logical rules, nevertheless is
logical inference, asserting its conclusion
only problematically, or conjecturally, it is








true, but nevertheless having a perfectly
definite logical form. (Peirce 5.188)

A crucial feature of abduction is that it is originary

in the sense of starting a new idea. It inclines, rather

than compels, one toward a new hypothesis.

At a certain stage of Kepler's eternal
exemplar of scientific reasoning, he found
that the observed longitudes of Mars, which
he had long tried in vain to get fitted with
an orbit, were (within the possible limits of
error of the observations) such as they would
be if Mars moved in an ellipse. The facts
were thus, in so far, a likeness of those of
motion in an elliptic orbit. Kepler did not
conclude from this that the orbit really was
an ellipse; but it did incline him to that
idea so much as to decide him to undertake to
ascertain whether virtual predictions about
the latitudes and parallaxes based on this
hypothesis would be verified or not. This
probational adoption of the hypothesis was an
abduction. An abduction is Originary in
respect of being the only kind of argument
which starts a new idea. (Buchler, 1955, p.
156)

A very simple way of expressing the anomalous orbit of Mars

and the resulting abductive hypothesis is indicated by

Hanson. It is relevant to note that it begins with an

interrogation. "Why does Mars appear to accelerate at 90

[degrees] and 270 [degrees]? Because its orbit is

elliptical" (Hanson, 1972, p. 87). Again putting the

formula into its simplest form, we may say: the surprising

fact R is observed, but what hypothesis H could be true

that would make R follow as a matter of course? It is the

upward reach for H that is fundamental to the notion of


abduction.








Apagoge

How then is abduction related to Plato? The initial

clue is given in a statement by Peirce.

There are in science three fundamentally
different kinds of reasoning, Deduction
(called by Aristotle sunagoge or anagoge),
Induction (Aristotle's and Plato's epagoge)
and Retroduction [abduction] (Aristotle's
apagoge). (Peirce 1.65)

Apagoge is defined as "I. a leading or dragging away.

II. a taking home. III. payment of tribute. IV. as a

law-term, a bringing before the magistrate" (Liddell and

Scott, 1972, p. 76). There is thus the underlying notion

of moving away from, or a return or reversion of direction.

Peirce's reference is to Aristotle's use of the term,

apagoge. It is generally translated as reduction.

By reduction we mean an argument in which the
first term clearly belongs to the middle, but
the relation of the middle to the last term
is uncertain though equally or more probable
than the conclusion; or again an argument in
which the terms intermediate between the last
term and the middle are few. For in any of
these cases it turns out that we approach
more nearly to knowledge. For example let A
stand for what can be taught, B for
knowledge, C for justice. Now it is clear
+hat knowledge can be taught [AB]: but it is
uncertain whether virtue is knowledge [BC].
If now the statement BC [virtue is knowledge]
is equally or more probable than AC [virtue
can be taught], we have a reduction: for we
are nearer to knowledge, since we have taken
a new term [B which gives premises AB and BC,
on which the inquiry now turns], being so far
without knowledge that A [what can be taught]
belongs to C [virtue]. (Prior Analytics
69a20-30)

On this view then, reduction is the grasping of a new term

which transforms the inquiry onto a new footing. According








to Aristotle we are nearer knowledge because by reducing

the problem to something simpler, we are closer to solving

it. By solving the new reduced problem, the solution to

the original problem will follow.

The evidence is that the Aristotelian term apagoge has

its roots in geometrical reduction. Thus Proclus says:

Reduction is a transition from one problem or
theorem to another, the solution or proof of
which makes that which is propounded manifest
also. For example, after the doubling of the
cube had been investigated, they transformed
the investigation into another upon which it
follows, namely the finding of two means; and
from that time forward they inquired how
between two given straight lines two mean
proportionals could be discovered. And they
say that the first to effect the reduction of
difficult constructions was Hippocrates of
Chios, who also squared a lune and discovered
many other things in geometry, being second
to none in ingenuity as regards
constructions. (Heath, 1956, vol. 1, p. 135)

Thus, we see the basic movement as later described by

Peirce, in which a problem is solved or an anomaly

explained, by the backwards reasoning movement to a

hypothesis from which the anomalous phenomenon or solution

would follow as a matter of course. The difference here is

that in reduction there is an initial step toward arriving

at a hypothesis from which the phenomenon or solution would

follow, but the hypothesis is such that it still must be

established. However, by selecting the hypothesis one has

succeeded in reducing the problem to another, but simpler,

problem. Hence, the Delian problem of doubling the cube

was reduced to the problem of finding two mean

proportionals between two given straight lines. As we








shall see subsequently, Archytas performed the initial step

of reduction, and Eudoxus performed the final step of

solution.

In a footnote to the Proclus passage above, Heath

makes the following relevant remarks:

This passage has frequently been taken as
crediting Hippocrates with the discovery of
the method of geomtrical reduction. As
Tannery remarks, if the particular reduction
of the duplication problem to that to the two
means is the first noted in history, it is
difficult to suppose that it was really the
first; for Hippocrates must have found
instances of it in the Pythagorean geometry.
but, when Proclus speaks vaguely of
"difficult constructions," he probably means
to say simply that "this first recorded
instance of a reduction of a difficult
construction is attributed to Hippocrates."
(Heath, 1956, vol.1, pp. 135-136)

This suggests that the real source of reduction or apagoge

is the Pythagoreans. I will return to this point later.

It is also interesting to note that in the Proclus

quotation above there is reference to the squaring of

lunes. Aristotle, in the Prior Analytics passage cited

above, goes on to refer to the squaring of the circle with

the aid of lunules.

Or again suppose that the terms intermediate
between B [knowledge] and C [virtue] are few:
for thus too we are nearer knowledge. For
example let D stand for squaring, E for
rectilinear figure, F for circle. If there
were only one term intermediate between E
[squaring] and F [circle] (viz. that the
circle made equal to a rectilinear figure by
the help of lunules), we should be near to
knowledge. But when BC [virtue is knowledge]
is not more probable than AC [virtue can be
taught], and the intermediate terms are not
few, I do not call this reduction: nor again
when the statement BC [virtue is knowledge]








is immediate: for such a statement is
knowledge. (Prior Analytics 69a30-37)

My own view is that reduction as expressed by

Aristotle is really a special limiting case of what Peirce

termed abduction. There is a more general model of the

abductive process available amongst the Greeks. And

further, reduction does not precisely fit the basic formula

Peirce has presented.

The surprising fact, C, is observed; But if A
were true, C would be a matter of course,
Hence, there is reason to suspect that A is
true. (Peirce 5.5189)

Reduction appears to be a species of this formula.

However, there appears to be a more apt generic concept

available amongst the Greeks. This I contend is the

ancient method of analysis. In the end reduction may be

seen to be closely allied to analysis. But successful

analysis or abduction requires the discovery of an adequate

hypothesis. It is possible that this is achieved through a

series of reductions.

In reference to the discovery of lemmas, Proclus says,

certain methods have been handed down.
The finest is the method which by means of
analysis carries the thing sought up to an
acknowledged principle, a method which Plato,
as they say, communicated to Leodamas, and by
which the latter, too, is said to have
discovered many things in geometry. (Heath,
1956, vol. 1, p. 134)

Heath, in some insightful remarks, sees this analysis

as similar to the dialectician's method of ascent. Thus he

says:


This passage and another from Diogenes








Laertius to the effect that "He [Plato]
explained (eisegasato) to Leodamos of Thasos
the method of inquiry by analysis" have been
commonly understood as ascribing to Plato the
invention of the method of analysis; but
Tannery points out forcibly how difficult it
is to explain in what Plato's discovery could
have consisted if analysis be taken in the
sense attributed to it in Pappus, where we
can see no more than a series of successive,
reductions of a problem until it is finally
reduced to a known problem. On the other
hand, Proclus' words about carrying up the
thing sought to an "acknowledged principle"
suggest that what he had in mind was the
process described at the end of Book VI of
the Republic by which the dialectician
(unlike the mathematician) uses hypotheses as
stepping-stones up to a principle which is
not hypothetical, and then is able to descend
step by step verifying every one of the
hypotheses by which he ascended. (Heath,
1956, vol. 1, p. 134, fn.1)

There is both some insight and some glossing over by.

Heath here. Heath is correct that there is a definite

relation here between analysis and what Plato describes as

the upward path in Book VI of the Republic. But he is

mistaken when he tries to divorce Platonic analysis from

mathematical analysis. They are very closely related.

Part of the confusion stems from the fact that in the

Republic Plato distinguishes the mathematician's acceptance

of hypotheses and subsequent deductions flowing from them,

from the hypotheses shattering upward ascent of the

dialectician. However, the mathematician also employs the

dialectical procedure when he employs reduction and

mathematical analysis. In these instances, unlike his

deductive descent, the mathematician reasons backwards (or

upwards) to other hypotheses, from the truth of which the








solution of his original problem will follow. This basic

process is common to both mathematician and dialectician.

The common denominator is the process of reasoning

backwards.

A further problem resulting in the confusion is that

it is not clear what is meant by references to an ancient

method of analysis. Heath is perplexed as well. Thus he

writes,

It will be seen from the note on Eucl. XIII.
1 that the MSS. of the Elements contain
definitions of Analysis and Synthesis
followed by alternative proofs of XIII. 1-5
after that method. The definitions and
alternative proofs are interpolated, but they
have great historical interest because of the
possibility that they represent an ancient
method of dealing with propositions, anterior
to Euclid. The propositions give properties
of a line cut "in extreme and mean ratio,"
and they are preliminary to the construction
and comparison of the five regular solids.
Now Pappus, in the section of his Collection
[Treasury of Analysis] dealing with the
latter subject, says that he will give the
comparisons between the five figures, the
pyramid, cube, octahedron, dodecahedron and
icosahedron, which have equal surfaces, "not
by means of the so-called analytical inquiry,
by which some of the ancients worked out the
proofs, but by the synthetical method." The
conjecture of Bretschneider that the matter
interpolated in Eucl. XIII is a survival of
investigations due to Eudoxus has at first
sight much to commend it. In the first
place, we are told by Proclus that Eudoxus
"greatly added to the number of the theorems
which Plato originated regarding the section,
and employed in them the method of analysis."
(Heath, 1956, vol. 1, p. 137)

This is an extremely interesting passage. Is this the

same method of analysis that was earlier attributed to the

discovery of Plato? However, if the method is ancient,








then at best Plato could only have discovered it in the

work of his predecessors, presumably the Pythagoreans. And

what about "the section" (tome), and the theorems that

Plato originated (and Eudoxus extended) regarding it?

It is obvious that "the section" was some
particular section which by the time of Plato
had assumed great importance; and the one
section of which this can safely be said is
that which was called the "golden section,"
namely the division of a straight line in
extreme and mean ratio which appears in Eucl.
II. 11 and is therefore most probably
Pythagorean. (Heath, 1956, vol. 1, p. 137)

If Plato had done so much work on this Pythagorean

subject, the golden section, and further, his pupil Eudoxus

was busy developing theorems regarding it, and further,

both were using a method of analysis that may have ancient

Pythagorean origins as well, then why is there no

straightforward mention of this in the dialogues? Could

the actual practice of what was occurring within the

Academy have been so far removed from what is in the

dialogues? Why was there such a discrepancy between

practice and dialogue? These are some of the questions

that will be answered in the course of this dissertation.

Focussing upon the question of analysis for the

moment, there are interpolated definitions of analysis and

synthesis in Book XIII of Euclid's Elements. Regarding the

language employed, Heath says that it "is by no means clear

and has, at the best, to be filled out" (Heath, 1956, vol.

1, p. 138).

Analysis is an assumption of that which is
sought as if it were admitted [and the








passage] through its consequences
antecedents] to something admitted (to be)
true. Synthesis is an assumption of that
which is admitted [and the passage] through
its consequences to the finishing or
attainment of what is sought. (Heath, 1956,
vol. 1, p. 138)

Unfortunately this passage is quite obscure.

Fortunately Pappus has preserved a fuller account.

However, it too is a difficult passage. One might even

speculate as to whether these passages have been

purposefully distorted.

The so-called Treasury of Analysis is, to put
it shortly, a special body of doctrine
provided for the use of those who, after
finishing the ordinary Elements [i.e.,
Euclid'sJ, are desirous of acquiring the
power of solving problems which may be set
them involving (the construction of) lines
and proceeds by way of analysis and
synthesis. Analysis then takes that which is
sought as if it were admitted and passes from
it through its successive consequences5
[antecedents] to something which is admitted
as the result of synthesis: for in analysis
we assume that which is sought as if it were
(already) done (gegonos), and we inquire what
it is from which this results, and again what
is the antecedent cause of the latter, and so
on, until by so retracing our steps we come
upon something already known or belonging to
the class of first principles, and such a
method we call analysis as being solution
backwards (anapalin lusin). (Heath, 1956,
vol. 1, p. 138)

Thomas translates the last two words of the former

passage, anapalin lusin, as "reverse solution' (Thomas,

1957, vol. 2, p. 597). It is this "solution backwards" or

"reverse solution" that I contend lay at the center of


Plato's dialectical method.










Cornford is one of the few commentators to have any

real insight into the passage from Pappus.

modern historians of
mathematics--"careful studies" by Hankel,
Duhamel, and Zeuthen, and others by
Ofterdinger and Cantor--have made nonsense of
much of it by misunderstanding the phrase,
"the succession of sequent steps" (TyV Lt5 aK)oovuJL
as meaning logical "consequences," as if it
were Ta O'U1#a.tVOV 7- Some may have been
misled by Gerhardt (Pappus, vii, viii, Halle,
1871), who renders it "Folgerungen." They
have been at great pains to show how the
premisses of a demonstration can be the
consequences of the conclusion. The whole is
clear when we see--what Pappus says--that the
same sequence of steps is followed in both
processes--upwards in Analysis, from the
consequence to premisses implied in that
consequence, and downwards in synthesis, when
the steps are reversed to frame the theorem
or demonstrate the construction "in the
natural (logical) order." You cannot follow
the same series of steps first one way, then
the opposite way, and arrive at logical
consequences in both directions., And Pappus
never said you could. He added ES to
indicate that the steps "follow in succession
"but are not, as kKPXAOUG< alone would
suggest, logically "consequent" in the upward
direction. (Cornford, 1965, p. 72, fn.1)

On the "abduction" view I am maintaining, Cornford has

hit upon an acceptable interpretation of the Pappus

passage. It is this reverse inference from conclusion to

premise that was at the center of Plato's method of

discovery. As Cornford goes on to say,

Plato realized that the mind must possess the
power of taking a step or leap upwards from
the conclusion to the premiss implied in it.
The prior truth cannot, of course, be deduced
or proved from the conclusion; it must be
grasped (Aae0-t. Republic 511b) by an act
of analytical penetration. Such an act is
involved in the solution "by way of








hypothesis" at Meno 86. The geometer
directly perceives, without discursive
argument, that a prior condition must be
satisfied if the desired construction is to
follow. (Cornford, 1965, p. 67)

But Cornford is not without opposition to his account.

Robinson takes direct issue with him on the matter,

claiming that all historians of Greek mathematics agree

with the non-Cornford interpretation.

The historians of Greek mathematics are at
one about the method that the Greek geometers
called analysis. Professor Cornford,
however, has recently rejected their account
and offered a new Qne. Professor
Cornford is mistaken and the usual view
correct. (Robinson, 1969, p. 1)

But if this is true, then why is there such a mystery

around the interpretation of the Pappus passage? And why

the mystery surrounding what was meant by Plato's discovery

of analysis? My view is that Cornford has gone far in

uncovering part of an old mystery about analysis. The

Cornford interpretation clearly lends support to the

abduction view of Plato's method that I am advocating.

Actually, the medieval philosopher John Scotus

Eriugena captured some of the underlying meaning of

analysis when he distinguished the upward and downward

movements of dialectic. In the Dialectic of Nature, he

refers to this dual aspect of dialectic

which divides genera into species and
resolves species into genera once more .
There is no rational division which
cannot be retraced through the same set of
steps by which unity was diversified until
one arrives again at that initial unit which
remains inseperable in itself. Analytic
comes from the verb analyo meaning "I return"








or "I am dissolved." From this the term
analysis is derived. It too can be
translated "dissolution" or "return," but
properly speaking, analysis refers to the
solution of questions that have been
proposed, whereas analytic refers to the
retracing of the divisions of forms back to
the source of their division. For all
division, which was called "merismos" by the
Greeks', can be viewed as a downward descent
from a certain definite unit to an indefinite
number of things, that is, it proceeds from
the most general towards the most special.
But all recollecting, as it were is a return
again and this begins from the most special
and moves towards the most general.
Consequently, there is a "return" or
"resolution" of individuals into forms, forms
into genera. (Whippel & Wolter, 1969,
pp. 116-117)

Now the earlier remark cited by Heath (supra. pp.24-25),

though missing the mark, may be insightful as to what

analysis is.

Tannery points out forcibly how difficult it
is to explain in what Plato's discovery could
have consisted if analysis be taken in the
sense attributed to it in Pappus, where we
see no more than a series of successive,
reductions of a problem until it is finally
reduced to a known problem. (Heath, 1956,
vol. 1, p. 134, fn.1)

But this "series of reductions" may be fundamentally what

was involved. Knowledge would be arrived at through a

series of apagoges. Plato may have discovered unique uses

of analysis, that extended beyond his predecessors. It may

be that he discovered that analysis, or backward reasoning,
7
may apply to propositions other than mathematical. On the

other hand, he may have simply "discovered" this more

esoteric technique in the ancient geometrical tradition of

the Pythagoreans. However, its true significance should be








considered within the context in which it is openly stated

it was used. That is, in particular it should be

considered in terms of what is said about Plato and Eudoxus

as to the theorems regarding the section, and their

discovery by analysis.

As Cantor points out, Eudoxus was the founder
of the theory of proportions in the form in
which we find it in Euclid V., VI., and it
was no doubt through meeting, in the course
of his investigations, with proportions not
expressible by whole numbers that he came to
realise the necessity for a new theory of
proportions which should be applicable to
incommensurable as well as commensurable
magnitudes. The "golden section" would
furnish such a case. And it is even
mentioned by Proclus in this connexion. He
is explaining that it is only in aritmetic
that all quantities bear "rational" ratios
(ratos logos) to one another, while in
geometry there are "irrational" ones
(arratos) as well. "Theorems about sections
like those in Euclid's second Book are common
to both [arithmetic and geometry] except that
in which the straight line is cut in extreme
and mean ratio. (Heath, 1956, vol. I, p. 137)

This mention of the golden section in conjunction with

analysis provides some clues, and foreshadows some of the

argument to come.

Dialectic

It is difficult to give a satisfactory account of the

views of both Aristotle and Plato regarding dialectic.

Whereas Aristotle refers to dialectic as less than

philosophy, Plato contends that it is the highest method

available to philosophy. In the end, however, their

methods are essentially the same, and dialectic can be

viewed as having various stages. At the bottom level








dialectic is used as a means of refutation. At the top

level it is a means for acquiring knowledge of real

essences.

Plato openly espouses dialectic as the finest tool

available in the acquisition of knowledge. "Dialectic is

the coping-stone that tops our educational system"

(Republic 534e).

It is a method quite easy to indicate, but
very far from easy to employ. It is indeed
the instrument through which every discovery
ever made in the sphere of arts and sciences
has been brought to light. [It] is a
gift of the gods and it was through
Prometheus, or one like him [Pythagoras],
that it reached mankind, together with a fire
exceeding bright. (Philebus 16c)

Dialectic is the greatest of knowledge (Philebus 57e-58a).

Through dialectic

we must train ourselves to give and to
understand a rational account of every
existent thing. For the existents which
have no visible embodiment, the existents
which are of highest value and chief
importance [Forms], are demonstrable only by
reason and are not to be apprehended by any
other means. (Statesman 286a)

Dialectic is the prime test of a man and is to be

studied by the astronomers (Epinomis 991c). A dialectician

is one who can "discern an objective unity and plurality"

(Phaedrus 266b). It is the dialectician "who can take

account of the essential nature of each thing" (Republic

534b). Dialectic leads one to the vision of the Good.

When one tries to get at what each thing is
in itself by the exercise of dialectic,
relying on reason without any aid from the
sense, and refuses to give up until one has
grasped by pure thought what the good is in









itself, one is at the summit of the
intellectual realm, as the man who looked at
the sun was of the visual realm. And
isn't this process what we call dialectic?
(Republic 532a-b)

Dialectic "sets out systematically to determine what

each thing essentially is in itself" (Republic 533b). For

Plato, it "is the only procedure which proceeds by the

destruction of assumptions to the very first principle, so

as to give itself a firm base" (Republic 533c-d). And

finally, in the art of dialectic,

the dialectician selects a soul of the right
type, and in it he plants and sows his words
founded on knowledge, words which can defend
both themselves and him who planted them,
words which instead of remaining barren
contain a seed whence new words grow up in
new characters, whereby the seed is
vouchsafed immortality, and its possessor the
fullest measure of blessedness that man can
attain unto. (Phaedrus 276e-277a)

Thus, Plato indicates that dialectic is the highest

tool of philosophy. Aristotle, on the other hand, appears

to have a more mundane account. His passages in the Topics

seem to indicate that dialectic is a tool for students

involved in disputation. In fact, both the Topics and

Sophistical Refutations give the impression of introductory

logic texts. His references to dialectic give the

appearance that dialectic does not ascend to the level of

philosophy. Thus, Aristotle says,

sophistic and dialectic turn on the same
class of things as philosophy, but this
differs from dialectic in the nature of the
faculty required and from sophistic in









respect of the purpose of philosophic life.
Dialectic is merely critical where philosophy
claims to know, and sophistic is what appears
to be philosophy but is not. (Metaphysics
1004b22-27)

However, these considerations are somewhat misleading.

Aristotle's own works seem to be much more in the line of

dialectical procedure that Plato has referred to. Mayer

has brought this point home forcefully through an

examination of Aristotle's arguments in Metaphysics, Book

IV. Referring to these arguments Mayer says,

One would hardly expect the argument that
Aristotle employs in the sections of
Metaphysics IV. to be dialectical. And
it is true he does not call it dialectical,
but rather a kind of "negative
demonstration." [i.e., at Metaphysics
1006al2] Yet .. [upon examination] the
arguments do appear to be dialectical.
(Mayer, 1978, p. 24)

Mayer has made a very useful classification of

dialectic into 3 types and their corresponding uses.

I.Eristic dialectic uses confusion and equivocation to

create the illusion of contradiction. II. Pedagogic

dialectic is used for refutation and the practice of

purification, as it leads to contradiction. III. Clarific

dialectic uses criticism, revision, discovery, and

clarification to dispel contradiction (Mayer, 1978, p. 1).

The eristic type would be that used by a sophist. The

pedagogic type is that seen in the early Socratic dialogues

where the respondent is subjected to cross examination

(elenchus). The third and highest type, clarific, is

closer to the level of dialectic that Plato so reveres.








What Aristotle has done is

limit the usage of the term "dialectic" to
the critical pedagogic] phase only, i.e. he
sees its purpose as entirely negative, and
the philosopher must go beyond this to "treat
of things according to their truth". By
limiting "dialectic" to the negative, or
pedagogic, phase of dialectic,
Aristotle is merely changing terminology, not
method. Clarific dialectic, for Aristotle,
is (or is part of ) philosophical method.
(Mayer, 1978, p. 1)

On this view, which I share with Mayer, Plato and

Aristotle do not really differ in method. The difference

is merely terminological, not substantive. This position

is even more strongly supported when one considers

Aristotle's response to the "eristic argument" or Meno's

paradox. There Plato writes,

A man cannot try to discover either what he
knows or what he does not know. He would not
seek what he knows, for since he knows it
there is no need of the inquiry, nor what he
does not know, for in that case he does not
even know what he is to look for. (Meno 80e)

Aristotle's response is to establish a halfway house 8

between not knowing and knowing in the full sense. The

crucial distinction is between the weak claim of knowing

"that" something is, and the strong claim of knowing "what"

something is. The former is simply to know or acknowledge

the existence of a kind. But to know "what" something is,

is to know the real essence of the thing. Of course this

latter knowledge is the goal of dialectic according to

Plato. The solution lies in the distinction Aristotle

draws between nominal and real essences (or definitions).

Thus, Aristotle argues that a nominal definition is a








statement of the meaning of a term, that is, meaning in the

sense of empirical attributes which appear to attach to the

thing referred to by the term. An example he uses of a

nominal essence (or definition), is the case of thunder as

"a sort of noise in the clouds" (Posterior Analytics

93b8-14). Another nominal definition is that of a lunar

eclipse as a kind of privation of the moon's light.

A real definition (or essence), on the other hand, is

a "formula exhibiting the cause of a thing's existence"

(Posterior Analytics 93b39). For both Plato and Aristotle

we ultimately must seek to know the real definition to

fully know the essence of the "kind." That is to say, to

have scientific knowledge of a thing one must ascend to

knowledge of its real essence. However, one must begin

with the former, the nominal essence. This is nothing more

than to acknowledge the fact that a kind of thing exists,

through an enumeration of its defining attributes. The

*setting forth of a nominal essence presupposes the

existence of actual samples (or instances, events) in the

world answering to the description contained in the

definition. In other words it is not just a matter of

knowing or not knowing. We begin by knowing "that"

something exists. Then we proceed to seek to discover its

real essence, "what" it is. The real essence of, for

example, lunar eclipses (as correctly set forth by

Aristotle) is the interposition of the earth between the

sun and the moon. It is this "interposition" which is the








real essence which gives rise to (i.e., causes) that

phenomenon we have described in our nominal definition as a

lunar eclipse.

The important point here is that whether one adopts

the Platonic notion of knowledge through reminiscence or

the Aristotelian distinction between nominal and real

essences, in both cases there is an analytic ascent to the

real essence. Thus, Plato proceeds by dialectic to the

real essences and first principles. Likewise, Aristotle

seeks the real defining essence through an analytic ascent

from things more knowable to us to things more knowable in

themselves, that is, from the nominal to the real essence.

Aristotle may use different terminology, but his method has

its roots in the Academy. At one point Aristotle makes it

clear that he sees this underlying ascent to essences and

first principles in dialectic. "Dialectic is a process of

criticism wherein lies the path to the principles of all

inquiries"9 (Topics 101b4). Elsewhere Aristotle makes it

manifestly clear that he appreciated Plato's thought on the

upward path of analysis to real essences and the first

principles.

Let us not fail to notice, however, that
there is a difference between arguments from
and those to the first principles. For
Plato, too, was right in raising this
question and asking, as he used to do, "are
we on the way from or to the first
principles?"O(Nicomachean Ethics 1095a30-35).

On the other hand, the beginning stages of dialectic occur

for Plato when one attempts to tether a true belief to its








higher level cause through the giving of an account

(logos). Only when one reaches the highest level of noesis

is there no reflection upon sensible things.



Meno and Theaetetus

In the Theaetetus, Plato acts as midwife to his

readers, just as, Socrates acts as midwife to Theaetetus.

I believe the dialogue is consistent with the views

unfolded in the Meno (&s well as the Phaedo and Republic).

In the Meno, Socrates indicates to Meno that true beliefs

are like the statues of Daedalus, "they too, if no one ties

them down, run away and escape. If tied, they stay where

they are put" (Meno 97d). Socrates then goes on to explain

how it is the tether which transmutes mere true belief into

knowledge.

If you have one of his works untethered, it
is not worth much; it gives you the slip like
a runaway slave. But a tethered specimen is
very valuable, for they are magnificent
creations. And that, I may say, has a
bearing on the matter of true opinions. True
opinions are a fine thing and do all sorts of
good so long as they stay in their place, but
they will not stay long. They run away from
a man's mind; so they are not worth much
until you tether them by working out the
reason. That process, my dear Meno, is
recollection, as we agreed earlier. Once
they are tied down, they become knowledge,
and are stable. That is why knowledge is
something more valuable than right opinion.
What distinguishes the one from the other is
the tether. (Meno 97e-98a)

It is this providing of a tether or causal reason that

assimilates a true belief of the realm of pistis, to

knowledge of the realm of dianoia. The tether is arrived








at by an upward grasp of the causal objects of the next

higher ontological and epistemological level. My view is

that the Phaedo and Republic expand upon this upward climb

using the method of hypothesis on the way. The Republic

goes beyond the tentative stopping points of the Phaedo.

maintaining that knowledge in the highest sense can only be

attained by arriving at the unhypothetical first principle,

the Good. Nevertheless, in the Phaedo, the basic method of

tethering by reasoning backwards to a higher premise is

maintained.

When you had to substantiate the hypothesis
itself, you would proceed in the same way,
assuming whatever more ultimate hypothesis
commended itself most to you, until you
reached the one which was satisfactory.
(Phaedo 10ld-e)

This method is, of course, depicted by Socrates as

being second best. The best method which continues until

arrival at the Good, or One, is depicted in the Republic.

The Theaetetus then is a critique of the relative level of

knowledge arrived at in the state of mind of dianoia. The

question is asked as to what is an adequate logos (or

tether). Thus as Stenzel says,

even Ao/05 is included in the general
skepticism. The word may indicate three
things: (a) speech or vocal expression, as
contrasted with the inner speech of the mind,
(b) the complete description of a thing by an
enumeration of its elements, (c) the
definition of a thing by discovery of its
distinctive nature, Ad apoTD7~ The third of
these meanings seems at first to promise a
positive criterion of knowledge. Butlook
more closely: it is ,not ,the thing's ACoL3OpoT5,
but knowledge of its AAAOfOTtSwhich will
constitute knowledge; and this is circular.
(Stenzel, 1940, pp. xv-xvi)








This has, in general, been taken as an indication of

the Theaetetus' negative ending. However, it appears to me

to be another case of Plato's obstetric method. Even if

one arrives at a satsifactory account or logos, and hence,

arrives at a tether, it does not follow that there would be

absolute certainty. "If we are ignorant of it [the Good or

One] the rest of our knowledge, however perfect, can be of

no benefit to us" (Republic 505a).

Thus, in the Theaetetus, Socrates indicates that, "One

who holds opinions which are not true, will think falsely

no matter the state of dianoias" (Theaetetus 188d). But

this is a critique only relative to certainty. To arrive

at tethered true opinions at the state of mind of dianoia

is, nevertheless, a kind of knowledge, albeit less than the

highest kind of knowledge of the Republic. On my view

Haring has moved in the correct direction of interpretation

when she writes:

[the Theaetetus has] a partly
successful ending. The latter has to be
discovered by readers. However the dialogue
itself licenses and encourages active
interpretation. The last ten pages of
discourse contain so many specific clues that
the text can be read as a single development
ending in an affirmative conclusion. There
is indeed a way to construe "true opinion
with logos" so it applies to a cognition
worthy of "episteme." (Haring, 1982, p. 510)

As stated in the previous section (supra pp. 36-38 ),

dialectic is a movement from nominal essences to real

essences. From the fact that a particular "kind" exists,

one reasons backwards to the real essence, causal








explanation, or tether of that kind. From the nominal

existence of surds, Theaetetus reasons backwards to the

grounds or causal tether. By doing so, Theaetetus arrives

at (or certainly approaches) a real definition. The

relevant passage in the Theaetetus begins:

Theaetetus: Theodorus here was proving to us
something about square roots, namely, that
the sides [or roots] of squares representing
three square feet and five square feet are
not commensurable in length with the line
representing one foot, and he went on in this
way, taking all the separate cases up to the
root of seventeen square feet. There for
some reason he stopped. The idea occurred to
us, seeing that these square roots were
evidently infinite in number, to try to
arrive at a single collective term by which
we could designate all these roots. We
divided number in general into two classes.
Any number which is the product of a number
multiplied by itself we likened to a square
figure, and we called such a number "square"
or "equilateral."
Socrates: Well done!
Theaetetus: Any intermediate number, such as
three or five or any number that cannot be
obtained by multiplying a number by itself,
but has one factor either greater or less
than the other, so that the sides containing
the corresponding figure are always unequal,
we likened to the oblong figure, and we
called it an oblong number.
Socrates: Excellent. And what next?
Theaetetus: All the lines which form the
four equal sides of the plane figure
representing the equilateral number we
defined as length, while those which form the
sides of squares equal in area to the oblongs
we called roots surdss] as not being
commensurable with the others in length, but
only in the plane areas to which their
squares are equal. And there is another
distinction of the same sort in the case of
solids. (Theaetetus 147d-148b)

What Theaetetus has managed to do is analogous to

Aristotle's discussion of lunar eclipses (supra p. 37).








Just as one reasons backwards to the real essence of lunar

eclipses (i.e., the interposition of the earth between the

sun and moon), so one reasons backwards to the real essence

of surds. They are not commensurablee with the others in

length [first dimension], but only in the plane areas

[second dimension] to which their squares are equal"

(Theaetetus 148b).

Theaetetus has successfully tethered the true belief

of Theodorus regarding surds, by working out the reason.

This is hinted at in Socrates' reply: "Nothing could be

better, my young friends; I am sure there will be no

prosecuting Theodorus for false witness" (Theaetetus 148b).

Socrates' statement, on the one hand, implies success, and,

on the other hand, suggests that Theodorus had initially

stated a true belief. Thus, there is the indication of a

possibly successful tethering (by Theaetetus) of a true

belief (that of Theodorus).

But Theaetetus is then asked to "discover" the nature

of knowledge. A very hard question indeed. But Socrates

suggests that he use his definition of surds as a model.

Forward, then, on the way you have just shown
so well. Take as a model your answer about
the roots. Just as you found a single
character to embrace all that multitude, so
now try to find a single formula that applies
to the many kinds of knowledge. (Theaetetus
148d)
Notes

References are to Charles S. Peirce, 1931-1958,
Collected Papers, 8 volumes, edited by Charles Hartshorne,
Paul Weiss, and Arthur Banks, Cambridge: Harvard University
Press. All references to the Collected Papers are in the








standard form, citing only the volume number, decimal
point, and paragraph number.

2It is interesting to note here that Kepler also became
very interested in the golden section. In 1596 he wrote,
"geometry has two great treasures: one is the theorem of
Pythagoras; the other, the division of a line into extreme
and mean ratio [golden section]. The first we may compare
to a measure of gold; the second we may have a precious
jewel." Citation in Dan Pedoe's Geometry and the Liberal
Arts, 1975, unpublished MS., Universiy of Minnesota, p. i15.

3One of the objections to abduction is that it is not
really a formal logic like deduction. However, by
weakening the conditions of validity and soundness it can
be given a weak formalism. Furthermore, it more than makes
up for lack of formalism in its creative discovery aspects.

4All citations to Aristotle are in the standard
numbering form for each of his works. These will include
the Topics, Prior Analytics, Posterior Analytics, DeAnima,
Metaphysics, Nicomachean Ethics, Metaphysics, and Physics.
All quotations are taken from Richard McKeon, ed., 1941,
The Basic Works of Aristotle. New York: Random House.

5Consequences are generally considered in logic as
proceeding deductively downwards. However, the essence of
analysis or abduction is that it proceeds upwards (or
backwards) ascending to antecedent principles.

Reverse solution is the common factor above all else
between Plato and Peirce.

It might apply for example to the moral virtues.
However, Plato most likely identifies the virtues with
proportion. Hence, justice is identified as a
proportionate harmony in the soul (Republic).

In this way he is able to escape the horns of the
"eristic" dilemma.

Here Aristotle is in complete agreement with Plato.

10The basic movements are up from the sensible world to
principles, and down from principles to particulars.
















CHAPTER III

THE PYTHAGOREAN PLATO



The Quadrivium1

In the Republic, Plato describes the method by which

the aspiring philosopher-statesman is to prepare himself to

govern the state. Ultimately, after a series of

conversions through succeeding states of awareness brought

about by the apprehension of corresponding levels of

subject-matter (Republic 513e), the philosopher must arrive

at the "highest form of knowledge and its object," the Good

(Republic 504e). As Plato points out, the Good is "the end

of all endeavour" (Republic 505d). And furthermore, "if we

are ignorant of it the rest of our knowledge, however

perfect, can be of no benefit to us" (Republic 505a). It

alone is the foundation of certainty, and is arrived at

finally through the process of dialectic, what Plato calls,

"the coping-stone that tops our educational system"

(Republic 534e). "Anyone who is going to act rationally

either in public or private life must have sight of it"

(Republic 517c). "Our society will be properly regulated

only if it is in the charge of a guardian who has this

knowledge" (Republic 506a-b).










However, it is not until about the age of 50 that the

philosopher is to attain this "vision" of the Good. Prior

to this time, particular virtues must be both apparent and

cultivated. Along with these virtues, a rigorous

educational program must be undertaken. The object of this

education is to assist the philosopher's mind in its ascent

through the relative levels of awareness and their

corresponding levels of subject-matter, being converted at

each stage to the comprehension of a greater degree of

clarity and reality.

Plato has spelled out the qualities that this

individual must possess:

A man must combine in his nature good memory,
readiness to learn, breadth of vision, grace,
and be a friend of truth, justice, courage,
and self-control. Grant then education
and maturity to round them off, and aren't
they the only people to whom you would
entrust your state? (Republic 487a)

Thus, besides the inherent abilities, requisite virtues,

and eventual maturity, it is the education that will

prepare one for the role of philosopher-statesman. Of what

is this education to consist? Leaving aside for the moment

the final dialectical procedure, Plato's answer is clearly

mathematics.

It is the thesis of this dissertation that Plato was

more intimately involved with a mathematical doctrine

throughout his career than is generally recognized by most

modern commentators. This mathematical doctrine is at the








very foundation of Plato's epistemology and ontology.

Through a careful analysis of his work, two very important

features emerge: Plato's expert use of a method, later to

be called by Charles Sanders Peirce, abduction, and his

reverence for the golden section. The former is the real

forerunner of the method of scientific discovery. The

latter is a very important mathematical construct, the

significance of which has been generally ignored by

Platonic scholarship. But to arrive at these points I

must consider carefully Plato's famous Divided Line and the

"notorious question of mathematical" (Cherniss, 1945).

From out of these I contend that the real significance of

abductive inference and the golden section in Plato's

philosophy will become apparent. In point of fact, I will

be defending the assertion that Plato, following the

Pythagorean mathematicoi,made mathematics the underlying

structure of his philosophy, with the golden section being

the basic modulor upon which space is given form. Further,

I will be arguing that the mathematical may be found in

the middle dialogues. And, therefore, they are not merely

a construction of Plato's later period, as some have

contended. Additionally, the Good of the Republic and the

Receptacle3of the Timaeus are to be identified respectively

with the One and the Indefinite Dyad. The Indefinite Dyad

in turn may have something to do with the golden section.

Finally, I will argue that it is the logical method of

abduction that lies at the center of Plato's reasoning








process. It is the very foundation of scientific

discovery. Hence, I will be attempting to place Plato more

firmly in the scientific tradition.

In one of the early "Socratic dialogues," the

Gorgias, Plato plants the seed of a view that will blossom

forth in the Republic. It is the view that mathematical

study, geometry in particular, is closely allied to the

establishment of a just and virtuous nature. There

Socrates chides Callicles, saying:

You are unaware that geometric equality is of
great importance among gods and men alike,
and you think we should practice overreaching
others, for you neglect geometry. (Gorgias
508a)

In one of the middle dialogues, the Republic, an even

stronger stand is taken. There mathematical study is not

only allied to a virtuous and orderly life, but is the very

"bridge-study" by which one may cross from the lower level

of mere belief (pistis) to the higher level of reason

(noesis). The bridge is understanding (dianoia), sometimes

translated, mathematical reasoning, and is the intermediate

level of awareness of the mathematician.

The components of this bridge-study are set out at

Republic 524d-530e. These are the five mathematical

sciences which include: (1) arithmetic, (2) plane

geometry, (3) solid geometry, (4) harmonics, and (5)

spherics (or astronomy). This is actually the Pythagorean

quadrivium, but Plato has seen fit to subdivide geometry

into "plane" and "solid," representing respectively number









in the 2nd and the 3rd dimensions. The intensive study of

these mathematical disciplines is to occur between the ages

of twenty and thirty. This mathematical study will prepare

one for the study of dialectic, which will in turn

ultimately lead to the Good. Plato explains that

the whole study of the [mathematical]
sciences we have described has the effect of
leading the best element in the mind up
towards the vision of the best among
realities [i.e.,the Good]. (Republic 532c)

However, mathematical study is to begin well before this

later intensive training.

Arithmetic and geometry and all the other
studies leading to dialectic should be
introduced in childhood. (Republic 536d)

These elementary mathematical studies are to be combined

with emphasis on music and gymnastic until the age of 17 or

18. Then, until the age of 20, emphasis is placed on

gymnastics alone. According to Plato, this combination of

intellectual and physical training will produce concord

between the three parts of the soul, which will in turn

help to establish justice in the nature of the individual.

And this concord between them [i.e.,
rational, spirited, and appetitive] is
effected by a combination of
intellectual and physical training, which
tunes up the reason by training in rational
argument and higher studies [i.e.,
mathematical], and tones down and soothes the
element of spirit by harmony and rhythm.
(Republic 44te-442a)

This training in gymnastics and music not only stabilizes

the equilibrium of the soul, promoting a just nature in the








individual (and analogously a just nature in the soul of

the polls), but also indicates that Plato may conceive of

mathematical proportion (such as that expressed in

harmonics or music) as underlying both the sensible and

intelligible aspects of reality. This may be the reason

why he draws music and gymnastics together in the proposed

curriculum. Plato wants the individual to comprehend the

proportional relationship common to both music and bodily

movements. And because the soul participates in both the
4
intelligible and sensible realms, lying betwixt the two

and yet binding them together into one whole, it is

natural that the soul might contain this implicit

knowledge. Then, upon making it explicit, the soul would

benefit the most through the establishment of concord.

It is also evident in one of the last dialogues, the

Philebus, that Plato feels the knowledge brought about by

this study (i.e., of proportions in harmony and bodily

movements), may have important implications in one's grasp,

and possible solution, of the ancient and most difficult

problem of the one and many.5 Plato says,

When you have grasped the number and
nature of the intervals formed by high pitch
and low pitch in sound, and the notes that
bound those intervals, and all the systems of
notes that result from them [i.e., scales] .
and when, further, you have grasped
certain corresponding features that must, so
we are told, be numerically determined and be
called "figures" and "measures" bearing in
mind all the time that this is the right way
to deal with the one and many problem--only
then, when you have grasped all this, have
you gained real understanding. (Philebus
17d-e)









I contend that this Pythagorean notion of discovering

the role of number in things is one of the most crucial

things that Plato wants his pupils to learn. Further, if

one can discern how Plato saw the role of number in the

Cosmos, and the ontological and epistemological

consequences of his doctrine of the intermediate soul, then

it may be possible to determine how he saw the sensible

world participating in the Forms. It is most relevant that

the question of participation was left an open question in

the Academy (see Aristotle in Metaphysics 987b). Given the

puzzle, each member was allowed to abduct his own

hypothesis.



The Academy and Its Members

Let us first look at the members of the Academy and

their interests. As Hackforth has pointed out, the Academy

was "designed primarily as a training school for

philosophic-statesmen'6 (Hackforth, 1972, p. 7). If the

Republic account is a correct indication of Plato's method

of preparing these individuals, then the primary work

carried out in the Academy would have been mathematical

study and research. Most of the information we have

concerning the associates of the Academy was copied down by

Proclus from a work by Eudemus of Rhodes, entitled History

of Geometry. Eudemus was a disciple of Aristotle. The

authenticity of this history by Proclus, what has come to

be called the "Eudemian summary," has been argued for by








Sir Thomas Heath. Heath says,

I agree with van Pesch that there is no
sufficient reason for doubting that the work
of Eudemus was accessible to Proclus at first
hand. For the later writers Simplicius and
Eutocius refer to it in terms such as leave
no room for doubt that they had it before
them. (Heath, 1956, vol. 1, p. 35)

Other information has been preserved in the works of

Diogenes Laertius and Simplicius. The important fragments

preserved by Simplicius are also from a lost work by

Eudemus, History of Astronomy. Unless otherwise indicated,

the following condensed summary account is taken from the

"Eudemian summary" preserved by Proclus (see Thomas, 1957,

vol. 1, pp. 144-161).

Now we do know that the two greatest mathematicians of

the 4th century B.C. frequented the Academy. These were

Eudoxus of Cnidus and Theaetetus of Athens. Eudoxus is

famous for having developed the "method of exhaustion for

measuring and comparing the areas and volumes of

curvalinear plane and solid surfaces" (Proclus, 1970,

p.55). Essentially, he solved the Delphic problem of

doubling the cube, developed a new theory of proportion

(adding the sub-contrary means) which is embodied in Euclid

Books V and VI, and hypothesized a theory of concentric

spheres to explain the phenomenal motion of the heavenly

bodies. Of this latter development Heath says:

Notwithstanding the imperfections of the
system of homocentric spheres, we cannot but
recognize in it a speculative achievement
which was worthy of the great reputation of
Eudoxus and all the more deserving of
admiration because it was the first attempt








at a scientific explanation of the apparent
irregularities of the motions of the planets.
(Heath, 1913, p. 211)

To this Thomas adds the comment:

Eudoxus believed that the motion of the sun,
moon and planets could be accounted for by a
combination of circular movements, a view
which remained unchallenged till Kepler.
(Thomas, 1957, vol. 1, p. 410, fn.b)

Eudoxus' homocentric hypothesis was set forth in direct

response to a problem formulated by Plato. This is

decidedly one of those instances where the role of

abduction entered into the philosophy of Plato. Plato

would present the problem by formulating what the puzzling

phenomena were that needed explanation. This fits very

neatly into the Peircean formula. The surprising fact of

the wandering motion of the planets is observed. What

hypothesis, if true, would make this anomalous phenomena

the expected? To solve this problem requires one to reason

backwards to a hypothesis adequate to explain the

conclusion.

We are told by Simplicius, on the authority
of Eudemus, that Plato set astronomers the
problem of finding what are the uniform and
ordered movements which will "save the
phenomena" of the planetary motions, and that
Eudoxus was the first of the Greeks to
concern himself [with this]. (Thomas, 1957,
vol. 1, p. 410, fn.b)

At this point it is relevant to consider what was

meant by the phrase "saving the phenomena." For this

purpose I will quote extensively from a passage in

Vlastos', Plato's Universe.










The phrase "saving the phenomena" does not
occur in the Platonic corpus nor yet in
Aristotle's works. In Plato "save a thesis
(or 'argument')" (Theaetetus 164a) or "save a
tale" (Laws 645b) and in Aristotle "save a
hypothesis" (de Caelo 306a30) and "preserve a
thesis" (Nicomachean Ethics 1096a2) occur in
contexts where "to save" is to preserve the
credibility of a statement by demonstrating
its consistency with apparently recalcitrant
logical or empirical considerations. The
phrase "saving the phenomena" must have been
coined to express the same
credibility-salvaging operation in a case
where phenomena, not a theory or an argument,
are being put on the defensive and have to be
rehabilitated by a rational account which
resolves the prima facie contradictions
besetting their uncritical acceptance. This
is a characteristically Platonic view of
phenomena. For Plato the phenomenal world,
symbolized by the shadow-world in the
Allegory of the Cave (Republic 517b) is full
of snares for the intellect. Thus, at the
simplest level of reflection, Plato refers us
(Republic 602c) to illusions of sense, like
the stick that looks bent when partly
immersed in water, or the large object that
looks tiny at a distance. Thrown into
turmoil by the contradictory data of sense,
the soul seeks a remedy in operations like
"measuring, numbering, weighing" (Republic
602d) so that it will no longer be at the
mercy of the phenomenon. For Plato, then,
the phenomena must be held suspect unless
they can be proved innocent ("saved") by
rational judgment. So it would not be
surprising if the phrase "saving the
phenomena"--showing that certain perceptual
data are intelligible after all--had
originated in the Academy. (Vlastos, 1975,
pp. 111-112)

Returning then to the problem Plato set the

astronomers, Eudoxus was not the only one to attempt to

abduct an adequate hypothesis or solution. Speusippus,

Plato, and Heraclides each developed a solution different

from that of Eudoxus. Menaechmus, in essential respects,








followed the solution of Eudoxus. Callipus then made

corrections on Menaechmus' version of Eudoxus' solution,

which was then adopted by Aristotle. Each attempted to

abduct an adequate hypothesis, or modification of a former

hypothesis, which would adequately explain and predict the

so-called anomalous phenomena.7

There is an even more telling reference to the

abductive approach of Plato and Eudoxus in the "Eudemian

summary." The reference is intriguing, because it refers

to both abductive inference, under the rubric of analysis,

and the golden section.

[Eudoxus] multiplied the number of
propositions concerning the "section" which
had their origin in Plato, applying the
method of analysis to them.8 (Thomas, 1957,
vol. 1, p. 153)

My contention is that this analysis is none other than

what Peirce referred to as abduction. The essential

feature of this method is that one reasons backwards to the

causal explanation. Once one has arrived at the

explanatory hypothesis, then one is able to deductively

predict how the original puzzling phenomenon follows from

that hypothesis. Analysis and synthesis were basic

movements to and from a principle or hypothesis. Proclus

was aware of these contrary, but mutually supportive

movements. Thus, Morrow, in the introduction of his

translation of Proclus' A Commentary of the First Book of

Euclid's Elements, says,

the cosmos of mathematical propositions
exhibits a double process: one is a movement








of "progression" (prodos), or going forth
from a source; the other is a process of
"reversion" (anodos) back to the origin of
this going forth. Thus Proclus remarks that
some mathematical procedures, such as
division, demonstration, and synthesis, are
concerned with explication or "unfolding" the
simple into its inherent complexities,
whereas others, like analysis and definition,
aim at coordinating and unifying these
diverse factors into a new integration, by
which they rejoin their original
starting-point, carrying with them added
content gained from their excursions into
plurality and multiplicity. For Proclus the
cosmos of mathematics is thus a replica of
the complex structure of the whole of being,
which is a progression from a unitrary, pure
source into a manifold of differentiated
parts and levels, and at the same time a
constant reversion of the multiple
derivatives back to their starting-points.
Like the cosmos of being, the cosmos of
mathematics is both a fundamental One and an
indefinite Many. (Proclus,1970, p. xxxviii)

My view is that Proclus has correctly preserved the

sense of analysis and synthesis as underlying the work of

Plato. This notion of analysis appears to have been

somewhat esoteric, not being clearly explicated in the

writings of Plato. However, as I will later argue, Plato's

notion of dialectic is closely allied to this concept. One

is either reasoning backwards from conclusions to

hypotheses, or forward from hypotheses to conclusions.

Thus, as Aristotle noted:

Let us not fail to notice that there is
a difference between arguments from and those
to the first principles. For Plato, too, was
right in raising this question and asking, as
he used to do, "are we on the way from or to
the first principles?" There is a
difference, as there is in a race-course
between the course from the judges to the
turning-point and the way back. For, while
we must begin with what is known, things are








objects of knowledge in two senses--some to
us, some without qualification. Presumably,
then, we must begin with things known to us.
Hence any one who is to listen intelligently
to lectures about what is noble and just and,
generally, about the subjects of political
science must have been brought up in good
habits. For the fact is the starting-point,
and if this is sufficiently plain to him, he
will not at the start need the reason as
well; and the man who has been well brought
up has or can easily get starting-points.
(Nicomachean Ethics 1095a30-b9)

Thus, in the movement towards first principles, it is a

motion opposite in direction to that of syllogism or

deduction. This is central to Plato's notion of dialectic.

Hence, Plato says,

That which the reason itself lays hold of by
the power of dialectic, treats its
assumptions not as absolute beginnings but
literally as hypotheses, underpinnings,
footings, and springboards so to speak, to
enable it to rise to that which requires no
assumption and is the starting point of all.
(Republic 511b)

It should be noted that the notion of dialectic

depicted here in the Intelligible world moves strictly from

one hypothesis as a springboard to a higher or more

primitive hypothesis. It does not begin from the

observation of sensible particulars. However, a central

theme throughout this dissertation will be that this same

movement in abductive explanation takes place at every

level for Plato, including the level of sensible

particulars. Therefore, the abductive movement from the

observation of irregular planetary motions to the

explanation in terms of the underlying regularity

discoverable in the Intelligible world is analogous to the








abductive movement purely within the Intelligible world.

When we later examine the Cave simile (Republic 514a-521b)

it will become apparent that each stage of conversion is an

abductive movement, and hence a kind of dialectic, or

analysis.

Now according to Plato, once a suitable explanatory

hypothesis has been abducted, one can deductively descend,

setting out the consequences. Of course in the Republic,

where one is seeking certainty, this means first arriving

at the ultimate hypothesis, the Good.

when it has grasped that principle [or
hypothesis] it can again descend, by keeping
to the consequences that follow from it, to a
conclusion. (Republic 511b)

Thus, one then proceeds in the downward direction with

synthesis (i.e., syllogism or deduction).

It is difficult to find clear statements about

analysis in either Plato or Aristotle. However, Aristotle

does liken deliberation to geometrical analysis in the

Nicomachean Ethics.

We deliberate not about ends but about means.
For a doctor does not deliberate whether he
shall heal, nor an orator whether he shall
persuade, nor a statesman whether he shall
produce law and order, nor does any one else
deliberate about his end. They assume the
end and consider how and by what means it is
to be attained; and if it seems to be
produced by several means they consider by
which it is most easily and best produced,
while if it is achieved by one only they
consider how it will be achieved by this and
by what means this will be achieved, till
they come to the first cause, which in the
order of discovery is last. For the person
who deliberates seems to investigate and
analyse in the way described as though he








were analysing a geometrical construction .
and what is last in the order of analysis
seems to be first in the order of becoming.
(Nichomachean Ethics 1112b12-24)



The other famous 4th century B.C. mathematician in the

Academy was Theaetetus of Athens, who set down the

foundations of a theory of irrationals which later found

its way into Book X of Euclid's Elements. As Furley has

pointed out, "Theaetetus worked on irrational numbers and

classified 'irrational lines' according to different types"

(Furley, 1967, p. 105). He furthered the work on the 5

regular solids, and is held to have contributed much to

Book XIII of Euclid's Elements.

Speusippus, the son of Plato's sister Potone,

succeeded Plato as head of the Academy,l0 and wrote a work

entitled, On the Pythagorean Numbers. Unfortunately only a

few fragments remain. According to lamblichus,

[Speusippus] was always full of zeal
for the teachings of the Pythagoreans, and
especially for the writings of Philolaus, and
he compiled a neat little book which he
entitled On the Pythagorean Numbers. From
the beginning up to half way he deals most
elegantly with linear and polygonal numbers
and with all the kinds of surfaces and solids
in numbers; with the five figures which he
attributes to the cosmic elements, both in
respect of their similarity one to another;
and with proportion and reciprocity. After
this he immediately devotes the other half of
the book to the decad, showing it to be the
most natural and most initiative of
realities, inasmuch as it is in itself (and
not because we have made it so or by chance)
an organizing idea of cosmic events, being a
foundation stone and lying before God the
Creator of the universe as a pattern complete
in all respects. (Thomas, 1957, vol. 1, p. 77)









Furthermore, it is Speusippus who apparently rejected the

Platonic Ideas but maintained the mathematical numbers.1

Xenocrates, who followed Speusippus as head of the

Academy, wrote six books on astronomy. He is also credited

with an immense calculation of the number of syllables that

one can form out of the letters of the Greek alphabet. The

number he derived is 1,002,000,000,000 (Sarton, 1970, vol.

1, p. 503, & icClain, 1978, p. 188, fn. 32). Unlike

Speusippus, he retained the Ideas, but identified them with

the mathematical.

Another one of Plato's pupils was Philippus of Opus,

who, according to the "Eudemian summary," was encouraged by

Plato to study mathematics. It appears that he may have

edited and published the Laws and possibly authored the
12
Epinomis. He wrote several mathematical treatises, the

titles of which are still preserved.

Another pupil was Leodamos of Thasos, who, according

to Diogenes Laertius, was taught the method of analysis by

Plato. Again the intriguing mention of the method of

analysis occurs. And it is important that it is Plato who

purportedly taught Leodamos this method. In another place

Proclus indicates that

certain methods have been handed down. The
finest is the method which by means of
analysis carries the thing sought up to an
acknowledged principle, a method which Plato,
as they say, communicated to Leodamas, and by
which the latter, too, is said to have
discovered many things in geometry. (Heath,
1956, vol. 1, p. 134)








In a footnote to the above statement by Proclus, Heath

makes the very interesting comment,

Proclus' words about carrying up the thing
sought to "an acknowledged principle"
suggests that what he had in mind was the
process described at the end of Book VI of
the Republic by which the dialectician
(unlike the mathematician) uses hypotheses as
stepping-stones up to a principle which is
not hypothetical, and then is able to descend
step by step verifying every one of the
hypotheses by which he ascended. (Heath,
1956, vol. 1, p. 134, fn.1)

This is a very insightful remark by Heath, and I will

return to this point when I consider the Divided Line in

the Republic.

Menaechmus, a pupil of both Eudoxus and Plato, wrote

on the methodology of mathematics. It is generally

inferred from Eratosthenes that he discovered the conic

sections. His brother, Dinostratus, applied Hippias'

quadratrix in an attempt to square the circle.

Both Leon, the pupil of Neoclides, and Theudius of

Magnesia wrote a "Book of Elements" in the Academy during

Plato's time. Heath, in fact, conjectures that the

elementary geometrical propositions cited by Aristotle were

derived from the work of Theudius. Thus, Heath says,

Fortunately for the historian of mathematics
Aristotle was fond of mathematical
illustrations; he refers to a considerable
number of geometrical propositions,
definitions, etc., in a way which shows that
his pupils must have had at hand some
textbook where they could find the things he
mentions; and this textbook must have been
that of Theudius. (Heath, 1956, vol. 1, p.
117)









Also living at this time was the Pythagorean,

Archytas of Tarentum. An older contemporary and

friend of Plato, there is little doubt that he had a

major influence on Plato, though he may have never

actually been in the Academy. It was Archytas who

reduced the Delphic problem of doubling the cube to

the problem of finding two mean proportionals. In

the "Eudemian Summary" it is stated that,

[Archytas] solved the problem of
finding two mean proportionals by a
remarkable construction in 3 dimensions.
(Thomas 1957, vol. 1, p. 285)

Thus, according to Van der Waerden, Archytas is responsible

for the material in Bk. VIII of Euclid's Elements.

Cicero tells us that

[it was during Plato's] first visit to South
Italy and Sicily, at about the age of forty,
that he became intimate with the famous
Pythagorean statesman and mathematician
Archytas. (Hackforth, 1972, p. 6)

And of course from Plato's 7th Letter (at 350b-c) we

find that it was Archytas who sent Lamiscus with an

embassy and 30-oared vessel to rescue Plato from the

tyrant Dionysius (7th Letter 350b-c, Cairns and

Hamilton, 1971, p. 1596).

As Thomas indicates in a footnote:

For seven years [Archytas] commanded the
forces of his city-state, though the law
forbade anyone to hold the post normally for
more than one year, and he was never
defeated. He is said to have been the first
to write on mechanics, and to have invented a
mechanical dove which would fly. (Thomas,
1957, vol. 1, p.4, fn.a)










It is clear that Archytas was probably a major source

of much of the Philolaic-Pythagorean doctrines that Plato

gained access to. He was also a prime example of what a

philosopher-statesman should be like. Furthermore, he is

generally considered to be a reliable source of information

on the early Pythagoreans. "No more trustworthy witness

could be found on this generation of Pythagoreans" (Kirk

and Raven, 1975, p. 314).

Many of the ideas of Archytas closely parallel those

of Plato. Porphyry indicates this when he quotes a

fragment of Archytas' lost book On Mathematics:

The mathematicians seem to me to have arrived
at true knowledge, and it is not surprising
that they rightly conceive each individual
thing; for having reached true knowledge
about the nature of the universe as a whole,
they were bound to see in its true light the
nature of the parts as well. Thus they have
handed down to us clear knowledge about .
geometry, arithmetic and sphaeric, and not
least, about music, for these studies appear
to be sisters. (Thomas, 1957, vol. 1, p.5)

It seems obvious that the four sister sciences

mentioned here refer to the Pythagorean quadrivium. But

Archytas' above statement that "they rightly conceive each

individual thing," should be contrasted with some of

Plato's remarks regarding dialectic. Thus, Plato says,

"dialectic sets out systematically to determine what each

thing essentially is in itself" (Republic 533b). And

further he says, the dialectician is one who "can take

account of the essential nature of each thing" (Republic

534b).








There was, of course, also Aristotle, who was a member

of the Academy for nineteen (perhaps twenty) years during

Plato's lifetime. "From his eighteenth year to his

thirty-seventh (367-348/7 B.C.) he was a member of the

school of Plato at Athens" (Ross, 1967, p. ix). Aristotle

considered mathematics to be one of the theoretical

sciences along with metaphysics and physics (Metaphysics

1026a 18-20). But he did not devote any writing strictly

to the subject itself, contending that he would leave it to

others more specialized in the area. Nevertheless, his

writings are interspersed with mathematical examples. And

what he has to say regarding Plato's treatment of

mathematics is of the utmost importance in trying to

properly interpret Plato. His remarks should not be swept

aside simply because one has difficulty tallying them with

the Platonic dialogues.

Aristotle was obviously subjected to mathematical

study in the Academy. Apparently he was not that pleased

with its extreme degree of emphasis there. Thus, he
14
indicates in a somewhat disgruntled tone, as Sorabji has

put it, that many of his modern cohorts in the Academy had

so over-emphasized the role of mathematics that it had

become not merely a propadeutic to philosophy, but the

subject-matter of philosophy itself.

Mathematics has come to be identical with
philosophy for modern thinkers, though they
say that it should be studied for the sake of
other things. (Metaphysics 992a 32-34)










Theodorus of Cyrene should also be mentioned here.

According to lamblichus he was a Pythagorean. And

according to Diogenes Laertius he was the mathematical

instructor of Plato (Thomas, 1957, Vol. 1, p. 380).

According to the "Eudemian summary," Theodorus "became

distinguished in geometry" (Thomas, 1957, Vol. 1, p. 151).

In Plato's dialogue, the Theaetetus, Theodorus appears with

the young Theaetetus. There Theaetetus is subjected to the

midwifery of Socrates. Theaetetus begins his account by

describing the mathematical nature of his training at the

hands of Theodorus. In the following passage it is

interesting to note that we again find the Pythagorean

quadrivium.

Socrates: Tell me, then, you are learning
some geometry from Theodorus?
Theaetetus: Yes.
Socrates: And astronomy and harmonics and
aritmetic?
Theaetetus: I certainly do my best to learn.
(Theaetetus 145c-d)

In the "Eudemian summary," Proclus also mentions

Amyclas of Heracleia, who is said to have improved the

subject of geometry in general; Hermotimus of Colophon, who

furthered the investigations of Eudoxus and Theaetetus; and

Athenaeus of Cyzicus, who "became eminent in other branches

of mathematics and especially in geometry" (Thomas, 1957,

vol. 1, p. 153). But it is not clear when these three

individuals appeared in the Academy, whether during Plato's

lifetime, or shortly thereafter.











Archytas, Theodorus, Amyclas, Hermotimus, and

Athenaeus aside, it is asserted of the others that "these

men lived together in the Academy, making their inquiries

in common" (Thomas, 1957, vol. 1, p. 153). However, it is

not suggested, and it certainly should not be inferred,

that they were all at the Academy simultaneously. It must

be remembered that Plato ran the Academy for some forty

years. And though Aristotle was there for nineteen of

those years, several of the other individuals may have come

and gone, appearing at the Academy at different times. But

one thing is clear; there was an overriding emphasis in

mathematical study and research. As Cherniss has correctly

pointed out,

If students were taught anything in the
Academy, they would certainly be taught
mathematics that their minds might be
trained and prepared for the dialectic; and
this inference from the slight external
tradition is supported by the dialogues,
especially the seventh book of the Republic.
(Cherniss, 1945, pp. 66-67)


On the Good

Further evidence in support of the mathematical nature

of Plato's philosophy may be found in the accounts of his

unwritten lecture, On the Good. Aristoxenus, like Eudemus,

a disciple of Aristotle, writes in his Elements of Harmony,

Bk. 2,

Plato's arguments were of mathematics and
numbers and geometry and astronomy and in the
end he declared the One to be the Good.
(Thomas, 1957, vol. 1, p. 389)










Thus, although the title of the lecture indicated that

it was about the Good, the ultimate object of knowledge as

expressed in the Republic, it nevertheless dealt with the

mathematical subject-matter involved in the ascent there.

And furthermore, the One was somehow identified with the

ultimate object of knowledge, the Good. This tallies with

what Aristotle has to say. Referring to Plato's use of two

causes, the essential cause, the One, and the material

cause, the Indefinite Ddyad or the Great and Small,

Aristotle says,

Further, he has assigned the cause of good
and evil to the elements, one to each of the
two. (Metaphysics 988a 13-15)

At another point Aristotle explains that

the objection arises not from their ascribing
goodness to the first principle as an
attribute, but from their making the One a
principle--and a principle in the sense of an
element--and generating number from the One.
(Metaphysics 1091b 1-4)

Aristotle objects elsewhere, fortunately for us in a

very telling way. He says,

to say that the first principle is good
is probably correct; but that this principle
should be the One or an element of
numbers, is impossible. For on this
view all the elements become identical with
species of good, and there is a great
profusion of goods. Again, if the Forms are
numbers, all the Forms are identical with
species of good. (Metaphysics 1091b 19-27)

Aristotle goes on to argue that if evil is identified

with the Dyad, Plato's Great and Small, it follows that

all things partake of the bad except one--the
One itself, and that numbers partake of it in








a more undiluted form than spatial
magnitudes, and that the Bad is the space in
which the Good is realized. (Metaphysics
1092a 1-3)

The statement of Aristoxenus and the passages in

Aristotle's Metaphysics appear to indicate that Plato held

the One and Indefinite Dyad to be the principles of all

entities, and furthermore gave them the attributes

respectively of good and evil. It follows that if the

Forms (as principles of all other entities) are derived

from the One and Indefinite Dyad, then the elements from

which they are derived are numerical and, hence, the Forms

themselves are of a numerical nature. As the Forms are

principles of all other entities, it would follow that

number would be perpetuated throughout the Cosmos down into

the sensible things as well.

Now Cherniss admits that

Alexander himself says that in Aristotle's
report of the lecture [On the Goodl, "the
One" and "the great and small" were
represented as the principles of number and
the principles of all entities. (Cherniss,
1945, p. 28)

How then can the numerical principles of the One and

Indefinite Dyad be the principles of all entities,

including all sensible entities, unless the crucial feature

is that numbers are in fact the essential characters of

those entities? Aristotle makes this more explicit, saying,

Since the Forms were causes of all other
things, he thought their elements were the
elements of all things. As matter, the great
and small were principles; as essential
reality, the One; for from the great and
small, by participation in the One, come the








Numbers. [And] he agreed with the
Pythagoreans in saying that the One is
substance and not a predicate of something
else; and in saying that the Numbers are the
causes of the reality of other things he
agreed with them. (Metaphysics 987b 19-25)

There is an interesting parallel between what Aristotle

says about the One in the Metaphysics, and what Plato says

about the Good in the Republic. Aristotle says,

the Forms are the causes of the essence
of all other things, and the One is the cause
of the essence of the Forms. (Metaphysics
988a 9-11)

The latter part of this statement should be compared with

what Plato has Socrates say about the Good.

The Good therefore may be said to be the
source not only of the intelligibility of the
objects of knowledge [the Forms], but also of
their being and reality; yet it is not itself
that reality but is beyond it and superior to
it in dignity and power. (Republic 508b)


Thus, there appears to be a definite identification of

the Good with the One, and evil with the Dyad. And hence,

for Plato, the two basic elements of the Cosmos are of a

numerical nature. The further implication then is that the

Forms are also numbers. This is in fact what Aristotle

suggests,

the numbers are by him [Plato]
expressly identified with the Forms
themselves or principles, and are formed out
of the elements. (De Anima. 404b24)

At another point Aristotle unequivocally asserts that,

"those who speak of Ideas say the Ideas are numbers"

(Metaphysics 1073a18-20). And in fact, not only are the

Forms to be identified with numbers, but so are the








sensibles, although as numbers of a different class. This

emerges in a passage in which Aristotle is discussing the

Pythagoreans.

When in one particular region they place
opinion and opportunity, and, a little above
or below, injustice and decision or mixture,
and allege, as proof, that each of these is a
number, and that there happens to be already
in this place a plurality of the extended
bodies composed of numbers, because these
attributes of number attach to the various
places--this being so, is this number, which
we must suppose each of these abstractions to
be, the same number which is exhibited in the
material universe, or is it another than
this? Plato says it is different; yet even
he thinks that both these bodies and their
causes are numbers, but that the intelligible
numbers are causes, while the others are
sensible. (Metaphysics 990a23-32)

Thus, numbers are not only the crucial feature of Forms,

but also of sensible particulars.

Returning then to Plato's unwritten lecture, On the

Good, Cherniss notes:

It is said that Aristotle, Speusippus,
Xenocrates, Heraclides, Hestiaeus, and other
pupils attended the lecture and recorded
Plato's remarks in the enigmatic fashion in
which he made them (see Simplicius).
Moreover, most of them apparently published
their notes or transcripts of the lecture.
Aristotle's notes were certainly
published under the title, On the Good [peri
tagathou]. (Cherniss, 1945, p. 12)

Why then was the lecture delivered (and recorded) in

this so-called "enigmatic fashion?" A clue to this may lie

in the Phaedrus where the Egyptian King Thamus (Ammon)

reprimands the god Theuth. The latter has claimed that his

discovery of writing "provides a recipe for memory and











wisdom" (Phaedrus 274e). Thamus replies that it only leads

to the "conceit of wisdom" (Phaedrus 275b).

If men learn this [writing], it will implant
forgetfulness in their souls; they will cease
to exercise memory because they rely on that
which is written, calling things to
remembrance no longer from within themselves,
but by means of external marks. What you
have discovered is a recipe not for memory,
but for reminder. And it is no true wisdom
that you offer your disciples, but only its
semblance, for by telling them of many things
without teaching them you will make them seem
to know much, while for the most part they
know nothing, and as men filled, not with
wisdom, but with the conceit of wisdom, they
will be a burden to their fellows. (Phaedrus
274e-275b)

Then Plato has Socrates follow this with an analogy

of writing to painting.

The painter's products stand before us as
though they were alive, but if you question
them, they maintain a most majestic silence.
It is the same with written words; they seem
to talk to you as though they were
intelligent, but if you ask them anything
about what they say, from a'desire to be
instructed, they go on telling you just the
same thing forever. And once a thing is put
in writing, the composition, whatever it may
be, drifts all over the place, getting into
the hands not only of those who understand
it, but equally those who have no business
with it; it doesn't know how to address the
right people, and not address the wrong. And
when it is ill-treated and unfairly abused it
always needs its parent to come to its help,
being unable to defend or help itself.
(Phaedrus 275d-e)

This tends to show a negative view on the part of Plato

toward the publishing of one's doctrines. The reason being

that written doctrines may be either misunderstood or

abused by falling into the wrong hands. And if either of









these situations occur, the architect of the doctrine must

be present to defend and rectify the situation. But too

often this attendance is impossible.

This position is clearly consistent with what Plato

has to say in the 7th Letter. There, referring to his most

complete doctrine, Plato says,

I certainly have composed no work in regard
to it, nor shall I ever do so in the future,
for there is no way of putting it in words
like other studies. (7th Letter 341c)

Here, of course, Plato is not as concerned with

misunderstanding or abuse, as he is with what appears

to be a somewhat more mystical doctrine. As he goes

15
on to say regarding this subject,

Acquaintance with it must come rather after a
long period of attendance on instruction in
the subject itself and of close
compainioship, when, suddenly, like a blaze
kindled by a leaping spark, it is generated
in the soul and at once becomes
self-sustaining. (7th Letter 341c-d)

Why then did so many of Plato's pupils copy down the

lecture and, apparently, publish it? Why was it so

important? I contend that it set forth, though still in an

enigmatic fashion, the essentially mathematical underlying

structure of Plato's philosophy. The parallel between the

lecture On the Good and the Republic, especially Books 6

and 7, has already been touched upon. The Good of the

Republic was identified with the One. And Aristotle goes

on to contrast the account given in the Timaeus of the

participant with the account in the so-called unwritten

teaching. In the latter, the participant-receptacle of the








Timaeus is identified with the great and small(Physics

209b11-15 and 209b33-210a2). As already indicated from

Alexander's account (following Aristotle) of the lecture Dn

the Good. "the One and the Great and the Small were

represented as the principles of number and the principles

of all entities" (Cherniss, 1945, p. 28).

If further we accept the argument of G.E.L. Owen

(Allen, 1965), that the Timaeus is to be grouped along with
16
the Phaedo and Republic as a middle dialogue, then it

appears that the lecture On the Good may provide some

mathematical keys to the interpretation of the middle

dialogues. This is strictly inferential. However, I

.maintain that it is a most plausible inference that allows

one to advocate a very consistent approach to Plato's

thought.

When this is conjoined with the question of

mathematical (intermediate, separate, immutable, and

plural) and their relation to the ontological and

epistemological function of the soul, a slightly revised

view of Plato's middle dialogues will emerge. This, in

turn, may shed some entirely different light upon the

paradoxidcal problems posed in the Parmenides, Theaetetus,

and Sophist,17 and their possible solution.



The Pythagorean Influence

This mathematical structuring of Plato's philosophy

suggests that he may have strongly adhered to, and further developed








developed, some of the doctrines of the Pythagoreans. And,

if I am correct, he especially followed the matheimatikoi.

It should be noted that in contrast to these matheimatikoi,

Plato critiques the more exoteric Pythagorean akousmatikoi

for getting caught up with the sensory aspects of harmonics.

They look for numerical relationships in
audible concords, and never get as far as
formulating problems and examining which
numerical relations are concordant, which
not, and why. (Republic 531c)

That is to say, the akousmatikoi never rise above the more

mundane features of harmonics. They do not formulate

problems for themselves from which they can abduct

hypotheses as solutions. These notions of setting a

problem and formulating the conditions for solution

(diorismos) are critical to the abductive movement,

referred to by the ancients as analysis. In this regard,

note especially Plato's Meno (see supra, pp. 36 & 39), where

he refers to the method "by way of hypothesis" (Meno

86e-87b).

In the Metaphysics, Aristotle indicates that Plato

was, in fact, a follower of the philosophy of the

Pythagoreans, but also differed from them in some respects.

He says,

After the systems we have named came the
philosophy of Plato, which in most respects
followed [akolouthousa] these thinkers [i.e.,
Pythagoreans], but had peculiarities that
distinguished it from the philosophy of the
Italians. (Metaphysics 987a29-31)










Entirely too much emphasis has been placed upon the

subsequent "differences" which were due to the Heraclitean

influence of Cratylus, and not enough weight placed on the

former words. Hackforth contends that the similarity

between the Pythagoreans and Plato is much stronger than

generally acknowledged. Of course, this is one of my

contentions here as well. Referring to Aristotle's account

of the relation between Platonic Forms and Pythagorean

Numbers, Hackforth says:

Despite the important divergences there
noted, one of which is the transcendence of
the Forms as against the Pythagorean
identification of things with numbers, it
seems clear that he regarded their general
resemblance as more fundamental. Moreover
the word akolouthousa [Metaphysics 987a30] is
more naturally understood as implying
conscious following of Pythagorean doctrine
than mere factual resemblance. (Hackforth,
1972, p. 6)

Unfortunately, an extremely dualistic picture of Plato

has been painted by those who accept the strict separation

of the Intelligible and Sensible worlds in Plato's

philosophy. This has resulted from too heavy of an

emphasis being placed upon the Heraclitean influence of

Cratylus upon Plato. This has led to a tendency by

scholars to get stumped by the problems Plato sets in the

dialogues, rather than solve them. If my Pythagorean

hypothesis about Plato is correct, then many of the

Platonic dialogue problems should be, if not actually

soluble, then at least reasonably understandable.











My own view is that it was probably Philolaus who had

the greatest impact upon the views of Plato. This

influence may well have been channeled through Plato's

friend, Archytas. It is important to recall that Plato's

nephew, Speusippus, "was always full of zeal for the

teachings of the Pythagoreans, and especially for the

writings of Philolaus" (Thomas, 1957, vol. 1, p. 77).

Along this line, it is interesting that the contents of

Speusippus' book On Pythagorean Numbers (see supra p. 59)

holds a close resemblance to material in Plato's Timaeus

regarding the five cosmic elements and their harmonious

relation in terms of ratio and proportion. And there is

the assertion of Diogenes Laertius, presumably following

Aristoxenus, that Plato copied the Timaeus out of a work by

Philolaus.

Philolaus of Croton, [was] a
Pythagorean. It was from him that Plato, in
a letter, told Dion to buy the Pythagorean
books. He wrote one book. Hermippus
says that according to one writer the
philosopher Plato went to Sicily, to the
court of Dionysius, bought this book from
Philolaus' relatives and from it copied
out the Timaeus. Others say that Plato
acquired the books by securing fom Dionysius
the release from prison of a young man who
had been one of Philolaus' pupils. (Kirk and
Raven, 1975, p. 308)

Though we need not assert plagiarism, it is entirely

reasonable to suppose that a work of Philolaus' acted as a

source book for Plato's Timaeus.










It is also noteworthy that Plato, in the Phaedo, refers

to Philolaus. He has Socrates ask the Pythagoreans, Cebes

and Simmias, whether they had not heard Philolaus, whom

they had been staying with, talk about suicide.

Why, Cebes, have you and Simmias never heard
about these things while you have been with
Philolaus [at Thebes]? (Phaedo 61d)

Plutarch hints that Plato in fact studied Pythagorean

philosophy at Memphis with Simmias.

Simmias appears as a speaker in Plutarch's
dialogue De genio Socratis, where he says
[578fj that he was a fellow-student of
philosophy with Plato at Memphis--an
interesting remark and conceivably true.
(Hackforth, 1972, pp. 13-14)

It is probable then that the unnamed authority in

Socrates' last tale (Phaedo 107d-115a) is Philolaus,

especially with the reference to the dodecahedron. Thus,

Socrates says to Simmias:

The real earth, viewed from above, is
supposed to look like one of these balls made
of twelve pieces of skin, variegated and
marked out in different colors, of which the
colors which we know are only limited
samples, like the paints which artists use,
but there the whole earth is made up of such
colors, and others far brighter and purer
still. One section is marvelously beautiful
purple, and another is golden. (Phaedo 1lOb-c)

Of course the dodecahedron reappears in the Timaeus as

the foundation of the structure of the Cosmos.

There was yet a fifth combination which God
used in the delineation of the universe with
figures of animals. (Timaeus 55c)










Very few of the Philolaic fragments remain. However,

what fragments do remain provide a clue as to why

Speusippus was so enthusiastic about his writings. It may

also indicate why Speusippus' uncle, Plato, found his work

so interesting, as well. Philolaus' fragment 12 appears as

though it could have come straight out of the Timaeus.

In the sphere there are five elements, those
inside the sphere, fire, and water and earth
and air, and what is the hull of the sphere,
the fifth. (Santillana and von Dechend, 1969,
p. 232)

It is difficult to adequately ascertain the thought of

the early Pythagoreans, including Philolaus. They

maintained an oral tradition in which their major tenets

were guarded with great secrecy. Substantial fragments of

a book on Pythagoreanism by Aristotle's pupil, Aristoxenus

of Tarentum, preserved by lamblichus, remain to bear this

fact out.

The strictness of their secrecy is
astonishing; for in so many generations
evidently nobody ever encountered any
Pythagorean notes before the time of
Philolaus. (Kirk and Raven, 1975, p. 221)

Furthermore, Porphyry, quoting another pupil of

Aristotle, Dicaearchus of Messene, indicates the same

thing.

What he [Pythagoras] said to his associates,
nobody can say for certain; for silence with
them was of no ordinary kind. (Kirk and
Raven, 1975, p. 221)

Thus, secrecy was the rule. He who would reveal the

Pythagorean tenets on number faced punishment.








There was apparently a rule of secrecy in the
community, by which the offence of divulging
Pythagorean doctrine to the uninitiated is
said by later authorities to have been
severely punished. (Kirk and Raven, 1975, p.
220)

Hence, lamblichus maintained the tradition that

the Divine Power always felt indignant with
those who rendered manifest the composition
of the icostagonus, viz., who delivered the
method of inscribing in a sphere the
dodecahedron (Blavatsky, 1972, vol. 1 p.
xxi).

This may well be why Plato was so cryptic in his

discussion of the construction of the four elements and the

nature of the fifth element in the Timaeus. He is

discussing the formation of the tetrahedron, icosahedron,

and octahedron out of the right-angled scalene triangles.

Of the infinite forms we must again select
the most beautiful, if we are to proceed in
due order, and anyone who can point out a
more beautiful form than ours for the
construction of these bodies, shall carry off
the palm, not as an enemy, but as a friend.
(Timaeus 54a)

Plato may well have been concerned with not being too

explicit about this Pythagorean doctrine. As he points out

in the 7th Letter,

I do not think the attempt to tell
mankind of these matters a good thing,
except in the case of some few who are
capable of discovering the truth for
themselves with a little guidance. .
There is a true doctrine, which I have often
stated before, that stands in the way of the
man who would dare to write even the least
thing on such matters. (7th Letter 341e-342a)

Nevertheless, there is a tradition amongst the

Neoplatonists that Plato was an initiate of various mystery









schools, including the Pythagorean school, and that he

incurred much wrath for "revealing to the public many of

the secret philosophic principles of the Mysteries" (Hall,

1928, p. 21).

Plato was an initiate of the State Mysteries.
He had intended to follow in the footsteps of
Pythagoras by journeying into Asia to study
with the Brahmins. But the wars of the time
made such a trip impractical, so Plato turned
to the Egyptians, and, according to the
ancient accounts, was initiated at Sais by
the priests of the Osirian rites. There
is a record in the British Museum that Plato
received the Egyptian rites of Isis and
Osiris in Egypt when he was forty-seven years
old. (Hall, 1967, pp. 1&5)

Whatever truth there is in this matter, it is clear

that Plato was greatly influenced by the Pythagoreans (and

possibly the Egyptians). See Figure # 1, p.-122, for a

projected chronological outline of Plato's life. Plato is,

throughout the dialogues, obstetric with his readers,

continually formulating problems and leaving hints for

their solution. The reader is left to ponder these

problems and, hopefully, abduct adequate solutions to them.

As stated earlier, Plato followed the Pythagoreans in

maintaining that the principle elements of things, the One

and Indefinite Dyad, are numbers. Likewise, as Aristotle

has indicated, Plato also identified Forms and sensible

particulars with numbers, though each with a different

class of numbers. The Pythagorean influence of Philolaus

upon Plato should become clear when considering one of the

Philolaic fragments.










And all things that can be known contain
number; without this nothing could be thought
or known. (Kirk and Raven, 1975, p. 310)

But what might it mean for Forms and sensible objects

to be numbers? Thomas Taylor has preserved the later

testimony of the Neopythagore=ns Nichomachus, Theon of

Smyrna, lamblichus, and Boetius, regarding the early

Pythagorean identification of number with things. The

following condensed version is adapted from Taylors's book

(T.Taylor, 1983).

According to the later Pythagorean elucidations, the

earliest Pythagoreans subdivided the class of odd numbers

(associated with equality) into incomposite, composite, and

incomposite-composite numbers. The first and incomposite

numbers were seen to be the most perfect of the odd

numbers, comparable to the perfections seen in sensible

things. They have no divisor other than themselves and

unity. Examples are 3, 5, 7, 11 13, 17, 19, 23, 29, 31,

etc. The incomposite-composites are not actually a

separate class, but merely the relationship between two or

more composite numbers which are not divisible by the same

divisor (other than unity). That is, they are composite

numbers which are incommensurable (in a certain sense) with

one another. For example, 25 and 27 are composite numbers

which have no common factor other than unity. In

approximately 230 B.C., Eratosthenes (see supra pp. 17-20 ),

a later Pythagorean, developed his famous mathematical








sieve. It was a mechanical method by which the "subtle"

incomposite numbers could be separated from the "gross,"

secondary composite numbers. These subtle and gross

qualities were likened to the qualities in sensible things.

Likewise, the even numbers (associated with

inequality) were divided into superabundant, deficient, and

perfect numbers, the last of which is a geometrical mean

between instances of the other two kinds. A superabundant

even number is one in which the sum of its fractional parts

is greater than the number itself. For example, 24 is a

superabundant number: 1/2 x 24 = 12; 1/3 x 24 =8; 1/4 x 24

= 6; 1/6 x 24 =4; 1/12 x 24 = 2; 1/24 x 24 =1. The sum of

these parts, 12+8+6+4+2+1 = 33, is in excess of 24.

A deficient even number is one in which the sum of its

fractional parts is less than itself. For example, 14 is a

deficient number: 1/2 x 14 7; 1/7 x 14 = 2; 1/14 x 14 =

1. The sum of these parts, 7+2+1 = 10, is less than 14.

A perfect even number is one in which the sum of its

fractional parts is equal to itself. For example, 28 is a

perfect number: 1/2 x 28 = 14; 1/7 x 28 = 7; 1/7 x 28 = 4;

1/14 x 28 = 2; 1/28 x 28 = 1. The sum of these parts is

equal to the original number 28. These perfect numbers are

geometric mediums between superabundant and deficient

numbers. Any perfect number multiplied by 2 results in a

superabundant number. Any perfect number divided by 2

results in a deficient number. Furthermore, perfect

numbers are very rare, there being only four of them








between the numbers 1 and 10,000: 6, 28, 496, 8,128. The

Pythagoreans saw a "resemblance" between this division of

even numbers into perfect, superabundant, and deficient,

and the virtues and vices of sensible things. Thus, Taylor

records,

Perfect numbers, therefore, are beautiful
images of the virtues which are certain media
between excess and defect. And evil is
opposed to evil [i.e., superabundance to
deficiency] but both are opposed to one good.
Good, however, is never opposed to good, but
to two evils at one and the same time .
[Perfect numbers] also resemble the virtues
on another account; for they are rarely
found, as being few, and they are generated
in a very constant order. On the contrary,
an infinite multitude of superabundant and
deficient numbers may be found [and]
they have a great similitude to the vices,
which are numerous, inordinate, and
indefinite. (T.Taylor, 1983, p. 29)

Aristotle, in the Magna Moralia 1182a11, indicates

that "Pythagoras first attempted to discuss goodness .

by referring the virtues to numbers" (Kirk & Raven, 1975,

p. 248). But the above recorded link between numbers and

virtues appears to be limited to resemblance. Certainly

Plato (and the Pythagoreans before him) had something much

stronger in mind. This suggestion of an actual

identification between numbers and Forms, and thereby

sensible things, will become clearer as we proceed.

The Notorious Question
of Mathematicals

Maintaining the Pythagorean mathematical influence of

Plato's philosophy clearly in mind, we will now consider

what has been termed "the notorious question of








mathematical" (Cherniss, 1945, p. 75). In the Republic,

Plato indicates that the trait of the philosopher is "love

of any branch of learning that reveals eternal reality"

(Republic 485a). The reason then that the mathematical

sciences may be appropriate as a bridge to the Forms

(Idea-Numbers) and ultimately the Good (One), is that their

subject-matter may be eternal. In fact Plato says

precisely this. "The objects of geometrical knowledge are

eternal" (Republic 527b). The question then is whether

these mathematical objects are distinct from the Forms, as

a separate ontological class, or to be identified with the

Forms.

Aristotle indicates that the mathematical are, for

Plato, a separate ontological class. Most modern

commentators, however, have rejected this notion, at least

that it was in the dialogues. Such diverse schools of

interpretation as those of Cornford, Robinson, and Cherniss

have all agreed in the rejection of a separate class of

mathematical. For example, Cornford, in reference to the

intelligible section of the Divided Line in the Republic,

states:

Where the intelligible section is subdivided,
clearly some distinction of objects is meant.
I agree with critics who hold that nothing
here points to a class of mathematical
numbers and figures intermediate between
Ideas and sensible things. (Cornford, 1965,
p. 62)

Most of the attempts to find the mathematical in the

dialogues have centered around the Divided Line passage in









the Republic. Cherniss sees the whole question as simply a

matter of "misunderstanding and misrepresentation" on the

part of Aristotle (Cherniss, 1945, p. 25). Robinson sees

it as a deduction in the Republic possible only on the

assumed grounds of exact correspondence between the Cave

simile and the Divided Line. This is an exact

correspondence which he asserts cannot be maintained.

Before dealing with the arguments of Cherniss and Robinson

I will first examine what Aristotle, and then Plato

himself, had to say.

Aristotle clearly sets out Plato's position on

mathematical.

Further besides sensible things and Forms he
[Plato] says there are the objects of
mathematics, which occupy an intermediate
position, differing from sensible things in
being eternal and unchangeable, from Forms in
that there are many alike, while the Form
itself is in each case unique. (Metaphysics
987bl4-18)

Thus, the mathematical are eternal but partake of

plurality. These mathematical are given by Plato a

definite ontological status separate from Forms and

sensibles. Thus Aristotle states,

Some do not think there is anything
substantial besides sensible things, but
others think there are eternal substances
which are more in number and more real; e.g.,
Plato posited two kinds of substance--the
Forms and objects of mathematics--as well as
a third kind, viz. the substance of sensible
bodies. (Metaphysics 1028b17-21)

Further, Aristotle makes it clear that it is these

mathematical with which the mathematical sciences are








concerned. It is "the intermediates with which they say

the mathematical sciences deal" (Metaphysics 997b 1-3).

Now if the mathematical sciences do in fact deal with these

intermediate mathematical objects, then it simply follows

that when we study the mathematical sciences, the objects

of our enquiry are the mathematical.

Aristotle again makes reference to the Platonic notion

that the objects of mathematics are substances. He does

this by clearly distinguishing the three views on the

question of mathematical. These are the views of Plato,

Xenocrates, and Speusippus.

Two opinions are held on this subject; it is
said that the objects of mathematics--i.e.,
numbers and lines and the like--are
substances, and again that the Ideas are
substances. And since (1) some [Plato]
recognize these as two different classes--the
Ideas and the mathematical numbers, and (2)
some [Xenocrates] recognize both as having
one nature, while (3) some others
[Speusippus] say that the mathematical
substances are the only substances, we must
consider first the objects of mathematics.
(Metaphysics 1076al6-23)

It is important to note that Cherniss contends that one

of the reasons Aristotle ascribes a doctrine of

mathematical and Idea-Numbers to Plato is because he

(Aristotle) mistakenly confused the doctrines of Speusippus

and Xenocrates with those of Plato. And yet in the above

quoted passage there is a clear distinction made between

the doctrines of the three individuals. And furthermore,

it is apparent that Aristotle is writing here as a member

of the Academy, and with reference to doctrines debated









therein. Thus, in the same paragraph, six lines later, he

makes reference to "our school," and the fact that these

questions are also being raised outside the Academy. Thus

he says that, "most of the points have been repeatedly made

even by the discussions outside of our school" (Metaphysics

1076a28-29). And this latter is in contradistinction to

the previous discussion of the positions held within the

Academy.

Furthermore, and to the great discredit of Cherniss'

position, why would both Xenocrates and Speusippus uphold a

doctrine of mathematical in the Academy if this was

entirely foreign to Plato? Certainly, Aristotle as a

member of the Academy for nineteen years could not be so

mistaken in attributing to his master a doctrine that Plato

never held. The variance in doctrine occurred not with

Plato's mathematical, which both Xenocrates and Speusippus

maintained. Rather, the differences involved the Ideas.

Speusippus apparently rejected the Ideas altogether while

keeping the mathematical, and Xenocrates lowered the Ideas

down to the ontological status of mathematical, in the end

identifying the two.

Cherniss, in fact, maintains that Aristotle also held

a doctrine of mathematical.

Aristotle himself held a doctrine of
mathematical intermediate between pure Forms
and sensibles, most of the Forms and all the
mathematical being immanent in the sensible
objects and separable only by abstraction.
(Cherniss, 1945, p. 77)









Cherniss apparently finds part of the basis for this latter

assertion in a passage where Aristotle appears to be in

agreement with the Pythagoreans regarding the non-separate

aspect of mathematical.

It is evident that the objects of mathematics
do not exist apart; for if they existed apart
their attributes would not have been present
in bodies. Now the Pythagoreans in this
point are open to no objection. (Metaphysics
1090a28-31)

But if Cherniss is correct in his assertions, then

the absurd conclusion that follows is that Plato's three

leading pupils held a doctrine of mathematical in one form

or another, but Plato held none. This is a mistaken view.

What Xenocrates, Speusippus, and Aristotle altered in

attempts to overcome problems they may have perceived in

Plato's doctrine, were not the mathematical, but rather,

the Ideas. Aristotle saw the universals as abstractions

from sensible particulars, thereby denying separate

ontologial status to the Ideas. Speusippus rejected the

Ideas, while maintaining the mathematical. And Xenocrates

collapsed the Ideas and mathematical into one. Surely

Aristotle was correct when he attributed to Plato a

doctrine of intermediate mathematical.

In another passage Aristotle again clearly

distinguishes the positions of Plato and Speusippus, one

from the other, and from them the position of the

Pythagoreans.

[Plato] says both kinds of number exist, that
which has a before and after being identical








with the Ideas, and mathematical number being
different from the Ideas and from sensible
things; and others [Speusippus] say
mathematical number alone exists as the first
of realities, separate from sensible things.
And the Pythagoreans also believe in one kind
of number--the mathematical; only they say it
is not separate but sensible substances are
formed out of it. (Metaphysics 1080bid-18)

Aristotle then goes on in the same passage to

distinguish the view of another unknown Platonist from that

of Xenocrates.

Another thinker says the first kind of
number, that of the Forms, alone exists, and
some [Xenocrates] say mathematical number is
identical with this. (Metaphysics 1080b22-23)


The Divided Line

In the Republic, Plato sets forth three related

similes in an attempt to explicate (metaphorically) his

conception of the ascent of the mind (or soul) of the

philosopher-statesman through succeeding stages of

illumination, culminating with the vision of the Good.

These three similes are that of the Sun (Republic

502d-509c), the Divided Line (509d-511e), and the Cave

(514a-521b). They are actually analogies intended to

perpetuate the notion of proportion, which is the

underlying bond for Plato. Each simile is indicative of a

process of conversion. In the Sun and Cave, the conversion

is to greater degress of light. In the Divided Line it is

clearly conversion to higher levels of awareness. The

analogy is between body and soul (mind), for in the end

Plato links the similes, saying,








The whole study of the sciences [i.e.,
mathematical sciences] we have described has
the effect of leading the best element in the
mind [soul] up towards the vision of the best
among realities [the Good in the Divided
Line], just as the body's clearest organ was
led to the sight of the brightest of all
things in the material and visible world [in
the Sun and Cave similes]. (Republic 532c-d)

In each case the conversion is from an image to the

original, or cause, of that image. It is clear from these

similes that Plato intends the reader to grasp that he

holds a doctrine of degrees of reality of the

subject-matter apprehended. That is, at each level of

conversion the soul (or mind) apprehends an increased level

of reality, tethering the previous level, just as the eye

apprehends an increased degree of light at each level, even

though the brilliance is at first blinding. Analogously,

through the use of these similes, Plato intends the reader

to apprehend a doctine of degrees of clarity of mind (or

soul). At each level of conversion the mind increases its

degree of clarity. Thus, Plato uses the imagery of light

in these similes to indicate the process of illumination

taking place. Furthermore, each simile should be viewed as

an extension of that which preceded it. Plato makes this

point clear when in the Cave scene he has Socrates say to

Glaucon, "this simile [the Cave] must be connected

throughout with what preceded it [Sun and Line similes]"

(Republic 517a-b).

The Divided Line is the most important of the similes

with regard to the issues of mathematical, abduction, and








proportion. There Plato begins by saying, "suppose you

have a line divided into two unequal parts" (Republic

509d). What could Plato possibly mean by this? Here is

the anomalous phenomenon: a line divided unequally. How

do we explain or account for it? From this bare

appearance, what could he possibly mean? What hypothesis

could one possibly abduct that would make a line divided

unequally follow as a matter of course, at least in the

sense of some meaningfulness being conveyed?

Before answering this question, let us look at the

remainder of his sentence for a clue.

Suppose you have a line divided into two
unequal parts, to represent the visible and
intelligible orders, and then divide the two
parts again in the same ratio [logos] .
in terms of comparative clarity and
obscurity. (Republic 509d)

We have already established the Pythagorean emphasis of

mathematics in Plato. Up to now not much has been said

about the emphasis on ratio (logos) and proportion

(analogia). But as we proceed, it will become clear that

these notions are critical in Plato's philosophy. Ratio is

the relation of one number to another, for example 1:2.

However, proportion requires a repeating ratio that

involves four terms, for example, 1:2::4:8. Here the ratio

of 1:2 has been repeated in 4:8. Thus, in proportion

(analogia) we have a repeating ratio with four terms.

Standing between the two-termed ratio and the four-termed

proportion lies the three-termed mean.











The relations among ratio, mean, and
proportion can be brought out by
distinguishing two kinds of mean, the
arithmetic and the geometric. In an
aritmetic mean, the first term is exceeded by
the second by the same amount that the second
is exceeded by the third. One, two, and
three form an arithmetic mean. One, two, and
four, however, form a geometric mean. In a
geometric mean, the first term stands to the
second in the same ratio that the second
stands to the third. Either mean may be
broken down into two ratios, namely, that of
the first and second terms and that of the
second and third. But the geometric mean
alone is defined by ratio, being the case in
which the two ratios are the same; all other
means have two different ratios. Of all the
means, therefore, only the geometric can be
expanded into a proportion, by repeating the
middle term. Furthermore all proportions in
which the middle terms are the same can
be reduced to geomtric means, by taking out
one of the two identical terms. (Des Jardins,
1976, p. 495)

An example of this relation of a geometric mean to

proportion will help to illustrate this. The ratio 1:2 and

the ratio 2:4 can be related so that we have a proportion.

Thus, 1:2::2:4 is our proportion repeating the same ratio.

However, because the middle terms are identical, each being

2, one of the middle 2's can be dropped to establish a

geometric mean: 1:2:4. Here 2 is the geometric mean

between I and 4. Whenever a case like this arises in which

a proportion contains two identical middle terms, we can say

we have the peculiar instance of a proportion which contains

only three "different" terms, even though it is true that

one of these terms (i.e., the middle) gets repeated. We

shall nevertheless refer to such a creature as a three-termed


geometric proportion.




Full Text
THE PYTHAGOREAN
A STUDY
PLATO AND THE GOLDEN SECTION:
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
1 983

Copyright 1983
By
Scott A. Olsen

This dissertation is respec tfuLly dedicated to two
excellent teachers, the late Henry Mehlberg, and Dan Pedoe.
Henry Mehlberg set an impeccable example in the quest for
knowledge. And Dan Pedoe instilled in me a love for the
ancient geometry.

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.
iv

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 43
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. 119
CHAPTER IV THE GOLDEN SECTION 124
Timaeus 124
Proportion 129
Taylor & Thompson on the Epinomis 134
$ and the Fibonacci Series 149
The Regular Solids 158
Conclusion 201
Notes 203
BIBLIOGRAPHY 205
BIOGRAPHICAL SKETCH 216
v

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:\Z!T Right-angled Isosceles Triangle.,..,..,.., 163
8 l:\JT: 2 Right-angled Scalene Triangle 163
9 Monadic Equilateral Triangle 165
10 Stylometric Datings of Plato’s Dialogues 166
11 Pentagon 175
12 Pentagon & 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
21 Pentalpha 184
22 Golden Cut 186
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
30 Double Square 195
31 Construction of Golden Rectangle 195
32 Golden Rectangle 195
33 Icosahedron with Intersecting Golden Rectangles.. 197
34 Dodecahedron with Intersection Golden Rectangles. 197

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 t 983
Chairperson: Dr. Ellen S. Haring
Cochairperson: Dr. Thomas W. 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
vii

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
mathematicals 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.
viii

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 Charles 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 epistemologically 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
1

2
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 hi3 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'3 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.
2
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,

3
and hints to a suitable explanatory hypothesis. This
abductive requirement is even more evident in the later
dialogues, especially the Theaetetus, Parmenides, and
q
Sophist. There, if one accepts the arguments on their
surface, it appears that Plato is attacking what he has
suggested earlier regarding knowledge and the Forms. 3ut
this is not the case.
A further perplexing problem for many scholars enters
the picture when one considers what Aristotle has to say
about Plato's unwritten teachings. It has led to the
mistaken view that Plato changed his philosophy radically
in later life. However, my contention is that a careful
reading of what Aristotle has to say upon the matter helps
to unfold the real underlying nature of the dialogues.
Mathematical concepts are present in one way or another
throughout the dialogues. Possibly obscure at the
beginning, they become central in the middle dialogues,
especially the Republic. And the later Philebus and
Bpinomis attest to the retention of the doctrine.
Copleston, who I am in agreement with on this matter,
summed up the position as follows:
There is indeed plenty of evidence that Plato
continued to occupy himself throughout his
years of academic and literary activity with
problems arising from the theory of Forms,
but there is no real evidence that he ever
radically changed his doctrine, still less
that he abandoned it altogether. ... It has
sometimes been asserted that the
mathematisation of the Forms, which is
ascribed to Plato by Aristotle, was a
doctrine of Plato's old age, a relapse into
Pythagorean "mysticism," but Aristotle does

4
not say that Plato changed his doctrine, and
the only reasonable conclusion to be drawn
from Aristotle's words would appear to be
that Plato held more or less the same
doctrine, at least during the time that
Aristotle worked under him in the Academy.
(Copleston, 1962, p. 188)
Others, like Cherniss, get around the problem by
accusing Aristotle of "misinterpreting and misrepresenting"
Plato (Cherniss, 1945, p. 25 ). This is ludicrous. We need
only recall that Aristotle was in the Academy with Plato
(until the master’s death) for 19 (possibly 20) years.
Surely he should know quite well what Plato had to say.
Fortunately we have some record of what Plato had to say in
his unwritten lectures. This helps to fill the gap. But
unfortunately the remnant fragments are sparse, though very
telling. The view of Cherniss' only indicates the extrema
to which some scholars will move in an attempt to overcome
the apparent disparity. As Copleston goes on to say,
. . . though Plato continued to maintain the
doctrine of Ideas, and though he sought to
clarify his meaning and the ontological and
logical implications of his thought, it does
not follow that we can always grasp what he
actually meant. It is greatly to be
regretted that we have no adequate record of
his lectures in the Academy, since this would
doubtless throw great light on the
interpretation of his theories as put forward
in the dialogues, besides conferring on us
the inestimable benefit of knowing what
Plato's "real" opinions were, the opinions
that he transmitted only through oral
teaching and never published. (Copleston,
1 962 , pp. 138-1 89)
My own view on the matter is that if we look closely
enough at the extant fragments, in conjunction with the
Pythagorean background of Plato's thought, and using these

5
as keys, we can unlock some of the underlying features of
Plato's dialogues. But this is premised on the assumption
that Plato is in fact being obstetric in the dialogues. I
will argue that this is the case, and that further,
sufficient clues are available to evolve an adequate
reconstruction of his mathematical-philosophical doctrine.
Thus, a central theme running throughout this
dissertation is that the words of Aristotle will help to
clarify the position of Plato. Rather than disregard
Aristotle's comments, I will emphasize them. In this way I
hope to accurately explicate some of the features of
Plato's mathematical doctrine and the method of discovery
by analysis that he employed. Plato deserves an even
richer foundation in the philosophies of science and logic
than he has heretofore been credited with. His method of
analysis, of the upward path of reasoning backwards from
conclusion to premises (or from facts to hypothesis or
principle), lies at the very roots of scientific discovery.
The mistaken view of a strictly bifurcated Platonic Cosmos,
with utter disdain for the sensible world, has done unjust
damage to the reputation of Plato among those in science.
This is unfortunate and needs to be remedied.
I set for myself the following problem at the outset.
When we arrive at the Timaeus we will see how the elements,
the regular solids, are said to be constructed out of two
kinds of right-angled triangles, one isosceles and the
other scalene. But he goes on to say,

6
These then ... we assume to be the original
elements of fire and other bodies, but the
priciples which are prior to to these deity
only knows, and he of men who is a friend of
deity. (Timaeus 53d-e )
I contend that this is a cryptic passage designed by
the midwife Plato to evoke in the reader a desire to search
for the underlying Pythagorean doctrine. To some it
conceals the doctrine. However, to others it is intended
to reveal, if only one is willing to reason backwards to
something more primitive. Thus Plato goes on to say,
. . . anyone who can point out a more
beautiful form than ours for the construction
of these bodies shall carry off the palm, not
as an enemy, but as a friend. Now, the one
which we maintain to be the most beautiful of
all the many triangles . . . is that of which
the double forms a third triangle which is
equilateral. The reason of this would be too
long to tell; he who disproves what we are
saying, and shows that we are mistaken, may
claim a friendly victory. (Timaeus 54a-b)
This is the problem: what more beautiful or primitive form
could there be for the construction of these bodies?
My views are undeniably in the vein of the
Neopythagorean and Neoplatonic traditions. But my
contention is that it is to the Pythagorean Neopiatonists
that we must turn if we are to truly understand Plato. I
have found a much greater degree of insight into Plato in
the Neopythagoreans and Neopiatonists than in the detailed
work of the logic choppers and word mongers. As Blavatsky
once said regarding one of the Neopiatonists, Thomas
Taylor, the English Platonist,

7
the answer given by one of Thomas Taylor's
admirers to those scholars who criticized his
translations of Plato [was]: "Taylor might
have known less Greek than his critics, but
he knew more Plato." (Blavatsky, 1971, vol.
2, p. 172)
As Flew has written, the origins of the Neoplatonic
interpretation
go back to Plato's own lifetime. Its
starting-point was Plato's contrast between
eternal Ideas and the transient objects of
sense, a contrast suggesting two lines of
speculative enquiry. First, what is the
connection, or is there anything to mediate
between intelligibles and sensibles, the
worlds of Being and of Becoming? Second, is
there any principle beyond the Ideas, or are
they the ultimate reality? (Flew, 1979, p.
254)
This dissertation speaks directly to the former question,
although, I will have something to say about the latter as
well.
Notes
Most subsequent Greek words will be transliterated.
Although I will occasionally give the word in the original
Greek. The golden section, tome, was often referred to by
the Greeks as division in mean and extreme ratio.
?
Cross-examination, or eienchus, is an important stage
in the dialectical ascent to knowledge. It is employed to
purge one of false beliefs. Through interrogation one is
led to the assertion of contradictory beliefs. This
method is decidedly socratic. Plato emphasized a more
cooperative effort with his students in the Academy.
3
All citations to works of Plato are according to the
convention of dialogue and passage number. All citations
are to H. Cairns and E. Hamilton, eds., 1971, The Collected
Dialogues of Plato, Princeton: Princeton University Press.
The one major exception is that all Republic quotes are
from D. Lee, transí., 1974, The Republic, London: Penquin.

CHAPTER II
ABDUCTION
Peirce
I choose to begin with Charles Sanders Peirce, because
better than anyone else he seems to have grasped the
significance of the logic of backwards reasoning, or
abduction. Once the position of Peirce is set out, with
some explicit examples, I will return to the examination of
Plato's philosophy.
What Peirce termed abduction (or alternatively,
reduction, retroduction, presumption, hypothesis, or novel
reasoning) is essentially a process of reasoning backwards
from an anomalous phenomenon to a hypothesis which would
adequately explain and predict the existence of the
phenomenon in question. It lies at the center of the
creative discovery process. Abduction occurs whenever our
observations lead to perplexity. Abduction is the initial
grasping at explanation. It is a process by which one
normalizes that which was previously anomalous or
surprising. Peirce's basic formula is very simple:
The surprising fact, C, is observed; But if A
were true, C would be a matter of course,
Hence, there is reason to suspect that A is
true. (Peirce 5*139)^
8

9
Abduction follows upon the initiation of a problem or
puzzling occurrence. The great positive feature about
*
abduction is that it can lead to very rapid solutions. On
this view discovery takes place through a series of leaps,
rather than a gradual series of developments. As Peirce
said, gradual progression
is not the way in which science mainly
progresses. It advances by leaps; and the
impulse for each leap is either some new
observational resource, or some novel way of
reasoning about the observations. Such novel
way of reasoning might, perhaps, be
considered as a new observational means,
since it draws attention to relations between
facts which should previously have been
passed by unperceived. (Buchler, 1955, p. 51)
Whenever present theories cannot adequately explain a
fact, then the door for abduction opens. Most major
scientific discoveries can be correctly viewed as an
abductive response to perplexing, or anomalous, phenomena.
Thus, Einstein was struck by certain perplexing, seemingly
unaccountable features of the world. Sometimes these
anomalous features cluster about a particular problem.
When this occurs, and is perceived by the individual, great
creative abductive solutions become possible. Thus, as
Kuhn points out,
Einstein wrote that before he had any
substitute for classical mechanics, he could
see the interrelation between the known
anomalies of black-body radiation, the
photoelectric effect and specific heats.
(Kuhn, 1970, p. 89)
Thus, on the basis of these perplexing facts, Einstein was

10
able to reason backwards to a suitable hypothesis that
would reconcile and adapt each of these puzzling features.
Abduction is the process by which surprising facts
invoke an explanatory hypothesis to account for them.
Thus, abduction "consists in studying facts and devising a
theory to explain them" (Peirce 5.145). It "consists in
examining a mass of facts and in allowing these facts to
suggest a theory" (Peirce 8.209). And it is "the logic by
which we get new ideas" (Peirce 7.98).
For example, Maslow was doing abduction when he
surveyed the data and observations, and, reasoning
backwards, inferred the hypothesis of self-actualization.
The fact that he called it "partly deductive," not knowing
the correct label, does not affect the nature of his
abduction.
I have published in another place a survey of
all the evidence that forces us in the
direction of a concept of healthy growth or
of self-actualizing tendencies. This is
partly deductive evidence in the sense of
pointing out that unless we postulate such a
concept, much of human behavior makes no
sense. This is on the same scientific
principle that led to the discovery of a
hitherto unseen planet that had to be there
in order to make sense of a lot of other
observed data. (Maslow, 1962, pp. 146-147)
Abduction is to be clearly distinguished from
deduction and induction. Nevertheless, the three logical
methods are mutually complementary. However, deduction
only follows upon the initial abductive grasping of the new
hypothesis. The deductive consequences or predictions are
then set out. Induction then consists of the experimental

11
testing and observation to see if in fact the consequences
deduced from the new hypothesis are correct. If not, then
abduction begins again seeking a new or modified hypothesis
which will more adequately explain and predict the nature
of our observations.
Thus, as Peirce points out,
abduction is the process of forming an
explanatory hypothesis. It is the only
logical operation which introduces any new
ideas; for induction does nothing but
determine a value, and deduction merely
evolves the necessary consequences of a pure
hypothesis. (Peirce 5.171)
Abduction does not have the nature of validity that,
for example, deduction possesses. Abduction is actually a
form of the so-called fallacy of affirming the consequent.
The abducted hypothesis cannot in any way be apprehended as
necessary. It must be viewed as a tentative conjecture,
and at best may be viewed as likely. When we say, the
surprising fact C is observed, but if A were true, C would
follow as a matter of course, the very most that we can do
is say that we therefore have reason to suspect A.
Abduction is a logical method of hypothesis selection, and
it is extremely effective, especially when anomalies are
used to guide one toward the best explanation. But
abductions may turn out to have false results.
The function of hypothesis [abduction] is to
substitute for a great series of predicates
forming no unity in themselves, a single one
(or small number) which involves them all,
together (perhaps) with an indefinite number
of others. It is, therefore, also a
reduction of a manifold to a unity. Every
deductive syllogism may be put into the form:

12
If A, then 3; But A: Therefore, B. And as
the minor premiss in this form appears as
antecedent or reason of a hypothetical
proposition, hypothetic inference [abduction]
may be called reasoning from consequent to
antecedent. (Peirce 5.276)
This notion of reasoning backwards from consequent to
hypothesis is central to abduction. One thing it shares in
common with induction is that it is "rowing up the current
of deductive sequence" (Peirce, 1968, p. 133). But
abduction and induction are to be clearly distinguished.
Abduction is the first step of explanatory discovery, the
grasping of the hypothesis or account. Induction is the
testing of the hypothesis that follows the previous
abductive hypothesis selection and the deductive prediction
of consequences. The operation of testing a
hypothesis by experiment, which consists in
remarking that, if it is true, observations
made under certain conditions ought to have
certain results, and noting the results, and
if they are favourable, extending a certian
confidence to the hypothesis, I call
induction. (Buchler, 1955, p* 152)
There is also a sense in which abduction and induction can
be contrasted as opposing methods.
The induction adds nothing. At the very most
it corrects the value of a ratio or slightly
modifies a hypothesis in a way which had
already been contemplated as possible.
Abduction, on the other hand, is merely
preparatory. It is the first step of
scientific reasoning, as induction is the
concluding step. They are the opposite poles
of reason. . . . The method of either is
the very reverse of the other'3. Abduction
3eeks a theory. Induction seeks the facts.
(Peirce 7.127-7.218)
The important point here is that abduction occurs in
the process of discovery. It is distinct from the later

13
process of justification of the hypothesis. Hanson made
this point strongly when he stated that
the salient distinction of "The Logic of
Discovery" consisted in separating (1)
Reasons for accepting a hypothesis, H, from
(2) Reasons for suggesting H in the first
place. (Hanson, I960, p. 183)
Abduction, the logic of discovery, underlies the latter
above. The robust anomaly R provides reasons to suspect
hypothesis H is true. Thi3 is the case, simply because if
H were true, then R would follow as a matter of course.
Hence we have reasons for suggesting or selecting H.
Abduction is the mark of the great theoretical
scientists. Through contemplation of the observables,
especially the puzzling observables, the theoretician
reasoning backwards fastens upon a hypothesis adequate to
explain and predict the occurrence of the anomalies. This,
in effect, defuses the anomalous nature of the observables,
having the effect of normalizing them.
A crucial feature of the activity of abduction is the
role that is played by the anomaly, R. It has the function
of directing one to the type of hypothesis that is
required. As Hanson pointed out, "to a marked degree [the]
observations locate the type of hypothesis which it will be
reasonable ultimately to propose" (Hanson, 1960, p. 185).
The overall interplay of abduction, deduction, and
induction can be appreciated more fully when considering
the following passage by Peirce:
The Deductions which we base upon the
hypothesis which has resulted from Abduction

14
produce conditional predictions concerning
our future experience. That is to say, we
infer by Deduction that if the hypothesis be
true, any future phenomena of certain
descriptions must present such and such
characters. We now institute a course of
quasi-experimentation in order to bring these
predictions to the test, and thus to form our
final estimate of the value of the
hypothesis, and this whole proceeding I term
Induction. (Peirce 7.H5, in.21)
Abduction, or the logic of discovery, has
unfortunately been ignored for some time. As Paul Weiss
says, "it is regrettable that the logicians are not yet
ready to follow Peirce into this most promising field
[abduction]" (Bernstein, 1965, p. 125). Only recently has
there been a rebirth of interest.
In the case of abduction, Peirce singles out
as an independent form of inference the
formulation of hypotheses for inductive
testing. All this is well known, but, we
fear, too much ignored outside the
constricted space of Peirce scholarship.
Unfortunately, the notion of abductive
inference, which is peculiarly Peirce's, has
not exerted an influence proportionate to the
significance of its insight. (Harris &
Hoover, 1980, p. 329)
The only point where Harris and Hoover err in the
prior statement is in attributing abduction as solely
belonging to Peirce. But this is a mistake. Peirce
himself acknowledged his Greek sources of the abductive
logic. I will argue that the roots of abduction lie in
Plato, and his work in the Academy, and his Pythagorean
predecessors. But first we will consider some of the more
recent developments of Peircean abduction, and then some
actual historical examples.

15
Consider the following example,
I catch the glint of light on metal through
the trees by the drive, remark that I see the
family car is there, and go on to infer my
son is home. It may be said that taken
literally I have misdescribed things. What I
see, it may be said, is a flash of light
through the trees. Strictly I infer, but do
not see, that the car is there. ... I
reason backward from what I see, the flash of
light on metal, and my seeing it, to a cause
the presence of which I believe to be
sufficient to explain my experience. Knowing
the situation, and knowing the way things
look in curcumstances like these, I infer
that the car is in the drive. (Clark, 1982,
pp. 1-2)
Clark goes on to describe the argument form involved
Let q be the puzzling perceptual occurrence or anomaly,
this case it was the glint of light passing through the
trees. Let p stand for the car is in the drive. Let B
stand for the belief that if p (the car is in the drive),
and other things being equal, then q (the glint of light)
would occur. We can then reconstruct the argument as
follows.
1. q (puzzling glint of light),
2. But B (belief that poq),
3. Therefore, p (car in drive).
I conclude from my premises, q and B, that
[hypothesis] p. I conclude that the family
car is there, this being the hypothesis I
draw the truth of which I believe is
sufficient to account for that puzzling
perceptual happening, q. (Clark, 1982, p. 2)
But if this is abduction, are we not simply employing the
fallacy of affirming the consequent? Or is it something
In
more ?

16
This pattern of reasoning is quite common.
And it is after all a sort of reasoning.
There is here a texture of structured
thoughts leading to a conclusion. Moreover,
there's something sensible about it. It is
not just silly. But of course reasoning this
way, I have sinned deductively. My reasoning
is not deductively valid. (q and B might
after all quite well be true and yet
[hypothesis p] false. Perhaps it is not in
fact the car but a visiting neighbor's camper
whose flash of light on metal I catch).
Peirce insisted that all creativity has its
source in sin: reasoning of this general
sort is the only creative form of inference.
It is the only sort that yields as
conclusions new hypotheses not covertly
asserted in the premises; new hypotheses now
to be tested and examined; hypotheses which
may determine whole new lines of inquiry.
This reasoning is, he thought, quite
ubiquitous, present indeed in all perception
but in nearly every area of contingent
inquiry as well. (it is philosophical
commonplace, too. How frequently we reason
backward from an epistemological puzzle to an
ontological posit.) Peirce, in
characterizing this backward, abductive,
reasoning which runs from effects to
hypotheses about causes sufficient to ensure
them, has implicitly answered the title
question. When is a fallacy valid? Answer:
When it is a good abduction. (Clark, 1982, p.
2)
Clark proceeds admirably, struggling with abduction,
attempting to define its formal standards for validity and
soundness. "It is . . . the need to characterize abductive
soundness which forces the nontrivial nature of abduction
on us"(ciark, 1982, p. 3). In the very process of this
attempt, Clark has reasoned abductively. In a very
analagous manner, this dissertation is an exercise in
abduction, reasoning backwards from the Platonic puzzles
(i.e., the dialogues and the extraneous statements

17
regarding Plato's doctrines), to an explanatory hypothesis
regarding them.
As an especial philosophical application and
final example, it is perhaps worth remarking
that this account of the nature of abduction
is itself an exercise in abduction. We have
reasoned backward from a puzzling fact--the
widespread employment in philosophical
inquiry of arguments which are deductively
fallaciou3--to an attempt to characterize an
adequate explanation of the phenomenon. We
have tried to sketch minimal formal standards
by which abductions can be evaluated as valid
or sound, and their employment justified. I
wish I could say more about what is important
about abduction and the competition of
sufficient hypotheses. I wish I could
formulate an articulate formal system of
abduction. But even a sketch like this is
something. It seems to me at least to
override an obvious competitor to explaining
our ubiquitous use of these forms of
inference; the view that these are just
logical lapses--irrational applications of
the fallacy of asserting the consequent.
(Clark, 1982, p. 12)
Eratosthenes & Kepler
The great discovery of Eratosthenes, the Librarian at
Alexandria, provides a good example of abduction. He
pondered over the puzzling fact that on the summer solstice
at noonday the sun was at its zenith directly overhead in
Syene, Egypt, and yet 500 miles north at that precise
moment in Alexandria, the sun was not directly at its
zenith. He abducted that this must be due to the curvature
of the earth away from the sun. He went further and
reasoned that he could determine the amount of curvature of
the earth through geometrical calculation by measuring the
length of shadow cast at Alexandria at noon on the summer

18
solstice. Knowing the distance from Syene to Alexandria,
he was then able to quite accurately (circa 2403.C.)
calculate the diameter and circumference of the earth.
Eratosthenes worked out the answer (in Greek
units), and, as nearly as we can judge, his
figures in our units came out at about 8,000
miles for the diameter and 25,000 miles for
the circumference of the earth. This, as it
happens, is just about right. (Asimov, 1975,
vo 1. 1 , p. 2 2)
In view of the perplexing difference in the position
of the sun in the two cities on the summer solstice,
Eratosthenes was able to reason backwards to a hypothesis,
i.e., the earth is round and therefore curves away from the
rays of the sun, which would render that anomalous
phenomenon the expected.
2
Kepler is another example of brilliant abductive
inferences. Both Peirce and Hanson revere the work of
Kepler. Hanson a3ks,
was Kepler's struggle up from Tycho's data to
the proposal of the elliptical orbit
hypothesis really inferential at all? He
wrote De Motibus Stellae Martis in order to
set out his reason for suggesting the
ellipse. These were not deductive reasons;
he was working from explicanda to explicans
[reasoning backwards]. But neither were they
induetive--not, at least, in any form
advocated by the empiricists, statisticians
and probability theorists who have written on
induction. (Hanson, 1972, p. 85)
The scientific process of discovery may at times be
viewed as a series of explanatory approximations to the
observed facts. An abductively conjectured hypothesis will
often approximate to an adequate explanation of the facts.
One continues to attempt to abduct a more complete

19
hypothesis which more adequately explains the recalcitrant
facts. Hence there will occassionally occur the unfolding
of a series of hypotheses. Each hypothesis presumably
approximates more closely to an adequate explanation of the
observed facts. This was the case with Kepler's work, De
Motibus Stellae Martis. As Peirce points out,
. . . at each stage of his long
investigation, Kepler has a theory which is
approximately true, since it approximately
satisfies the observations . . . and he
proceeds to modify this theory, after the
most careful and judicious reflection, in
such a way as to render it more rational or
closer to the observed fact. (Buchler, 1955,
p. 155)
Although abduction does involve an element of
guess-work, nevertheless, it does not proceed capriciously.
Never modifying his theory capriciously, but
always with a sound and rational motive for
just the modification he makes, it follows
that when he finally reaches a
modification--of most striking simplicity and
rationality--which exactly satisfies the
observations, it stands upon a totally
different logical footing from what it would
if it had been struck out at random.
(Buchler, 1955, p. 155)
Hence, there is method to abduction. Rather than referring
to it as a case of the fallacy of affirming the consequent,
it would be better to term it directed affirmation of the
consequent. The arrived at hypothesis will still be viewed
a a tentative. But as Peirce indicated there is a logical
- ^ ^ 3
form to it.
Abduction, although it is very little
hampered by logical rules, nevertheless is
logical inference, asserting its conclusion
only problematically, or conjecturally, it is

20
true, but nevertheless having a perfectly
definite logical form. (Peirce 5*188)
A crucial feature of abduction is that it is originary
in the sense of starting a new idea. It inclines, rather
than compels, one toward a new hypothesis.
At a certain stage of Kepler's eternal
exemplar of scientific reasoning, he found
that the observed longitudes of Mars, which
he had long tried in vain to get fitted with
an orbit, were (within the possible limits of
error of the observations) such as they would
be if Mars moved in an ellipse. The facts
were thus, in so far, a likeness of those of
motion in an elliptic orbit. Kepler did not
conclude from this that the orbit really was
an ellipse; but it did incline him to that
idea so much as to decide him to undertake to
ascertain whether virtual predictions about
the latitudes and parallaxes based on this
hypothesis would be verified or not. This
probational adoption of the hypothesis was an
abduction. An abduction is Originary in
respect of being the only kind of argument
which starts a new idea. (Buchler, 1955, p.
1 56)
A very simple way of expressing the anomalous orbit of Mars
and the resulting abductive hypothesis is indicated by
Hanson. It is relevant to note that it begins with an
interrogation. "Why does Mars appear to accelerate at 90
[degrees] and 270 [degrees]? Because its orbit is
elliptical" (Hanson, 1972, p. 87). Again putting the
formula into its simplest form, we may say: the surprising
fact R is observed, but what hypothesis H could be true
that would make R follow as a matter of course? It is the
upward reach for H that is fundamental to the notion of
abduc tion.

21
Apagoge
How then is abduction related to Plato? The initial
clue is given in a statement by Peirce.
There are in science three fundamentally
different kinds of reasoning, Deduction
(called by Aristotle sunagoge or anagoge),
Induction (Aristotle’s and Plato's epagoge)
and Retroduction [abduction] (Aristotle's
apagoge). (Peirce 1.65)
Apagoge is defined as "I. a leading or dragging away.
II. a taking home. III. payment of tribute. IV. as a
law-term, a bringing before the magistrate" (Liddell and
Scott, 1972, p. 76). There is thus the underlying notion
of moving away from, or a return or reversion of direction.
Peirce’s reference is to Aristotle's use of the term,
apagoge. It is generally translated as reduction.
By reduction we mean an argument in which the
first term clearly belongs to the middle, but
the relation of the middle to the last term
is uncertain though equally or more probable
than the conclusion; or again an argument in
which the terms intermediate between the last
term and the middle are few. For in any of
these cases it turns out that we approach
more nearly to knowledge. For example let A
stand for what can be taught, B for
knowledge, C for justice. Now it is clear
that knowledge can be taught [ab]: but it is
uncertain whether virtue is knowledge [_ BC ].
If now the statement BC [virtue is knowledge]
is equally or more probable than AC [virtue
can be taught], we have a reduction: for we
are nearer to knowledge, since we have taken
a new term [B which gives premises AB and 3C,
on which the inquiry now turns], being so far
without knowledge that A [what can be taught]
belongs to C [virtue].^(Prior Analytics
6 9a2 0-30)
On this view then, reduction is the grasping of a new term
which transforms the inquiry onto a new footing. According

22
to Aristotle we are nearer knowledge because by reducing
the problem to something simpler, we are closer to solving
it. By solving the new reduced problem, the solution to
the original problem will follow.
The evidence is that the Aristotelian terra apagoge has
its roots in geometrical reduction. Thus Proclus says:
Reduction is a transition from one problem or
theorem to another, the solution or proof of
which makes that which is propounded manifest
also. For example, after the doubling of the
cube had been investigated, they transformed
the investigation into another upon which it
follows, namely the finding of two means; and
from that time forward they inquired how
between two given straight lines two mean
proportionals could be discovered. And they
say that the first to effect the reduction of
difficult constructions was Hippocrates of
Chios, who also squared a lune and discovered
many other things in geometry, being second
to none in ingenuity as regards
constructions. (Heath, 1956, vol. 1, p. 155)
Thus, we see the basic movement as later described by
Peirce, in which a problem is solved or an anomaly
explained, by the backwards reasoning movement to a
hypothesis from which the anomalous phenomenon or solution
would follow as a matter of course. The difference here is
that in reduction there is an initial step toward arriving
at a hypothesis from which the phenomenon or solution would
follow, but the hypothesis is such that it still must be
established. However, by selecting the hypothesis one has
succeeded in reducing the problem to another, but simpler,
problem. Hence, the Delian problem of doubling the cube
was reduced to the problem of finding two mean
proportionals between two given straight lines. As we

23
shall see subsequently, Archytas performed the initial step
of reduction, and Eudoxus performed the final step of
solution.
In a footnote to the Proclus passage above, Heath
makes the following relevant remarks:
This passage has frequently been taken as
crediting Hippocrates with the discovery of
the method of geomtrical reduction. ... As
Tannery remarks, if the particular reduction
of the duplication problem to that to the two
means is the first noted in history, it is
difficult to suppose that it was really the
first; for Hippocrates must have found
instances of it in the Pythagorean geometry.
. . . but, when Proclus speaks vaguely of
"difficult constructions," he probably means
to say simply that "this first recorded
instance of a reduction of a difficult
construction is attributed to Hippocrates."
(Heath, 1956, vol.1, pp. 135-136)
This suggests that the real source of reduction or apagoge
is the Pythagoreans. I will return to this point later.
It is also interesting to note that in the Proclus
quotation above there is reference to the squaring of
lunes. Aristotle, in the Prior Analytics passage cited
above, goes on to refer to the squaring of the circle with
the aid of lunules.
Or again suppose that the terms intermediate
between B [knowledge] and C [virtue] are few:
for thus too we are nearer knowledge. For
example let D stand for squaring, E for
rectilinear figure, F for circle. If there
were only one term intermediate between E
[squaring] and F [circle] (viz. that the
circle made equal to a rectilinear figure by
the help of lunules), we should be near to
knowledge. But when BC [virtue is knowledge]
is not more probable than AC [virtue can be
taught], and the intermediate terms are not
few, I do not call this reduction: nor again
when the statement BC [virtue is knowledge]

24
i3 immediate: for such a statement is
knowledge. (Prior Analytics 69a30-37)
My own view is that reduction as expressed by
Aristotle is really a special limiting case of what Peirce
termed abduction. There is a more general model of the
abductive process available amongst the Greeks. And
further, reduction does not precisely fit the basic formula
Peirce has presented.
The surprising fact, C, is observed; But if A
were true, C would be a matter of course,
Hence, there is reason to suspect that A is
true. (Peirce 5-5189)
Seduction appears to be a species of this formula.
However, there appears to be a more apt generic concept
available amongst the Greeks. This I contend is the
ancient method of analysis. In the end reduction may be
seen to be closely allied to analysis. But successful
analysis or abduction requires the discovery of an adequate
hypothesis. It is possible that this is achieved through a
series of reductions.
In reference to the discovery of lemmas, Proclus says,
. . . certain methods have been handed down.
The finest is the method which by means of
analysis carries the thing sought up to an
acknowledged principle, a method which Plato,
as they say, communicated to Leodamas, and by
which the latter, too, is said to have
discovered many things in geometry. (Heath,
1 956, vol. 1 , p. 134)
Heath, in some insightful remarks, sees this analysis
as similar to the dialectician's method of ascent. Thus he
say s:
This passage and another from Diogenes

25
Laertius to the effect that "He [Plato]
explained (eisegasato) to Leodamos of Thasos
the method of inquiry by analysis" have been
commonly understood as ascribing to Plato the
invention of the method of analysis; but
Tannery points out forcibly how difficult it
is to explain in what Plato'3 discovery could
have consisted if analysis be taken in the
sense attributed to it in Pappus, where we
can see no more than a series of successive,
reductions of a problem until it is finally
reduced to a known problem. On the other
hand, Proclus' words about carrying up the
thing sought to an "acknowledged principle"
suggest that what he had in mind was the
process described at the end of Book VI of
the Republic by which the dialectician
(unlike the mathematician) uses hypotheses as
stepping-stones up to a principle which is
not hypothetical, and then is able to descend
step by step verifying every one of the
hypotheses by which he ascended. (Heath,
1956 , vo 1. 1 , p. 134, fn.1 )
There is both some insight and some glossing over by.
Heath here. Heath is correct that there is a definite
relation here between analysis and what Plato describes as
the upward path in Book VI of the Republic. But he is
mistaken when he tries to divorce Platonic analysis from
mathematical analysis. They are very closely related.
Part of the confusion stems from the fact that in the
Republic Plato distinguishes the mathematician's acceptance
of hypotheses and subsequent deductions flowing from them,
from the hypotheses shattering upward ascent of the
dialectician. However, the mathematician also employs the
dialectical procedure when he employs reduction and
mathematical analysis. In these instances, unlike his
deductive descent, the mathematician reasons backwards (or
upwards) to other hypotheses, from the truth of which the

26
solution of his original problem will follow. This basic
process is common to both mathematician and dialectician.
The common denominator is the process of reasoning
backwa rds.
A further problem resulting in the confusion is that
it is not clear what is meant by references to an ancient
method of analysis. Heath is perplexed as well. Thus he
writes,
It will be seen from the note on Eucl. XIII.
1 that the MSS. of the Elements contain
definitions of Analysis and Synthesis
followed by alternative proofs of XIII. 1-5
after that method. The definitions and
alternative proofs are interpolated, but they
have great historical interest because of the
possibility that they represent an ancient
method of dealing with propositions, anterior
to Euclid. The propositions give properties
of a line cut "in extreme and mean ratio,"
and they are preliminary to the construction
and comparison of the five regular solids.
Now Pappus, in the section of hi3 Collection
[Treasury of Analysis! dealing with the
latter subject, says that he will give the
comparisons between the five figures, the
pyramid, cube, octahedron, dodecahedron and
icosahedron, which have equal surfaces, "not
by means of the so-called analytical inquiry,
by which some of the ancients worked out the
proofs, but by the synthetical method." The
conjecture of Bretschneider that the matter
interpolated in Eucl. XIII i3 a survival of
investigations due to Eudoxus has at first
sight much to commend it. In the first
place, we are told by Proclus that Eudoxus
"greatly added to the number of the theorems
which Plato originated regarding the section,
and employed in them the method of analysis."
(Heath, 1 956, vol. 1 , p. 137)
This is an extremely interesting passage. Is this the
same method of analysis that was earlier attributed to the
discovery of Plato? However, if the method is ancient,

27
then at best Plato could only have discovered it in the
work of hi3 predecessors, presumably the Pythagoreans. And
what about "the section" (tome), and the theorems that
Plato originated (and Eudoxus extended) regarding it?
It is obvious that "the section" was some
particular section which by the time of Plato
had assumed great importance; and the one
section of which this can safely be said is
that which was called the "golden section,"
namely the division of a straight line in
extreme and mean ratio which appears in Eucl.
II. 11 and is therefore most probably
Pythagorean. (Heath, 1956, vol. 1, p. 137)
If Plato had done so much work on this Pythagorean
subject, the golden section, and further, his pupil Eudoxus
was busy developing theorems regarding it, and further,
both were using a method of analysis that may have ancient
Pythagorean origins as well, then why is there no
straightforward mention of this in the dialogues? Could
the actual practice of what was occurring within the
Academy have been so far removed from what is in the
dialogues? Why was there such a discrepancy between
practice and dialogue? These are some of the questions
that will be answered in the course of this dissertation.
Focussing upon the question of analysis for the
moment, there are interpolated definitions of analysis and
synthesis in Book XIII of Euclid's Elements. Regarding the
language employed, Heath says that it "is by no means clear
and has, at the best, to be filled out" (Heath, 1956, vol.
1 , p. 133).
Analysis is an assumption of that which is
sought as if it were admitted [and the

28
fassage] through its consequences
antecedents] to something admitted (to be)
true. Synthesis is an assumption of that
which is admitted [and the passage] through
its consequences to the finishing or
attainment of what is sought. (Heath, 1956,
vo i. 1 , p . 138)
Unfortunately this passage is quite obscure.
Fortunately Pappus has preserved a fuller account.
However, it too is a difficult passage. One might even
speculate as to whether these passages have been
purposefully distorted.
The so-called Treasury of Analysis is, to put
it shortly, a special body o’? doctrine
provided for the use of those who, after
finishing the ordinary Elements [i.e.,
Euclid's], are desirous of acquiring the
power of solving problems which may be set
them involving (the construction of) lines .
. . and proceeds by way of analysis and
synthesis. Analysis then takes that which is
sought as if it were admitted and passes from
it through its successive consequences^
[antecedents] to something which is admitted
as the result of synthesis: for in analysis
we assume that which is sought as if it were
(already) done (gegonos), and we inquire what
it is from which this results, and again what
is the antecedent cause of the latter, and so
on, until by so retracing our steps we come
upon something already known or belonging to
the class of first principles, and such a
method we call analysis as being solution
backwards (anapalin iusin). (Heath, 1956,
vo1. 1 , p. 138)
Thomas translates the last two words of the former
passage, anapalin lusin, as "reverse solution'^ (Thomas,
1957, vol. 2, p. 597). It is this "solution backwards
"reverse solution" that I contend lay at the center of
Plato's dialectical method.
o r

29
Cornford is one of the few commentators to have any
real insight into the passage from Pappus.
. . . modern historians of
mathematics--"careful studies” by Hankel,
Duhamel, and Zeuthen, and others by
Ofterdinger and Cantor--have made nonsense of
much of it by misunderstanding the phrase, . »
"the succession of sequent steps" (T&V a.Kokov&'dV)
as meaning logical "consequences," as if it
wereTa O’UH^o^t.voVTa. . Some may have been
misled by Gerhardt (Pappus, vii, viii, Halle,
1871)» who renders it "Folgerungen." They
have been at great pains to show how the
premisses of a demonstration can be the
consequences of the conclusion. The whole is
clear when we see--what Pappus says--that the
same sequence of steps is followed in both
processes—upwards in Analysis, from the
consequence to premisses implied in that
consequence, and downwards in synthesis, when
the steps are reversed to frame the theorem
or demonstrate the construction "in the
natural (logical) order." You cannot follow
the same series of steps first one way, then
the opposite way, and arrive at logical
consequences in both directions.( And Pappus
never said you could. He added to
indicate that the steps "follow in succession
"but are not, as ¿VCoAoudoC alone would
suggest, logically "consequent" in the upward
direction. (Cornford, 1965, p. 72, fn.1)
On the "abduction" view I am maintaining, Cornford has
hit upon an acceptable interpretation of the Pappus
passage. It is this reverse inference from conclusion to
premise that was at the center of Plato’s method of
discovery. As Cornford goes on to say,
Plato realized that the mind must possess the
power of talcing a step or leap upwards from
the conclusion to the premiss implied in it.
The prior truth cannot, of course, be deduced
or proved(/from the conclusion; it must be
grasped (Alf>A of analytical penetration. Such an act is
involved in the solution "by way of

30
hypothesis" at Meno 86. . . . The geometer
directly perceives, without discursive
argument, that a prior condition must be
satisfied if the desired construction is to
follow. (Cornford, 1965, p. 67)
3ut Cornford is not without opposition to his account.
Robinson takes direct issue with him on the matter,
claiming that all historians of Greek mathematics agree
with the non-Cornford interpretation.
The historians of Greek mathematics are at
one about the method that the Greek geometers
called analysis. Professor Cornford,
however, has recently rejected their account
and offered a new one. . . . Professor
Cornford is mistaken and the usual view
correct. (Robinson, 1969, p. 1)
But if this is true, then why is there such a mystery
around the interpretation of the Pappus passage? And why
the mystery surrounding what was meant by Plato's discovery
of analysis? My view is that Cornford has gone far in
uncovering part of an old mystery about analysis. The
Cornford interpretation clearly lends support to the
abduction view of Plato's method that I am advocating.
Actually, the medieval philosopher John Scotus
Eriugena captured some of the underlying meaning of
analysis when he distinguished the upward and downward
movements of dialectic. In the Dialectic of Nature, he
refers to this dual aspect of dialectic
which divides genera into species and
resolves species into genera once more. . . .
There i3 no rational division . . . which
cannot be retraced through the same set of
steps by which unity was diversified until
one arrives again at that initial unit which
remains inseperable in itself. . . . Analytic
comes from the verb analyo meaning "I return"

31
or "I am dissolved.” From this the term
analysis is derived. It too can be
translated "dissolution" or "return," but
properly speaking, analysis refers to the
solution of questions that have been
proposed, whereas analytic refers to the
retracing of the divisions of forms back to
the source of their division. For all
division, which was called "merismos" by the
Greeks', can be viewed as a downward descent
from a certain definite unit to an indefinite
number of things, that is, it proceeds from
the most general towards the most special.
But all recollecting, as it were is a return
again and this begins from the most special
and moves towards the most general.
Consequently, there is a "return" or
"resolution" of individuals into forms, forms
into genera. . . . (Whippel & Volter, 1969,
pp. 116-117)
Now the earlier remark cited by Heath (supra, pp.24-25),
though missing the mark, may be insightful as to what
analysis is.
Tannery points out forcibly how difficult it
is to explain in what Plato's discovery could
have consisted if analysis be taken in the
sense attributed to it in Pappus, where we
see no more than a series of successive,
reductions of a problem until it is finally
reduced to a known problem. (Heath, 1956,
vol. 1, p. 134, fn.1)
But this "series of reductions" may be fundamentally what
was involved. Knowledge would be arrived at through a
series of apagoges. Plato may have discovered unique uses
of analysis, that extended beyond his predecessors. It may
be that he discovered that analysis, or backward reasoning,
may apply to propositions other than mathematical/ On the
other hand, he may have simply "discovered" this more
esoteric technique in the ancient geometrical tradition of
the Pythagoreans. However, its true significance should be

32
considered within the context in which it is openly stated
it was used. That is, in particular it should be
considered in terms of what is said about Plato and Eudoxus
as to the theorems regarding the section, and their
discovery by analysis.
As Cantor points out, Eudoxus was the founder
of the theory of proportions in the form in
which we find it in Euclid V., VI., and it
was no doubt through meeting, in the course
of his investigations, with proportions not
expressible by whole numbers that he came to
realise the necessity for a new theory of
proportions which should be applicable to
incommensurable as well as commensurable
magnitudes. The "golden section" would
furnish such a case. And it is even
mentioned by Proclus in this connexion. He
is explaining that it is only in aritmetic
that all quantities bear "rational" ratios
(ratos logos) to one another, while in
geometry there are "irrational" ones
(arratos) as well. "Theorems about sections
like those in Euclid's second Book are common
to both [arithmetic and geometry] except that
in which the straight line is cut in extreme
and mean ratio. (Heath, 1956, vol. I, p. 137)
This mention of the golden section in conjunction with
analysis provides some clues, and foreshadows some of the
argument to come.
Dialectic
It i3 difficult to give a satisfactory account of the
views of both Aristotle and Plato regarding dialectic.
Whereas Aristotle refers to dialectic as less than
philosophy, Plato contends that it is the highest method
available to philosophy. In the end, however, their
methods are essentially the same, and dialectic can be
viewed as having various stages. At the bottom level

33
dialectic is used as a means of refutation. At the top
level it is a means for acquiring knowledge of real
essences.
Plato openly espouses dialectic as the finest tool
available in the acquisition of knowledge. "Dialectic is
the coping-stone that tops our educational system"
(Republic 534e).
It is a method quite easy to indicate, but
very far from easy to employ. It is indeed
the instrument through which every discovery
ever made in the sphere of arts and sciences
has been brought to light. . . . [it] is a
gift of the gods . . . and it was through
Prometheus, or one like him [Pythagoras],
that it reached mankind, together with a fire
exceeding bright. (Philebus 16c)
Dialectic is the greatest of knowledges (Philebus 57e-58a).
Through dialectic
we must train ourselves to give and to
understand a rational account of every
existent thing. For the existents which
have no visible embodiment, the existents
which are of highest value and chief
importance [Forms], are demonstrable only by
reason and are not to be apprehended by any
other means. (S tatesman 286a)
Dialectic is the prime test of a man and is to be
studied by the astronomers (Epinomis 991c). A dialectician
is one who can "discern an objective unity and plurality"
(Phaedrus 266b). It is the dialectician "who can take
account of the essential nature of each thing" (Republic
534b). Dialectic leads one to the vision of the Good.
When one tries to get at what each thing i3
in itself by the exercise of dialectic,
relying on reason without any aid from the
sense, and refuses to give up until one has
grasped by pure thought what the good is in

34
itself, one is at the summit of the
intellectual realm, as the man who looked at
the sun was of the visual realm. . . . And
isn't this progess what we call dialectic?
(Republic 532a-b)
Dialectic "sets out systematically to determine what
each thing essentially is in itself" (Republic 533b). For
Plato, it "is the only procedure which proceeds by the
destruction of assumptions to the very first principle, 30
as to give itself a firm base" (Republic 533c-d). And
finally, in the art of dialectic,
the dialectician selects a soul of the right
type, and in it he plants and sows his words
founded on knowledge, words which can defend
both themselves and him who planted them,
words which instead of remaining barren
contain a 3eed whence new words grow up in
new characters, whereby the seed is
vouchsafed immortality, and its possessor the
fullest measure of blessedness that man can
attain unto. (Phaedrus 276e-277a)
Thus, Plato indicates that dialectic is the highest
tool of philosophy. Aristotle, on the other hand, appears
to have a more mundane account. His passages in the Topics
seem to indicate that dialectic is a tool for students
involved in disputation. In fact, both the Topics and
Sophistical Refutations give the impression of introductory
logic texts. His references to dialectic give the
appearance that dialectic does not ascend to the level of
philosophy. Thus, Aristotle says,
sophistic and dialectic turn on the same
class of things as philosophy, but this
differs from dialectic in the nature of the
faculty required and from sophistic in

35
respect of the purpose of philosophic life.
Dialectic is merely critical where philosophy
claims to know, and sophistic is what appears
to be philosophy but is not. (Metaphysics
1004b22-27)
However, these considerations are somewhat misleading.
Aristotle's own works seem to be much more in the line of
dialectical procedure that Plato has referred to. Mayer
has brought this point home forcefully through an
examination of Aristotle's arguments in Metaphysics, Book
IV. Referring to these arguments Mayer says,
One would hardly expect the argument that
Aristotle employs in the sections of
Metaphysics IV. . . to be dialectical. And
it is true he does not call it dialectical,
but rather a kind of "negative
demonstration." [i.e., at Metaphysics
1006a12] Yet . . . [upon examination] the
arguments do appear to be dialectical.
(Mayer, 1978, p. 24)
Mayer has made a very useful classification of
dialectic into 3 types and their corresponding uses.
I.Eristic dialectic uses confusion and equivocation to
create the illusion of contradiction. II. Pedagogic
dialectic is used for refutation and the practice of
purification, as it leads to contradiction. III. Clarific
dialectic uses criticism, revision, discovery, and
clarification to dispel contradiction (Mayer, 1978, p. 1).
The eristic type would be that used by a sophist. The
pedagogic type is that seen in the early Socratic dialogues
where the respondent is subjected to cross examination
(elenchus). The third and highest type, clarific, is
closer to the level of dialectic that Plato so reveres.

36
What Aristotle has done is
limit the usage of the term "dialectic" to
the critical [pedagogic] phase only, i.e. he
sees its purpose as entirely negative, and
the philosopher must go beyond this to "treat
of things according to their truth". ... By
limiting "dialectic" to the negative, or
pedagogic, phase of . . . dialectic,
Aristotle is merely changing terminology, not
method. Clarific dialectic, for Aristotle,
is (or is part of ) philosophical method.
(Mayer, 1978, p. 1)
On this view, which I share with Mayer, Plato and
Aristotle do not really differ in method. The difference
is merely terminological, not substantive. This position
is even more strongly supported when one considers
Aristotle’s response to the "eristic argument" or Meno's
paradox. There Plato writes,
A man cannot try to discover either what he
knows or what he does not know. He would not
seek what he knows, for since he knows it
there is no need of the inquiry, nor what he
does not know, for in that case he does not
even know what he is to look for. (Meno 80e)
Aristotle's response is to establish a halfway house ^
between not knowing and knowing in the full sense. The
crucial distinction is between the weak claim of knowing
"that" something is, and the strong claim of knowing "what
something is. The former is simply to know or acknowledge
the existence of a kind. But to know "what" something is,
is to know the real essence of the thing. Of course this
latter knowledge is the goal of dialectic according to
Plato. The solution lies in the distinction Aristotle
draws between nominal and real essences (or definitions).
Thus, Aristotle argues that a nominal definition is a

37
statement of the meaning of a term, that is, meaning in the
sense of empirical attributes which appear to attach to the
thing referred to by the term. An example he uses of a
nominal essence (or definition), is the case of thunder as
"a sort of noise in the clouds" (Posterior Analytics
93b8-J4). Another nominal definition is that of a lunar
eclipse as a kind of privation of the moon's light.
A real definition (or essence), on the other hand, is
a "formula exhibiting the cause of a thing's existence"
(Posterior Analytics 93b39)* Por both Plato and Aristotle
we ultimately must seek to know the real definition to
fully know the essence of the "kind." That is to say, to
have scientific knowledge of a thing one must ascend to
knowledge of its real essence. However, one must begin
with the former, the nominal essence. This is nothing more
than to acknowledge the fact that a kind of thing exists,
through an enumeration of its defining attributes. The
•setting forth of a nominal essence presupposes the
existence of actual samples (or instances, events) in the
world answering to the description contained in the
definition. In other words it is not just a matter of
knowing or not knowing. We begin by knowing "that"
something exists. Then we proceed to seek to discover its
real essence, "what" it is. The real essence of, for
example, lunar eclipses (as correctly set forth by
Aristotle) is the interposition of the earth between the
sun and the moon. It is this "interposition" which is the

38
real essence which gives rise to (i.e., causes) that
phenomenon we have described in our nominal definition as a
lunar eclipse.
The important point here is that whether one adopts
the Platonic notion of knowledge through reminiscence or
the Aristotelian distinction between nominal and real
essences, in both cases there is an analytic ascent to the
real essence. Thus, Plato proceeds by dialectic to the
real essences and first principles. Likewise, Aristotle
seeks the real defining essence through an analytic ascent
from things more knowable to us to things more knowable in
themselves, that is, from the nominal to the real essence.
Aristotle may use different terminology, but his method has
its roots in the Academy. At one point Aristotle makes it
clear that he sees this underlying ascent to essences and
first principles in dialectic. "Dialectic is a process of
criticism wherein lies the path to the principles of all
inquiries"^(Topics 101b4). Elsewhere Aristotle makes it
manifestly clear that he appreciated Plato's thought on the
upward path of analysis to real essences and the first
principles.
Let us not fail to notice, however, that
there is a difference between arguments from
and those to the first principles. For
Plato, too, was right in raising this
question and asking, as he used to do, "are
we on the wav from or to the first
principles?Nicomachean Ethics 1 095a30-35)•
On the other hand, the beginning stages of dialectic occur
for Plato when one attempts to tether a true belief to its

39
higher level cause through the giving of an account
(logos). Only when one reaches the highest level of noesis
is there no reflection upon sensible things.
Heno and Theaetetus
In the Theaetetus, Plato acts as midwife to his
readers, just as, Socrates acts as midwife to Theaetetus.
I believe the dialogue is consistent with the views
unfolded in the Meno (a=s well as the Phaedo and Republic).
In the Meno, Socrates indicates to Meno that true beliefs
are like the statues of Daedalus, "they too, if no one ties
them down, run away and escape. If tied, they stay where
they are put" (Meno 97d). Socrates then goes on to explain
how it is the tether which transmutes mere true belief into
knowledge.
If you have one of his works untethered, it
is not worth much; it gives you the slip like
a runaway slave. But a tethered specimen is
very valuable, for they are magnificent
creations. And that, I may say, has a
bearing on the matter of true opinions. True
opinions are a fine thing and do all sorts of
good so long as they stay in their place, but
they will not stay long. They run away from
a man's mind; so they are not worth much
until you tether them by working out the
reason. That process, my dear Meno, is
recollection, as we agreed earlier. Once
they are tied down, they become knowledge,
and are stable. That is why knowledge is
something more valuable than right opinion.
What distinguishes the one from the other is
the tether. (Meno 97e-98a)
It is this providing of a tether or causal reason that
assimilates a true belief of the realm of pistis, to
knowledge of the realm of dianoia. The tether is arrived

40
at by an upward grasp of the causal objects of the next
higher ontological and epistemological level. My view is
that the Phaedo and Republic expand upon this upward climb
using the method of hypothesis on the way. The Republic
goes beyond the tentative stopping points of the Phaedo.
maintaining that knowledge in the highest sense can only be
attained by arriving at the unhypo thetical first principle,
the Good. Nevertheless, in the Phaedo, the basic method of
tethering by reasoning backwards to a higher premise is
maintained.
When you had to substantiate the hypothesis
itself, you would proceed in the same way,
assuming whatever more ultimate hypothesis
commended itself most to you, until you
reached the one which was satisfactory.
(Phaedo 101d-e)
This method is, of course, depicted by Socrates as
being second best. The best method which continues until
arrival at the Good, or One, is depicted in the Republic.
The Theaetetus then is a critique of the relative level of
knowledge arrived at in the state of mind of dianoia. The
question is asked as to what is an adequate logos (or
tether). Thus as Stenzel says,
. . . even \oyos is included in the general
skepticism. The word may indicate three
things: (a) speech or vocal expression, as
contrasted with the inner speech of the mind,
(b) the complete description of a thing by an
enumeration of its elements, (c) the
definition of a thing by discovery of its
distinctive na tu re , K.C a.$opc>Tty£ . The third of
these meanings seems at first to promise a
positive criterion of knowledge. But,look t
more closely: it is ,not ,the thing's Á¿?poT¡qS,
but knowledge of i t s w h i c h will
constitute knowledge; and this is circular.
(Stenzel, 1940, pp. xv-xvi)

41
This has, in general, been taken as an indication of
the Theaetetus' negative ending. However, it appears to me
to be another case of Plato's obstetric method. Sven if
one arrives at a satsifactory account or logos, and hence,
arrives at a tether, it does not follow that there would be
absolute certainty. "If we are ignorant of it [the Good or
One] the rest of our knowledge, however perfect, can be of
no benefit to us" (Republic 505a). •
Thus, in the Theaetetus, Socrates indicates that, "One
who holds opinions which are not true, will think falsely
no matter the state of dianoias" (Theaetetus 188d). But
this is a critique only relative to certainty. To arrive
at tethered true opinions at the state of mind of dianoia
is, nevertheless, a kind of knowledge, albeit less than the
highest kind of knowledge of the Republic. On my view
Haring has moved in the correct direction of interpretation
when she writes:
[the Theaetetus has] ... a partly
successful ending. The latter has to be
discovered by readers. However the dialogue
itself licenses and encourages active
interpretation. The last ten pages of
discourse contain so many specific clues that
the text can be read as a single development
ending in an affirmative conclusion. There
is indeed a way to construe "true opinion
with logos" so it applies to a cognition
worthy of "episteme." (Haring, 1982, p. 510)
As stated in the previous section (supra pp. 36-38 ),
dialectic is a movement from nominal essences to real
essences. Prom the fact that a particular "kind" exists,
one reasons backwards to the real essence, causal

42
explanation, or tether of that kind. From the nominal
existence of surds, Theaetetus reasons backwards to the
grounds or causal tether. By doing so, Theaetetus arrives
at (or certainly approaches) a real definition. The
relevant passage in the Theaetetus begins:
Theaetetus: Theodorus here was proving to us
something about square roots, namely, that
the sides [or roots] of squares representing
three square feet and five square feet are
not commensurable in length with the line
representing one foot, and he went on in this
way, taking all the separate cases up to the
root of seventeen square feet. There for
some reason he stopped. The idea occurred to
us, seeing that these square roots were
evidently infinite in number, to try to
arrive at a single collective term by which
we could designate all these roots. . . . We
divided number in general into two classes.
Any number whicjh is the product of a number
multiplied by itself we likened to a square
figure, and we called such a number "square"
or "equilateral."
Socrates: Well done!
Theaetetus: Any intermediate number, such as
three or five or any number that cannot be
obtained by multiplying a number by itself,
but has one factor either greater or less
than the other, so that the sides containing
the corresponding figure are always unequal,
we likened to the oblong figure, and we
called it an oblong number.
Socrates: Excellent. And what next?
Theaetetus: All the lines which form the
four equal sides of the plane figure
representing the equilateral number we
defined as length, while those which form the
sides of squares equal in area to the oblongs
we called roots [surds] as not being
commensurable with the others in length, but
only in the plane areas to which their
squares are equal. And there is another
distinction of the sane 3ort in the case of
solids. (Theaetetus 147d-148b)
What Theaetetus has managed to do is analogous to
Aristotle's discussion of lunar eclipses (supra p. 37).

43
Just as one reasons backwards to the real essence of lunar
eclipses (i.e., the interposition of the earth between the
sun and moon), so one reasons backwards to the real essence
of surds. They are not "commensurable with the others in
length [first dimension], but only in the plane areas
[second dimension] to which their squares are equal"
^Theaetetus 148b).
Theaetetus has successfully tethered the true belief
of Theodorus regarding surds, by working out the reason.
This is hinted at in Socrates' reply: "Nothing could be
better, my young friends; I am sure there will be no
prosecuting Theodorus for false witness" (Theaetetus 148b).
Socrates' statement, on the one hand, implies success, and,
on the other hand, suggests that Theodorus had initially
stated a true belief. Thus, there is the indication of a
possibly successful tethering (by Theaetetus) of a true
belief (that of Theodorus).
But Theaetetus is then asked to "discover" the nature
of knowledge. A very hard question indeed. But Socrates
suggests that he use his definition of surds as a model.
Forward, then, on the way you have just shown
so well. Take as a model your answer about
the roots. Just as you found a single
character to embrace all that multitude, so
now try to find a single formula that applies
to the many kinds of knowledge. (Theaete tu3
1 48d)
Notes
References are to Charles S. Peirce, 1931-1958,
Collected Papers, 8 volumes, edited by Charles Hartshorne,
Paul Weiss, and Arthur Banks, Cambridge: Harvard University
Press. All references to the Collected Papers are in the

44
standard form, citing only the volume number, decimal
point, and paragraph number.
2
It is interesting to note here that Kepler also became
very interested in the golden section. In 1596 he wrote,
"geometry has two great treasures: one is the theorem of
Pythagoras; the other, the division of a line into extreme
and mean ratio [golden section]. The first we may compare
to a measure of gold; the second we may have a precious
jewel." Citation in Dan Pedoe's Geometry and the Liberal
Arts, 1 975 , unpublished MS., Universiy of Minnesota, pi ZT5 •
3
One of the objections to abduction is that it is not
really a formal logic like deduction. However, by
weakening the conditions of validity and soundness it can
be given a weak formalism. Furthermore, it more than makes
up for lack of formalism in its creative discovery aspects.
^All citations to Aristotle are in the standard
numbering form for each of his works. These will include
the Topics, Prior Analytics, Posterior Analytics, DeAnima,
Metaphysics, Nicomachean 3thies, Metaphysics, and Physics.
All quotations are taken from Richard McKeon, ed., 1941 ,
The Basic Works of Aristotle. New York: Random House.
^ Consequences are generally considered in logic as
proceeding deductively downwards. However, the essence of
analysis or abduction is that it proceeds upwards (or
backwards) ascending to antecedent principles.
^Reverse solution is the common factor above all else
between Plato and Peirce.
^It might apply for example to the moral virtues.
However, Plato most likely identifies the virtues with
proportion. Hence, justice is identified as a
proportionate harmony in the soul (Republic) .
g
In this way he is able to ejcape the horns of the
"eristic" dilemma.
9
Here Aristotle is in complete agreement with Plato.
â– ^The basic movements are up from the sensible world to
principles, and down from principles to particulars.

CHAPTER III
THE PYTHAGOREAN PLATO
The Quadrivium^
In the Republic, Plato describes the method by which
the aspiring philosopher-statesman is to prepare himself to
govern the state. Ultimately, after a series of
conversions through succeeding states of awareness brought
about by the apprehension of corresponding levels of
subject-matter (Republic 515e) , the philosopher must arrive
at the "highest form of knowledge and its object," the Good
(Republic 504e). As Plato points out, the Good is "the end
of all endeavour" (Republic 5 0 5 d ). And furthermore, "if we
are ignorant of it the rest of our knowledge, however
perfect, can be of no benefit to us" (Republic 505a). It
alone is the foundation of certainty, and is arrived at
finally through the process of dialectic, what Plato calls,
"the coping-stone that tops our educational system"
(Republic 554e). "Anyone who is going to act rationally
either in public or private life must have sight of it"
(Republic 517c). "Our society will be properly regulated
only if it is in the charge of a guardian who has this
knowledge" (Republic 506a-b).
45

46
However, it is not until about the age of 50 that the
philosopher is to attain this "vision" of the Good. Prior
to this time, particular virtues must be both apparent and
cultivated. Along with these virtues, a rigorous
educational program must be undertaken. The object of this
education is to assist the philosopher's mind in its ascent
through the relative levels of awareness and their
corresponding levels of subject-matter, being converted at
each stage to the comprehension of a greater degree of
clarity and reality.
Plato has spelled out the qualities that this
individual must possess:
A man must combine in his nature good memory,
readiness to learn, breadth of vision, grace,
and be a friend of truth, justice, courage,
and self-control. . . . Grant then education
and maturity to round them off, and aren't
they the only people to whom you would
entrust your state? (Republic 487a)
Thus, besides the inherent abilities, requisite virtues,
and eventual maturity, it is the education that will
prepare one for the role of philosopher-statesman. Of what
is this education to consist? Leaving aside for the moment
the final dialectical procedure, Plato's answer is clearly
mathematics.
It is the thesis of this dissertation that Plato was
more intimately involved with a mathematical doctrine
throughout his career than is generally recognized by most
modern commentators. This mathematical doctrine is at the

47
very foundation of Plato's epistemology and ontology.
Through a careful analysis of hi3 work, two very important
features emerge: Plato's expert use of a method, later to
be called by Charles Sanders Peirce, abduction, and his
reverance for the golden section. The former is the real
forerunner of the method of scientific discovery. The
latter is a very important mathematical construct, the
significance of which has been generally ignored by
2
Platonic scholarship. But to arrive at these points I
must consider carefully Plato's famous Divided Line and the
"notorious question of mathematicals" (Cherniss, 1945).
From out of these I contend that the real significance of
abductive inference and the golden section in Plato's
philosophy will become apparent. In point of fact, I will
be defending the assertion that Plato, following the
Pythagorean mathematicoi,made mathematics the underlying
structure of his philosophy, with the golden section being
the basic modulor upon which space is given form. Further,
I will be arguing that the mathematicals may be found in
the middle dialogues. And, therefore, they are not merely
a construction of Plato's later period, as some have
contended. Additionally, the Good of the Republic and the
Receptacle of the Timaeus are to be identified respectively
with the One and the Indefinite Dyad. The Indefinite Dyad
in turn may have something to do with the golden section.
Finally, I will argue that it is the logical method of
abduction that lies at the center of Plato's reasoning

48
process. It is the very foundation of scientific
discovery. Hence, I will be attempting to place Plato more
firmly in the scientific tradition.
In one of the early "Socratic dialogues," the
Gorgias, Plato plants the seed of a view that will blossom
forth in the Republic. It is the view that mathematical
study, geometry in particular, is closely allied to the
establishment of a just and virtuous nature. There
Socrates chides Callicles, saying:
You are unaware that geometric equality is of
great importance among gods and men alike,
and you think we should practice overreaching
others, for you neglect geometry. (Gorgias
508a)
In one of the middle dialogues, the Republic, an even
stronger stand is taken. There mathematical study is not
only allied to a virtuous and orderly life, but is the very
"bridge-study" by which one may cross from the lower level
of mere belief (pistis) to the higher level of reason
(noesis). The bridge is understanding (dianoia), sometimes
translated, mathematical reasoning, and is the intermediate
level of awareness of the mathematician.
The components of this bridge-study are set out at
Republic 524d-530e. These are the five mathematical
sciences which include: (1) arithmetic, (2) plane
geometry, (3) solid geometry, (4) harmonics, and (5)
spherics (or astronomy). This is actually the Pythagorean
quadrivium, but Plato haa seen fit to subdivide geometry
into "plane" and "solid," representing respectively number

49
in the 2nd and the 3rd dimensions. The intensive study of
these mathematical disciplines is to occur between the ages
of twenty and thirty. Thi3 mathematical study will prepare
one for the study of dialectic, which will in turn
ultimately lead to the Good. Plato explains that
the whole study of the [mathematical]
sciences we have described has the effect of
leading the best element in the mind up
towards the vision of the best among
realities [i.e.,the Good]. (Republic 532c)
However, mathematical study is to begin well before this
later intensive training.
Arithmetic and geometry and all the other
studies leading to dialectic should be
introduced in childhood. (Republic 536d)
These elementary mathematical studies are to be combined
with emphasis on music and gymnastic until the age of 17 or
18. Then, until the age of 20, emphasis is placed on
gymnastics alone. According to Plato, this combination of
intellectual and physical training will produce concord
between the three parts of the soul, which will in turn
help to establish justice in the nature of the individual.
And this concord between them [i.e.,
rational, spirited, and appetitive] is
effected ... by a combination of
intellectual and physical training, which
tunes up the reason by training in rational
argument and higher studies [i.e.,
mathematical], and tones down and soothes the
element of spirit by harmony and rhythm.
(Republic 441e-442a)
This training in gymnastics and music not only stabilizes
the equilibrium of the soul, promoting a just nature in the

50
individual (and analogously a just nature in the soul of
the polis), but also indicates that Plato may conceive of
mathematical proportion (such as that expressed in
harmonics or music) as underlying both the sensible and
intelligible aspects of reality. This may be the reason
why he draws music and gymnastics together in the proposed
curriculum. Plato wants the individual to comprehend the
proportional relationship common to both music and bodily
movements. And because the soul participates in both the
4
intelligible and sensible realms, lying betwixt the two
and yet binding them together into one whole, it is
natural that the soul might contain this implicit
knowledge. Then, upon making it explicit, the soul would
benefit the most through the establishment of concord.
It is also evident in one of the last dialogues, the
Philebus , that Plato feels the knowledge brought about by
this study (i.e., of proportions in harmony and bodily
movements), may have important implications in one's grasp,
and possible solution, of the ancient and most difficult
problem of the one and many Plato says,
When you have grasped . . . the number and
nature of the intervals formed by high pitch
and low pitch in sound, and the notes that
bound those intervals, and all the systems of
notes that result from them [i.e., scales] .
. . and when, further, you have grasped
certain corresponding features that must, so
we are told, be numerically determined and be
called "figures" and "measures" bearing in
mind all the time that this is the right way
to deal with the one and many problem--only
then, when you have grasped all this, have
you gained real understanding. (Philebus
17d-e)

51
I contend that this Pythagorean notion of discovering
the role of number in things is one of the most crucial
things that Plato wants his pupils to learn. Further, if
one can discern how Plato saw the role of number in the
Cosmos, and the ontological and epistemological
consequences of his doctrine of the intermediate soul, then
it may be possible to determine how he saw the sensible
world participating in the Forms. It is most relevant that
the question of participation was left an open question in
the Academy (see Aristotle in Metaphysics 987b). Given the
puzzle, each member was allowed to abduct his own
hypothesis.
The Academy and It3 Members
Let us first look at the members of the Academy and
their interests. As Hackforth has pointed out, the Academy
was "designed primarily as a training school for
philosophic-statesmen'^ (Hackforth, 1 972 , p. 7). If the
Republic account is a correct indication of Plato's method
of preparing these individuals, then the primary work
carried out in the Academy would have been mathematical
study and research. Most of the information we have
concerning the associates of the Academy was copied down by
Proclus from a work by Eudemus of Rhodes, entitled History
of Geometry. Eudemus was a disciple of Aristotle. The
authenticity of this history by Proclus, what has come to
be called the "Eudemian summary," ha3 been argued for by

52
Sir Thomas Heath. Heath says,
I agree with van Pesch that there is no
sufficient reason for doubting that the work
of Eudemus was accesible to Proclus at first
hand. For the later writers Simplicius and
Eutocius refer to it in terms such as leave
no room for doubt that they had it before
them. (Heath, 1 956, vol. 1 , p. 35)
Other information has been preserved in the works of
Diogenes Laertius and Simplicius. The important fragments
preserved by Simplicius are also from a lost work by
Eudemus, History of Astronomy. Unless otherwise indicated,
the following condensed summary account is taken from the
"Eudemian summary" preserved by Proclus (see Thomas, 1957,
vol. 1 , pp. 144-161).
Now we do know that the two greatest mathematicians of
the 4th century B.C. frequented the Academy. These were
Eudoxus of Cnidus and Theaetetus of Athens. Eudoxus is
famous for having developed the "method of exhaustion for
measuring and comparing the areas and volumes of
curvalinear plane and solid surfaces" (Proclus, 1970,
p.55). Essentially, he solved the Delphic problem of
doubling the cube, developed a new theory of proportion
(adding the sub-contrary means) which is embodied in Euclid
Books V and VI, and hypothesized a theory of concentric
spheres to explain the phenomenal motion of the heavenly
bodies. Of this latter development Heath says:
Notwithstanding the imperfections of the
system of homocentric spheres, we cannot but
recognize in it a speculative achievement
which was worthy of the great reputation of
Eudoxus and all the more deserving of
admiration because it was the first attempt

53
at a scientific explanation of the apparent
irregularities of the motions of the planets.
(Heath, 1913 , p. 211)
To this Thomas adds the comment:
Eudoxus believed that the motion of the sun,
moon and planets could be accounted for by a
combination of circular movements, a view
which remained unchallenged till Kepler.
(Thomas, 1 957, vol. 1 , p. 410, fn.b)
Eudoxus' homocentric hypothesis was set forth in direct
response to a problem formulated by Plato. This is
decidedly one of those instances where the role of
abduction entered into the philosophy of Plato. Plato
would present the problem by formulating what the puzzling
phenomena were that needed explanation. This fits very
neatly into the Peircean formula. The surprising fact of
the wandering motion of the planets is observed. What
hypothesis, if true, would make this anomalous phenomena
the expected? To solve this problem requires one to reason
backwards to a hypothesis adequate to explain the
conclusion.
We are told by Simplicius, on the authority
of Eudemus, that Plato set astronomers the
problem of finding what are the uniform and
ordered movements which will "save the
phenomena" of the planetary motions, and that
Eudoxus was the first of the Greeks to
concern himself [with this]. (Thomas, 1957,
vol. 1, p. 410, fn.b)
At this point it is relevant to consider what was
meant by the phrase "saving the phenomena." For this
purpose I will quote extensively from a passage in
Vlastos', Plato's Universe.

54
The phrase "saving the phenomena" does not
occur in the Platonic corpus nor yet in
Aristotle's works. In Plato "save a thesis
(or 'argument')" (Theaetetus 164a) or "save a
tale" (Laws 645b) and in Aristotle "save a
hypothesis" (de Cáelo 306a30) and "preserve a
thesis" (Nicomachean Ethics 1096a2) occur in
contexts where "to save" TÂ¥ to preserve the
credibility of a statement by demonstrating
its consistency with apparently recalcitrant
logical or empirical considerations. The
phrase "saving the phenomena" must have been
coined to express the same
credibility-salvaging operation in a case
where phenomena, net a theory or an argument,
are being put on the defensive and have to be
rehabilitated by a rational account which
resolves the prima facie contradictions
besetting their uncritical acceptance. This
is a characteristically Platonic view of
phenomena. For Plato the phenomenal world,
symbolized by the shadow-world in the
Allegory of the Cave (Republic 517b) is full
of snares for the intellect. Thus, at the
simplest level of reflection, Plato refers us
(Republic 602c) to illusions of sense, like
tFe stick that looks bent when partly
immersed in water, or the large object that
looks tiny at a distance. Thrown into
turmoil by the contradictory data of sense,
the soul seeks a remedy in operations like
"measuring, numbering, weighing" (Republic
602d) so that it will no longer be at the
mercy of the phenomenon. For Plato, then,
the phenomena must be held suspect unless
they can be proved innocent ("saved") by
rational judgment. So it would not be
surprising if the phrase "saving the
phenomena"--showing that certain perceptual
data are intelligible after all--had
originated in the Academy. (Vlastos, 1975,
pp. 111-112)
Returning then to the problem Plato set the
astronomers, Eudoxus was not the only one to attempt to
abduct an adequate hypothesis or solution. Speusippus,
Plato, and Heraclides each developed a solution different
from that of Eudoxus. Menaechmus, in essential respects,

55
followed the solution of Eudoxus. Callipus then made
corrections on Menaechraus’ version of Eudoxus’ solution,
which was then adopted by Aristotle. Each attempted to
abduct an adequate hypothesis, or modification of a former
hypothesis, which would adequately explain and predict the
so-called anomalous phenomena.^
There is an even more telling reference to the
abductive approach of Plato and Eudoxus in the "Eudemian
summary." The reference is intriguing, because it refers
to both abductive inference, under the rubric of analysis,
and the golden section.
[Eudoxus] multiplied the number of
propositions concerning the "section" which
had their origin in Plato, applying the
method of analysis to them.8 (Thomas, 1957,
vo1. 1 , p. 153)
My contention is that this analysis is none other than
what Peirce referred to as abduction. The essential
feature of this method is that one reasons backwards to the
causal explanation. Once one has arrived at the
explanatory hypothesis, then one is able to deductively
predict how the original puzzling phenomenon follows from
that hypothesis. Analysis and synthesis were basic
movements to and from a principle or hypothesis. Proclus
was aware of these contrary, but mutually supportive
movements. Thus, Morrow, in the introduction of hi3
translation of Proclus' A Commentary of the First Book of
Euclid's Elements, says,
. . . the cosmos of mathematical propositions
exhibits a double process: one is a movement

56
of "progression” (prodos), or going forth
from a source; the other is a process of
"reversion" (ánodos) back to the origin of
this going forth. Thus Proclus remarks that
some mathematical procedures, such as
division, demonstration, and synthesis, are
concerned with explication or "unfolding" the
simple into its inherent complexities,
whereas others, like analysis and definition,
aim at coordinating and unifying these
diverse factors into a new integration, by
which they rejoin their original
starting-point, carrying with them added
content gained from their excursions into
plurality and multiplicity. For Proclus the
cosmos of mathematics is thus a replica of
the complex structure of the whole of being,
which is a progression from a unitrary, pure
source into a manifold of differentiated
parts and levels, and at the same time a
constant reversion of the multiple
derivatives back to their starting-points.
Like the cosmos of being, the cosmos of
mathematics is both a fundamental One and an
indefinite Many. (Proclus,1970, p. xxxviii)
My view is that Proclus has correctly preserved the
sense of analysis and synthesis as underlying the work of
Plato. This notion of analysis appears to have been
somewhat esoteric, not being clearly explicated in the
writings of Plato. However, as I will later argue, Plato's
notion of dialectic is closely allied to this concept. One
is either reasoning backwards from conclusions to
hypotheses, or forward from hypotheses to conclusions.
Thus, a3 Aristotle noted:
Let us not fail to notice . . . that there is
a difference between arguments from and those
to the first principles. For Plato, too, was
right in raising this question and asking, as
he used to do, "are we on the way from or to
the first principles?"^ There is a
difference, as there is in a race-course
between the course from the judges to the
turning-point and the way back. For, while
we must begin with what is known, things are

57
objects of knowledge in two senses--some to
us, some without qualification. Presumably,
then, we must begin with things known to us.
Hence any one who is to listen intelligently
to lectures about what is noble and just and,
generally, about the subjects of political
science must have been brought up in good
habits. For the fact is the starting-point,
and if this is sufficiently plain to him, he
will not at the start need the reason as
well; and the man who has been well brought
up has or can easily get starting-points.
(Nicomachean Ethics 1095aJ0-b9)
Thus, in the movement towards first principles, it is a
motion opposite in direction to that of syllogism or
deduction. This is central to Plato's notion of dialectic.
Hence, Plato says,
That which the reason itself lays hold of by
the power of dialectic, treats its
assumptions not as absolute beginnings but
literally as hypotheses, underpinnings,
footings, and springboards so to speak, to
enable it to rise to that which requires no
assumption and is the starting point of all.
(Republic 511b)
It should be noted that the notion of dialectic
depicted here in the Intelligible world moves strictly from
one hypothesis as a springboard to a higher or more
primitive hypothesis. It does not begin from the
observation of sensible particulars. However, a central
theme throughout this dissertation will be that this same
movement in abductive explanation takes place at every
level for Plato, including the level of sensible
particulars. Therefore, the abductive movement from the
observation of irregular planetary motions to the
explanation in terms of the underlying regularity
discoverable in the Intelligible world is analogous to the

58
abductive movement purely within the Intelligible world.
When we later examine the Cave simile (Republic 5Ma-521b)
it will become apparent that each stage of conversion is an
abductive movement, and hence a kind of dialectic, or
analysis.
Now according to Plato, once a suitable explanatory
hypothesis has been abducted, one can deductively descend,
setting out the consequences. Of course in the Republic,
where one is seeking certainty, this means first arriving
at the ultimate hypothesis, the Good.
. . . when it has grasped that principle [or
hypothesis] it can again descend, by keeping
to the consequences that follow from it, to a
conclusion. (Republic 511b)
Thus, one then proceeds in the downward direction with
synthesis (i.e., syllogism or deduction).
It is difficult to find clear statements about
analysis in either Plato or Aristotle. However, Aristotle
does liken deliberation to geometrical analysis in the
Nicomachean Ethics.
We deliberate not about ends but about means.
For a doctor does not deliberate whether he
shall heal, nor an orator whether he shall
persuade, nor a statesman whether he shall
produce law and order, nor does any one else
deliberate about his end. They assume the
end and consider how and by what means it is
to be attained; and if it seems to be
produced by several means they consider by
which it is most easily and best produced,
while if it is achieved by one only they
consider how it will be achieved by this and
by what means this will be achieved, till
they come to the first cause, which in the
order of discovery is last. For the person
who deliberates seems to investigate and
analyse in the way described as though he

59
were analysing a geometrical construction . .
. and what is last in the order of analysis
seems to be first in the order of becoming.
(Nichomachean Ethics 1112b12-24)
The other famous 4th century B.C. mathematician in the
Academy was Theaetetus of Athens, who set down the
foundations of a theory of irrationals which later found
its way into Book X of Euclid’s Elements. As Furley has
pointed out, "Theaetetus worked on irrational numbers and
classified 'irrational lines' according to different types"
(Furley, 1967, p. 105). He furthered the work on the 5
regular solids, and is held to have contributed much to
Book XIII of Euclid’s Elements.
Speusippus, the son of Plato's sister Potone,
succeeded Plato as head of the Academy,^ and wrote a work
entitled, On the Pythagorean Numbers. Unfortunately only a
few fragments remain. According to Iamblichus,
. . . [Speusippus] was always full of zeal
for the teachings of the Pythagoreans, and
especially for the writings of Philolaus, and
he compiled a neat little book which he
entitled On the Pythagorean Numbers. From
the beginning up to half way he deals most
elegantly with linear and polygonal numbers
and with all the kinds of surfaces and solids
in numbers; with the five figures which he
attributes to the cosmic elements, both in
respect of their similarity one to another;
and with proportion and reciprocity. After
this he immediately devotes the other half of
the book to the decad, showing it to be the
most natural and most initiative of
realities, inasmuch as it is in itself (and
not because we have made it so or by chance)
an organizing idea of cosmic events, being a
foundation stone and lying before God the
Creator of the universe as a pattern complete
in all respects. (Thomas, 1 957, voi. 1 , p. 77)

60
Furthermore, it is Speusippus who apparently rejected the
Platonic Ideas but maintained the mathematical numbers.
Xenocrates, who followed Speusippus as head of the
Academy, wrote six books on astronomy. He is also credited
with an immense calculation of the number of syllables that
one can form out of the letters of the Greek alphabet. The
number he derived is 1,002,000,000,000 (Sarton, 1970, voi.
1, p. 503, Speusippus, he retained the Ideas, but identified them with
the mathematicals.
Another one of Plato's pupils was Philippus of Opus,
who, according to the "Eudemian summary," was encouraged by
Plato to study mathematics. It appears that he may have
edited and published the Laws and possibly authored the
12
Epinomis. He wrote several mathematical treatises, the
titles of which are still preserved.
Another pupil was Leodamos of Thasos, who, according
to Diogenes Laertius, was taught the method of analysis by
Plato. Again the intriguing mention of the method of
analysis occurs. And it is important that it is Plato who
purportedly taught Leodamos this method. In another place
Proclus indicates that
certain methods have been handed down. The
finest is the method which by means of
analysis carries the thing sought up to an
acknowledged principle, a method which Plato,
a3 they say, communicated to Leodamas, and by
which the latter, too, is said to have
discovered many things in geometry. (Heath,
1956, vol. 1, p. 134)

61
In a footnote to the above statement by Proclus, Heath
makes the very interesting comment,
Proclus' words about carrying up the thing
sought to "an acknowledged principle"
suggests that what he had in mind was the
process described at the end of Book VI of
the Republic by which the dialectician
(unlike the mathematician) uses hypotheses as
stepping-stones up to a principle which is
not hypothetical, and then is able to descend
step by step verifying every one of the
hypotheses by which he ascended. (Heath,
1956, vol. 1, p. 134, fn.1)
This is a very insightful remark by Heath, and I will
return to this point when I consider the Divided Line in
the Republic.
Menaechmus, a pupil of both Eudoxus and Plato, wrote
on the methodology of mathematics. It Í3 generally
inferred from Eratosthenes that he discovered the conic
sections. His brother, Dinostratus, applied Hippias’
quadratrix in an attempt to square the circle.
3oth Leon, the pupil of Neoclides, and Theudius of
Magnesia wrote a "Book of Elements" in the Academy during
Plato's time. Heath, in fact, conjectures that the
elementary geometrical propositions cited by Aristotle were
derived from the work of Theudius. Thus, Heath says,
Fortunately for the historian of mathematics
Aristotle was fond of mathematical
illustrations; he refers to a considerable
number of geometrical propositions,
definitions, etc., in a way which shows that
his pupils must have had at hand some
textbook where they could find the things he
mentions; and this textbook must have been
that of Theudius. (Heath, 1956, vol. 1, p.
117)

62
Also living at this time was the Pythagorean,
Archytas of Tarentum. An older contemporary and
friend of Plato, there is little doubt that he had a
major influence on Plato, though he may have never
actually been in the Academy. It was Archytas who
reduced the Delphic problem of doubling the cube to
the problem of finding two mean proportionals. In
the "Eudemian Summary" it is stated that,
. . . [Archytas] solved the problem of
finding two mean proportionals by a
remarkable construction in 3 dimensions.
(Thomas 1 957, vol. 1 , p. 285)
Thus, according to Van der Waerden, Archytas is responsible
for the material in Bk. VIII of Euclid's Elements.
Cicero tells us that
[it was during Plato's] first visit to South
Italy and Sicily, at about the age of forty,
that he became intimate with the famous
Pythagorean statesman and mathematician
Archytas. (Hackforth, 1972, p. 6)
And of course from Plato's 7th Letter (at 350b-c) we
find that it was Archytas who sent Lamiscus with an
embassy and 30-oared vessel to rescue Plato from the
tyrant Dionysius (7th Letter 350b-c, Cairns and
Hamilton, 1971, p. 1596).
As Thomas indicates in a footnote:
For seven years [Archytas] commanded the
forces of his city-state, though the law
forbade anyone to hold the post normally for
more than one year, and he was never
defeated. He is said to have been the first
to write on mechanics, and to have invented a
mechanical dove which would fly. (Thomas,
1957, vol. 1, p.4, fn.a)

63
It is clear that Archytas was probably a major source
of much of the Phi1o laic-Pytha gorean doctrines that Plato
gained access to. He was also a prime example of what a
philosopher-statesman should be like. Furthermore, he is
generally considered to be a reliable source of information
on the early Pythagoreans. "No more trustworthy witness
could be found on this generation of Pythagoreans" (Kirk
and Raven, 1975, p. 314)•
Many of the ideas of Archytas closely parallel those
of Plato. Porphyry indicates this when he quotes a
fragment of Archytas' lost book On Mathematics:
The mathematicians seem to me to have arrived
at true knowledge, and it is not surprising
that they rightly conceive each individual
thing; for having reached true knowledge
about the nature of the universe as a whole,
they were bound to see in its true light the
nature of the parts as well. Thus they have
handed down to us clear knowledge about . . .
geometry, arithmetic and sphaeric, and not
least, about music, for these studies appear
to be sisters. (Thomas, 1957, vo1. 1, p.5)
It seems obvious that the four sister sciences
mentioned here refer to the Pythagorean quadrivium. But
Archytas' above statement that "they rightly conceive each
individual thing," should be contrasted with some of
Plato's remarks regarding dialectic. Thus, Plato says,
"dialectic sets out systematically to determine what each
thing essentially is in itself" (Republic 533b). And
further he says, the dialectician is one who "can take
account of the essential nature of each thing" (Republic
534b) .

64
There was, of course, also Aristotle, who was a member
of the Academy for nineteen (perhaps twenty) years during
Plato's lifetime. "From his eighteenth year to his
thirty-seventh (367-343/7 B.C.) he was a member of the
school of Plato at Athens" (Ross, 1967, p. ix). Aristotle
considered mathematics to be one of the theoretical
sciences along with metaphysics and physics (Me taphysics
1026a 18-20). But he did not devote any writing strictly
to the subject itself, contending that he would leave it to
others more specialized in the area. Nevertneless, his
writings are interspersed with mathematical examples. And
what he has to say regarding Plato’s treatment of
mathematics is of the utmost importance in trying to
properly interpret Plato. His remarks should not be swept
aside simply because one has difficulty tallying them with
the Platonic dialogues.
Aristotle was obviously subjected to mathematical
study in the Academy. Apparently he was not that pleased
with its extreme degree of emphasis there. Thus, he
14
indicates in a somewhat disgruntled tone, as Sorabji has
put it, that many of his modern cohorts in the Academy had
so over-emphasized the role of mathematics that it had
become not merely a propadeutic to philosophy, but the
subject-matter of philosophy itself.
Mathematics has come to be identical with
philosophy for modern thinkers, though they
say that it should be studied for the sake of
other things. (Metaphysics 992a 32-34)

65
Theodorus of Cyrene should also be mentioned here.
According to Iamblichus he was a Pythagorean. And
according to Diogenes Laertius he was the mathematical
instructor of Plato (Thomas, 1957, Vol. 1, p. 380).
According to the "Eudemian summary," Theodorus "became
distinguished in geometry" (Thomas, 1957, Vol. 1, p. 151)*
In Plato's dialogue, the Theaetetus , Theodorus appears with
the young Theaetetus. There Theaetetus is subjected to the
midwifery of Socrates. Theaetetus begins his account by
describing the mathematical nature of his training at the
hands of Theodorus. In the following passage it is
interesting to note that we again find the Pythagorean
quadrivium.
Socrates: Tell me, then, you are learning
some geometry from Theodorus?
Theaetetus: Yes.
Socrates: And astronomy and harmonics and
aritmetic ?
Theaetetus: I certainly do my best to learn.
(Theaetetus 145c-d)
In the "Eudemian summary," Proclus also mentions
Arayclas of Heracleia, who is said to have improved the
subject of geometry in general; Hermotimus of Colophon, who
furthered the investigations of Eudoxus and Theaetetus; and
Athenaeus of Cyzicus, who "became eminent in other branches
of mathematics and especially in geometry" (Thomas, 1957,
vol. 1, p. 153)- But it is not clear when these three
individuals appeared in the Academy, whether during Plato's
lifetime, or shortly thereafter.

66
Archytas, Theodorus, Amyclas, Hermotimus, and
Athenaeus aside, it is asserted of the others that "these
men lived together in the Academy, making their inquiries
in common" (Thomas, 1957, vol. 1, p. 153)* However, it is
not suggested, and it certainly should not be inferred,
that they were all at the Academy simultaneously. It must
be remembered that Plato ran the Academy for some forty
years. And though Aristotle was there for nineteen of
those years, several of the other individuals may have come
and gone, appearing at the Academy at different times. But
one thing is clear; there was an overriding emphasis in
mathematical study and research. As Cherniss has correctly
pointed out,
If students were taught anything in the
Academy, they would certainly be taught
mathematics . . . that their minds might be
trained and prepared for the dialectic; and
this inference from the slight external
tradition is supported by the dialogues,
especially the seventh book of the Republic.
(Cherniss, 1945, pp. 66-67)
On the Good
Further evidence in support of the mathematical nature
of Plato's philosophy may be found in the accounts of his
unwritten lecture, On the Good. Aristoxenus, like Sudemus,
a disciple of Aristotle, writes in his Elements of Harmony,
Bk. 2 ,
Plato's arguments were of mathematics and
members and geometry and astronomy and in the
end he declared the One to be the Good.
(Thomas, 1 957, vol. 1 , p. 389)

67
Thus, although the title of the lecture indicated that
it was about the Good, the ultimate object of knowledge as
expressed in the Republic, it nevertheless dealt with the
mathematical subject-matter involved in the ascent there.
And furthermore, the One was somehow identified with the
ultimate object of knowledge, the Good. This tallies with
what Aristotle has to say. Referring to Plato's use of two
causes, the essential cause, the One, and the material
cause, the Indefinite Ddyad or the Great and Small,
Aristotle says,
Further, he has assigned the cause of good
and evil to the elements, one to each of the
two. (Metaphysics 988a 13-15)
At another point Aristotle explains that
the objection arises not from their ascribing
goodness to the first principle as an
attribute, but from their making the One a
principle--and a principle in the sense of an
element--and generating number from the One.
(Metaphysics 1091b 1-4)
Aristotle objects elsewhere, fortunately for us in a
very telling way. He says,
. . . to say that the first principle is good
is probably correct; but that this principle
should be the One or ... an element of
numbers, is impossible. . . . For on this
view all the elements become identical with
species of good, and there is a great
profusion of goods. Again, if the Forms are
numbers, ail the Forms are identical with
species of good. (Metaphysics 1091b 19-27)
-#
Aristotle goes on to argue that if evil is identified
with the Dyad, Plato's Great and Small, it follows that
all things partake of the bad except one--the
One itself, and that numbers partake of it in

68
a more undiluted form than spatial
magnitudes, and that the Bad is the space in
which the Good is realized. (Metaphysics
1 0 92a 1-3)
The statement of Aristoxenus and the passages in
Aristotle’s Metaphysics appear to indicate that Plato held
the One and Indefinite Dyad to be the principles of all
entities, and furthermore gave them the attributes
respectively of good and evil. It follows that if the
Forms (as principles of all other entities) are derived
from the One and Indefinite Dyad, then the elements from
which they are derived are numerical and, hence, the Forms
themselves are of a numerical nature. As the Forms are
principles of all other entities, it would follow that
number would be perpetuated throughout the Cosmos down into
the sensible things as well.
Now Cherniss admits that
Alexander himself says that in Aristotle's
report of the lecture |~0n the Good] , "the
One" and "the great and small" were
represented as the principles of number and
the principles of all entities. (Cherniss,
1945, p. 23)
How then can the numerical principles of the One and
Indefinite Dyad be the principles of all entities,
including all sensible entities, unless the crucial feature
is that numbers are in fact the essential characters of
those entities? Aristotle makes this more explicit, saying,
Since the Forms were causes of all other
things, he thought their elements were the
elements of all things. As matter, the great
and small were principles; as essential
reality, the One; for from the great and
small, by participation in the One,, come the

69
Numbers. [And] he agreed with the
Pythagoreans in saying that the One is
substance and not a predicate of something
else; and in saying that the Numbers are the
causes of the reality of other things he
agreed with them. (Metaphysics 987b 19-25)
There is an interesting parallel between what Aristotle
says about the One in the Metaphysics, and what Plato says
about the Good in the Republic. Aristotle says,
. . . the Forms are the causes of the essence
of all other things, and the One is the cause
of the essence of the Forms. (Metaphysics
988a 9-11)
The latter part of this statement should be compared with
what Plato has Socrates say about the Good.
The Good therefore may be said to be the
source not only of the intelligibility of the
objects of knowledge [the Forms], but also of
their being and reality; yet it is not itself
that reality but is beyond it and superior to
it in dignity and power. (Republic 508b)
Thus, there appears to be a definite identification of
the Good with the One, and evil with the Dyad. And hence,
for Plato, the two basic elements of the Cosmos are of a
numerical nature. The further implication then is that the
Forms are also numbers. This is in fact what Aristotle
suggests,
. . . the numbers are by him [Plato]
expressly identified with the Forms
themselves or principles, and are formed out
of the elements. (De Anima. 404b24)
At another point Aristotle unequivocally asserts that,
"those who speak of Ideas say the Ideas are numbers"
(Metaphysics 1073a18-20). And in fact, not only are the
Forms to be identified with numbers, but so are the

70
sensibles, although as numbers of a different class. This
emerges in a passage in which Aristotle is discussing the
Pythagoreans.
When in one particular region they place
opinion and opportunity, and, a little above
or below, injustice and decision or mixture,
and allege, as proof, that each of these is a
number, and that there happens to be already
in this place a plurality of the extended
bodies composed of numbers, because these
attributes of number attach to the various
places--this being so, is this number, which
we must suppose each of these abstractions to
be, the same number which is exhibited in the
material universe, or is it another than
this? Plato says it is different; yet even
he thinks that both these bodies and their
causes are numbers, but that the intelligible
numbers are causes, while the others are
sensible. (Metaphysics 990a23-32)
Thu3, numbers are not only the crucial feature of Forms,
but also of sensible particulars.
Returning then to Plato's unwritten lecture, On the
Good, Cherniss notes:
It is said that Aristotle, Speusippus,
Xenocrates, Heraclides, Hestiaeus, and other
pupils attended the lecture and recorded
Plato's remarks in the enigmatic fashion in
which he made them (see Simplicius).
Moreover, most of them apparently published
their notes or transcripts of the lecture. .
. . Aristotle's notes were certainly
published under the title, On the Good [peri
tagathou]. (Cherniss, t 945, pü 12 )
Why then was the lecture delivered (and recorded) in
this 30-called "enigmatic fashion?" A clue to this may lie
in the Phaedrus where the Egyptian King Tharaus (Ammon)
reprimands the god Theuth. The latter has claimed that his
discovery of writing "provides a recipe for memory and

71
wisdom" (Phaedrus 274e). Thamus replies that it only leads
to the "conceit of wisdom" (Phaedrus 275b).
If men learn this [writing], it will implant
forgetfulness in their souls; they will cease
to exercise memory because they rely on that
which is written, calling things to
remembrance no longer from within themselves,
but by means of external marks. What you
have discovered is a recipe not for memory,
but for reminder. And it is no true wisdom
that you offer your disciples, but only its
semblance, for by telling them of many things
without teaching them you will make them seem
to know much, while for the most part they
know nothing, and as men filled, not with
wisdom, but with the conceit of wisdom, they
will be a burden to their fellows. (Phaedrus
274e-275b)
Then Plato has Socrates follow this with an analogy
of writing to painting.
The painter's products stand before us as
though they were alive, but if you question
them, they maintain a most majestic silence.
It is the same with written words; they seem
to talk to you as though they were
intelligent, but if you ask them anything
about what they say, from a'desire to be
instructed, they go on telling you just the
same thing forever. And once a thing is put
in writing, the composition, whatever it may
be, drifts all over the place, getting into
the hands not only of those who understand
it, but equally those who have no business
with it; it doesn't know how to address the
right people, and not address the wrong. And
when it is ill-treated and unfairly abused it
always needs its parent to come to its help,
being unable to defend or help itself.
(Phaedrus 2 75d-e)
This tends to show a negative view on the part of Plato
toward the publishing of one's doctrines. The reason being
that written doctrines may be either misunderstood or
abused by falling into the wrong hands. And if either of

72
these situations occur, the architect of the doctrine must
be present to defend and rectify the situation. But too
often this attendance is impossible.
This position is clearly consistent with what Plato
has to say in the 7th Le tter. There, referring to his most
complete doctrine, Plato says,
I certainly have composed no work in regard
to it, nor shall I ever do so in the future,
for there is no way of putting it in words
like other studies. (7th Letter 341c)
Here, of course, Plato is not as concerned with
misunderstanding or abuse, as he is with what appears
to be a somewhat more mystical doctrine. As he goes
on to say regarding this subject^
Acquaintance with it must come rather after a
long period of attendance on instruction in
the subject itself and of close
compainioship, when, suddenly, like a blaze
kindled by a leaping spark, it is generated
in the soul and at once becomes
self-sustaining. (7th Letter 341c-d)
Why then did so many of Plato's pupils copy down the
lecture and, apparently, publish it? Why was it so
important? I contend that it set forth, though still in an
enigmatic fashion, the essentially mathematical underlying
structure of Plato's philosophy. The parallel between the
lecture On the Good and the Republic, especially Books 6
and 7, has already been touched upon. The Good of the
Republic was identified with the One. And Aristotle goes
on to contrast the account given in the Timaeus of the
participant with the account in the so-called unwritten
teaching. In the latter, the participant-receptac1e of the

73
Timaeus is identified with the great and small(Physics
209bl1-15 and 209b33-21Oa2 ) . As already indicated from
Alexander's account (following Aristotle) of the lecture He
the Good, "the One and the Great and the Small were
represented as the principles of number and the principles
of all entities" (Cherniss, 1945, p. 28).
If further we accept the argument of G.E.L. Owen
(Allen, 1965), that the Timaeus is to be grouped along with
16
the Phaedo and Republic as a middle dialogue, then it
appears that the lecture On the Good may provide some
mathematical keys to the interpretation of the middle
dialogues. This is strictly inferential. However, I
maintain that it is a most plausible inference that allows
one to advocate a very consistent approach to Plato's
thought.
When this is conjoined with the question of
mathematicais (intermediate, seperate, immutable, and
plural) and their relation to the ontological and
epistemological function of the soul, a slightly revised
view of Plato's middle dialogues will emerge. This, in
turn, may shed some entirely different light upon the
paradoxidcal problems posed in the Parmenides, Theaetetus,
and Sophist ^ and their possible solution.
The Pythagorean Influence
This mathematical structuring of Plato's philosophy
suggests that he may have strongly adhered to, and further developed

74
developed, some of the doctrines of the Pythagoreans. And,
if I am correct, he especially followed the matheimatikoi.
It should be noted that in contrast to these matheimatikoi,
Plato critiques the more exoteric Pythagorean akousmatikoi
for getting caught up with the sensory aspects of harmonics.
They look for numerical relationships in
audible concords, and never get as far as
formulating problems and examining which
numerical relations are concordant, which
not, and why. (Republic 531c)
That is to say, the akousmatikoi never rise above the more
mundane features of harmonics. They do not formulate
problems for themselves from which they can abduct
hypotheses as solutions. These notions of setting a
problem and formulating the conditions for solution
(diorismos) are critical to the abductive movement,
referred to by the ancients as analysis. In this regard,
note especially Plato's Meno (see supra, pp. 36 & 39), where
he refers to the method "by way of hypothesis" (Meno
86e-37b).
In the Metaphysics, Aristotle indicates that Plato
was, in fact, a follower of the philosophy of the
Pythagoreans, but also differed from them in some respects.
He says ,
After the systems we have named came the
philosophy of Plato, which in most respects
followed Lakolouthousa ] these thinkers [i.e.,
Pythagoreans], but had peculiarities that
distinguished it from the philosophy of the
Italians. (Metaphysics 987a29-3l)

75
Entirely too much emphasis has been placed upon the
subsequent "differences" which were due to the Heraclitean
influence of Cratylus, and not enough weight placed on the
former words. Hackforth contends that the similarity
between the Pythagoreans and Plato is much stronger than
generally acknowledged. Of course, this is one of my
contentions here as well. Referring to Aristotle's account
of the relation between Platonic Forms and Pythagorean
Numbers, Hackforth says:
Despite the important divergences there
noted, one of which is the transcendence of
the Forms as against the Pythagorean
identification of things with numbers, it
seems clear that he regarded their general
resemblance as more fundamental. Moreover
the word akolouthousa [Metaphysics 987a30] i s
more naturally understood as implying
conscious following of Pythagorean doctrine
than mere factual resemblance. (Hackforth,
1972, p. 6)
Unfortunately, an extemely dualistic picture of Plato
has been painted by those who accept the strict separation
of the Intelligible and Sensible worlds in Plato's
philosophy. This has resulted from too heavy of an
emphasis being placed upon the Heraclitean influence of
Cratylus upon Plato. This has led to a tendency by
scholars to get stumped by the problems Plato sets in the
dialogues, rather than solve them. If my Pythagorean
hypothesis about Plato is correct, then many of the
Platonic dialogue problems should be, if not actually
soluble, then at least reasonably understandable.

76
My own view is that it was probably Philolaus who had
the greatest impact upon the views of Plato. This
influence may well have been channeled through Plato's
friend, Archytas. It is important to recall that Plato's
nephew, Speusippus, "was always full of zeal for the
teachings of the Pythagoreans, and especially for the
writings of Philolaus" (Thomas, 1 957, vol. 1 , p. 77).
Along this line, it is interesting that the contents of
Speusippus' book On Pythagorean Numbers (see supra p. 59)
holds a close resemblance to material in Plato's Timaeus
regarding the five cosmic elements and their harmonious
relation in terms of ratio and proportion. And there is
the assertion of Diogenes Laertius, presumably following
Aristoxenus, that Plato copied the Timaeus out of a work by
Philolaus.
. . . Philolaus of Croton, [was] a
Pythagorean. It was from him that Plato, in
a letter, told Dion to buy the Pythagorean
books. ... He wrote one book. Hermippus
says that according to one writer the
philosopher Plato went to Sicily, to the
court of Dionysius, bought this book from
Philolaus' relatives . . . and from it copied
out the Timaeus. Others say that Plato
acquired the bo’oks by securing fom Dionysius
the release from prison of a young man who
had been one of Philolaus' pupils. (Kirk and
Raven, 1975, p. 508)
Though we need not assert plagiarism, it is entirely
reasonable to suppose that a work of Philolaus' acted as a
source book for Plato's Timaeus.

77
It is also noteworthy that Plato, in the Phaedo, refers
to Philolaus. He has Socrates ask the Pythagoreans, Cebes
and Siramias, whether they had not heard Philolaus, whom
they had been staying with, talk about suicide.
Why, Cebes, have you and Siramias never heard
about these things while you have been with
Philolaus [at Thebes]? (Phaedo 61 d)
Plutarch hints that Plato in fact studied Pythagorean
philosophy at Memphis with Simmias.
Simmias appears as a speaker in Plutarch's
dialogue De genio Soeratis, where he says
[578f] that he was a fellow-student of
philosophy with Plato at Memphis--an
interesting remark and conceivably true.
(Hackforth, 1972, pp. 13-14)
It is probable then that the unnamed authority in
Socrates' last tale (Phaedo 107d-115a) is Philolaus,
especially with the reference to the dodecahedron. Thus,
Socrates says to Simmias:
The real earth, viewed from above, is
supposed to look like one of these balls made
of twelve pieces of skin, variegated and
marked out in different colors, of which the
colors which we know are only limited
samples, like the paints which artists use,
but there the whole earth is made up of such
colors, and others far brighter and purer
still. One section is marvelously beautiful
purple, and another is golden. (Phaedo I10b-c)
Of course the dodecahedron reappears in the Timaeus as
the foundation of the structure of the Cosmos.
There was yet a fifth combination which God
used in the delineation of the universe with
figures of animals. (Timaeus 55c)

78
Very few of the Philolaic fragments remain. However,
what fragments do remain provide a clue as to why
Speusippus was so enthusiastic about his writings. It may
also indicate why Speusippus' uncle, Plato, found his work
so interesting, as well. Philolaus' fragment 12 appears as
though it could have come straight out of the Timaeus.
In the sphere there are five elements, those
inside the sphere, fire, and water and earth
and air, and what is the hull of the sphere,
the fifths (Santillana and von Dechend, 1969,
p. 232)
It is difficult to adequately ascertain the thought of
the early Pythagoreans, including Philolaus. They
maintained an oral tradition in which their major tenets
were guarded with great secrecy. Substantial fragments of
a book on Pythagoreanism by Aristotle's pupil, Aristoxenus
of Tarentum, preserved by Iamblichus, remain to bear this
fact out.
The strictness of their secrecy is
astonishing; for in so many generations
evidently nobody ever encountered any
Pythagorean notes before the time of
Philolaus. (Kirk and Raven, 1975, p. 221)
Furthermore, Porphyry, quoting another pupil of
Aristotle, Dicaearchus of Messene, indicates the same
thing.
What he [Pythagoras] said to his associates,
nobody can say for certain; for silence with
them was of no ordinary kind. (Kirk and
Raven, 1975, p. 221)
Thus, secrecy was the rule. He who would reveal the
Pythagorean tenets on number faced punishment.

79
There was apparently a rule of secrecy in the
community, by which the offence of divulging
Pythagorean doctrine to the uninitiated is
said by later authorities to have been
severely punished. (Kirk and Haven, 1975, p.
220)
Hence, Iamblichus maintained the tradition that
the Divine Power always felt indignant with
those who rendered manifest the composition
of the icostagonus, viz., who delivered the
method of inscribing in a sphere the
dodecahedron (Blavatsky, 1972, vol. 1 , p.
xxi).
This may well be why Plato was so cryptic in his
discussion of the construction of the four elements and the
nature of the fifth element in the Timaeus. He is
discussing the formation of the tetrahedron, icosahedron,
and octahedron out of the right-angled scalene triangles.
Of the infinite forms we must again select
the most beautiful, if we are to proceed in
due order, and anyone who can point out a
more beautiful form than ours for the
construction of these bodies, shall carry off
the palm, not as an enemy, but as a friend.
(Timaeus 54a)
Plato may well have been concerned with not being too
explicit about this Pythagorean doctrine. As he points out
in the 7th Letter,
I do not . . . think the attempt to tell
mankind of these matters a good thing,
except in the case of some few who are
capable of discovering the truth for
themselves with a little guidance. . . .
There is a true doctrine, which I have often
stated before, that stands in the way of the
man who would dare to write even the least
thing on such matters. (7th Letter 341e-342a)
Nevertheless, there is a tradition amongst the
Neoplatonists that Plato was an initiate of various mystery

schools, including the Pythagorean school, and that he
incurred much wrath for "revealing to the public many of
the secret philosophic principles of the Mysteries" (Hail,
1 92 8, p. 21 ) .
Plato was an initiate of the State Mysteries.
He had intended to follow in the footsteps of
Pythagoras by journeying into Asia to study
with the Brahmins. But the wars of the time
made such a trip impractical, so Plato turned
to the Egyptians, and, according to the
ancient accounts, was initiated at Sais by
the priests of the Osirian rites. . . . There
is a record in the British Museum that Plato
received the Egyptian rites of Isis and
Osiris in Egypt when he was forty-seven years
old. (Hall, 1967, pp. 145)
Whatever truth there is in this matter, it is clear
that Plato was greatly influenced by the Pythagoreans (and
possibly the Egyptians). See Figure # 1, p.'122, for a
projected chronological outline of Plato's life. Plato is
throughout the dialogues, obstetric with his readers,
continually formulating problems and leaving hints for
their solution. The reader is left to ponder these
problems and, hopefully, abduct adequate solutions to them
As stated earlier, Plato followed the Pythagoreans in
maintaining that the principle elements of things, the One
and Indefinite Dyad, are numbers. Likewise, as Aristotle
has indicated, Plato also identified Forms and sensible
particulars with numbers, though each with a different
class of numbers. The Pythagorean influence of Philolaus
upon Plato should become clear when considering one of the
Philolaic fragments.

81
And all things that can be known contain
number; without this nothing could be thought
or known. (Kirk and Raven, 1975, p. 310)
But what might it mean for Forms and sensible objects
to be numbers? Thomas Taylor has preserved the later
testimony of the Neopythago re = ns Nichomachus, Theon of
Smyrna, Iamblichus, and Boetius, regarding the early
Pythagorean identification of number with things. The
following condensed version is adapted from Taylors'3 book
(T.Taylor, 1983).
According to the later Pythagorean elucidations, the
earliest Pythagoreans subdivided the class of odd numbers
(associated with equality) into incomposite, composite, and
incomposite-composite numbers. The first and incomposite
numbers were seen to be the most perfect of the odd
numbers, comparable to the perfections seen in sensible
things. They have no divisor other than themselves and
unity. Examples are 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,
etc. The incomposite-composites are not actually a
separate class, but merely the relationship between two or
more composite numbers which are not divisible by the same
divisor (other than unity). That is, they are composite
numbers which are incommensurable (in a certain sense) with
one another. For example, 25 and 27 are composite numbers
which have no common factor other than unity. In
approximately 230 B.C., Eratosthenes (see supra pp. 17-20 ),
a later Pythagorean, developed his famous mathematical

82
sieve. It was a mechanical method by which the "subtle"
incomposite numbers could be separated from the "gross,"
secondary composite numbers. These subtle and gross
qualities were likened to the qualities in sensible things.
Likewise, the even numbers (associated with
inequality) were divided into superabundant, deficient, and
perfect numbers, the last of which is a geometrical mean
between instances of the other two kinds. A superabundant
even number is one in which the sum of its fractional parts
is greater than the number itself. For example, 24 is a
superabundant number: 1/2 x 24 = 12; 1/3 x 24 =8; 1/4 x 24
= 6; 1/6x24 =4; 1/12 x 24 = 2; 1 /24 x 24 =1 . The sura of
these parts, 12+8+6+4+2+1 = 33, is in excess of 24.
A deficient even number is one in which the sum of its
fractional parts is less than itself. For example, 14 is a
deficient number: 1/2 x 1 4 3 7; 1/7 x 14 = 2; 1/14 x 14 =
1 . The sum of these parts, 7 + 2 + 1 =10, is less than 14.
A perfect even number is one in which the sum of its
fractional parts is equal to itself. For example, 28 is a
perfect number: 1 /2 x 28 = 1 A ; 1/7 x 28 = 7; 1/7 x 28 = 4;
1/14 x 28 = 2 ; 1 /28 x 28 = 1 . The sum of these parts is
equal to the original number 28. These perfect numbers are
geometric mediums between superabundant and deficient
numbers. Any perfect number multiplied by 2 results in a
superabundant number. Any perfect number divided by 2
results in a deficient number. Furthermore, perfect
numbers are very rare, there being only four of them

83
between the numbers 1 and 10,000: 6, 28, 496, 8,128. The
Pythagoreans saw a "resemblance” between this division of
even numbers into perfect, superabundant, and deficient,
and the virtues and vices of sensible things. Thus, Taylor
records,
Perfect numbers, therefore, are beautiful
images of the virtues which are certain media
between excess and defect. And evil is
opposed to evil [i.e., superabundance to
deficiency] but both are opposed to one good.
Good, however, is never opposed to good, but
to two evils at one and the same time. . . .
[Perfect numbers] also resemble the virtues
on another account; for they are rarely
found, as being few, and they are generated
in a very constant order. On the contrary,
an infinite multitude of superabundant and
deficient numbers may be found . . . [and]
they have a great similitude to the vices,
which are numerous, inordinate, and
indefinite. (T.Taylor, 1983, p. 29)
Aristotle, in the Magna Moralia 1182a11, indicates
that "Pythagoras first attempted to discuss goodness . . .
by referring the virtues to numbers" (Kirk & Raven, 1975,
p. 248). But the above recorded link between numbers and
virtues appears to be limited to resemblance. Certainly
Plato (and the Pythagoreans before him) had something much
stronger in mind. This suggestion of an actual
identification between numbers and Forms, and thereby
sensible things, will become clearer as we proceed.
The Notorious Question
of Mathematicals
Maintaining the Pythagorean mathematical influence of
Plato's philosophy clearly in mind, we will now consider
what ha3 been termed "the notorious question of

84
mathematicals" (Cherniss, 1 94 5 » p. 75). In the Republic,
Plato indicates that the trait of the philosopher is "love
of any branch of learning that reveals eternal reality"
(Republic 485a). The reason then that the mathematical
sciences may be appropriate as a bridge to the Forms
(Idea-Numbers) and ultimately the Good (One), is that their
subject-matter may be eternal. In fact Plato says
precisely this. "The objects of geometrical knowledge are
eternal" (Republic 527b). The question then is whether
these mathematical objects are distinct from the Forms, as
a separate ontological class, or to be identified with the
Forms.
Aristotle indicates that the mathematicals are, for
Plato, a separate ontological class. Most modern
commentators, however, have rejected this notion, at least
that it was in the dialogues. Such diverse schools of
interpretation as those of Cornford, Robinson, and Cherniss
have all agreed in the rejection of a separate class of
mathematicals. For example, Cornford, in reference to the
intelligible section of the Divided Line in the Republic,
states:
Where the intelligible section is subdivided,
clearly some distinction of objects is meant.
I agree with critics who hold that nothing
here points to a class of mathematical
numbers and figures intermediate between
Ideas and sensible things. (Cornford, 1965,
P • 62 )
Most of the attempts to find the mathematicals in the
dialogues have centered around the Divided Line passage in

85
the Republic. Chernies sees the whole question as simply a
matter of "misunderstanding and misrepresentation" on the
part of Aristotle (Cherniss, 1945, p. 25). Robinson sees
it as a deduction in the Republic possible only on the
assumed grounds of exact correspondence between the Cave
simile and the Divided Line. This is an exact
correspondence which he asserts cannot be maintained.
Before dealing with the arguments of Cherniss and Robinson
I will first examine what Aristotle, and then Plato
himself, had to say.
Aristotle clearly sets out Plato's position on
mathematicals.
Further besides sensible things and Forms he
[Plato] says there are the objects of
mathematics, which occupy an intermediate
position, differing from sensible things in
being eternal and unchangeable, from Forms in
that there are many alike, while the Form
itself is in each case unique. (Metaphysics
987b14-18)
Thus, the mathematicals are eternal but partake of
plurality. These mathematicals are given by Plato a
definite ontological status separate from Forms and
sensibles. Thus Aristotle states,
Some do not think there is anything
substantial besides sensible things, but
others think there are eternal substances
which are more in number and more real; e.g.,
Plato posited two kinds of substance--the
Forms and objects of mathematics--as well as
a third kind, viz. the substance of sensible
bodies. (Metaphysics 1028b17-2i)
Further, Aristotle makes it clear that it is these
mathematicals with which the mathematical sciences are

86
concerned. It is "the intermediates with which they say
the mathematical sciences deal" (Metaphysics 997b 1-3).
Now if the mathematical sciences do in fact deal with these
intermediate mathematical objects, then it simply follows
that when we study the mathematical sciences, the objects
of our enquiry are the mathematicals.
Aristotle again makes reference to the Platonic notion
that the objects of mathematics are substances. He does
this by clearly distinguishing the three views on the
question of mathematicals. These are the views of Plato,
Xenocrates, and Speusippus.
Two opinions are held on this subject; it is
said that the objects of matheraatics--i.e.,
numbers and lines and the like--are
substances, and again that the Ideas are
substances. And since (t) some [Plato]
recognize these as two different classes--the
Ideasand the mathematical numbers, and (2)
some [Xenocrates] recognize both as having
one nature, while (3) some others
[Speusippus] say that the mathematical
substances are the only substances, we must
consider first the objects of mathematics.
(Metaphysics "l076a16-23)
It is important to note that Cherniss contends that one
of the reasons Aristotle ascribes a doctrine of
mathematicals and Idea-Numbers to Plato is because he
(Aristotle) mistakenly confused the doctrines of Speusippus
and Xenocrates with those of Plato. And yet in the above
quoted passage there is a clear distinction made between
the doctrines of the three individuals. And furthermore,
it is apparent that Aristotle is writing here as a member
of the Academy, and with reference to doctrines debated

87
therein. Thus, in the same paragraph, six lines later, he
makes reference to "our school," and the fact that these
questions are also being raised outside the Academy. Thus
he says that, "most of the points have been repeatedly made
even by the discussions outside of our school" (Metaphysics
1076a28-29). And this latter is in contradistinction to
the previous discussion of the positions held within the
Academy.
Furthermore, and to the great discredit of Cherniss'
position, why would both Xenocrates and Speusippus uphold a
doctrine of mathematicals in the Academy if this was
entirely foreign to Plato? Certainly, Aristotle as a
member of the Academy for nineteen years could not be so
mistaken in attributing to his master a doctrine that Plato
never held. The variance in doctrine occurred not with
Plato's mathematicals, which both Xenocrates and Speusippus
maintained. Rather, the differences involved the Ideas.
Speusippus apparently rejected the Ideas altogether while
keeping the mathematicals, and Xenocrates lowered the Ideas
down to the ontological status of mathematicals, in the end
identifying the two.
Cherniss, in fact, maintains that Aristotle also held
a doctrine of mathematicals.
Aristotle himself held a doctrine of
mathematicals intermediate between pure Forms
and sensibles, most of the Forms and all the
mathematicals being immanent in the sensible
objects and separable only by abstraction.
(Cherniss, 1945, p. 77)

88
Cherniss apparently finds part of the basis for this latter
assertion in a passage where Aristotle appears to be in
agreement with the Pythagoreans regarding the non-separate
aspect of mathematicals.
It is evident that the objects of mathematics
do not exist apart; for if they existed apart
their attributes would not have been present
in bodies. Now the Pythagoreans in this
point are open to no objection. (Metaphysics
1090a28-3O
But if Cherniss is correct in his assertions, then
the absurd conclusion that follows is that Plato's three
leading pupils held a doctrine of mathematicals in one form
or another, but Plato held none. This is a mistaken view.
What Xenocrates, Speusippus, and Aristotle altered in
attempts to overcome problems they may have perceived in
Plato's doctrine, were not the mathematicals, but rather,
the Ideas. Aristotle saw the universals as abstractions
from sensible particulars, thereby denying separate
ontologial status to the Ideas. Speusippus rejected the
Ideas, while maintaining the mathematicals. And Xenocrates
collapsed the Ideas and mathematicals into one. Surely
Aristotle was correct when he attributed to Plato a
doctrine of intermediate mathematicals.
In another passage Aristotle again clearly
distinguishes the positions of Plato and Speusippus, one
from the other, and from them the position of the
Py thagoreans.
[Plato] says both kinds of number exist, that
which has a before and after being identical

89
with the Ideas, and mathematical number being
different from the Ideas and from sensible
things; and others [Speusippus] say-
mathematical number alone exists as the first
of realities, separate from sensible things.
And the Pythagoreans also believe in one kind
of number--the mathematical; only they say it
is not separate but sensible substances are
formed out of it. (Metaphysics 1080bt1-18)
Aristotle then goes on in the same passage to
distinguish the view of another unknown Platonist from that
of Xenocrates.
Another thinker says the first kind of
number, that of the Forms, alone exists, and
some [Xenocrates] say mathematical number is
identical with this. (Metaphysics 1080b22-23)
The Divided Line
In the Republic, Plato sets forth three related
similes in an attempt to explicate (metaphorically) his
conception of the ascent of the mind (or soul) of the
philosopher-statesman through succeeding stages of
illumination, culminating with the vision of the Good.
These three similes are that of the Sun (Republic
502d-509c), the Divided Line (509d-511e) , and the Cave
(514a-521b ). They are actually analogies intended to
perpetuate the notion of proportion, which is the
underlying bond for Plato. Each simile is indicative of a
process cf conversion. In the Sun and Cave, the conversion
is to greater degress of light. In the Divided Line it is
clearly conversion to higher levels of awareness. The
analogy is between body and soul (mind), for in the end
Plato links the similes, saying,

90
The whole study of the sciences [i.e.,
mathematical sciences] we have described has
the effect of leading the best element in the
mind [soul] up towards the vision of the best
among realities [the Good in the Divided
Line], just as the body's clearest organ was
led to the sight of the brightest of all
things in the material and visible world [in
the Sun and Cave similes]. (Republic 532c-d)
In each case the conversion is from an image to the
original, or cause, of that image. It is clear from these
- •
similes that Plato intends the reader to grasp that he
holds a doctrine of degrees of reality of the
subject-matter apprehended. That is, at each level of
conversion the soul (or mind) apprehends an increased level
of reality, tethering the previous level, just as the eye
apprehends an increased degree of light at each level, even
though the brilliance is at first blinding. Analogously,
through the use of these similes, Plato intends the reader
to apprehend a doctine of degrees of clarity of mind (or
soul). At each level of conversion the mind increases its
degree of clarity. Thus, Plato uses the imagery of light
in these similes to indicate the process of illumination
taking place. Furthermore, each simile should be viewed as
an extension of that which preceded it. Plato makes this
point clear when in the Cave scene he has Socrates 3ay to
Glaucon, "this simile [the Cave] must be connected
throughout with what preceded it [Sun and Line similes]"
(Republic 5^7a-b).
The Divided Line is the most important of the similes
with regard to the issues of mathematicals, abduction, and

91
proportion. There Plato begins by saying, "suppose you
have a line divided into two unequal parts" (Republic
509d). What could Plato possibly mean by this? Here is
the anomalous phenomenon: a line divided unequally. How
do we explain or account for it? From this bare
appearance, what could he possibly mean? What hypothesis
could one possibly abduct that would make a line divided
unequally follow as a matter of course, at least in the
sense of some meaningfulness being conveyed?
Before answering this question, let us look at the
remainder of his sentence for a clue.
Suppose you have a line divided into two
unequal parts, to represent the visible and
intelligible orders, and then divide the two
parts again in the same ratio [logos] . . .
in terms of comparative clarity and
obscurity. (Republic 509d)
We have already established the Pythagorean emphasis of
mathematics in Plato. Up to now not much has been said
about the emphasis on ratio (logos) and proportion
(analogia). But as we proceed, it will become clear that
these notions are critical in Plato’s philosophy. Ratio is
the relation of one number to another, for example 1:2.
However, proportion requires a repeating ratio that
involves four terms, for example, \:2::4:8. Here the ratio
of f:2 has been repeated in 4:8. Thus, in proportion
(analogia) we have a repeating ratio with four terms.
Standing between the two-termed ratio and the four-termed
proportion lies the three-termed mean.

92
The relations among ratio, mean, and
proportion can be brought out by
distinguishing two kinds of mean, the
arithmetic and the geometric. In an
aritmetic mean, the first term is exceeded by
the second by the same amount that the second
is exceeded by the third. One, two, and
three form an arithmetic mean. One, two, and
four, however, form a geometric mean. In a
geometric mean, the first term stands to the
second in the same ratio that the second
stands to the third. Either mean may be
broken down into two ratios, namely, that of
the first and second terms and that of the
second and third. But the geometric mean
alone is defined by ratio, being the case in
which the two ratios are the same; all other
means have two different ratios. Of all the
means, therfore, only the geometric can be
expanded into a proportion, by repeating the
middle term. Furthermore all proportions in
which the middle terms are the same . . . can
be reduced to geomtric means, by taking out
one of the two identical terms. (Des Jardins,
1976, p. 495)
An example of this relation of a geometric mean to
proportion will help to illustrate this. The ratio 1 :2 and
the ratio 2:4 can be related so that we have a proportion.
Thus, t:2::2:4 is our proportion repeating the same ratio.
However, because the middle terms are identical, each being
2, one of the middle 2's can be dropped to establish a
geometric mean: 1:2:4. Here 2 is the geometric mean
between 1 and 4. Whenever a case like this arises in which
a proportion contains two identical middle terms, we can say
we have the peculiar instance of a proportion which contains
only three "different" terms, even though it is true that
one of these terms (i.e., the middle) gets repeated. We
shall nevertheless refer to such a creature as a three-termed
geometric proportion.

93
Now given that we apprehend how three different terms
can make up a proportion, a geometric proportion, is it
possible to somehow arrive at such a proportion with only
two terms? Let us again refer to the Divided Line. Here
we have a line divided into two unequal parts. Hence it
would be reasonable to say that we have two different terms
here in the sense that we have two different lengths of
line. For example, it might be that we have one segment
being 2 units in length and another segment being 4 units
in length. But of course we do not know the number of
units of each segment. All we are told initially is that
we have two unequal segments of line.
Let us inquire before going any further, as to why we
are dealing with a line in the first place. Wouldn't it
have been just as simple to deal strictly in terms of
number, without magnitude entering into the discussion?
Did not the Pythagoreans, whom Plato (as I have argued)
relied so heavily upon, find number (i.e., arithmetic) to
be the mother of the other mathematical sciences? As such,
wouldn't it be easier to work just with primitive numbers
rather than number extended in one dimension? The answer
lies in our first abduction. Why a line? Because lines
alone can be used to represent incommensurable magnitudes.
Des Jardins has also recognized this fact.
Lines as such are neither commensurable nor
incommensurable, but only the incommensurable
magnitudes have to be represented by lines,
for by definition the commensurables can also

94
be represented by numbers. It is only if the
ratio of kinds indicates an incommensurable
division that one needs lines instead of
numbers. (Des Jardins, 1976, p. 485)
This then may be our tentative conjecture, or
18
abduction. It need not be true, but it is a good working
hypothesis. Let us merely keep it in the back of our minds
while we pursue the former question I posed. Is there a
way in which out of the two line segments we can arrive at
not merely some ratio, but a geometric proportion of three
different terms as defined above? The only way to achieve
this would be to consider the whole line as one magnitude,
in relation to the greater segment, and the greater segment
in relation to the lesser segment. Thus we could say as
the lesser is to the greater, so the greater is to the
whole. If we used the magnitudes originally suggested, 2
and 4, we would have 2:4: :4:6. But this is a false
assertion of a geometric proportion. If we had 2:4::4:8,
we would have a geometric proportion, because the two
middle terms are identical. However, the two segments of 2
and 4 when added together do not equal 8, but rather 6.
The sad fact is that no matter what commensurable whole
numbers we attach to the Line, the proportion will not come
out correct. Although, in a later section (ifra p.149 ff)
we will see how the terms of the Fibonacci series
approximate it.
rfhat do we do? tfe return to our original abduction,
i.e., that Plato has chosen a line because lines are

95
necessary to represent incommensurable magnitudes. Wow the
answer should spring into our minds. There is absolutely
one and only one way to divide a line, such that the lesser
is to the greater in the same ratio as the greater is to
the whole. This is the division in mean and extreme ratio.
This is none other than that which has come to be called
the golden cut or golden section. The full abduction then
is: given the mere fact that we are presented with a line
divided, that is, a line divided in a particular ratio
which will then be cut twice again by Glaucon to perpetuate
the porportion, the hypothesis that the original cut is the
golden cut, is the most reasonable one to set forth.
Furthermore, the fact that Plato is essentially a
Pythagorean interested in the ratios, proportions, and
harmonies of the Cosmos, makes it ludicrous to suggest that
he intended that the line merely have an initial
indiscriminate division. As I have taken pains to show, we
are dealing in the Academy with some of the most refined
mathematician-philosophers of all time. We add to this the
fact that Plato is obstetric in all that he does. He may
not have clearly set out his deepest doctrines, but they
are there for those who have eyes to see. His method was
to formulate the problem, the object of his pupils was then
to reason backwards to, or abduct, a hypothesis that would
make the anomalous feature (set out in the problem) follow
as a matter of course. I will consider this matter of the

96
golden section more extensively later, but for now let us
return to the Divided Line.
Referring to Figure # 2(p* 123 ), the initial division
at C separates the Intelligible sphere of knowing
(episteme) BC, from the sensible or Visible sphere of
opinion (doxa) AC. The lower line (doxa) is then itself
divided into belief (pistis) DC and conjecture (eikasia)
AD. Furthermore, this last division is expressly
distinguished by different objects. These objects are then
purported to be in the
relation of image [AD] to origininal [CD] . .
. the same as that of the realm of opinion
[A C ] to that of knowledge [bc]. (Republic
5J0A)
Thus, just as the reality of the visible world is of a
lesser degree than, and stands in relation to, the greater
reality of the intelligible world, in like manner the
reality of the subject-matter of eikasia stands to the
subject-matter of pistis. By analogy then, dianoia will
stand in a similar relation to noesis.
Plato then goes on to distinguish the subsections of
the upper line, not expressly in terms of subject-matter,
but only inferentiaily. Rather, the distinction is made as
to how the mind operates in the two subsections, though it
nevertheless is true that he infers a separate set of
objects for CE, the mathematicals, distinct from those of
BE. The gist of the matter is that in the state of
dianoia, the "students of geometry and calculation and the

97
like," use the objects of pistis such as visible diagrams
as images (images of what is not clearly stated), and using
various assumptions proceed downwards "through a series of
consistent steps to the conclusion which they set out to
find" (Republic 5t0d).
However, in the state of noesis (BE), the assumptions
are treated
not as principles, but as assumptions in the
true sense, that is, as starting points and
steps in the ascent to something which
involves no assumption and is the first
principle of everything. (Republic 51fb)
Plato has Socrates then go on to say,
. . . the whole procedure involves nothing in
the sensible world; but moves solely through
forms to forms and finishes with forms.
(Republic 5tfb-c)
Thus, the objects of noesis are clearly Forms.
Although Plato does not specifically assert the
existence of a separate class of objects for dianoia, he
does have Socrates say two important things. First, the
geometers who operate at the intermediate level of awareness
make use of and argue about visible figures
[objects of CD], though they are not really
thinking about them but the originals
[objects of CE]which they resemble. . . . The
real objects of their investigation being
invisible except to the eye of reason
[dianoia]. (Republic 510d)
Secondly,
. . . you may arrange them [the four states
of mind] in a scale, and assume that they
have degrees of clarity corresponding to the
degree of truth possessed by their
subject-matter. (Republic 5*te)

98
The big question is whether the subject-matter of
dianoia and noesis are different, or whether they are the
same, i.e., Forms, but simply "seen" in light of differing
degrees of intensity at the two levels of awareness. When
the analogy (proportion) between the four levels is
compounded with the assumption of correspondence between
Line and Cave, it is reasonable to abduct the hypothesis of
a separate set of objects (presumably the matheraaticals)
corresponding to the intermediate level of awareness,
dianoia. Robinson has most succinctly summarized the
reasoning as follows:
[in the Cave simile] the unreleased prisoners
regard the shadows and echoes as originals.
They take them, not as means of knowing a
reality beyond them, but as themselves the
only reality to be known. It follows that
'conjecture' [eikasia] is not trying to
apprehend originals through their images but
taking the image for the original. Adding
this result to Plato's statement that thought
[dianoia] is to intelligence [noesis] as
conjecture [eikasia] is to conviction
[pistis], we obtain the probable conclusion
that thought or dianoia is also a form of
taking the image for the original. Since the
original is in this case the Idea (for the
object of intelligence or noesis is said to
be Ideas), we infer that thought [dianoia]
takes images of the Ideas for the realities
themselves. But these images cannot be what
Plato usually points to as images of the
Ideas, namely the world of Becoming, for that
is clearly the object of "conviction." What
can they be? And how tantalizing of Plato
not to say! (Robinson, t953, P* 181)
If we follow the prior evidence of Plato's emphasis
upon mathematics within the Academy and its crucial role as
a bridge-study for the philosopher-statesman in the

99
Republic, and further, consider Aristotle's assertion that
Plato held a doctrine of intermediate mathematical which
are like the Forms in being eternal; and then, combine this
with Plato's remarks that the state of awareness of dianoia
corresponds to that of the mathematician, it would appear
that this set of objects corresponding to dianoia would be
the mathematicals. This would be the obvious abductive
conclusion, though one might ask the question as to why
Plato does not explicitly state it as such within the
dialogues, especially the Republic. For there he simply
says ,
The relation of the realities correspoding to
knowledge [episteme] and opinion [doxa] and
the twofold divisions into which they fall we
had better omit if we're not to involve
ourselves in an argument even longer than
we've already had. (Republic 534a)
Why the mystery? But may it not be the case that this
is at the center of the oral tradition within the Academy,
a tradition that has its roots in Plato's Pythagoreanism?
Again, taking into consideration Plato's statements in the
Phaedrus and 7th Letter regarding written doctrine, along
with the one place where Aristotle clearly expresses a
difference in doctrine between the dialogues and Plato’s
oral presentation over the question of participation (a
difference based upon mathematical principles), may we not
very plausibly infer that Plato's doctrine of mathematicals
is a central doctrine not to be explicitly, though
inferentially, exposed within the dialogues? In fact, even

100
in the later period dialogues, the mathematicals are not
explicitly exposed.
If Plato is acting as a midwife in the dialogues, as I
contend he is, then he has planted sufficient seeds for the
reader to reasonably abduct a hypothesis of intermediate
mathematicals. Robinson mistakenly maintains that the
mathematicals must be deduced from an exact correspondence
between Line and Cave. And further, he argues that if the
exactness of the correspondence breaks down, then any
supposed deduction of mathematicals is invalid (Robinson,
1953). But the requirement of deduction is too onorous.
The conjectural notion of abduction is much more
appropriate, and is consistent with Plato's underlying
method. Furthermore, deduction or synthesis follows only
upon the result of arriving at the hypothesis through
abduction or analysis. Hence, "exact" correspondence
between Line and Cave is not necessary. However, the
question of correspondence of Line and Cave is relevant,
and therefore, it is appropriate to consider Robinson's
argument.
Robinson's argument is essentially that if there were
an "exact" correspondence between Line and Cave, then the
state of the unreleased prisoners in the Cave would be
equivalent to the initial stage of the Line, eikasia, and
the indiviual would at that stage be taking images for
originals. And furthermore, the move from eikasia to

101
¿.istia would be analogous to the initial release of the
prisoners, turning from the shadows to the objects wnich
have cast them. If this were not the case, Robinson
argues, then it would break the back of the correspondence
argument from which the mathematicals may be "deduced"
(Robinson, 1 953, p« 183). Regarding this latter point,
Robinson argues as follows:
There is one point at which the correlation
must break down completely whatever
interpretation we assume of the debated
questions; and Professor Ferguson has shown
very forcibly what it is [see Ferguson, 1 934,
p. 203]. If there were a precise
correlation, the state of the unreleased
prisoner would have to be "conjecture"
(eikasia), and the state immediately
succeeding his release would have to be the
adjacent state in the Line, namely pistis.
But pistis which means conviction or
confidence and refers at least primarily to
our ordinary attitude to "the animals about
us and all that grows and everything that is
made" [Republic 510a], bears no resemblance
to the prisoner's condition immediately after
his release; for the latter is expressly
described as bewilderment and as the belief
that his present objects are less real than
his previous objects [Rebublic 515d]. In
view of this observation we must say that
Plato's Cave is not parallel to his Line,
even if he himself asserts that it is.
(Robinson, 1953* p« 183)
Is Robinson's contention true? To answer this let us
first look at the state of eikasia. Plato intends that we
understand by eikasia (AD), in addition to mere perceptual
illusions, a purposefully distorted set of objects or
notions which create a false conjecture, or illusion in the
mind of the recipient. It is the realm of false opinion in
which the copies are distortions of the facts (originals)

102
of the realm of pistis (CD). This level is one of
corruption. It is, in fact, like the bound prisoners of
the Cave simile, who see only shadows and never the causes
of those shadows (Republic 514a). It is the state of mind
of those who have been corruped by sophists, rhetoricians,
politicans, imitative artists, false poets and dramatists.
It is the realm of those involved in "their shadow battles
and struggles for political power" (Republic 520c). Here
imitative art deludes the individual, as Plato points out
in Book 10 of the Republic. It is an "inferior child
[level AD] born of inferior parents [level CD] (Republic
602c ).
Hamlyn links eikasia with the state of the bound
prisoners of the Cave scene, and further, identifies the
move from eikasia to pistis with the freeing of the
prisoners from chains as a result of Socratic enquiry
[elenchus] (Hamlyn, 1958, p. 22). However, I would add to
this that the initial move, as well as containing the
negative Socratic element, also contains a positive
Platonic element, and is not completed until the individual
apprehends a new object of reference, an object
corresponding to pistis (CD).
Hamlyn points out that AD is a caricature of the
philosophical views of a Protagoras or a Hume.
•
The position of the prisoners in the Cave
[bound to AD] is not that of the ordinary
man but of a sophist like, Protagoras, as
Plato sees him . . . [or] those who have been
corrupted by the sophist. . . . this view
makes more plausible the view that the first

103
stage in the ascent towards illumination is
being freed from chains. It is not chains
that bind the ordinary man unless he has been
corrupted by others; the most xhat can be
said of him is that he is lacking
sophistication. The trouble with the
sophists and their disciples is, according to
Plato, that they do possess sophistication,
but sophistication of the wrong sort.
Consequently the first stage in their
treatment is freeing them from their false
beliefs so that they can return to the
ordinary unsophistication which can itself be
a prelude to the acquisiton of "true
sophistication." The first stage is ordinary
Socratic practice. (Hamlyn, t 958, p. 22)
These false beliefs do have their objects of
reference. They are distortions of facts, like their
originals in that they pertain to factual material, but
unlike them in that they are purposefully distorted. The
individual bound to the level of awareness of eikasia
accepts these false beliefs as though they were true
beliefs based on real facts. Thus, distortions or illusory
images of the originals are accepted in place of the
originals. Hence, contrary to Robinson's first point (see
supra), the state of eikasia is to be identified with the
state of the unreleased prisoners in the Cave. In both
cases the individual is accepting images for the originals.
The acceptance of these false beliefs is overturned by
breaking the hold that the illusory images have on the mind
of the individual. This is done through the Socratic
elenchus. Sntailments of one's belief are drawn into
question, as are the supposed factual underpinnings upon
which it is based. The more positive Platonic aspect is
begun when a new set of objects of reference, i.e., the

104
originals, are displayed to the individual. It is the
apprehension of these new objects of reference (and all of
their logical relations) which brings about a new state of
awareness, pistis. This is depicted in the Cave simile
when the formerly bound prisoner turns toward the objects
which cast the shadows. Though at first "he would be too
dazzled to see properly the objects of which he used to see
only the shadows" (Republic 5T5d).
Robinson's second point (following Ferguson) is that
the initial move in the Line from eikasia to pistis cannot
lead to bewilderment as it does in the Cave. But is this
true? Is it not the case that at each level of conversion
there is an initial state of bewilderment and confusion,
both before the eyes (in the case of the visible realm) or
the mind's eye (in the case of the intelligible realm) can
be adjusted to the new level of reality? Take as a purely
hypothetical example the case in which an individual has
been corrupted by the influence of sophistical politicians.
These politicians have actually instilled in the
individual's mind (through falsified evidence) the false
belief that the assassination of a former President of the
United States was the personal effort of an extreme
left-wing advocate of communist policy with Russian
anti-capitalistic connections. Further, our subject has
been persuaded that the assassin acted alone, shooting the
President with a single shot. And further assume for the
moment that other corrupt individuals (politicians and

105
judges) falsified and distorted evidence which enhanced
the conviction in the mind of the individual that these
were actual facts and that his corresponding opinions,
doxa, were unquestionably true. If our subject is then
presented with the "real facts" that indicate that the
assassin was actually a member of a group of assassins who
were solicited by high government and military officials,
it will not lead simply to a change of opinion. Further,
assume that sufficient data are revealed in terms of
government documents, tape recordings, films, public
testimonies and admissions. Is not the individual going to
be pressed into an initial state of bewilderment? Surely,
if we may assume for the moment that sufficient tangible
facts are revealed and the truth of the matter is unfolded,
rhe individual will abandon his false conjecture and take
on a new belief. But it is not a mere change of opinion;
rather, it is a completely new situation. In the first
instance, the individual was purposefully given falsified,
illusory data (objects of eikasia), actual distortions of
the real facts. And, correspondingly, his mind held a
distorted and illusory conjecture. In the second instance,
entailments and underpinnings of the false conjecture were
drawn into question by a new set of "real facts" or
evidence. This, in turn, led to the rejection of the false
hypothesis which the individual had been led to accept by
the sophistical influence of the corrupt politicians. This
results in an initial state of perplexity, as prior

106
convictions are broken down with an associated destruction
of faith in politicians and certain governmental
institutions. The perplexity precedes the actual
acceptance of a new hypothesis (or belief) based on the new
set of "real facts," i.e., the originals of pistis. The
important point is that any move from eikasia to pistis
involves this initial state of bewilderment , contrary to
the aforementioned contention of Robinson (Robinson, 1953)
following Ferguson.
This initial state of confusion is, in fact, evident
at each level of conversion, whether one is considering the
perception of the visible realm through the eyes or the
apprehension of intelligibles by the mind's eye. It
appears that both Robinson and Ferguson have ignored a very
relevant passage where Plato explicitly makes this point:
Anyone with any sense will remember that the
eyes may be unsighted in two ways, by a
transition either from light to darkness or
from darkness to light, and will recognize
that the same thing applies to the mind. So
when he sees a mind confused and unable to
see clearly he will not laugh without
thinking, but will ask himself whether it has
come from a clearer world and is confused by
the unaccustomed darkness, or whether it is
dazzled by the stronger light of the clearer
world to which it has escaped from its
previous ignorance. (Republic 518a-b)
Thus, Robinson is incorrect in both of his assertions.
But he further argues that it will be impossible to discern
in the Cave the four states found in the Line. But is this
notion of "exact correspondence" necessary? If I am

107
correct that Plato is concerned with ratio and proportion,
then we should examine the relation of Sun to Line to see
if there is a ratio which is then perpetuated in terms of
the relation between Line and Cave. Referring to Figure #2
we notice that the Sun simile contains an initial twofold
division into the Intelligible and Visible realms. In the
Line, each of these realms is then further bifurcated.
This gives us a fourfold division into noesis, dianoia,
pistis, and eikasia. This gives the ratio 2:4. If this
ratio is to be extended into the Cave scene to create a
geometric proportion then we would expect there to be an
eightfold division there. This would give the geometric
proportion of 2:4::4:8. However, because the middle term
repeats itself, we can collapse the formula to 2:4:8.
Hence, this would mean that the ratio of 2:4 is perpetuated
through the similes. But 2:4 is really reducible to 1:2.
or the relation of the One to the Dyad.
Looking at the Cave scene, there does appear to be a
plausible eightfold division. The first stage would be
that of the shackled prisoners who view only the shadows
projected onto the wail, and take them to be the real
(Republic 514b-515c). The second stage would occur when
the prisoner is unshackled, and turning, views the objects
themselves which had cast the shadows (515c-d). The third
stage would occur when he looks directly at the light of
the fire (515e). The fourth stage would occur when he is
forcibly dragged out of the cave into the sunlight.

108
Although
his eyes would be so dazzled by the glare of
it that he wouldn't be able to see a single
one of the things he was now told were real
(515e-5t 6a ).
The fifth stage would occur when he views the shadows
(516a). The sixth stage would occur when he looks "at the
reflections of men and other objects in water (516a). The
seventh stage would occur when he looks at the objects
themselves (516a). And the eighth stage would occur when
he looks at the planets and stars in the light of the moon
at night ( 516a-b ).
One might legitimately ask at this point as to why
there is no stage corresponding to the vision of the sun
itself? The answer lies in the fact that if we return to
the Line, we notice that even though there is a final
vision of the Good, it does not act so as to make a fifth
division of the Line. Rather, the Good (or One) 3its at
the summit of noesis. Analogously, the sun in the Cave
scene sits at the summit of the upward ascent. And
conformably to the nature of the Good in the Line, there is
no extra stage set out in the Cave. Just as the Good is
the last upward move in the Line and the foundation for
certainty (505 a&d), so the vision of the sun in the Cave
scene, "must come last" (516b). Hence, on this view we
have the ratio of the One to the Dyad, 1 :2 , perpetuated
through the three similes in the geometric proportion 2:4:8.
Robinson, however, goes on to assert that the changes
in the Cave do not

109 .
fit precisely . . . the states enumerated in
the Line. Plato's intention seems rather to
describe a single continuous change
terminated in both directions. (Robinson,
1953, P- 182)
To this he adds the remark that the change is "infinitely
divisible within those bounds" (Robinson, 1953, p• 182).
The notion that the change is terminated in both directions
is correct. However, the whole process of conversion must
be conceived of as a finite number of discrete steps ending
with the contemplation of that which sits at the summit.
This is supported by the evidence that Plato abhorred the
concept of an infinitely divisible continuum. His response
was to postulate the existence of indivisible lines.
Otherwise one would be continually going through
conversions without ever reaching the summit, if one could
even get started in the first place, much as in Zeno's
paradox of the stadium (see Physics 239b11-12 & Topics
160b7-8). Thus, Aristotle says,
Further, from what principle will the
presence of the points in the line be
derived? Plato even used to object to this
class of things as being a geometrical
fiction. He gave the name of principle of
the line--and this he often posited--to
indivisible lines. (Metaphysics 992a19-22)
And as Furley has argued,
Aristotle did attribute to Plato a theory of
indivisible lines, and he did understand this
to mean that there is a lower limit to the
divisibility of extended magnitude. (Furley,
1967, p. 107)
What does Plato himself say about the correspondence
between Sun, Line, and Gave. Plato has Socrates say to

110
This simile [the Cave] must be connected
throughout with what preceded it. The realm
revealed by sight corresponds to the prison,
and the light of the fire in the prison to
the power of the sun. And you won’t go wrong
if you connect the ascent into the upper
world and the sight of the objects there with
the upward progress of the mind into the
intelligible region. (Republic 517b-c)
This is clearly an unequivocal connection of Cave and Line,
placing them in correspondence.
Now, there is a very big difference between denying
"exact correspondence" of Line and Cave, on the one hand,
and straightforward correspondence of Line and Cave, on the
other. Our first effort is surely to understand the
intentions of Plato, not to construct a theory and then
forcibly fit him into it. Sometimes commentators will go
so far as to literally deny Plato's very words. Thus,
Robinson says, "we must say that Plato's Cave is not
parallel to his Line, even if he himself asserts that it
is" (Robinson, 1953, p. 183)! At other times they will
reject the words of Plato's most prolific disciple,
Aristotle, as being mere misunderstanding and
misrepresentation. This was the case with Cherniss, as
cited earlier (see supra., p. 85).
Plato's Cave and Line may not be in exact
correspondence, but they are in correspondence, as Plato
intends them to be, and in fact, states that they are.
This is true irrespective of whether or not ray eightfold
division of the Cave is correct. This correspondence

Ill
provides sufficient grounds upon which to base the
abductive inference to the hypothesis of mathematicals. In
the first place both Cave and Line are terminated in both
directions (up and down). In both similes the
starting-point and ending-point are the same. That is to
say, the state of the bound prisoner in the Cave taking the
shadows as the real objects is the analogue of the state of
eikasia in the Line in which the individual holds false
conjectures having accepted as real the distorted images of
supposed facts presented him by corrupt politicians,
sophists, rhetoricians, etc. And the culminating vision
of the sun in the Cave scene is the analogue of the final
vision of the Good in the Line. In the second place, in
both Line and Cave the ascent is made by a series of
conversions from what is in each case a copy of less
reality (that has been accepted as the real) to an original
of greater reality. In particular we have seen that the
conversion from eikasia to pistis may be identified with
the release and turning around of the prisoner to face a
new set of objects of reference, the originals of the
formerly accepted distorted images.
The important point is that there is adequate
correspondence between Line and Cave to assert that each
level of awareness accepts a copy of a higher reality with
the conviction that it is the real, right on up until we
arrive at the truly real, the unhypothetical first

112
principle, the Good. This results primarily from the
identification of eikasia with the unreleased prisoners
watching the shadows, on the one hand, and the geometrical
proportion existing between the sections of the Divided
Line.
The argument may be stated in the following way.
Where L is eikasia, M is the state of the bound prisoners,
N is taking a copy for the original:
t. L=M,
2. M has property N,
3. Therefore, L has property N.
Then assume pistis is X, dianoia is Y, and noesis is Z.
Taking a copy as "the real" is relative to the original of
which it is a copy. Thus, eikasia takes its object of
reference relative to the original in pistis. The
proportion involved is L:X::Y:Z. If L has the property N
in relation to X, then Y has the property N in relation to
Z. Eikasia does in fact have this property N in relation
to pistis, hence, it follows that dianoia has the property
N in relation to noesis. What the argument leads to is the
inference that there is a set of objects of reference for
dianoia which are copies of the Forms of noesis, and which
thereby are distinct ontologically from the Forms in that
they partake of a lesser degree of reality. Prior to the
conversion to the state of noesis, these copies are
accepted as the real. What objects occupy this space?
They must be eternal like the Forms because they also exist
in the Intelligible region. There must be some other

113
criterion of distinction. If we now consider that the
state of awareness of dianoia is identified with that of
the mathematician, then it is reasonable to infer that the
objects we are looking for would be those with which the
mathematician is concerned. These are not the diagrams
which he draws (the objects of pistis). These diagrams are
mere images of the intelligible mathematical objects upon
which the mathematician is now focussed. Thus, dianoia is
most probably concerned with non-sensible mathematial
objects. What the argument leads to is the need for a set
of objects which are intelligible (hence eternal), yet
which possess less reality than their originals the Forms.
If I am correct in asserting that mathematics provides
the scaffolding for Plato's philosophy, and further that
the Forms are numbers and mathematical concepts (i.e.,
numbers, lines, planes, solids, equality, etc.), and if the
objects of dianoia are images of these Idea-Numbers, then
they will themselves be mathematical objects or concepts.
The question then is whether the mathematicals
attributed to Plato by Aristotle fulfill the requirements
of this class of objects corresponding to dianoia.
Aristotle's assertions seem to indicate that they do.
Further besides sensible things and Forms he
says there are the objects of mathematics,
which occupy an intermediate position,
differing from sensible things in being
eternal and unchangeable, from Forms in that
there are many alike, while the Form itself
is in each case unique. (Metaphysics
987bt 4-Í 3)

114
Thus, both Forms and mathematicals being eternal, the
latter differ from the former in their plurality. On the
hypothesis that the Forms are Idea-Numbers, there is no
difficulty in conceiving of the mathematicals as images of
the Forms. The mathematicals simply participate to a
greater degree in plurality, possibly in the Indefinite
Dyad. It is their plurality which distinguishes the
mathematicals from the Forms. They clearly fill the role
left open by the inference of a less real class of objects
which are images of the Forms.
Furthermore, Plato clearly indicates that "the objects
of geometrical knowledge are eternal" (Republic 527b).
Geometrical knowledge is a subclass of mathematical
knowledge. Plato says that the state of awareness of a
mathematician is dianoia. Thus, the mathematician
operating at the level of dianoia has eternal objects as
the referents of his mathematical knowledge. Now Plato
says that "different faculties have different natural
fields" (Republic 487a-b). But the objects of dianoia are
eternal, though they can't be the Forms, for the reasons
just suggested. Again the argument leads to the abductive
suggestion of a separate class of eternal objects,
differing from the Forms. Furthermore, it follows from the
fact that the state of dianoia is the state of awareness of
the mathematician, that the appropriate objects of
reference will be mathematical. Thus, there appears to be
the need for a class of eternal mathematical objects,

115
distinct from the Forms. Aristotle attributes this
ontologically intermediate class of objects to Plato, and
it appears to satisfy all of the requirements brought out
in the Republic.
Alternatively one can arrive at the need for an
ontologically separate class of mathematicals without
appealing to any correspondence between Line and Cave at
all. The derivation is contained within the Line itself in
relation to Plato’s statement about the lover of knowledge.
Our true lover of knowledge naturally strives
for reality, and will not rest content with
each set of particulars which opinion takes
for reality. (Republic 490b)
Thus, doxa, in general, tends to take its objects
(images relative to the objects of episteme in general) as
reality. One of the primary purposes of the Divided Line
is to transmit the basic features of the relation between
Itelligible and Visible worlds (and of course man's
epistemological relation to those worlds) into the
subsections of this major division. Thus, just as man
tends to take the sensible particulars (images of the
objects of the Intelligible region) as real, so within the
subsections he at different stages takes the objects in
eikasia (images of the objects of pistis) for the real, and
analagously takes the objects of dianoia (images of the
Forms) for the real. It must be remembered that in each
set of circumstances what is "the real" is relative to the
level of reality of the prior and posterior ontological and

116
epistemological levels. Employing the notation used in the
Divided Line Figure #2 (p. 123), we have,
AC:BC::AD:CD::CE:BE. That is to say, as doxa is to
episteme, so eikasia is to pistis, and so dianoia is to
noesis. Thus, just as doxa is taking an image (relative to
the original of episteme) for the real, so dianoia is
taking an image (relative to the original of noesis) for
the real.
Plato employs other metaphors to set forth the
analogous positions of eikasia and dianoia, thereby
indicating that in both cases one is taking an image of the
original for the real. Thus, as regards dianoia he says,
. . . as for the rest, geometry and the like,
though they have some hold on reality, we can
see that they are only dreaming about it.
(Republic 533b-c)
Analogously, those tied to the world of conjecture, more
specifically to the sophistical contentions involved in the
state of eikasia, are also dreaming. Only when the
philosopher-king descends from his vision of the Good will
the spell of eikasia be broken, and the polis awaken from
its slumber and not be "merely dreaming like most societies
today, with their shadow battles and their struggles for
political power" (Republic 520c).
If my argument is correct, there is sufficient
material within the Republic to abduct the hypothesis of an
ontologicaliy intermediate level of mathematical objects.
But there is really no clear statement regarding them.

117
There is one passage though that is very suggestive, where
Socrates says to Glaucon, if one were to ask the
arithmeticians
"this is very extraordinary--what are these
numbers you are arguing about whose
constituent units are, so you claim, all
precisely equal to each other, and at the
same time not divisible into parts?" What do
fou think their answer would be to that?
Glaucon to Socrates:] I suppose they would
say that the numbers they mean can be
apprehended by reason [dianoia], but that
there is no other way of handling them.
(Republic 526a)
In a footnote to Repbulic 526a, Lee makes the
following remarks,
The language of the previous paragraph
[Republic 524d-e] ("number themselves," "the
unit itself") is that of the theory of Forms.
It is less clear what are the numbers
referred to in this sentence. Some have
supposed them to be entities intermediate
between forms and particulars. . . . But
though Plato did hold some such view later in
his life, this sentence is very slender
evidence for them in the Republic. (Lee,
T980, p. 333)
This is not an atypical reaction. But where does one
distinguish between the earlier and later view as regards
the mathematicals. Aristotle makes no such distinction in
his writings.^ However, he dose provide a further clue as
to what these numbers made up of precisely equal units and
yet indivisible may be. In contrasting ideal and
mathematical number he says, "if all units are associable
and without difference, we get mathematical number"
(Metaphysics 108ta5-6). Then in the same passage thirteen

118
lines later he says, “mathematical number consists of
undifferentiated units" (Metaphysics t081 at 9)» Thus, Plato
indicates that these numbers are the referential objects of
the state of dianoia (Republic 526a). and Aristotle
unequivocally indicates that they are to be identified with
mathematical number (i.e., mathematicals) in contrast to
Ideal-Numbers. When these two points are combined with the
abduction of mathematical objects in the Republic, we have
what appears to be further support that Plato had a
doctrine of mathematicals operative at this stage of his
work, though he does not openly pronounce it in writing as
such. But then does he ever do so in the dialogues? If
Plato was to make such a dramatic change in later life,
then why doesn't he indicate it in the last dialogues?
There is no radical change. On the other hand, he does not
give up the mathematicals later either. Thus the talk of
the intermediate in the Philebus (I5a-17a) and the role of
the soul in the Laws (396e) does lend support to a doctrine
of intermediate mathematicals. Nevertheless, there is no
straightforward explicit statement regarding them in any of
the dialogues?® On the other hand, Plato has not left the
matter out of his dialogues entirely. Reference to them
can be found in other middle dialogues besides the
Republic. References occur in both the Phaedo and Timaeus.
Of course, not all agree that the Timaeus is to be dated as
a middle dialogue. However, irrespective of its dating, it
does refer to the mathematicals. In my view, the more

119
critical later dialogues, including Theaetetus, Parmenides,
and the Sophist, invoke a need for the intermediate
mathematicais and the corresponding epistemological state
to help solve some of the paradoxical problems posed
therein. The crucial point is that the Republic inferences
appear to provide very strong support for the hypothesis
that Plato, at least as early as the Phaedo and Republic,
had an underlying doctrine of intermediate mathematicais.
Combining this evidence further with Plato's statements in
the 7th Letter regarding disparagement of written doctrine,
and the mathematical nature of his unwritten lecture, On
the Good, it appears reasonable to abductively conjecture
that this was in the main part of an unwritten, esoteric
tradition within the Academy. Hence, Plato was following
the procedure of silence of his earlier Pythagorean
predecessors. But it is a doctrine not inconsistent with
the subject-matter of the dialogues, Cherniss' arguments to
the contrary, but rather a means of elucidating,
clarifying, and possibly solving some of the perplexing
problems contained therein.
Notes
^The Quadrivium was the four-fold Pythagorean division
of mathematics into arithmetic, music, geometry, and
spherics (or astronomy). Plato divided geometry into plane
and solid, thereby making the Quadrivium into a Pentrivium.
2
The exceptions are few. They include D'Arcy Wentworth
Thompson and Gregory Des Jardins.
3
Additionally, the Receptacle is identified as the
Great and Small, Space, and the Bad where Good is realized.

120
^This will be seen to be central to the understanding
of Plato's epistemology and ontology.
^This is still a problem in physics today, and is
referred to as the "one body problem" and the "many body
problem."
^Although it is clear that the members of the Academy
were primarily mathematicans, with the possible exception
of Aristotle who did not enjoy the the emphasis on
mathematics.
^The great leaps of science are due to creative
abduction initiated by the presence of anomalous phenomena
or theoretical dilemmas.
g
It is the combination of these two factors, analysis
and the golden section, that is central to Plato’s actual
concerns.
9
This shows how closely Aristotle agreed with Plato on
this matter.
^It is probable that the reason Speusippus, rather than
Aristotle, succeeded Plato, is because the former loved
mathematics whereas the latter somewhat disparaged it.
â– 'â– â– ''Mathematical numbers will generally be referred to as
the intermediate "raathematicals" throughout the
dissertation.
â– '"^Heath disputes this (1 957).
13
It is important to note that Proclus refers to
analysis rather than division (dihairesis) as the finest
method handed down. Some commentators, for example Gadaraer
(1980), place too much emphasis on division as being the
essence of dialectic. To the contrary, it is an important
tool to be employed in dialectic, but not a substitute for
analysis. As Cherniss correctly notes, "[diahairesisJ
appears to be only an aid to reminiscence of the Ideas"
(Cherniss, 1945, p. 55).
14
In lectures I attended at the University of London in
1 976.
â– 'â– ^What follows sounds somewhat like an eastern doctrine
of yoga. Of course, the Neoplatonic tradition maintains
that Plato travelled widely, as Pythagoras had done before
him.
^The Timaeus is generally considered late because it
appears to make reference xo problems already brought up in

121
the Sophist, a dialogue certainly later than the middle
period group. My own view is that the Timaeus is late, and
hence, shows that the mathematicals were maintained into
Plato's later life.
^I take these dialogues to consist of problems
presented to the reader that he might abduct the Platonic
solutions.
1 ft
This is a very simple but powerful abduction.
1 Q
Any distinction is strictly the concoction of some
commentators.
90
These are merely allusions and inferences to them.

122
Birth of Plato, 429-427
Charmides
Laches
Euthyphro
Hippias Major
Meno
First visit to Sicily, 389-388
[Plato begins teaching at Academy, 388-387]
?Cratylus
Symposium, 385 or later
[Plato's initiation at Sais, Egypt at age of 47, 382-380]
Phaedo
Republic
Phaedrus
Parmenides
Theaetetus, 369 or later
[Aristotle enters Academy]
Second visit to Sicily, 367-366
Sophistes
Politicus
Third visit to Sicily, 361-360
Timaeus
Critias
Philebus
Seventh Letter
Laws
[Epinomis]
Death of Plato, 348-347
Figure # 1: Plato Chronology (after Ross, 1951, p, 10)

123
INTELLIGIBLE SPHERE
(episteme)
VISIBLE SPHERE
(doxa)
A
State of Mind Subj ect-Matter
B
NOESIS FORMS
DIANOIA
MATHEMATICALS
VISIBLE SPHERE
(doxa)
PISTIS
EIKASIA
ORIGINALS
DISTORTED
COPIES
Figure # 2
Divided Line

CHAPTER IV
THE GOLDEN SECTION
Timaeus
Plato's doctrine of the intermediate nature of the soul
is very closely related to his doctrine of the intermediate
mathematicals with their corresponding state of dianoia.
It is, in fact, the mathematicals embedded at this
intermediate position in the soul that are infused into the
bodily receptacle or space, providing it with number,
ratio, and form. It is also the reason the philosopher can
recollect, from within, the Forms. That is, the nature of
the Idea-Numbers are embedded in the soul via the
mathematicals. Hence, by turning to the mathematical
disciplines, as shown in the Republic, the soul (or mind)
is able to reawaken this knowledge, and ascends up the path
of dianoia to noesis, where through a series of conversions
it finally contemplates the Idea-Numbers, and ultimately
the One Itself.
The dual functions of the mathematicals in the soul
are then apparent. Ontologicaily, they solve the problem
of how the sensible world participates in the intelligible
world. The mathematicals in the world 30ul are infused
into the sensible world. Each individual thing
124

125
participates in, and is assimilated toward, the Forms
through this intermediate soul. Epistemologically, the
mathematicals in the soul solve the problem of how we can
come to know the Forms. It is through contemplation of the
likenesses, i.e., the mathematicals within, that we are
able to attain knowledge of the Idea-Numbers. This then is
how the "worst difficulty" argument of the Parmenides is
o ve rcome.
In the Parmenides an argument is set forth which has
the consequences of making it impossible for an individual
to have knowledge of the Forms. The argument may be
reconstructed as follows:
I. Each Form has real Being just by itself.
II. No such real Being exists in our world
(of sensible things).
III. These Forms have their Being in
reference to one another, but not with
reference to the sensible likenesses in our
world.
IV. Things in our world which bear the same
names as the Forms, are related among
themselves, but not to the Forms.
V. Knowledge itself is knowledge of the real
Being in Forms.
VI. Knowledge in our world is knowledge of
sensible things, and has no relation to
Knowledge itself or the other Forms.
Therefore, we cannot have knowledge of
the Forms. (Parmenides 133a-134e)
The argument gains its strength on the view that
sensible things and Forms are completely separate. But
this only follows upon a superficial view of Plato's
philosophy. It is dependent upon the completely bifurcated
picture of the Platonic worlds of Being and Becoming. As
Cornford points out, "it is this separation of the Forms

126
from their instances which threatens to isolate them in a
world of their own, inaccesible to our knowledge”
(Cornford, t940, p. 95). But Plato hints that there may be
something wrong with the premises of the argument.
The worst difficulty will be this. . . .
Suppose someone should say that the forms, if
they are such as we are saying they must be,
cannot even be known. One could not convince
him that he was mistaken in that objection,
unless he chanced to be a man of wide
experience and natural ability, and were
willing to follow one through a long and
remote train of argument. Otherwise there
would be no way of convincing a man who
maintained that the forms were unknowable.
(Parmenides 133b)
The essence of the solution revolves around the
intermediate nature of the soul and raathematicals.
However, the solution has already been suggested in the
decidedly earlier middle dialogue, the Phaedo. There we
find that man is "part body, part soul" (Phaedo 79b). The
"soul is more like the invisible, and body more like the
visible" (Phaedo 79b).
The soul is most like that which is divine,
immortal, intelligible, uniform,
indissoluble, and ever self-consistent and
invariable, whereas body is most like that
which is human, mortal, multiform,
unintelligible, dissoluble, and never
self-consistent. (Phaedo 80b)
Thus, because man is part soul and part body, it follows
that man has access to the Forms via the soul. Thus, the
premises of complete separation of the intelligible and
sensible worlds are suspect.
In the Timaeus the doctrine of the intermediate soul
is established more fully
And in the center [of the

127
Cosmos] he put the soul, which he diffused throughout the
body" (Tjjnaeus 34b). Thus, the entire body of the Cosmos
participates in the Forms through the world soul which is
"diffused throughout" it.
[Deity] made the soul in origin and
excellence prior to and older than the body,
to be the ruler and mistress, of whom the
body was to be the subject. . . . From the
being which is indivisible and unchangeable
[i.e., ousia], and from that kind of being
which is distributed among bodies [i.e.,
genesis] he compounded a third and
intermediate kind of being. He did likewise
with the same and the different, blending
together the indivisible kind of each with
that which is portioned out in bodies. Then
taking the three new elements, he mingled
them all into one form, compressing by force
the reluctant and unsociable nature of the
different into the same. When he had mingled
them with the intermediate kind of being and
out of three made one, he again divided this
whole into as many potions as was fitting,
each portion being a compound of the same,
the different, and being. (Timaeus 34c-35b)
Thus, the soul in its intermediate position is
compounded out of the material of both the intelligible and
sensible worlds. As such it touchs, so to speak, both of
the worlds. Because of this the individual has access to
knowledge of the Forms. This process of assimilation to
the higher knowledge begins to occur at the intermediate
state of awareness of dianoia, and the contemplation of the
mathematicals. "A silent inner conversation of the the
soul with itself, has been given the special name of
dianoia" (Sophist 263d). Dialectic then takes one through
noesis to the direct contemplation of the Forms.

128
However, besides the epistemological role of dianoia
in relation to the intermediate soul and the raathematicals,
the soul and mathematicals also play a very important
ontological role. The soul is responsible for the infusion
of number, ratio, and proportion into the bodily receptacle
of space. My argument is that the soul infuses the
mathematicals as copies of the Forms, into the bodily
receptacle of the Cosmos, producing ordered extension,
ordered movement, and the various qualities related to
number. This is how the sensible particulars participate
in the Forms. It is through the mathematicals infused by
the soul into the visible world to give it form.
Ross, at one point in his work, recognized this
intermediate role of the soul and its mathematicals.
It would be a mistake to describe Plato as
having ... at any stage of his development,
made a complete bifurcation of the Universe
into Ideas and sensible things. For one
thing we have the casual reference to "equals
themselves" j~ P h a e d o 74c]--an allusion to
mathematical entities which are neither ideas
nor sensible things, an allusion which paves
the way for the doctrine of the
Intermediates. (Ross, 1 951 , pp. 25-26)
There is an additional suggestion of mathematicals in
the Timaeus. There referring to the receptacle or space,
Plato says:
She is the natural recipient of all
impressions, and is stirred and informed by
them, and appears different from time to time
by reason of them. But the forms which enter
into and go out of her are the likeness of
eternal realities modeled after their

129
patterns in a wonderful and mysterious
manner, which we will hereafter investigate.
(Timaeus 50c)
Ross also sees this passage as a reference to
mathematicals. He explains that for Plato sensible things
were moulded in space, "by the entrance into it of shapes
which are likensses of the eternal existents, the Forms"
(Ross, Aristotle's Metaphysics, vol. 1, pp. 167-168).
Cherniss, of course, takes issue with this interpretation.
He claims that Ross, "mistakenly takes ta eisionta kai
exionta of Timaeus 50c to be a class distinct from
sensibles as well as Ideas" (Cherniss, 1945, p. 76, fn.79).
But this must be argued by Cherniss because he has rejected
Aristotle's remarks about Plato's intermediate
mathematicals doctrine as mere misinterpretation and
misunderstanding on Aristotle's part. To the contrary, on
my view, this passage is very suggestive of Plato's
underlying mathematical doctrine.
Proportion
What then were these mathematicals infused via the
world soul into the spatial receptacle? The answer is
ratios (logos) and proportions (analogia). This was the
Pythagorean conception of Plato, that the order and harmony
of number was infused into the sensible world.
[Deity] made the body of the universe to
consist of fire and earth. But two things
cannot be rightly put together without a
third; there must be some bond of union
between them. And the fairest bond is that

130
which makes the most complete fusion of
itself and the things which it combines, and
[geometrical] proportion is best adapted to
effect such a union. For whenever in any
three numbers, whether cube or square, there
is a mean, which is to the last term what the
first term is to it, and again, when the mean
is to the first term as the last term is to
the mean--then the mean becoming first and
last, and the first and last both becoming
means, they will all of them of necessity
come to be the same, and having become the
same with one another will be all one.
(Timaeus 3tb-32a)
What Plato is referring to is none other than a
geometrical proportion. He is depicting the relation
A:B::B:C. As an example we might have, 1:2::2:4, or
alternatively, 1:3::3:9« This is the kind of proportion he
feels is best adapted to be the bond of union making the
Cosmos one. But this single mean is sufficient only for
plane figures or square numbers. If one proceeds to solid
figures and cube numbers, then two means are required.
If the universal frame had been created a
surface only and having no depth, a single
mean would have sufficed to bind together
itself and the other terms, but now, as the
world must be solid, and solid bodies are
always compacted not by one mean but by two,
God placed water and air in the mean between
fire and earth, and made them to have the
same proportion so far as was possible--as
fire is to air so is air to water, and as air
is to water so is water to earth--and thus he
bound and put together a visible and tangible
heaven. And for these reasons, and out of
such elements which are in number four, the
body of the world was created, and it wa3
harmonized by proportion. (Timaeus 32a-c)
Thus, the resulting continued geometric proportion has
the following relation: fire : air :: air : water :: water
: earth. Expressing this numerically we have:

131
1 :2 : : 2 : 4 : : 4 : 8. Eight is of course a cube number, being the
cube of 2. Alternatively, we could have: 1:3::3:9::9:27.
Again we end with a solid as 27 is the cube of 3-
Geometric proportion was considerd to be the primary
proportion because the other proportions (i.e., harmonic
and arithmetic) require it, but it does not require them.
Thus Corford writes,
[Adrastus] says that geometrical proportion
is the only proportion in the full and proper
sense and the primary one, because all the
others require it, but it does not require
them. The first ratio is equality (1/1), the
element of all other ratios and the
proportions they yield. He then derives a
whole series of geometrical proportions from
"the proportion with equal terms" (t,t ,1)
according to the following law: "given three
terms in continued proportion, if you take
three other terms formed of these, one equal
to the first, another composed of the first
and the second, and another composed of the
first and twice the second and the third,
these new terms will be in continued
proportion." (Cornford, 1956, PP« 47-48)
The double and triple proportions that Plato
establishes in the world soul can be seen immediately to
follow from this formula. Thus, we get 1 for the first
term, 1+1=2 for the second term, and 1+2+1=4 for the third
term. This gives the proportion of 1:2:4, or the double
proportion through the second power or square. To extend
the double proportion further into the third power or cube
number, we 3imply multiply the resulting second and third
terms, 2x4=8.
In a similar manner Adrastus' formula can be employed
to arrive at Plato's triple proportion, by beginning with

132
the first three units of the double proportion (i.e.,
t ,2,4). Thus, 1 is the first term, 1+2=3 is the second
term, and 1+(2x2)+4=9 is the third term. This gives the
triple proportion through the second power or square
number, 1:3:9* By multiplying the resulting second and
third terms (3x9), we arrive at the third power or cube
number, 27. Hence, we arrive at Plato's triple proportion
of I : 3 : 9 :27.
The doctrines of the Timaeus are clearly a
perpetuation of Pythagorean doctrine.
It is well known that the mathematics of
Plato's Timaeus is essentially Pythagorean. .
. . [in passage 32a-b] by planes and solids
Plato certainly meant square and solid
numbers respectively, so that the allusion
must be to the theorems established in Sucl.
VIII, 11, 12, that between two square numbers
there is one mean proportional number and
between two cube numbers there are two mean
proportionals. (Heath, 1956, vo1. 2, p. 294)
Then the world soul which has been compounded out of
being and becoming is divided into a harmonious relation
very much along the same lines as Plato mentioned earlier
regarding the proportions of the 4 elements (Timaeus
3tb-32c). However, it is left to the reader to make the
connection (abductive leap) between the geometric
proportion of the four elements at Timaeus 32a-c, on the
one hand, and the proportionate division of the world soul
at Timaeus 35b-36b, on the other.
And he [Deity] proceeded to divide after this
manner. First of all, he took away one part
of the whole [l], and then he separated a
second part which was double the first [2],
and then he took away a third part which was

133
half as much again as the second and three
times as much as the first [3], and then he
took a fourth part which was twice as much as
the second [4], and a fifth part which was
three times the third [9], and a sixth part
which was eight times the first [a], and a
seventh part which was twenty-seven times the
first [27]. After this he filled up the
double intervals [that is, between 1,2,4,8]
and the triple [that is, between 1,3,9,27],
cutting off yet other portions from the
mixture and placing them in the intervals, so
that in each interval there were two kinds of
means, the one exceeding and exceeded by
equal parts of its extremes [as for example,
1 , 4/3, 2, in which the mean 4/3 is one third
of 1 more than 1, and one third of 2 less
than 2], the other being that kind of mean
which exceeds and is exceeded by an equal
number. Where there were intervals of 3/2 1 a
fifth] and of 4/3 [fourth] and of 9/8 [tone],
made by the connecting terms in the former
intervals, he filled up all the intervals of
4/3 with the interval of 9/8, leaving a
fraction over, and the interval which this
fraction expressed was in the ratio of 256 to
243. And thus the whole mixture out of which
he cut these portions was all exhausted by
him. (Timaeus 35b-36b)
In this manner Plato has placed £n the soul the odd
and even numbers, and their perfect relationships in terms
of ratio and proportion. In this way he has included the
harmonic and arithmetic means within the primary geometric
proportion. Thus, in terms of the even numbers we have the
following geometric proportion, embodying two geometric
means and ending in a solid number: 1, 4/3(harmonic),
3/2(arithmetic), 2, 8/3, 3, 4, 16/3, 6, 8. In this series,
1,2,4, and 8, are in continued geometric proportion.
Likewise, the odd numbers find their expression as follows:
1, 3/2, 2, 3, 9/2, 6, 9, 21/2, 18, 27. Here the numbers,
1,3,9, and 27, are in continued geometric proportion. In

134
this way the soul comprehends within itself the fundamental
proportions of the Cosmos. Furthermore, the soul, by being
infused into the bodily receptacle, provides sensible
things with number, ratio, and proportion. These numbers
cover a range of 4 octaves and a major 6th. It is of note
that the major 6th will be seen to involve a Fibonacci
approximation to the golden section. As Adrastus puts it,
Plato is looking to the nature of things.
The soul must be composed according to a
harmonia and advance as far as solid numbers
and be harmonised by two means, in order that
extending throughout the whole solid body of
the world, it may grasp all the things that
exist. (Cornford, 1956, p. 68)
It is in this way that the body of the world was
"harmonized by proportion" (Timaeus 32c). But these
numbers in the soul are linked to the proportional
relationship of the four elements, which he has alluded to
earlier (Timaeus 31b-32c). To this point we have only
considered the proportions relating to commensurable
numbers. Does Plato have a similar embodiment of the
golden section in the Timaeus as I have claimed he does in
the Divided Line of the Republic? Let us turn first for a
moment to a consideration of the Epinomis.
Taylor & Thompson on the Spinomis
This argument will require a somewhat long and
circuitous route. However, it will be valuable in aiding
the strength of my position concerning the relevance of the
golden section to Plato's philosophy of number and

135
proportion. I will be considering, in particular, the
arguments of A.E. Taylor and D.W. Thompson. In the
Epinomis Plato suggests that "the supreme difficulty is to
know how we are to become good men [people]" (Epinomis
979c). Furthermore, it is the good man who is happy. To
be good, a soul must be wise (Epinomis 979c). Thus Plato
asks, "What are the studies which will lead a mortal man to
wisdom" (Epinomis 973b)? The answer is "the knowledge of
number" (Epinomis 976e). He then argues that if one were
utterly unacquainted with number, it would be impossible to
give a rational account of things. But if no rational
account is possible, then it is impossible to be wise.
But if not wise, then it is impossible to become perfectly
good. And if one is not perfectly good, then he is not
happy (Epinomis 977c-d).
Thus there is every necessity for number as a
foundation . . . all is utterly evacuated, if
the art of number is destroyed. (Epinomis
977d-e ) .
This passage from the Epinomis appears to contain
echoes of the Pythagorean Philolaus.
And all things that can be known contain
number; without this nothing could be thought
or known. (Philolaus in Kirk and Raven, 1975,
p. 310)
It is therefore evident to the very last dialogue that
Plato is essentially a Pythagorean.
It should be noted here that I take the Epinomis to be
one of Plato's works. Even it it were true that it was
edited or written by Plato's pupil Philip, it nevertheless

136
is Platonic doctrine. Taylor brings this out when he says
. . . my own conviction is that it is genuine
and is an integral part of the Laws. Those
who have adopted, on the slenderes t of
grounds, the ascription to Philippus of Opus
[or of Medma] at least recognise that the
author is an immediate scholar of Plato,
specially competent in mathematical matters,
and that the work was issued from the first
along with the Laws. ... We may accept the
matter of a mathematical passage from the
dialogue as genuinely Platonic with
reasonable confidence. (Taylor, 1926, p. 422)
Now the "art of number" referred to at Epinomis 977e
is not limited to knowledge of the commensurable integers.
The philosopher must also attain knowledge of the
incommensurabies. Referring now to the Thomas translation
of the Epinomis:
There will therefore be need of studies
[matharaata]: the first and most important is
of numbers in themselves, not of corporeal
numbers, but of the whole genesis of the odd
and even, and the greatness of their
influence on the nature of things. When the
student has learnt these matters there comes
next in order after them what they call by
the very ridiculpus name of geometry, though
it proves to be an evident likening, with
reference to planes, of numbers not like one
another by nature; and that this is a marvel
not of human but of divine origin will be
clear to him who is able to understand.
(Epinomis 990c-99tb, Thomas, 1957, vol. 1, p.
4 err)
Again there is the cryptic suggestion that he who has
assimilated his mind near to deity, will be able to
apprehend the deeper meaning involved. The reference to
"numbers not like one another by nature" is to
incommensurabies. As Thomas has pointed out:
The most likely explanation of "numbers not
like one another by nature" is "numbers

137
incommensurable with each other"; drawn as
two lines in a plane, e.g. as the side and
diagonal of a square, they are made like to
one another by the geometer's art, in that
there is no outward difference between them
as there is between an integer and an
irrational number. (Thomas, 1 957, vol. 1 , p.
401, fn.b)
Taylor places great emphasis on this passage from the
Spinomis. He feels that it supplies a clue as to why Plato
made the apeiron (the infinite or space)' of the
Pythagoreans into an indefinite dyad (or greater and
smaller). In the Philebus (24a ff.), Plato has Socrates
use the old Pythagorean antithesis of apeiron and peras
(limit or bound). However, Aristotle indicates that the
apeiron came to be the greater and smaller for Plato. As
Taylor says,
Milhaud, Burnet and Stenzel are all, rightly
. . . agreed, that the Platonic formula is
somehow connected with the doctrine of
"irrational" numbers. . . . Again, when
Milhaud, Burnet, Stenzel, all look for the
explanation of the formula in the conception
of the value of an "irrational," they are
plainly on the right track, as the passage of
the Epinomis [990c-99tb] . . . demonstrates.
(Taylor, 1 926 , p. 421)
They were on the right track but they did not go deep
enough in their interpretations. Taylor does go deeper:
When Plato replaced the Pythagorean apeiron
by the "duality" of the "great and small," he
was thinking of a specific way of
constructing infinite convergent series which
his interpreters seem not to have identified.
(Taylor, 1926, p. 422)
The fifth century B.C. philosophers had been concerned
with the value of the square root of 2, whereas the fourth
century B.C. was concerned with the Delian problem and the

138
cube root of 2. The method which was employed, and alluded
to in the Epinomis, was the technique of making increasingly
more accurate rational approximations to the number's
irrational value. As Taylor points out,
. . . there is a general rule given by Theon
of Smyrna (Hiller, p. 43f • ) for finding all
the integral solutions to the equation, or,
as the Greek expression was, for finding an
unending succession of "rational diameters,"
that is, of increasingly accurate rational
approximations to the square root of 2, the
"ratio of the diagonal to the side." The
rule as given by Theon is this. We form two
columns of integers called respectively
"sides" and "diagonals." In either column we
start with 1 as the first term; to get the
rest of the "sides," we add together the nth
"side" and the nth "diagonal" to form the
(n+t)th "side";in the column of "diagonals,"
the (n+t)th "diagonal" is made by adding the
nth "diagonal" to twice the nth side.
Fortunately also Proclus . . . has preserved
the recognised demonstration of this rule; it
is a simple piece of geometry depending only
on the identity (a+b)^ + bl = 2(a/2) + 2(a /2+b)
which forms Euclid's proposition II, 10.
(Taylor, 1926, pp. 428-429)
In effect by dividing the diagonal number by the side
number, one obtains successively closer approximations to,
or a "rational value" for, the square root of 2. Some
interesting features emerge when one considers the side and
diagonal method of approximation. First is the fact that
at each step of convergence, the new convergent term is
nearer to the desired limit value. Second, the convergents
alternate in being "rather less and rather greater than the
value to which they are approximations" (Taylor, 1926, p.
430). To illustrate, 7/5 is less than the square root of
2. 2.
2, since 7 = 2x5 - 1; 17/12 is greater than the square

139
root of 2, 172* = 2x1 ¿ + 1 (Taylor, 1 926, p. 430). Third,
. . . the interval, or absolute distance,
between two successive "convergents" steadily
decreases, and by taking n sufficiently
large, we can make the interval between the
nth and (n+t)th convergent less than any
assigned rational fraction s, however small,
and can therefore make the interval between
the nth convergent and the required
"irrational" smaller still than s. (Taylor,
1926, p. 431)
Fourth, ". . . the method is manifestly applicable to any
'quadratic' surd, since it rests on the general formula
-b) = (a-b^ ) /(^a + b) " (Taylor, 1 926, p. 431). Fifth,
. . . we are not merely approximating to a
"limit," we are approximating to it from both
sides at once ; & is at once the upper limit
to which the series of values which are too
small, 1 , 7/5, • • • are tending, and the
lower limit to which the values which are too
large, 3/2, 17/12, . . . are tending. This,
as it seems to me, is manifestly the original
reason why Plato requires us to substitute
for the apeiron as one thing, a "duality" of
the great and small. f2 is an apeiron,
because you may go on endlessly making closer
and closer approximations to it without ever
reaching it; it never quite turns into a
rational number, though it seems to be on the
way to do so. But also it is a "great and
small" because it is the limit to which one
series of values, all too large, tends to
decrease, and also the limit to which another
series, all too small, tends to increase.
(Taylor, 1926, pp. 431-432)
On this view Taylor concludes that the meaning of the
Epinomis passage (referred to) is to evaluate all quadratic
surds, and in so doing discover their appropriate side and
diagonal converging series. D.W. Thompson accepts Taylor's
view to this point and then elaborates upon it. Thompson
argues that "ho arithmos" is to be taken in its technical
sense as surd.

140
. . . ho arithmos may be used here in its
technical sense, meaning a surd or
"irrational number," especially lT2; and the
general problem of Number may never have been
in question at all. It was the irrational
number, the numerical ratio (if any) between
two incommensurable segments, which was a
constant object of search, whose nature as a
number was continually in question, and whose
genesis as a number cried aloud for
explanation or justification. (Thompson,
1929, PP- 43-44)
Thompson's contribution is to go beyond Taylor's
insight into the nature of the great and small. Thompson
shows how the One acted as an equalizer relative to the
greater and smaller.
. . . that the side and diagonal numbers show
us what Plato means by the Great-and-Small,
or Aristotle by his Excess-and-Defect, is
certain; Prof. Taylor [Taylor 1926 & 1927]
has made it seem clear and obvious. But
Prof. Taylor has not by any( ^ana made it
clear what Plato meant by T& £V .... it is
another name for that Unit or "Monad" which
we continually subtract from the "Great" or
add to the "Small," and which so constructs
for us the real number. (Thompson, 1929, p.
47)
This will become more evident by examining the side
and diagonal numbers for the approximation to {?.
Sides Diagonals
1 1
2 3
5 7
12 17
etc.
What Thompson noticed was that
the striking and beautiful fact appears that
this "excess or defect" is always capable of
being expressed by a difference of 1 . The
square of the diagonal number (i.e., of what

141
Socrates calls the "rational diagonal"
[rational diameter, Republic 546c] is
alternately less or more by one than the sum
of the square of the sides." (Thompson, 1929,
p. 46)
This may be depicted in the following table
2x1* = lZ+t
2X22, = 3Z-1
2x5a‘ = 72+1
2x122 =172-1
etc.
Thus, the alternating excess and defect is in each case
measured by one unit. Taylor continues:
For Unity then comes into the case in a
twofold capacity. It is the beginning arch?
[*P **] of the whole series. Then again as
the series proceeds, the "One" has to be
imported ^ntp each succeeding Dyad, where it
defines the amount of excess or
defect, and equates or equalises ( t two incompatible quantities. (Thompson, 1929,
pp. 46-47)
Thus, Thompson sees "the One as the continual
'equaliser' of the never-ending Dyad" (Thompson, 1929, p.
50). The use of side and diagonal numbers were therefore
employed by the Greeks to gain rational approximations to
the value of surds. It is true that the successive
convergente never end. However, the difference between
successive convergents can be made to be less than any
predetermined value. Taylor writes:
They never actually meet, since none of the
"convergents" is ever the same as its
successor, but, by proceeding far enough with
the series we can make the interval between
two successive "convergents" less than any
assigned difference, however small. (Taylor
quoted in Thompson, 1929, p. 47)

142
The use of side and diagonal numbers appears to have
been central to the Greek quest for a rational account of
irrational numbers. The relation of the One and Indefinite
Dyad on the hybrid Taylor-Thompson account does much to
clarify the matter. However, Thompson proceeds further.
He argues that there is another table "which may be just as
easily, or indeed still more easily derived from the first,
and which is of very great importance" (Thompson, 1929, p.
50). But there is no mention of it in the history of Greek
mathematics. My view is that the reason there is no
mention of it is because it lay at the center of the early
unwritten Pythagorean tradition, at a time when silence was
taken very seriously.
We remember that, to form our table of side
and diagonal numbers, we added each
side-number to its own predecessor, that is
to say, to the number standing immediately
over it in the table, and so we obtained the
next diagonal; thus we add 5 to 2 to get 7.
(Thompson, 1929, pp. 50-51)
Hence we would get the following Í2 table:
side diagonal
1 1
2 5
5 7
12 17
etc.
But now instead of adding the two side numbers 5 and 2
to get the diagonal 7, we add each side number to the
previous diagonal to get the new diagonal. Thus, for
example, we add 5 to 5 to get 8. The new side numbers are
arrived at in the same manner as the previous method.

143
However, we now come up with a very interesting table.
1
1
2
3
5
8
1 3
21
34
55
89
144
etc.
This is none other than the Fibonacci series. This is
the famous series . . . supposed to have been
"discovered" or first recorded by Leonardo of
Pisa, nicknamed the Son of the Buffalo, or
"Fi Bonacci." This series has more points of
interest than we can even touch upon. It is
the simplest of all additive series, for each
number is merely the sum of its two
predecessors. It has no longer to do with
sides and diagonals, and indeed we need no
longer write it in columns, but a single
series, 1,1,2, 3 ,5 ,8, "I 3 ,21, etc. (Thompson,
1 929, p. 50
But Thompson the morphologist saw even more deeply
into the significance of this series. Thus he said:
It is identical with the simplest of ail
continued fractions. . . . Its successive
pairs of numbers, or fractions, as 5/3, 8/5,
etc., are the number of spirals which may be
counted to right and to left, on a fir-cone
or any other complicated inflorescence. . . .
But the main property, the essential
characteristic, of these pairs of numbers, or
fractions, is that they approximate rapidly,
and by alternate excess and defect, to the
value of the Golden Mean. (Thompson, 1929,
pp. 51-52)
Thus far Thompson has penetrated into the Platonic
arcanum. The positive value of the golden section is
(/5 + !)/2. This incommensurable, placed in abbreviated
decimal notation is 1 . 61 80339 ••• • If we take the Fibonacci
series, 1 ,1 ,2 , 3,5,8,1 3,21 , 34,55,89,1 44,233,etc. and form a

144
fraction out of each set of two succeeding terms, placing
the first term in the denominator, and the second term in
the numerator, we get the following series of fractions:
1/1,2/1,3/2,5/3*8/5,13/8,21/13, etc. i/hen looking at the
corresponding decimal values, note that the first and
succeeding alternate terms are deficient, the second and
succeeding even terms are in excess. The terms of the
series asymptotically continue to get closer to the limit
value of the golden section. The values are 1, 2, 1.5,
1.666..., 1.6, 1.625, 1.6153846..., 1.6190476...,
1.617647..., 1.6181818..., 1.6179775..., 1.6180555....
Thus, by the time we arrive at the value of 233/144,
1.6180555••., we are very near to the value of the golden
section. In other words, these fractions derived from this
simplest of additive series, the Fibonacci series,
asymptotically converge upon the golden cut in the limit.
I will again quote Thompson extensively because I
believe that he has penetrated deeply into the mystery.
The Golden Mean itself is, of course, only
the numerical equivalent, the
"arithmetisation," of Euclid Elements 1
11.11 ; where we are shown how to divide a
line in "extreme and mean ratio," as a
preliminary to the construction of a regular
pentagon: that again being the half-way
house to the final triumph, perhaps the
ultimate aim, of Euclidean or Pythagorean
geometry, the construction of the regular
dodecahedron, Plato's symbol of the Cosmos
itself. Euclid himself is giving us a sort
of algebraic geometry, or rather perhaps a
geometrical algebra; and the [Fibonacci]
series we are now speaking of
"arithmeticises" that geometry and that
algebra. It is surely much more than
coincidence that this [Fibonacci] series is

145
closely related to Euclid 11.11, and the
other [V"2 series] (as Theon expressed it), to
the immediately preceding proposition [Euclid
II.to]. (Thompson, 1929, p. 52)
It is my contention that it certainly is not a
coincidence for Euclid 11.10 and 11 to be grouped together,
and further for each of these propositions to correspond
respectively to the 12 and Fibonacci series. This is the
Pythagoreanism of Plato extended through the writings of
one of his Academic descendente, Euclid. And it is not
insignificant that the 11th proposition of Book II of
Euclid is preparatory to the two dimensional construction
of the pentagon, and ultimately to the three dimensional
construction of the embodiment of the binding proportion of
the Cosmos itself, the dodecahedron.
Since the students of the history of Greek
geometry seem agreed that the contents of
Euclid II are all early Pythagorean, there is
no reason why the rule given by Theon [side
and diagonals for V"2 ] should not have been
familiar not only to Plato, but to Socrates
and his friends in the fifth century. The
probability is that they were acquainted with
it, and thus knew how to form an endless
series of increasingly close approximations
to one "irrational," ^2*. (Taylor, 1 926, p.
429)
But by extension of this argument regarding the early
Pythagorean contents of Book II of Euclid's Elements, the
case of 11.11 , is covered as well. It is also early
Pythagorean in origin. As such it was part of the
esoteric, unwritten tradition. The so-called discovery by
Leonardo of Pisa regarding the Fibonacci series is at best
a rediscovery. And most probably he is given credit

146
because he is the first to expose openly the esoteric
tradition by putting it in writing.
In Euclid 11.11, we have a line AG divided in extreme
and mean ratio at B. See Figure # 3 , p. 147. We thus
have the three magnitudes AC, AB, and BC. The relations
2,
are such that (AB) = ACxBC. From the Fibonacci series we
can replace the geometrical magnitudes with any three
consecutive numbers from the series. As Thompson points
out:
. . . the square of the intermediate number .
. . [isJ equivalent--approximately
equivalent--to the product of the other two.
Observe that, precisely as in the former case
j_ i. e. , side and diagonal numbers for 2], the
approximation gets closer and closer; there
is alternate excess and defect; and (above
all) the "One" is needed in every case, to
equate the terms, or remedy the defective
approximation. (Thompson, 1929, p. 52)
Thus the "One" plays the role of modifying the excess
and deficiency of the successive squares. This will become
more clear through the following illustration. Where
AC = 21 , AB=13, and BC=8, we have the approximate proportion
of 21 ; 13 : : I 3:8. But 21/13 = 1 .61 53846..., and 13/8 =
1.625. Hence the relation is not exact. How then does the
"one" act as an equalizer in the view of Thompson? Taking
2.
the formula AB = ACxBC, we get the following:
1 32’ = (21x8) + 1
169 = 168 + 1
Again we can substitute any three consecutive numbers of
the Fibonacci series. In each case we will discover that
the "One" acts as an equalizer. But again it does so

147
i
21
Figure # 3: Golden Cut 4 .-ibonaeci Approximation

148
through the equalizing alternately of a unit of excess and
a unit of deficiency.
= ( 5x2 ) -1
51 = (8x3)+1
QZ = (13x5)-1
132 = (21x8)+1
etc.
It is in this manner that we discover more meaning in
the latent suggestions in the Epinomis of the method of the
geometer's art in discovering the nature of surds or
incommensurables. There are echoes of it in the Meno and
the 'Theaetetus. And it is not a coincidence that these
Pythagorean doctrines were later published in sequence
(i.e., Elements, Book 11.10 & 11) by the Platonic disciple,
Euclid.
It is inconceivable that the Greeks should
have been unacquainted with so simple, so
interesting and so important a series
[Fibonacci series]; so closely connected
with, so similar in its properties to, that
table of side and diagonal numbers which they
knew familiarly. Between them they
"arithmeticise" what is admittedly the
greatest theorem, and what is probably the
most important construction, in all Greek
geometry. Both of them hark back to themes
which were the chief topics of discussion
among Pythagorean mathematicians from the
days of the Master himself; and both alike
are based on the arithmetic of fractions,
with which the early Egyptian mathematicians,
and the Greek also, were especially familiar.
Depend upon it, the series which has its
limit in the Golden Mean was ju3t as familiar
to them as that other series whose limit is
l/T. (Thompson, 1 929, p. 53)

149
This provides further evidence that Plato and the
other Pythagoreans of the time were familiar with the
golden section. But we must inquire more fully to see just
what significance the golden section may have had for
Plato's philosophy.
*
and the Fibonacci Series
Why then might Plato feel that the golden section was
so important? We shall look at some of the properties of
the golden mean for the answer.
The "Golden Section," (1 + \/5*)/2 = 1 . 61 80339• • • ,
positive root of the equation x2- = x + t , has a
certain number of geometrical properties
which make it the most remarkable algebraical
number, in the same way asTT and £ are the
most remarkable transcendendent numbers.
(Ghyka, 1 946, p. 2 )
Here are some of the golden section's properties.
í> = (/5 + t)/2 = 1 . 61 80339875....
1.618... is a very accurate approximation to 1/0 = (ÍT-O/2 = 0.618... = ^-1
<$>*â–  = (V5+3)/2 = 2.618... = |) + 1
= j) +j)
The geometrical series known as the "golden series,"
in which the golden section ratio is the fundamental
module , 1 , <|>, J)*’ , , f)** , . . . $)*', possesses the remarkable
property of being at the same time both additive and
geometrical. It is an additive series in that each term is
the sum of the two preceding terms, for example =4^ +|) •

150
The Fibonacci series
tends asymptotically towards the {)
progression with which it identifies itself
very quickly; and it has also the remarkable
property of producing "gnomonic growth" (in
which the growing surface or volume remains
homothetic, similar to itself) by a simple
process of accretion of discrete elements, of
integer multiples of the unit of accretion,
hence the capital role in botany of the
Fibonacci series. For example, the
fractionary series 1/1, 1/2, 2/3, 3/5, 5/8,
8/13, 13/21, 21 /34 , 34/55 , 55/89, 89/1 44,...
appears continually in phyliotaxis (the
section of botany dealing with the
distribution of branches, leaves, seeds),
specially in the arrangements of seeds. A
classical example is shown in the two series
of intersecting curves appearing in a ripe
sunflower (the ratios 13/21,21/34,34/55, or
89/144 appear here, the latter for the best
variety). The ratios 5/8,8/13, appear in the
seed-cones of fir-trees, the ratio 21/34 in
normal daisies. If we consider the
disposition of leaves round the stems of
plants, we will find that the characteristic
angles or divergencies are generally found in
the series 1/2, 1/3, 2/5, 3/8, 5/13, 8/21,
1 3/34, 21 /55 (Ghyka, 1 946, pp. 1 3 — 1 4)
As seen earlier, the golden section ratio existing
between the two parts of a whole, such as the two segments
of a line, determines between the whole and its two parts a
continuous geometrical proportion. The ratio between the
whole to the greater part is equal to the ratio between the
greater to the smaller part. We saw this earlier in the
Divided Line. Thus, the simplest asymmetrical section and
the corresponding continuous proportion is that of the
golden section. Of course the successive terms of the
Fibonacci series can be employed as an approximation to the
golden cut.

151
To illustrate and review, the typical continuous
geometrical proportion is a:b::b:c, in which b is the
proportional (geometrical mean) between a and c. In the
geometrical proportion b becomes the "analogical invariant
which besides the measurement brings an ordering principle"
(Ghyka, 1 946, p. 2). This analogical invariant transmits
itself through the entire progression as a characteristic
ratio, acting as a module.
But one can obtain a greater simplification by
reducing to two the number of terms, and making c=a+b.
Thus, taking the Divided Line, where the two segments are
designated a and b, and the whole line is of length c, if
they are fit into the formula, then the continuous
proportion becomes: a:b::b:(a+b). This may be translated
2,
into (b/a) = (b/a)+1. If one makes b/a=x, it is quickly
seen that x, positive root of the equation xa = x+1, is equal
to (V^+0/2, the golden section. Thus, the golden section
is the most logical and therefore the simplest asymmetrical
division of a line into two segments. This result may also
be found by employing "Ockham's Razor," for which the
reader may consult Ghyka's work (Ghyka, 1946, p. 3).
It is interesting to note that the golden section (and
series), and its corresponding approximation, the Fibonacci
series, are found throughout nature.
The reason for the appearance in botany of
the golden section and the related Fibonacci
series is to be found not only in the fact
that the {) series and the Fibonacci series

152
are the only ones which by simple accretion,
by additive steps, can produce a "gnomonic,"
homothetic, growth (these growths, where the
shapes have to remain similar, have always a
logarithmic spiral as directing curve), but
also in the fact that the "ideal angle"
(constant angle between leaves or branches on
a stem producing the maximum exposition to
vertical light) is given by [the formula]
(a/b)=b/(a+b), (a + b ) =360" ; one sees that b
divides the angular circumference (360*2
according to the golden section. b = 360 /

2 2 0* 29’ 32"; a = b /\ = 1 37* 30' 27" 95. The
name of "ideal angle" was given to "a" by
Church, who first discovered that it
corresponds to the best distribution of the
leaves; the mathematical confirmation was
given by Wiesner in 1875. (Ghyka, 1946, pp.
14-16)
The dynamic growth in nature based on the golden
section and Fibonacci series is evident in the logarithmic
spirals (see Figure # 4, p. 153), so preponderant in
plants, insects, and animals. As Thompson has pointed out,
it is sometimes difficult to pick out these logarithmic
spirals, or what he terms genetic spirals. However, like
the clues scattered throughout Plato's works, they are
there for those with eyes to see.
It is seldom that this primary, genetic
spiral catches the eye, for the leaves which
immediately succeed one another in this
genetic order are usually far apart on the
circumference of the stem, and it is only in
close-packed arrangements that the eye
readily apprehends the continuous series.
Accordingly in such a case as a fir-cone, for
instance, it is certain of the secondary
spirals or "parastichies" which catch the
eye; and among fir-cones, we can easily count
these, and we find them to be on the whole
very constant in number. . . . Thus,*in many
cones, such as those of the Norway spruce, we
can trace five rows of scales winding steeply
up the cone in one direction, and the three
rows winding less steeply the other way; in
certain other species, such as the common

153

154
larch, the normal number is eight rows in the
one direction and five in the other; while in
the American larch we have again three in the
one direction and five in the other. It not
seldom happens that two arrangements grade
into one another on different parts of one
and the same cone. Among other cases in
which such spiral series are readily visible
we have, for instance, the crowded leaves of
the stone-crops and mesembryanthemums, and .
. . the crowded florets of the composites.
Among these we may find plenty of examples in
which the numbers of the serial rows are
similar to those of the fir-cones; but in
some cases, as in the daisy and others of the
smaller composites, we shall be able to trace
thirteen rows in one direction and twenty-one
in the other, or perhaps twenty-one and
thirty-four; while in a great big sunflower
we may find (in one and the same species)
thirty-four and fifty-five, fifty-five and
eighty-nine, or even as many as eighty-nine
and one hundred and forty-four. On the other
hand, in an ordinary "pentamerous" flower,
such as a ranunculus, we may be able to
trace, in the arrangement of its sepals,
petals and stamens, shorter spiral series,
three in one direction and two in the other;
and the scales on the little cone of a
Cypress shew the same numerical simplicity.
It will be at once observed that these
arrangements manifest themselves in
connection with very different things, in the
orderly interspacing of single leaves and of
entire florets, and among all kinds of
leaf-like structures, foliage-leaves, bracts,
cone-scales, and the various parts or members
of the flower. (Thompson, 1968, vol. 2, pp.
921-922)
Descartes became fascinated with the logarithmic
spiral based on the golden section (see Figure # 5, p.153).
"The first [modern exoteric] discussions of this spiral
occur in letters written by Descartes to Mersenne in 1638"
(Hambidge, 1920, p. 146). It should also be noted that
three of the propositions in Newton's Principia are based

155
on the logarithmic spiral. These are 3ook I, proposition
9,and Book II, propositions 15 and 16.
Kepler was fascinated by the golden section. In his
writings he calls it the "sectio divina" and the "proportio
divina." He was presumably following the appellation given
it by Da Vinci's friend, Fra Luca Pacioli, who in 1509
referred to it as the "divine proportion." Kepler went on
to write:
Geometry has two great treasures, one is the
Theorem of Pythagoras, the other the division
of a line into extreme and mean ratio; the
first we may compare to a measure of gold,
the second we may name a precious jewel.
(quoted in Hambidge, 1920, p. 153)
Also the Fibonacci "series was well known to Kepler,
who discusses and connects it with [the] golden section and
growth, in a passage of his 'De nive sexángula', 1611"
(Hambidge, 1920, p. 155). Not only did Kepler employ the
abductive, or analytic techniques so wonderfully developed
by Plato, but he also followed the latter into the
significance of the golden section and its approximating
whole numbers, the Fibonacci series.
After discussions of the form of the bees'
cells and of the rhombo-dodecahedral form of
the seeds of the pomegranite (caused by
equalizing pressure) he [Kepler] turns to the
structure of flowers whose peculiarities,
especially in connection with quincuncial
arrangement he looks upon as an emanation of
sense of form, and feeling for beauty, from
the soul of the plant. He then "unfolds some
other reflections" on two regular solids the
dodecagon and icosahedron "the former of
which is made up entirely of pentagons, the
latter of triangles arranged in pentagonal

156
form. The structure of these solids in a
form so strikingly pentagonal could not come
to pass apart from that proportion which
geometers to-day pronounce divine." In
discussing this divine proportion he arrives
at the series of numbers 1 , 1 , 2, 3, 5, 8,
13, 21 and concludes: "For we will always
have as 5 is to 8 so is 8 to 13, practically,
and as 8 is to 13, so is 13 to 21 almost. I
think that the seminal faculty is developed
in a way analogous to this proportion which
perpetuates itself, and so in the flower is
displayed a pentagonal standard, so to speak.
I let pass all other considerations which
might be adduced by the most delightful study
to establish this truth." (Hambidge, 1920, p.
1 55)
Goethe discovered the golden section directed
logarithmic spiral in the growth of the tails and horns of
animals. X-rays of shell growth and horn development
(antelopes, wild goats, sheep) provide astounding examples
of logarithmic spiral growth based on the golden section.
. . . in the shells Murex, Fusus Antiquus,
Scalaria Pretiosa, Solarium Trochleare, and
many fossil Ammonites . . .the specific
spiral has the 4th root of p as
characteristic rectangle or ratio, or
quadrantal pulsation, and therefore 'J) as
radial pulsation. (Ghyka, 1946, p. 94)
Of course the golden section directed logarithmic
spiral is evident in the shell of the chambered nautilus
(Nautilus pompilius). This is especially true if one is
able to observe a radiograph of the shell. It is clear
that as the shell grows the chambers increase in size, but
the shape remains the same.
The golden section also plays a dominant role in the
proportions of the human body. According to Ghyka this
fact was recognized by the Greek sculptors

157
who liked to put into evidence a parallelism
between the proportions of the ideal Temple
and of the human body. . . . [Thus] the bones
of the fingers form a diminishing series of
three terms, a continuous proportion 1 , 1 /(j) ,
1 /(J> (in which the first, longest term, is
equal to the sum of the two following ones),
but the most important appearance of the
golden section is in the ratio of the total
height to the (vertical) height of the navel;
this in a well-built body is always
0=1.618... or a near approximation like
8/5=1.6 or 15/8=1.625. ... if one measures
this ratio for a great number of male and
female bodies, the average ratio obtained
will be 1.618. It is probable that the
famous canon of Polycletes (of which his
"Doryphoros" was supposed to be an example),
was based on this dominant role of the golden
section in the proportions of the human body;
this role was rediscovered about 1850 by
Zeysing, who also recognized its importance
in the morphology of the animal world in
general, in botany, in Greek Architecture
(Parthenon) and in music. The American Jay
Hambidge . . . guided by a line in Plato's
Theaetetus about "dynamic symmetry,"
established carefully the proportions and
probable designs not only of many Greek
temples and of the best Greek vases in the
Boston Museum, but also measured hundreds of
skeletons, including "ideal" specimens from
American medical colleges, and confirmed
Zeysing's results. . . . (Ghyka, 1946, pp.
16-17)
It is not at all surprising that Hambidge found his
clue to "dynamic symmetry" in the Theaetetus. My own view
is that Plato probably saw the interplay and infusion of
the golden section and Fibonacci series in nature. The
ideal numbers, in terms of ratios and proportions, were
infused into the Cosmos by the world-soul for Plato. But
he was not referring to some mere abstract mathematics.
Rather, Plato "saw" the concrete realization of these
numbers in the sensible world. He saw this interplay of

158
proportion in the doctrine of the five regular solids,
which his Pythagorean mentors had transmitted to him. Next
we will consider these regular solids, the only ones
perfectly inscribable in a sphere, to see if they really do
embody these proportions.
The Regular Solids
It is recorded that over the doors of his Academy,
Plato wrote: "let noone unversed in geometry come under my
roof [madeis ageometratos eisito mou tan stegan]" (Thomas,
1957, vol. 1, p. 387). The Pythagorean mathematical
disciplines were the essence of his doctrine. The care
with which geometry and the other dianoetic disciplines
were treated in the Academy is unequalled in our time.
Long before Saccheri, Lobachevsky, and Riemann began to
consider the nature of non-Euclidean geometries, the
Academy in Plato and Aristotle's time had extensively
considered them. But the hypothesis of the right angle was
adopted over the acute and obtuse angle hypotheses (see
Toth, 1969, pp. 87-93).
Likewise the nature of the five Platonic solids (see
Figure # 6, p. 159 ), was thoroughly investigated. In fact
the later work, Euclid's Elements, was viewed as designed
to show the construction of the five solids out of points,
lines, and planes.
A'regular solid is one having all its faces
equal polygons and all its solid angles
equal. The term is usually restricted to
those regular solids in which the centre is

159
Figure # 6: Five Regular Solids

160
singly enclosed. They are five, and only
five, such figures--the pyramid, cube
octahedron, dodecahedron and icosahedron.
They can all be inscribed in a sphere. Owing
to the use made of them in Plato's Timaeus
for the construction of the universe they
were often called by the Greeks the cosmic or
Platonic figures. (Thomas, 1957, vol. 1, p.
216, f n.a)
The tradition is that Plato received his ideas
regarding the regular solids from the Pythagoreans. It is
not clear whether his source was to have been Philolaus,
Archytas, or Alcmaeon.
Proclus attributes the construction of the
cosmic figures to Pythagoras, but Suidas says
Theaetetus was the first to write on them.
The theoretical construction of the regular
solids and the calculation of their sides in
terms of the circumscribed sphere occupies
Book XIII of Euclid's Elements. (Thomas,
1957, vol. 1, p. 216, fn.a)
It is recorded by Aetius, presumably upon the
authority of Theophrastus, that,
Pythagoras, seeing that there are five solid
figures, which are also called the
mathematical figures, says that the earth
arose from the cube, fire from the pyramid,
air from the octahedron, water from the
icosahedron, and the sphere of the universe
from the dodecahedron. (Thomas, 1957, vol. 1,
P- 217)
This is precisely the characterization given to the
elements by Plato. But Plato does something surprising in
the Timaeus in the construction of the solids (i.e.,
Timaeus 53c-55c). He constructs them out of two kinds of
triangles. Why is there not a further reduction from
triangular planes, to lines, and from lines to ratios of
numbers? He appears to stop prematurely.'*'

161
God now fashioned them by form and number. .
. . God made them as far as possible the
fairest and best, out of things which were
not fair and good. . . . fire and earth and
water and air are bodies. And every sort of
body possesses volume, and every volume must
necessarily be bounded by surfaces, and every
rectilinear surface is composed of triangles,
and all triangles are originally of two
kinds, both of which are made up of one right
and two acute angles. . . . These, then,
proceeding by a combination of probability
with demonstration, we assume to be the
original elements of fire and the other
bodies, but the principles which are prior to
these God only knows, and he of men who is
the friend of God. And next we have to
determine what are the four most beautiful
bodies which could be formed, unlike one
another, yet in some instances capable of
resolution into one another, for having
discovered thus much, we shall know the true
origin of earth and fire and of the
proportionate and intermediate elements.
Wherefore we must endeavor to construct the
four forms of bodies which excel in beauty,
and secure the right to 3ay that we have
sufficiently apprehended their nature.
(Timaeus 53b-e )
This is a difficult and at times cryptic passage. It
should be noted that it is "put into the mouth of Timaeus
of Locri, a Pythagorean leader, and in it Plato is
generally held to be reproducing Pythagorean ideas"
(Thomas, 1957, vol. 1, p.218, fn.a). There is the hint
that he who is a friend of God, and therefore one on the
upward path of analytic ascent near to the Forms, may know
the principles prior to the triangular components of the
solids.
But first, what can be said about Plato's primitive
triangles? The first triangle is the isosceles
right-angled triangle, or half-square, whose sides are in

162
the ratio 1:1 :/2\( see Figure # 7, p. 163)* Out of this
triangle is constructed the cube or element of earth. Out
of the many right-angled scalene triangles Plato chooses as
his second primitive, the one which is a half-equilateral
triangle. He calls it the most bequtiful of the scalene
triangles. Its sides are in the ratio of 1: 1^3:2 (see
Figure # 8 , p. 163 ). This latter triangle is to be used in
the construction of the elements of fire (tetrahedron), air
(octahedron), and water (icosahedron).
Plato typifies these triangles as the most beautiful,
making it evident that he is equally concerned with
aesthetics here. As Vlastos notes:
It would be hard to think of a physical
theory in which aesthetic considerations have
been more prominent: [Vlastos quoting Plato
passages] "Now the question to be considered
is this: What are the most beautiful
(kallista) bodies that can be constructed
[Timaeus] (53d7-et). ..." "If we can hit
upon the answer to this question we have the
truth concerning the generation of earth and
fire (e3-4). ..." "For we will concede to
no one that there are visible bodies more
beautiful (kallio) than these (e4-5)." "This
then we must bestir ourselves to do:
construct four kinds of body of surpassing
beauty (diaferonta kallei) and declare that
we have reached a sufficient grasp of their
nature (e6-8). ..." "Of the infinitely
many [scalene triangles] we must prefer the
most beautiful (to kalliston) (54a2-3)."
(Vlastos, 1975, p. 93, fn.4t)
We know from Speusippus that Plato's isosceles
right-angled triangle represented the dyad and even
numbers. Plato's scalene right-angled triangle (the
half-equilateral) represented the triad and odd numbers.
But there was also, according to Speusippus, the

163
Figure # 7
A
1;1:\Í2 Right-angled Isosceles Triangle

164
equilateral triangle which represented the monad (see
Figure ff 9 , p. 165). This triangle occurs in the Timaeus.
But rather than being a primitive there, it is constructed
out of the scalene right-angled triangles. It is then
employed in the construction of the elements. Referring to
triangular plane figures from the point of view of number
in his book, On Pythagorean Numbers, Speusippus says:
For the first triangle is the equilateral
which has one side and angle; I say one
because they are equal; for the equal is
always indivisible and uniform. The second
triangle is the half-square; for with one
difference in the sides and angles it
corresponds to the dyad. The third is the
half-triangle which is half of the
equilateral triangle; for being completely
unequal in every respect, its elements number
three. In the case of solids, you find this
property also; but going up to four, so that
the decad is reached in this way also.
(Thomas, 1957, vol. t , pp. 80-83)
It is interesting that here the equilateral triangle
is the first primitive, whereas, in the Timaeus the
equilateral triangle is built up out of the
half-equilaterals (i.e., out of the scalene right-angled
primitives). Nevertheless, this work of Speusippus' On
Pythagorean Numbers tends to reinforce the argument for the
Pythagorean origins of Plato's doctrine in the Timaeus.
But note further that this Pythagorean "awareness" of Plato
dates back to the earliest dialogues. Reference to these
primitive triangles is made as early as the Suthyphro (note
Figure # 10, p . 166 ) .
Suppose . . . you had asked me what part of
number is the even, and which the even number
is. I would have said that it is the one

165
Figure # 9: Monadic Equlateral Triangle

Amim
Ion
Prot.
Lach.
Rep. 1
Lysis
Charm.
Euthyph.
Euthvd.
Gorg.
Meno
Hipp. Mi.
Crat.
Symp.
Hipp. Ma.
Phaedo
Crito
Rep. 2-10
Theaet.
Parm.
Phaedr.
Soph.
Pol.
Phil.
Laws
Figure
Lutoslawski
Raeder
Apol.
Apol.
Ion
Hipp. Mi
Euthyph.
Lach.
Crito
Charm.
Charm.
Crito
Hipp. Ma
Lach.
Prot.
Prot.
Gor§-
Menex.
Meno
Euthyph.
Euthyd.
Meno
Gorg.
Euthyd.
Rep. 1
Crat.
Crat.
Lysis
Symp.
Symp.
Phaedo
Phaedo
Rep. 2-10
R-eP:.
Phaedr.
Phaedr.
Theaet.
Theaet.
Parm.
Parm.
Soph.
Soph.
Pol.
Pol.
Phil.
Phil.
Tim.
Tim.
Critias
Critias
Laws
Laws
Epin.
Ritter
Wilamowitz
Hipp. Mi.
Ion
Hipp. Mi.
Lach. Prot.
Prot.
Apol.
Charm.
Crito
Euthyph.
Apol.
Lach.
Crito
Lysis
Gorg.
Charm.
Hipp. Ma.
Euthyph.
Euthyd.
Gorg.
Crat.
Menex.
Meno
Meno
Menex.
Crat.
Lysis
Euthyd.
Phaedo
Phaedo
Symp.
Rep.
Rep.
Phaedr.
Phaedr.
Theaet.
Parm.
Parm.
Theaet.
Soph.
Soph.
Pol.
Pol.
Tim.
Tim.
Critias
Critias
Phil.
Phil.
Laws
Laws
10: Stylometric Datings of Plato's Dialogues
(after Ross, 1951, p, 2)

167
that corresponds to the isosceles, and not
the scalene. (Euthyphro 12d)
There appear, however, to be difficulties with the
doctrine in the Timaeus. The beauty of the Pythagorean
number theory is manifest. But why is the cube presumably
not transformable into the other elements? And why is the
fifth element which is to stand as the foundation of the
Cosmos, the dodecahedron, not transformable into any of the
other four elements?
Vlastos brings out this typical criticism when he says:
I am not suggesting that the aesthetics of
the Platonic theory are flawless. The
exclusion of earth from the combinatorial
scheme (necessitated by the
noninterchangeability of its cubical faces
with the triangular faces of the solids
assigned to fire [tetrahedron], air
[octahedron], and water [icosahedron]) is
awkward. Worse yet is the role of the fifth
regular solid, the dodecahedron, whose
properties would also shut it out of the
combinatorial cycle. The hasty reference to
it (55c) suggests embarrassed uncertainty.
What could he mean by saying that "the god
used it for the whole?" The commentators
have taken him to mean that the Demiurge made
the shape of the Universe a dodecahedron;
this unhappily contradicts the firm and
unambiguous doctrine of 33b (reaffirmed in
43d and 62d) that the shape of the Universe
is spherical. (Vlastos, 1975, p. 94, fn.43)
Irrespective of what Vlastos claims, it is evident
that Plato intended the dodecahedron to be of fundamental
importance in the structure of the Cosmos. And
furthermore, as I will show, the terse reference at Timaeus
55c is not due to "embarrassed uncertainty," but to
reticence and midwifery on the part of Plato. Plato is
reticent because he is discussing what are probably some of

168
the most cherished of Pythagorean doctrines. He is playing
midwife because while again suggesting that the friend of
deity can penetrate deeper, he at the same time presents
the problem: an analysis into primitives which has not
been carried far enough. I will answer Vlastos' charges
directly, and return to this question of primitive
triangles shortly.
The regular bodies are avowed to be the most
beautiful. And then we are told that we must "secure the
right to say that we have sufficiently apprehended their
nature" (Timaeus 53e). But why does Plato stop short in
the analysis to the most beautiful component? The answer
lies in the fact that Plato must use some form of blind or
cover. The deepest Pythagorean discoveries were not to be
openly revealed. And as Plato has pointed out in the
Phaedrus and the 7th Letter, he is opposed to openly
exposing one's doctrine in writing. And yet Plato provides
sufficient hints, as he continues to be obstetric in the
Timaeus, as he has been in other dialogues.
We must also inquire as to why the fifth figure, or
dodecahedron, was singled out as the element of the Cosmos?
"There was yet a fifth combination which God used in the
delineation of the universe with figures of animals"
(Timaeus 55c). Was there something special about the
dodecahedron? We do have one story preserved by Iamblichus
in his On the Pythagorean Life.
It is related of Hippasus that he was a
Pythagorean, and that, owing to his being the

169
first to publish and describe the sphere from
the twelve pentagons, he perished at sea for
his impiety, but he received credit for the
discovery, though really it ail belonged to
HIM (for in this way they refer to
Pythagoras, and they do not call him by his
name). (Thomas, 1957, vol. 1, pp. 223-225)
We do know that a dodecahedron of course has 12
pentagonal faces. The pentagram which is created by
connecting opposing vertices of the pentagon was the sacred
symbol of the Pythagorean brotherhood.
The triple interlaced triangle, the
pentagram, which they (the Pythagoreans) used
as a password among members of the same
school, was called by them Health. (Lucian in
Thomas, 1957, vol. 1, p. 225)
Are there any other clues? Yes, there are some very
relevant clues. Returning to Heath's comments upon the
ancient methods of analysis and synthesis, we may begin to
be able to begin tieing together, the notions of analysis
and that of the golden section. This begins to emerge when
we consider Book XIII of Euclid's Elements.
It will be seen from the note on Sucl. XIII.
1 that the MSS. of the Elements contain
definitions of Analysis and Synthesis
followed by alternative proofs of XIII 1-5
after that method. The definitions and
alternative proofs are interpolated because
of the possibility that they represent an
ancient method of dealing with these
propositions, anterior to Euclid. The
propositions give properties of a line cut
"in extreme and mean ratio," and they are
preliminary to the construction and
comparison of the five regular solids.
(Heath, 1 956 , vol. 1 , p. 137)
My contention, simply put, is that this lies at the
center of what was going on in Plato's Academy. The line

170
cut in extreme and mean ratio was central to these
endeavours. In fact, if one takes the fifth solid, the
dodecahedron, it is discovered that its analysis into plane
surfaces, or the second dimension, yields the pentagons.
The further analysis into one dimension produces the
relation of a line cut in mean and extreme ratio. And
finally, reduced to the level of number we have the
relation of ^ to 1 .
Now Pappus, in the section of his Collection
dealing with [the construction and comparison
of the regular solids] says that he will give
the comparisons between the five figures, the
pyramid, cube, octahedron, dodecahedron and
icosahedron, which have equal surfaces, "not
by means of the so-called analytical inquiry,
by which some of the ancients worked out the
proofs, but by the synthetical method." The
conjecture of Bretschneider that the matter
interpolated in Eucl. XIII. is a survival of
investigations due to Eudoxus has at first
sight much to commend it. In the first
place, we are told by Proclus that Eudoxus
"greatly added to the number of the theorems
which Plato originated regarding the section,
and employed in them the method of analysis."
It is obvious that "the section" was some
particular section which by the time of Plato
had assumed great importance; and the one
section of which this can safely be said is
that which was called the "golden section,"
namely, the division of a straight line in
extreme and mean ratio which appears in Eucl.
II.11 and is therefore most probably
Pythagorean. (Heath, 1956, vo1. 1, P. 137)
Here is the anomalous evidence that the Pythagorean
Platonists, namely Plato, Eudoxus, and perhaps Theaetetus,
Speusippus, and Xenocrates, were busying themselves with a
consideration of the golden section through a dialectical
technique of analysis, which was essentially a series of
reductions. This golden section was the mathematical (par

171
excellence) of the inconmensurables. It not only is
essential to an understanding of the relationships of the
regular solids, but is also fundamental to a theory of
proportions based upon inconmensurables. Just as the
regular solids and the world soul were shown to embody the
geometrical proportions of the whole numbers (e.g.,
2:4::4:8), in the same way that the Sun, Divided Line, and
Cave similea were related one to another, so the regular
solids embody in their relations the ratio of the golden
section, just as the Divided Line embodied this geometrical
proportion of incommensurables.
As Cantor points out, Eudoxus was the founder
of the theory of proportions in the form in
which we find it in Euclid V., VI., and it
was no doubt through meeting, in the course
of his investigations, with proportions not
expressible by whole numbers that he came to
realise the necessity for a new theory of
proportions which should be applicable to
incommensurable as well as commensurable
magnitudes. The "golden section" would
furnish such a case. And it is even
mentioned by Proclus in this connexion. He
is explaining that it is only in arithmetic
that all quantities bear "rational" ratios
(ratos logos) to one another, while in
geometry there are "irrational" ones
(arratos) as well. "Theorems about sections
like those in Euclid's second Book are common
to both [arithmetic and geometry] except that
in which the straight line is cut in extreme
and mean ratio. (Heath, 1956, vol. 1, p. 137)
Now it appears that Theaetetus was also intimately
involved with this consideration of the golden section and
its relation to the five regular solids.
It is true that the author of the scholium
No. 1 to Eucl. XIII. says that the Book is
about "the five so-called Platonic figures,
which however do not belong to Plato, three

172
of the aforesaid five figures being due to
the Pythagoreans, namely the cube, the
pyramid and the dodecahedron, while the
octahedron and the icosahedron are due to
Theaetetus." This statement (taken probably
from Geminus) may perhaps rest on the fact
that Theaetetus was the first to write at any
length about the two last-mentioned solids.
tVe are told indeed by Suidas (s.v.
Theaitatos ) that Theaetetus "first wrote on
the 'five solids' as they are called." This
no doubt means that Theaetetus was the first
to write a complete and systematic treatise
on all the regular solids; it does not
exclude the possibility that Hippasus or
others had already written on the
dodecahedron. The fact that Theaetetus wrote
upon the regular solids agrees well with the
evidence which we possess of his
contributions to the theory of irrationals,
the connexion between which and the
investigation of the regular solids is seen
in Euclid's Book XIII. (Heath, 1956, vol. 3,
p. 438)
Santillana is one of the few modern scholars who has
sincerely tried to penetrate the Platonic mystery. He
argues as follows:
. . . that world [the Cosmos] is a
dodecahedron. This is what the sphere of
twelve pieces [_in Phaedo 107d-115a] stands
for: there is the same simile in the
Timaeus (55c), and then it is said further
that the Demiurge had the twelve faces
docorated with figures (diazographon) which
certainly stand for the signs of the zodiac.
A.E. Taylor insisted rather prosily that one
cannot suppose the zodiacal band uniformly
distributed on a spherical surface, and
suggested that Plato (and Plutarch after him)
had a dodecagon in mind and they did not know
what they were talking about. This is an
unsafe way of dealing with Plato, and
Professor Taylor's suffisance soon led him to
grief. Yet Plutarch had warned him: the
dodecahedron "seems to resemble both the
Zodiac and the year." (Santillana and von
Dechend, 1969, p. 187)

173
Santiilana has in mind a passage where Plutarch
relates the pentagons of the dodecahedron into triangles.
Plutarch asks:
. . . is their opinion true who think that he
ascribed a dodecahedron to the globe, when he
says that God made use of its bases and the
obtuseness of its angles, avoiding all
rectitude, it is flexible, and by
circumtension, like globes made of twelve
skins, it becomes circular and comprehensive.
For it has twenty solid angles, each of which
is contained by three obtuse planes, and each
of these contains one and the fifth part of a
right angle. Now it is made up of twelve
equilateral and equangular quinquangles (or
pentagons), each of which consists of thirty
of the first scalene triangles. Therefore it
seems to resemble both the Zodiac and the
year, it being divided into the same number
of parts as these. (Plutarch in Santiilana
and von Dechend, 1969, p. 187)
It is in this way that Santiilana sees the golden section
emerge from it. Santiilana remarks:
In other words, it is stereometrically the
number 12, also the number 30, the number 360
("the elements which are produced when each
pentagon is divided into 5 isosceles
triangles and each of the latter into 6
scalene triangles") -- the golden section
itself. This is what it means to think like
a Pythagorean. (Santiilana and von Dechend,
1 969, p. 187)
But has Santiilana really discovered how the golden
section is involved? When Plutarch says above that each
pentagon may be divided into "thirty of the first scalene
triangles," it is a blind. They are right-angled scalene
triangles, but they are not Plato's primitive right-angled
scalene triangles. Furthermore, when Santiilana says
above, that to arrive at this number, the pentagon is first

174
"divided into 5 isosceles triangles," it is again
misleading. Yes, they are isosceles triangles, but they
are not Plato's primitive right-angled isosceles triangles.
However, by examination of these geometrical reductions, we
may get a better grasp of what actually was primitive for
Plato.
Looking at Figure # 11, p. 175 , we have a pentagon
ABODE. If we then connect each of the corners to the
center. F, we have the pentagon divided into five isosceles
triangles (see Figure #12» p. 176 )• If we then bisect each
isosceles triangle by drawing a line, for example, from A
through F projecting it until it meets line CD at N, and
doing the same for points B, C, D, and E, we have ten
scalene right-angled triangles (see Figure # 13 , p. 177 )•
However, they are not Plato’s primitive scalene triangles.
If we next proceed around the pentagon, first drawing a
line from A to C, and then from A to D, then from 3 to D,
and then from B to E, continuing with points C, D, E, in
like manner, we arrive at Figure # 14 » p. 178- It will now
be noticed that each of the five isosceles triangles is
composed of six scalene triangles. This results in thirty
scalene triangles within the pentagon. However, ten of
these thirty triangles are different from the others.
I will try to illustrate the point slightly
differently. Beginning with our bare regular pentagon
ABODE (see Figure # 11, p. 175), let us draw lines AG and
CE (Figure # 15, p. 179). The resulting triangle is termed

175
C
Figure # 11:
Pentagon

176
C
Figure // 12; Pentagon & Isoscles Triangle

177
C
Figure # 13; Pentagon & 10 Scalene Triangles

178
Figure # 14: Pentágona & 30 Scalene Triangles

179
C
Figure # 15: Pentagon & Pentalpha
C
Figure # 16: Pentagon & Two Pentalphas

180
the pentalpha or golden triangle. If line AC = then
line AE = 1. If we then draw lines BD and AD, we find two
intersecting pentalphas (Figure # 16, p. 179). If we then
draw line BE we discover the figure of the pentagram
(Figure # 17, p. 181). In effect we have drawn three
intersecting pentalphas. The resulting pentagram inscribed
in the pentagon can also easily be viewed as five
intersecting pentalphas. Next locating the center point F,
let us draw lines AF and EF. Then we will draw a line from
F perpendicular to AE meeting AE in J. J will be midpoint
between A and E (see again Figure # 18, p. 181 ). Let us
then eliminate all internal lines except those within the
isosceles triangle AFE (Figure # 19 , p. 182). Let us then
extract triangle AFE (with its internal lines intact) from
our pentagon (again see Figure # 19 , p. 182 ).
The first thing to be noticed is that triangles AJI,
EJI, FGI, and FHI are all scalene right-angled triangles.
However, triangles AGI and EHI are scalene right-angled
triangles of a different kind. As already noted, none of
these are Plato's primitive scalene triangles. If we then
extract the two triangles AGI and EHI, we quickly discover
that they are ha 1 f-p en ta lp ha s (Figure # 20 > p. 183) . If we
then place the two triangles together, back to back, we
find that we have a golden triangle or pentalpha (Figure #21,
, p. 184 )• Thus, the ha 1f-pentalpha emerges through a
division of the pentagram. Because of this we can say that
the pentalpha is basic to the construction of the pentagon,

181
C
Figure // 17; Pentagon & Pentagram
C

F
Figure # 19: Pentagon & Isosceles Triangle (top)
(latter extracted at bottom)

133
A E
Figure // 20: Two Half-Pentalphas

184
Figure # 21: Pentalpha

185
and hence, the dodecahedron. If the pentalpha is reduced
from a two dimensional plane figure, to its one dimensional
depiction in a line, we discover the golden section. This
is the fundamental modular of Plato's cosmos.
Next we shall view the same thing but strictly from
the perspective of the golden section. Let us begin with a
new line AB. Then we will draw BD which is equal to 1 /2
AB, perpendicular to AB (see Figure #22, p. 186 ). Next we
will draw line AD. Then with a compass at center D and
with radius DB we will draw an arc cutting DA in E. Next
with center A, and radius AE, we will draw an arc cutting
AB at C (Figure #22, p. 186 ). Line AB is then cut in the
golden section at C.
Next let us place the compass at center C and with the
radius CA, draw an arc. Then again with radius CA, but
with the compass at center B, we draw another arc which
cuts the earlier arc at F (see Figure # 23, p. 186). We
then draw the lines AF. , CF, and BF. We then have
constructed the pentalpha BCF. Next by bisecting angles
BAF and CFB, we are able to locate the point of
intersection I (see Figure # 24* p. 187 ). Then with the
compass at center I and radius IB, we construct the circle
BCF (Figure # 25, p. 188). The circle also intersects at
points C and J, as well as B and F. Then we extend a line
through AI projected to meet the circle BCF in K (again see
Figure # 25. ?• 188 )• Next we draw lines BC, CJ, JF, FK,
and K3. This gives us pentagon BCJFK inscribed in circle

186
D
Figure # 22: Golden Cut
•D
Figure # 23: Golden Cut & Pentalpha

187
Figure # 24: Pentalpha Bisection

188
Figure # 25; Circle & Pentalpha

189
BCF (Figure # 26» P* 190 ) • It would be correct to say at
this point we are half-way towards inscribing a
dodecahedron in a sphere.
O
But now let us rotate the entire figure 180 . Point F
will now be uppermost (Figure #27» P* 191 )• Now let us
remove everything except the circle , pentagon, and
pentalpha. Then we will drop a line from F through I
meeting line BC at G (Figure # 28, p. 192 ). Here we find
the two half-pentalphas.
It has certainly been recognized before that the
pentalpha is basic to the construction of the pentagon, and
hence the dodecahedron. Thus, Cornford remarks: "[the
Dodecahedron] was in fact constructed by means of an
isosceles triangle having each of its base angles double of
the vertical angle" (Cornford, 1956, p. 218). In
conjunction with Plato's two primitive triangles, and the
equilateral triangle, representing the monad, odd, and
even, the pentalpha (or half-pentalpha) adds the fourth
triangle which completes the tetraktys. The Timaeus opens
with Socrates saying: "One, two, three, but where my dear
Timaeus, is the fourth of those who were yesterday ray
guests and are to be my entertainers today" (Timaeus 17a)?
Of course, Critias, Timaeus of Locri, and Hermocrates of
Syracuse are present. But the unnamed guest is absent due
to indisposition. However, I think this is also a cryptic
suggestion to discover the fourth triangle. This I would
argue is the half-pentalpha or pentalpha.

190
Figure # 26; Pentagon in Circle

191
F
Figure # 27: 180° Rotation of Figure # 26

Figure i1 28: Circle, Pentagon, & Half-Pentalphas

193
Now if we take the pentagram BKFJC inscribed in a
circle (see Figure ff 29, p. 194 ), we find the following
properties:
a
KJ = d>
KL = Í
LM = 1
IS/IR = J>/2
IK /IS -'2 If T is the intersection point of two
diagonals MR and LP, then:
PT/TL = 4>
RT/TM = j)
and FT/TQ = |).
IF PL produced meets KF in N, since NLP is
parallel to KC, then:
FN/NK = i>
FL/LR = Í
FP/PÍ = Nhat then can be said in general about the
transformability of the regular solids? Or was Vlastos
correct in supposing the transformability breaks down? To
answer these questions let us first construct the golden
rectangle. For this we will consider Figure #’s 30, 31,
& 32, p. 195. Let us begin with the construction of the
rectangle ABCD, which is composed of two squares, ABFE and
EFCD. Next let us divide one of these squares at GH. Then
with compass at center H and radius HF, draw an arc from F
meeting ED in I. Then draw the line IJ parallel to CD.
The resulting rectangle ABJI is a golden rectangle. If JI
is 1, then AI is Also the smaller rectangle EFJI is a
golden rectangle. Now a gnomon is the smallest surface
which can be added to a given surface to produce a similar
surface. One of the features of the golden rectangle is
that it is the only rectangle of which the gnomon is a

194
F
10
Figure // 29: Golden Section in Pentagram
11

195
E D
Figure if 30: Double Square
(T F J C
Figure if 31: Construction of Golden Rectangle
B.
1
I
Figure if 32: Golden Rectangle
A

196
square. Like the pentalpha or golden triangle, the golden
rectangle is another example of the embodiment of the
golden ratio in two dimensions. As Ghyka writes,
This most "logical" asymmetrical division of
a line, or of a surface, is also the most
satisfactory to the eye; this has been tested
by Fechner (in 1876) for the "golden
rectangle," for which the ratio between the
longer and the shorter side is J) = 1.618....
In a sort of "Gallup Poll" asking a great
number of participants to choose the most
(aestheticall) pleasant rectangle, this
golden rectangle or $ rectangle obtained the
great majority of votes. (Ghyka, 1946, pp.
9-10)
These golden rectangles are very important in
considering the transformability and relations of the five
regular solids.
. . . two pairs of the Platonic solids are
reciprocal and the fifth is self-reciprocating
in this sense: if the face centers of the
cube are joined, an octahedron is formed,
while the joins of the centroids of the
octahedron surfaces form a cube. Similar
relationship holds between the icosahedron and
the dodecahedron. The join of the four
centroids of the tetrahedron’s faces makes
another tetrahedron. . . . The twelve vertices
of a regular icosahedron are divisible into
three coplanar groups of four. These lie at
the corners of three golden rectangles which
are symmetrically situated with respect to
each other, being mutually perpendicular,
their one common point being the centroid of
the icosahedron [see Figure # 33, p. 197], An
icosahedron can be inscribed in an octahedron
so that each vertex of the former divides an
edge of the latter in th golden section. The
centroids of the twelve pentagonal faces of a
dodecahedron are divisible into three coplanar
groups of four. These quadrads lie at the
corners of three mutually perpendicular,
symmetrically placed golden rectangles, their
one common point being the centroid of the
dodecahedron [see Figure # 34, p. 197 ].
(Huntley, 1970, pp. 33-34)

197
Figure # 33: Icosahedron with Intersecting
Golden Rectangles

198
Consider for a moment Figures if 33 and #34, p. 197 .
The former is an icosahedron embodying three intersecting
golden rectangles. The latter figure is a dodecahedron,
embodying the same intersecting golden rectangles. These
are the propotional interrelations that are relevant in
Plato's cosmos. Thus, the five regular solids are
transformable in terms of their mutually reciprocating
properties. The relation of each to one another when
inscribed within a sphere brings to the forefront the
predominance of the fundamental invariant, the divine
ratio, the golden section. It is the fundamental invariant
transmitted into the cosmos by the world soul.
This is how number, ratio, and proportion get infused
into the world. According to Plato, prior to this the
world is not separated into elements. Only when proportion
is imposed upon it do we get the elements and various
qualities. Thus Plato says,
. . . when all things were in disorder, God
created in each thing in relation to itself,
and in all things in relation to each other,
all the measures and harmonies which they
could possibly receive. For in those days
nothing had proportion except by accident,
nor was there anything deserving to be called
by the names which we now use--as, for
example, fire, water, and the rest of the
elements. All these [the elements] the
creator first set in order, and out of them
he constructed the universe. . . . (Timaeus
69b-c )
My own view is that the relevance of the golden
section to the regular solids can be most easily seen by
examining Euclid's Book XIII of the Elements. Heath

199
maintains that the first five propositions of this Book are
due to Eudoxus. He says,
it will be remembered that, according to
Proclus, Eudoxus "greatly added to the number
of theorems which originated with Plato
regarding the section" (i.e., presumably the
"golden section"); and it is therefore
probable that the five theorems are due to
Eudoxus. (Heath, 1956, voi. 3, p. 441)
The five propositions are in order as follows:
I. If a straight line be cut in extreme and
mean ratio, the square on the greater segment
added to the half of the whole is five times
the square on the half.
II. If the square on a straight line be five
times the square on a segment of it, then,
when the double of the said segment is cut in
exteme and mean ratio, the greater segment is
the remaining part of the original straight
line.
III. If a straight line be cut in extreme and
mean ratio, the square on the lesser segment
added to the half of the greater segment is
five times the square on the half of the
greater segment.
IV. If a straight line be cut in extreme and
mean ratio, the square on the whole and
square on the lesser segment together are
triple of the square on the greater segment.
V. If a straight line be cut in externe and
mean ratio, and there be added to it a
straight line equal to the greater segment,
the whole straight line has been cut in
extreme and mean ratio, and the orginal
straight line is the greater segment. (Heath,
1956, vo1. 3, pp. 440-451)
These first 5 propositions are only preparatory for
the later propositions, especially propositions 17 and 18.
XVII. To construct a dodecahedron and
comprehend it in a sphere, like the aforesaid
figures, and to prove that the side of the
dodecahedron is the irrational line called
apo tome.
XVIII. To set out the sides of the five
figures and to compare them with one another.

200
It is very interesting that the first five
propositions of Book XIII contain alternative proofs using
an ancient method of analysis. As Heath remarks,
Heiberg (after Bretschneider) suggested in
his edition (Vol. v. p. Ixxxiv.) that it
might be a relic of analytical investigations
by Theaetetus or Eudoxus, and he cited the
remark of Pappus (v. p. 410) at the beginning
of his "comparisons of the five [regular
solid] figures which have an equal surface,"
to the effect that he will not use "the
so-called analytical investigation by means
of which some of the ancients effected their
demonstrations."(Heath, 1956, vol. 3, p. 442)
The proof follows the abduction pattern that I have
suggested Plato employed. The analysis part of the proof
begins by assuming what is to be proved and then moves
backwards to an evident hypothesis. Then in the synthesis,
the direction is reversed. And beginning with the result
which was arrived at last in the analysis, one moves
downwards deductively, deriving as a conclusion that which
was set out initially to be proven. It seems that this is
a remnant of the analytic or abductive investigations
carried on by Plato and the members of the Academy. The
proof is even more intriguing because not only does it
contain a remnant of the analytic method, but it also
involves the golden section. When the analysis of the
regular solids is carried to the first dimension of lines,
and thence to the ratio of number, we discover that it is
the golden section that is fundamental for Plato. The
method of analysis and the golden section appear to
mutually complement one another. They are, I contend, at
the center of Plato's philosophy.

201
Conclusion
In this dissertation I have accomplished three things.
First, through argument and example, I have exposed the
plethora of evidence that Plato was primarily a
Pythagorean, concerning himself, and the members of his
Academy, with the discovery and application of mathematical
principles to philosophy. Secondly, I have traced the
roots of Perircean abduction back to Plato's (and his
pupils') use of an ancient method of analysis. This
"anapalin lusin," or reasoning backwards, was employed in
the Academy to solve problems, and to provide rational
explanations for anomalous phenomena. And thirdly, through
an examination of the evidence for Plato's "intermediate
mathematicals," the role of proportion, and the interplay
of the One (Good) and Indefinite Dyad, I have set forth the
premiere importance of the golden section () for
Plato.
I also accomplished the task I set for myself in the
Introduction (supra p.5&6 ). That is, I set out to find
the "more beautiful form than ours for the construction of
these l re 1a r] bodies" (Timaeus 54a). I discovered the
fourth triangle in the half-pentalpha. It completes the
tetraktys. I suggest that the monad then is the
equilateral triangle. The dyad is Plato's right-angled
isosceles triangle, the half-square. The triad is Plato's

202
right-angled scalene triangle, the half-equilateral. And I
have added the tetrad as the haif-pentalpha.
But the most beautiful form is even more fundamental
than this. When one takes the fifth solid, the
dodecahedron reserved for the whole, and reduces it from
the third to the second dimension, the result is either the
plane figure of the golden rectangle or the golden triangle
(pentalpha). The further reduction to the first dimension
yields the line divided in mean and extreme ratio. And
finally, the reduction to number yields the relation of <|)
(the golden section) to the One (unit or monad). It is,
therefore, the golden section which is the most beautiful
form. And we discovered it embodied in the Republic's
Divided Line, in the continued geometrical proportion ({l*-
Throughout this dissertation I have employed the very
method of analysis (or abduction) that I have attributed to
Plato. Consistent with this, I have contended that Plato
is obstetric in his dialogues. Accordingly, a good
abduction must make what was previously anomalous, the
expected. It must, so to speak, normalize the paranormal.
The overall hypothesis (and the lesser hypotheses) that I
have abducted, provide a basis for making many of the
Platonic riddles and puzzles easily understandable. My
position goes far to eliminate the mystery surrounding
Plato's identification of numbers with Forms, the
references to intermediate mathematicals, the open question

203
of participation in the Academy, the nature of the
unwritten lectures, the superficial bifurcation of the
Forms and the sensible world, and the discrepancies between
the dialogues and Aristotle's account of Plato's
philosophy. Also, the suggestions that Plato discovered
analysis, taught it to Leodamos, and started the theorems
regarding the golden section (which Eudoxus then multiplied
using analysis), become much more reasonable.
Many of the elements I have woven together in this
%
dissertation have been touched upon by others before me. I
have but managed to tie together some loose ends, and
provide a consistent overall framework for Plato's
philosophy. I only hope that enough groundwork has been
set for another to penetrate even more deeply into the
Platonic arcanum. So much has been written on Plato, and
yet we have only begun to dip beneath the veil to view the
real essence of his work.
Notes
Plato's termination of the analysis seems too obvious
to overlook. It is relevant to consider here a passage by
Aristotle:
Empedocles declares that it (soul) is formed
out of ail the elements, each of them also
being soul; in the same way Plato in the
Timaeus fashions the soul out of his
elements. . . . Similarly also in his
lectures "On Philosophy": it was set forth
that the Animal-itself is compounded of the
Idea itself of the One together with the
primary length, breadth and depth, everything
else, the objects of its perception, being
similarly constituted. Again he puts his
view in yet other terms: Mind is the monad,
science or knowledge the dyad (because it

204
goes undeviatingly from one point to
another), opinion the number of the plane
(the triad), sensation the number of the
solid (the tetrad). (DeAnima 404bT2-24)
why then does Plato stop with the triad or opinion? Just
as one ascends the Divided Line of the Republic, so one
should ascend this analogous scale. The triangle should be
analyzed into lines, and from thence into numbers and their
ratios.

205
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216
BIOGRAPHICAL SKETCH
Scott Anthony Olsen, a native of Minnesota, received
his Bachelor of Arts degree cum laude in philosophy and
sociology from the University of Minnesota in 1975. He
then received his Master of Arts degree from the University
of London in 1977. He received his Juris Doctorate from
the University of Florida College of Law in 1982. Scott's
extracurricular activities include racquetball and
bodybuilding. He is presently the University of Florida,
mixed doubles racquetball champion. And following in the
Platonic tradition of a healthy mind in a healthy body,
Scott is the 1979 Mr. Northeast Florida, and 1979 Mr.
Gainesville.

I certify that
opinion it conforms
presentation and is
as a dissertation f
I have read this study and that in my
to acceptable standards of scholarly
fully adequate, in scope and quality,
or the degree of Doctor of Philosophy.
3,
Ellen 3 . Haring, C
Professor of Philo
‘airperson
ophy
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, ir. scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
33KTQOSj Ij)-
Thomas Simon, Co chairperson
Associate Professor of Philosophy
I certify that
opinion it conforms
presentation and is
as a dissertation fo
I have read this study and that in my
to acceptable standards of scholarly
fully adequate, in scope and quality,
r the degree of Doctor of Philosophy.
o i
I certify that I have read this study and that in ny
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree, of Doctor of Philosophy.
^Philip £■. Callahan
Professor of Entomology
This dissertation was submitted
the Department of Philosophy in
& Sciences and to the Graduate 3
partial fulfillment of the requi
Philosophy.
to the Graduate Faculty of
the College of Liberal Arts
choo2, and was accepted for
r e m e r. t s of D o c t o r o f
August 19S3
Dean for Graduate Studies arid Lesearen

UNIVERSITY OF FLORIDA
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