The Pythagorean Plato and the golden section

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The Pythagorean Plato and the golden section a study in abductive inference
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Thesis (Ph. D.)--University of Florida, 1983.
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Includes bibliographical references (leaves 205-215).
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by Scott Anthony Olsen.
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THE PYTHAGOREAN PLATO AND THE GOLDEN SECTION:
A STUDY IN ABDUCTIVE INFERENCE







BY

SCOTT ANTHONY OLSEN


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


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.