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Dictionary of the History of Ideas

Studies of Selected Pivotal Ideas

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TO 300 B.C.

The historian of Pythagoreanism in the sixth and fifth
centuries must make his bricks without straw. He can
give us an account not of persons, events, doctrines,
but only of more or less plausible reconstructions and
of controversies between disparate positions. The
historian of ideas is in a more fortunate position. He
may accept what no one doubts, that Pythagoras lived


and that he played in his time a role analogous to that
of founders of religions or sects whose impact and
whose message survive their persons. Nor need he
question the general nature of Pythagoras' message:
first, a doctrine of the soul and, second, an arithmologi-
cal theory of the physical world.

These two themes, soul and number, characterized
Pythagoreanism throughout antiquity. We have no
reason to believe that a connection was established
between them in the early period. The soul doctrine
determined moral conduct. The number theory
purported to explain aspects of the structure of the
physical world. They were first related by Plato, when
he made of soul an intermediate between intelligibles
and sensibles, both of these mathematically deter-
mined. We may therefore pursue the two themes sepa-
rately up to the time when they are conjoined.

Until a generation ago Erwin Rohde's account of
Greek doctrines of the soul in his great study, Psyche
(1925), found very general acceptance. It was believed
that when Pythagoras migrated from Ionia to Magna
Graecia he found there a flourishing sect of the
worshippers of Dionysus known as Orphics (Rohde, pp.
335-61; Jaeger, pp. 55-89), and that of this Orphic
sect Pythagoras and his followers formed an offshoot
observance. But in 1941 Linforth reexamined critically
all the evidence for this belief; and since then skepti-
cism about it has grown, until now Dodds (p. 147) can
remark: “I must confess that I know very little about
early Orphism, and the more I read about it the more
my knowledge diminishes.” Whatever our assessment
of the evidence may be, it seems clear that if
Pythagoras found beliefs about the soul abroad in
Magna Graecia he transformed them for his purpose.
Nothing ecstatic or dionysiac remained. Even a con-
nection with the worship of Dionysus seems im-
probable. Pythagoras is said to have claimed divine
status (Aristotle, frag. 1) as an incarnation of Apollo,
not of Dionysus, and our whole tradition connects him
with Apollo. For him the soul which lodges in our
bodies has come from a previous existence and is pro-
ceeding to further existences. It may pass into an ani-
mal body (Xenophanes, Vors. 21B 7). It may in the end
achieve divinity. Its migrations are linked to rewards
and punishments. It is held responsible for the deeds
and the fate of the person.

We do not know what were the formal and external
aspects of Pythagoreanism during the life of Pythagoras
and, subsequently, through the fifth century. In the late
sixth and in the first half of the fifth century
Pythagoreans engaged in political activities in Magna
Graecia, and apparently became a dominant faction
in Croton, Metapontum, and elsewhere. But there is
no evidence for the brotherhood having the quasi
monastic character of which we hear so much from
Neo-Pythagorean sources (Philip [1966a], pp. 24-34).
The Symbola or Tokens which Aristotle (frags. 5-7)
has preserved for us are not any rule of a community
but merely a collection of superstitious injunctions not
peculiar to any one sect, nor indeed to any one country.

When at the end of the fifth century B.C. the
Pythagoreans emerge in the light of history, Plato
(Republic 600A) and Isocrates (Busiris 28) regard them
with respect for their conduct and discipline. However,
the New Comedy (Vors. I, 278-80) treats them as
figures of fun because of their rigors in diet and cloth-
ing. We may assume that the asceticism (askesis) they
practiced arose from beliefs about the soul: that it was
entombed in the body (Plato, Gorgias 493A) in the
sense in which Socrates says (Phaedo 64A) that “philos-
ophy is a practice of dying and of the state of being
dead.” If the body is nothing but a temporary habita-
tion for a soul pilgriming through earthly and other
than earthly existences towards a final reward or pun-
ishment, then our concern must be to “care for the
soul.” A frequent corollary of care for the soul is ne-
glect of the body. It would appear that early
Pythagoreanism was not altogether exempt from this
excess. Neglect of the body characterized also the
Orphics of the fourth century, but what marked off
Pythagoreans from Orphics was that the Pythagorean
askesis had an intellectual character. Care for the soul
implied for them a discipline not only of the body but
also of the mind.

The revolutionary aspect of Pythagoras' doctrine of
the soul was not transmigration, nor a system of re-
wards and punishments—characteristics we find in
other sects—but the notion of personal and intellectual
askesis. It may be that Pythagoras as an Ionian found
the ecstatic practices of the mysteries uncongenial. He
cultivated pursuits of an intellectual character
(Heraclitus, Vors. 22B 40) and a serious demeanor
(gravitas), and these were a part of the heritage of

Among the thinkers of the fifth century Empedocles
alone taught a similar doctrine of the soul and its
migrations. Although he was only a generation older
than Socrates, his thought has the characteristics of an
earlier period in which each thinker faced ex novo the
problems of the universe, making use of the ideas of
his predecessors only as stimuli. Empedocles' great
poem On Nature falls into two parts so disparate that
scholars have sought to explain the yawning gap be-
tween them by the hypothesis of a conversion. One
part presents a theory of the physical world, the other
concerns Purifications. The purifications deal with the
soul, its migrations through plant and animal bodies,
and the means we may take to escape from this cycle


to divine status. So in Empedocles as in Pythagoras
we have two concomitant but apparently unrelated
doctrines, one of the physical world and one concern-
ing the soul.

That Empedocles was influenced by some Pythago-
rean tradition we may assume if a famous fragment
(Vors. 31B 129) alludes, as is generally thought, to
Pythagoras. But the influence is remote. Empedocles
is animated by a horror of the pollution incurred in
eating flesh. His aim is, by avoiding such pollution,
to achieve again the godhead from which he is fallen
(Vors. 32B 115). There is no trace of the personal
askesis that is a mark of Pythagoreanism, nor is there
in the poem On Nature any trace of arithmological
speculation except for two forces, four elements, and
similar hints such as we find also in Homer. We know
that in Empedocles' city, Acragas, Orphic ideas were
current (Pindar, Ol. 2, 62-83)—ideas to which we may
reasonably suppose that Empedocles subscribed.
Though his notions of soul derive from the same
cultural milieu as those of Pythagoras, his poems do
not reveal Pythagorean influence. The peculiarly
Pythagorean notion of soul, with its corollary for con-
duct, did not so far as we know find philosophical
expression in the fifth century, and no more did the
arithmological doctrines. Pythagoras' message was
“Care for your soul, and endeavour to penetrate the
mysteries of the universe by observing numerical cor-
respondences.” The time was not yet ripe for a philo-
sophical development of such teachings. It was only
with the emergence of mathematics as a science that
number mysticism could acquire philosophical impor-
tance; and it was only when the investigation of ethical
concepts began that thinkers had to concern themselves
with the soul that enjoined respect of those principles.

The fifth-century vacuum in Pythagoreanism has
excited the attention of scholars since antiquity. The
Neo-Pythagoreans sought either by invention or adop-
tion to people that vacuum, and Iamblichus gives us
a list of their names. A century ago an attempt was
made to construct for a hypothetical Pythagorean
brotherhood a theory of number atomism by reaction
to which the monism of Parmenides, as a dissident
Pythagorean, could be explained. Tannery, the
originator of this thesis, has been followed by several
scholars, among them Cornford (p. 62). Von Fritz fur-
ther argues that Hippasus of Metapontum, a dubious
and shadowy Pythagorean, discovered incommensura-
bility. But increasingly of late, doubt has been cast on
these and similar theses on the grounds that they are
built on questionable hypotheses (Philip [1966a], p. 2).
They assume a formal development of the mathe-
matical disciplines earlier than is historically possible,
and they impute a mathematical character to the
thought of Parmenides and Zeno for which there is
no evidence. It is much more probable that
Parmenides' “One” was a revolutionary elaboration of
the nous of Xenophanes that “sees as a whole, thinks
as a whole, hears as a whole” (Vors. 21B 24). But we
may not assume that this nous/mind is simply a cosmic
extension of the psyche/soul of Pythagoras. Even when
later nous is situated within or as a function of soul,
the two concepts remain distinct. Psyche for Homer
and Hesiod is what imparts life to bodies. Its later role
as the cause of motion develops from its more primitive
function, motion being the primary manifestation of
life. Nous on the other hand is from the beginning
intellective or cognitive, expressive of seeing as
knowing. Plato would appear to have been the first
to relate the two by locating, as he does, nous within
soul both in the macrocosm and the microcosm.

Our tradition then suggests that the twin doctrines
of Pythagoras, in spite of their potential importance,
remained philosophically dormant throughout the fifth
century, professed as a way of life rather than as theory
by whatever persons may have espoused them. They
acquired philosophical importance and the form in
which they were transmitted to later antiquity only
when Plato encountered them. But what of Philolaus?
He is the one Pythagorean of the fifth century having
a clear title to philosophical stature and of whom we
have extensive fragments. It is therefore not surprising
that controversies regarding early Pythagoreanism
center around Philolaus. His fragments have had their
defenders, but a majority of scholars has pronounced
them falsifications (Burkert, p. 206, n. 17). Recently
however Burkert has presented an able defense of a
group of them, and the question is again open. How-
ever the question may be decided, we must observe,
first, that Philolaus' main interests, astronomy and
physiology, are peripheral to Pythagoreanism; second,
that having apparently lived in exile from his youth
until advanced years, his knowledge of Pythagoreanism
must have been slight; and, third, that the fragments,
if genuine, do not inspire any high opinion of Philolaus'
philosophical grasp. They are no foundation for a
reconstruction of fifth-century Pythagoreanism.

The Pythagoreans of the fifth century are insub-
stantial figures owing such character as they exhibit
to Neo-Platonic fabulation (Guthrie, I, 319-33).
Unlike them Archytas is an historical person, known
repeatedly to have held the highest office in Tarentum.
He was a contemporary of Plato's and a friend cer-
tainly from the time of Plato's second voyage to Sicily
(366 B.C.) and probably from the time of his first voyage
(388 B.C.). It is often suggested that the Republic,
written between the two journeys, was influenced by
Plato's encounter with Archytas and the Pythagoreans
of Magna Graecia. If the encounter provoked a signifi-
cant change in the complexion of Plato's thought, we


must ask ourselves what new ideas he may have met
with. It is unlikely that he found current in Tarentum
doctrines of the soul unfamiliar to him. There were
Orphic adepts in his own Attica (Republic 364E) and
a few persons, some of them probably known to him,
calling themselves Pythagoreans (Vors. 52, 1). He may
have been impressed by the mathematical abilities of
Archytas, but the fragments suggest not an adept of
arithmology, of which there is no trace, but an original
mind engaged in mathematical investigations. Plato
had about him in Athens mathematicians of genius who
were the peers of Archytas. But Archytas may have
communicated to Plato arcane doctrines of the
Pythagoreans. If this had been the case Aristotle, who
wrote a treatise on the philosophy of Archytas in three
books (Diogenes Laërtius, V. 25), could hardly have
failed to have pointed out Plato's dependence on him
when in the Metaphysics he was discussing the origins
of Plato's thought. There however (987a 29-31) he tells
us simply that Plato's thought was in the main a devel-
opment of Pythagorean doctrines. If we can discover
no individual Pythagorean whom Plato could have
regarded as a master, and we have no evidence for
a “brotherhood” professing a body of doctrine, what
can Aristotle mean?

We can attempt to answer this question only when
we have first ascertained what doctrines Aristotle
imputes to the Pythagoreans. He tells us that from
Limit and Unlimited as first principles there proceeds
a One, and from the One somehow the numbers that
are things (Metaphysics 986a 17-21). Further, he
ascribes to the Pythagoreans doctrines, not of a philo-
sophical nature, about the soul (De anima 404a 17).
That there were in Plato's time Pythagoreans who
professed such doctrines, and that these derived
ultimately from Pythagoras himself we may easily
believe. But this seems an inadequate basis for what
passes as Pythagoreanism at the end of the fourth
century, after Plato's death, unless we assume that
Plato himself, in rethinking, transformed them. By
the end of the century Pythagoreanism had acquired
important tenets of which we have no trace in the early
period but which continue to characterize Pythago-
reanism thereafter. In particular the two principal
tenets are substantially modified. Soul now no longer
animates only the human person, plants, and animals.
It also animates the universe. It has acquired mathe-
matical characteristics, the souls of macrocosm and
microcosm both being structured according to mathe-
matical ratios. Further, numbers have ceased to be
things. For the universe is now divided into the realm
of the intelligible and the sensible, and numbers apper-
tain primarily to the former.

We still have the two basic Pythagorean doctrines
of soul and of number-things. But soul has been given
a metaphysical role, and number-things have had to
accommodate themselves to a chorismos or separation
of intelligibles and sensibles. This is that mathematiza-
tion of philosophy against which Aristotle protests so
vigorously. It represents in its main lines the thought
not only of Speusippus and Xenocrates but also, as far
as we can discern, that of Plato in the agrapha dogmata
or “unwritten doctrines” (Ross, pp. 142-53; Merlan,
pp. 11-33; Krämer, pp. 1-2; Gaiser with frags.).

Can Plato himself have transformed in this manner
the simpler tenets ascribed by Aristotle to “the
Pythagoreans”? We observe in the dialogues how he
moves from the Socratic form of refutation (elenchus)
and a Socratic form of definition, especially of ethical
ideas or concepts, towards a metaphysical structure in
which these ideas can, as it were, be anchored. We
observe how he differentiates between the knowledge
we can have of such intelligibles and the knowledge,
or better belief, we can have of sensibles. We observe
his struggle to clarify the relation between intelligibles
and sensibles. But can we believe that Plato was ready
to adopt a whole metaphysical structure, at best im-
plicit darkly in the dialogues, and that his dialogues
serve to link and explain that structure in a way the
dialogues hardly adumbrate?

We can attempt to answer that question in two ways;
first, by asking whether the history of Greek thought,
of which we have from Plato himself the first percep-
tive and imaginative accounts, would authorize any
such hypothesis and, second, whether Aristotle's ac-
count of the manner in which Plato's thought evolved
either confirms or at least does not contradict our
hypothesis. If this proves to be the case we may then
consider Speusippus and Xenocrates, as to whose spec-
ulation we are perhaps better informed, to see if their
metaphysics may reasonably be supposed to have
evolved from Plato's agrapha dogmata.

We are imputing to Plato a revolutionary rethinking
of the two basic Pythagorean tenets. That someone in
the early fourth century should rethink their number
notions is entirely plausible. For at about that time
mathematics became an organized discipline and,
shortly thereafter, made tremendous strides in all its
branches. In the Academy of Plato there were a num-
ber of eminent mathematicians (Proclus, in Euclid.
64-68, Friedlein), and our tradition tells us that Plato
furthered their pursuits. His own works are a testimony
to his mathematical interests. What could have been
more natural than that he should modify the crude
notion of number-things to adapt it to his two-realm
theory? How he did so is still a matter of debate, but
it would appear that he recognized a class of mathe-
maticals intermediate between intelligibles and sensi-
bles, and that in the Timaeus he regarded sensibles as
somehow consisting of geometrical configurations.


How his two ultimate principles—the One or unity
and the indefinite dyad or the great-and-small—gen-
erated this universe and what was the relation between
them we need not attempt to determine here. Nor need
we hazard a guess whether the atomism of Democritus
in part suggested such a development.

The Pythagorean soul-tenet we find similarly ex-
tended and transformed. For Pythagoras the soul was
what animates the body. It apparently had no cosmic
function nor was it a first principle of motion. Aristotle
tells us (De anima 405a 29) that Alcmaeon saw soul
as immortal because it was in motion, like the heavenly
bodies. Motion is a property of everything ensouled,
but soul is not therefore a principle of motion (cf.
Skemp, pp. 36-64), and Plato is the first thinker known
to us who explicitly regards soul as the first principle
of motion (kinesis). But he extends the notion of soul
also in another direction. Soul animates not only all
living creatures but also that living creature par excel-
lence, the universe. How did Plato move from the
individual migrating soul to the soul of the universe?
He may have taken a hint from Anaxagoras. For in
the account Socrates gives of his own intellectual de-
velopment in the Phaedo, nous is the cause of motion.
The relations between nous and psyche in Plato are
a problem fraught with difficulties; but it is clear that
the passage from nous as cosmic mover to the world
soul of the Timaeus is an easy one and, if the theory
is to leave room not only for intellection but also for
perception, a necessary one.

Let us concede for the moment that, within the
historical framework of preceding thought, a modifica-
tion of Pythagorean tenets in the sense we have sug-
gested is a possible one, and let us ask ourselves how
Aristotle's account of the development of Plato's
thought is consonant with this scheme.

In the Metaphysics (987a 29-988a 17) he gives us
a concise account of Plato's philosophy, in the context
of a survey of preceding views about first principles
or causes. In the course of his survey Aristotle discusses
“those who go by the name of Pythagoreans.” These,
he says, regard number as first principle and as material
substrate (986a 16-21). Number they hold to proceed
from an ultimate duality, Limited and Unlimited, from
which derives the One, and thence come the numbers
that constitute the physical universe. Apparently some
Pythagoreans differed from this account in that they
recognized not one pair of opposites but a table of
opposite pairs. As their table begins with Limited and
Unlimited, as its pairs are not logically or derivatively
related, and as they are obviously padded to reach the
number of the decad Aristotle is doubtless correct in
thinking theirs only a later variant of the original
doctrine. So we have for the Pythagoreans a duality
of first principles, and numbers constituting things.
There is no mention of soul, probably because Pythag-
orean notions of soul are not at the level of theory.

Plato's philosophy, says Aristotle, in most respects
followed after and conformed to Pythagorean doc-
trines. But it had also its own peculiar characteristic.
Plato believed, with the Heracliteans, that there was
no knowledge of sensibles because they exhibited no
constant state, but that there was knowledge of uni-
versals and in particular of the sort of universals that
Socrates sought to define. His universals Plato called
“ideas.” (Aristotle's polemic against the ideas arises out
of this separation or chorismos of intelligibles and
sensibles, Metaphysics 1040b 28 and passim.) Aristotle
goes on to explain that there were differences between
Plato and the Pythagoreans in matters of immanence
and transcendence (how the Pythagoreans with their
number-things could hold any such doctrines he does
not suggest) and that though they agreed on the One
as being, Plato modified the initial contrariety, substi-
tuting for the Unlimited the indefinite dyad.

Another important difference Aristotle sees is the
fact that, whereas for the Pythagoreans things are
numbers, for Platonists there exist mathematicals in-
termediate between ideas and sensibles and these me-
diate the reality we find in sensibles in respect of their
idea paradigms. So Plato, according to Aristotle, has
adopted the theory of dual first principles and the
number-substance notion of the Pythagoreans. But as
his Heraclitean views compelled Plato to recognize a
gradational reality, downwards to sensibles, he modi-
fied their theories so as to achieve a procession of being
from first principles and the mediation of mathe-
maticals between ideas and particulars.

The general remarks on the origin of Platonic
theories in Book A of the Metaphysics has as its com-
plement in Books M and N a long and involved argu-
ment against the doctrine of idea numbers propounded
by Plato, Speusippus, and Xenocrates. Though Aristotle
makes little reference to persons it is possible to distin-
guish the aspects of the doctrine we must ascribe to
the latter two (Ross, p. 152), and its substance remains
imputable to Plato. Unless therefore we are willing to
maintain that the whole complex structure is a fantasy
of Aristotle's built on a few tentative hints in the
dialogues, we must concede that Plato, at least in his
later years, professed a number doctrine which was
a modification of more primitive Pythagorean teaching,
and that he was indeed the author of the astonishing
synthesis—a unique product of historical, mathe-
matical, and metaphysical imagination or insight.

If we find Speusippus and Xenocrates taking as their
point of departure a doctrine such as we have ascribed
to Plato, and if we discover that their teachings were


reputed to be Pythagorean, we may regard this as
substantially confirming the hypothesis of a Platonic
rethinking of basic Pythagorean doctrines. Let us then
turn to these two thinkers, successors to Plato in the
headship of the Academy: the first a nephew who
succeeded probably by Plato's nomination, the second
regarded as his most faithful disciple. We find to our
surprise that both of them abandon basic Platonic
tenets, including belief in the ideas, and that they
apparently do so under the pressure of criticism (Meta-
1086a 2). That some of this criticism came from
Aristotle we cannot doubt. He was a member of the
Academy until his departure for Assos on Plato's death.
If he ever subscribed to the theory of ideas he had
long ceased to do so, and the Metaphysics give us
repeatedly and on many counts résumés of his objec-
tions which must have been worked out in discussion
within the Academy. That Aristotle extorted conces-
sions is however less surprising than the fact that both
men appear to have retreated to Pythagorean positions.
Let us consider the two thinkers separately, observing
both the ground they yield and the positions they take.

To Speusippus (ca. 395-339 B.C.) and his Encomium
of Plato (Diogenes Laërtius 3.2), we owe the curious
tale that Plato's father, Ariston, attempted to force his
own wife Perictione, but to no avail. When he desisted
he had a vision of Apollo, and thereupon refrained from
intercourse with her until Plato was born. As
Wilamowitz has observed, this tale strikes us, and
probably would strike an Athenian of the fourth cen-
tury, as bizarre and not in the best of taste. It might
even strike an Athenian as ludicrous. For Plato was
the youngest child of four, and gods traditionally
favored virgins. However the point of Speusippus' tale
may be quite a different one. Iamblichus (Vita Pythag-
4.6) and probably also Prophyry (Vita Pythagorae
2) recount similar tales of Pythagoras' paternity, that
he was fathered by Apollo. If this tradition goes back
to the fourth century or earlier, then what Speusippus
means us to infer is that Plato is a Pythagoras redivivus.
In any event Apollinian paternity connects him with
the Pythagorean tradition.

That Speusippus had for Aristotle peculiarly
Pythagorean associations we see from the way in which
he couples the names “Speusippus and the Pythago-
reans” (Metaphysics 1072b 30, Nicomachean Ethics
1096b 5) and frequently alludes to common doctrine.
We need not question Aristotle's testimony here, for
our longest fragment (Lang, frag. 4) comes from
Speusippus' treatise On Pythagorean Numbers. The first
part deals with the numbers involved in the derivation
of solids and with the five cosmic figures which, as Eva
Sachs (pp. 42-48) has shown, are to be ascribed as
mathematical constructions to Theaetetus. The second
part of the treatise is more interesting for our purposes,
and lamblichus paraphrases the words of Speusippus.
Speusippus' theme is the decad. He treats it not, as
we might expect, by seeking for “correspondences”
after the Pythagorean manner familiar to us from
Aristotle (Metaphysics 1092b 8-1093b 29) and by find-
ing mystical significances. His is rather an essay in the
theory of number that we later encounter in Nico-
machus of Gerasa the mathematician (end of first cen-
tury A.D.) and the Neo-Pythagoreans. In this he may
be their precursor.

If then we have ample warrant for regarding
Speusippus as a Platonist having marked and confessed
Pythagorean leanings, let us now consider how he
modified Plato's doctrines, and whether his modifica-
tions may be regarded as “Pythagorean.” It is notorious
that he abandoned the ideas (Lang, frag. 30) and to-
gether with them one of the two Platonic first princi-
ples, the indefinite dyad. In its place he recognized
plurality as the companion principle of the One. From
these two principles proceeded number, the whole
cosmos of intelligibles and sensibles consisting of
number. For Speusippus did not abandon the two-
realm theory of Plato (Lang, frag. 29). But the “com-
mon ground” (Lang, frag. 4) on which he established
association of intelligible universals and sensible par-
ticulars was number. Now if particulars too were
essentially number it could no longer be denied that
they also were knowable. So Speusippus conceded that,
as a scientific rationality (ratio) in us enables us to know
intelligibles, so a scientific perception—a judging fac-
ulty or criterion—enables us to know sensibles. So we
have of them not belief (doxa) as Plato taught, but
knowledge in the full sense.

Why then did Speusippus not abandon the notion
of separate substance? We are told (Metaphysics 1090a
3-37) that he held that as objects of science they must
be separate. But if he was prepared to use Ockham's
razor on the ideas why should he cling to a realm of
transcendent number? Much of his philosophical
activity was devoted to the discovery of similarities
(homoia), and his longest work, in ten books, bore that
title. Though his classifications bore a strong resem-
blance to, and in some instances anticipated, Aristotle's
biological classification, they were not in intent bio-
logical but were apparently meant to exhibit the struc-
ture of reality, to show how a plurality of particulars
constitutes a unity and class. The numbers constituting
classes and exemplified in particulars are not them-
selves subject to process as are the particulars, and this
was the criterion on which Plato established his two
realms (Timaeus 27D). But whereas for Plato soul
mediated between these two realms, for Speusippus
soul became “the form of the everywhere extended”


(Lang, frag. 40). This formula seems to imply a mathe-
matical penetration of both realms, the fact of process
being the only difference between them. It could be
made to apply to microcosm as well as macrocosm,
but it is difficult to see how such a soul could be the
cause of motion; and indeed when Diogenes Laërtius
attributes the definition of Speusippus to Plato he
interprets this soul as pneuma (Diogenes Laërtius, 3.

Ingenious but unsuccessful attempts have been made,
in particular by Frank (pp. 130-34) to reconstruct the
system of Speusippus. For this the fragments do not
afford us sufficient knowledge of detail. But one aspect
of this system is of special interest to us. Aristotle
(Nicomachean Ethics 1096b 5) tells us that “the
Pythagoreans seem to me to give a more convincing
account (of the Good) when they situate the One on
the good side of their column of opposites. Speusippus
apparently conforms to this doctrine of theirs.”
Aristotle recognizes (Metaphysics 1091a 36) that
Speusippus is meeting a real difficulty here, but a
difficulty which, according to Aristotle, arises from mak-
ing the One a first principle and principle of number,
not from equating it with the Good. The Pythagoreans
and Speusippus did not predicate goodness of their
One, Aristotle says (Metaphysics 1072b, 30), because
they observed that in plants and animals the good was
a telos (“goal” or “end”) achieved only in the course
of development. But Theophrastus' account (Meta-
11a 24) suggests that this was only an argument
to buttress their case. The real reason, as Aristotle
recognized (Metaphysics 1091b 30), was that if one of
a pair of opposed first principles was said to be good
or the Good, then the other must be recognized as evil.
“So he [Speusippus] used to avoid predicating good
of the One, on the grounds that, since process occurs
from opposites, then necessarily plurality would be evil
itself.” (For the problems here see L. G. P., pp. 25-27.)

Speusippus (and the Pythagoreans) may have
thought that this difficulty was adequately met by a
pair of number first principles from which numbers
proceeded, their physical manifestations achieving
their good only in the course of development. But it
is possible that he met it also by recognizing a One
above and beyond his primary contrariety, as Plato's
Good was epekeina (“above and beyond”). Proclus
(Comm. in Parmen., Klibansky 38.34) tells us that
Speusippus held a doctrine he ascribed to “the an-
cients” (Pythagoreans?)—a doctrine differentiating
between a first One not participating in being and a
One in which existents participate, we assume as one
of a pair of first principles. If there is a One from which
the two first principles proceed, it is easier to think
of them as purely mathematical and having no value
connotations. Such a theory however might lead to
consequences such as those on which Aristotle remarks
(Metaphysics 1028b 18), that differing first principles
have to be recognized for numbers, magnitudes, soul,
and so forth. Instead of a coherent system we have
episodes, as in a bad tragedy (Metaphysics 1075b 37).

So we may conclude that in abandoning Platonic
doctrines Speusippus reverts to simpler Pythagorean
ones, especially towards the thesis that “things are
number”—a position of which Aristotle says that
though it is an impossible one it has the merit of being
consistent. If he had abandoned it to achieve some new
synthesis of his own he would no doubt have professed
himself a Platonist, as Plotinus did later. But it would
appear that Speusippus was driven from his Platonic
positions and took refuge in a profession of Pythago-
reanism, a Pythagoreanism to which he himself had

With Xenocrates (ca. 406-315 B.C.) we are on more
difficult ground. He pythagorized, as Speusippus and
Plato himself had pythagorized before him. But
whereas Speusippus, in departing from Platonic doc-
trines, was an overt pythagorizer, Xenocrates professed
orthodox Platonism but modified Platonic metaphysics
in order to render some positions less vulnerable, and
to systematize. His modifications may have been natu-
ral developments of the “unwritten doctrines.” In part
they may have been countenanced by Plato himself.

What positions did Xenocrates take up in respect
of the twin themes of soul and number? He defined
soul, in a phrase that became widely current, as “self-
moving number.” This definition Plutarch attributes to
Xenocrates (Heinze, frag. 68), but according to Proclus
(Heinze, frag. 62) Xenocrates himself attributes it to
Plato. (It is attributed by the doxography, charac-
teristically, to Pythagoras—Dox. Gr. 386a 13, 651.11.)
If it was meant as a summary definition of the world
soul of the Timaeus, it was an inadequate one. Its
difficulties Aristotle does not fail to point out (De
408b 32), and some of the commentators, in
particular Philoponus (Heinze, frag. 65), argue in
Xenocrates' defense that no one who had “dipped into
mathematics with the tips of a finger” could define soul
so naively.

We may relate his definition of soul as self-moving
number to his other famous definition of the ideas as
“paradigmatic causes of physical objects as and when
they occur” (Heinze, frag. 30). These ideas he identified
with the idea numbers (Heinze, frag. 48). So both soul
and intelligibles are number-structured. We know fur-
ther that he derived physical magnitudes numerically
(Heinze, frags. 37-39; cf. Philip [1996a], p. 2), indivis-
ible lines being the atomic unit of derivation (Heinze,
frags. 41-49; Pines, passim.). That he should have


modified and developed Platonic doctrines seems cred-
ible. He survived Plato by thirty-three years, Aristotle
by seven. So it seems clear that his modifications were
in the direction of greater mathematization, and so of

How his system of derivation, from first principles
to physical particulars, was articulated is still a matter
of controversy (Krämer, 2, 21-101). On one vital issue
he seems to have been an innovator (Heinze, frags.
140-42). Plato had been singularly reticent about the
relations of nous and psyche. That they were to be
regarded as distinct is implied in the Sophistes (249a
2), and the Philebus (28c 7) in speaking of nous as “king
of heaven and earth” assigns it some exalted though
undefined status, as does the Laws. Xenocrates has a
single supreme first principle, nous, above the One and
the dyad. This nous is also the monad and is identified
with the Zeus of popular religion. This new doctrine,
which had for later Platonism consequences Xenocrates
could not foresee, may derive in part from Plato's One
and the Good. It had no Pythagorean origins. Never-
theless we may say of Xenocrates that, faithful Platonist
though he may have held himself to be, such modifica-
tions of Plato's later metaphysics as he admitted were
in the direction of pythagorizing. He commanded the
respect of his contemporaries as much for his manners
and morals as for his thought, thus conforming to the
Pythagorean ideal of the sage.

To sum up, we have in Pythagoras a great precursor
to whom is ascribed a doctrine of soul having implica-
tions for ethical conduct and a doctrine of number as
the nature of the universe. It was not to these teachings
however that he owed his authority. Instead, his moral
dominance lent credit to his teachings. After his death
his partisans formed the ruling clique of many city
states of Magna Graecia and remained in power for
half a century. During this period we hear little of
his doctrines and know of no Pythagorean of philo-
sophical eminence. Some time about the middle of the
fifth century a wave of opposition swept the Pythago-
reans out of office and into exile, if they came off with
their lives. As a political party they never returned
to power and it appears to have been only towards
the end of the century that their return from exile was
tolerated. It was under a democratic regime that
Archytas, a Pythagorean, became chief magistrate of
Tarentum about 375 B.C.

The fifth century was productive of great thinkers
in the West—Parmenides, Zeno, Empedocles, perhaps
Leucippus. But the only Pythagorean of some stature
is Philolaus. He enjoys a dubious reputation as an
astronomer, and for physiological speculation. His
teaching was as peripheral to what we must regard
as central Pythagorean doctrines as that of Archytas
was later. It is only with Plato that we find the notion
of soul achieving philosophical importance and cosmic
functions. It does so in a universe where things are
not numbers but number-structured. We conclude that
Plato seized on two basic Pythagorean notions, soul
and number, and used them (together with other hints
from earlier thinkers) in the construction of his own
metaphysics. Plato however did not publish in detail
and as a system his metaphysical doctrines. They
remained largely “unwritten” partly because he
believed they could not and should not be communi-
cated by the written word.

Plato's closest disciples were less reticent. They
published notes on a lecture or lectures of Plato's On
the Good,
and they developed in treatises their own
metaphysical doctrines, modifying what may be
inferred to have been Plato's position. They did so in
the direction of a Pythagoreanism Plato himself had
not merely espoused but in a measure had endowed
with its characteristic tenets. When with Arcesilaus (ca.
250 B.C.) there occurred in the Academy a reaction
towards a supposedly Socratic skepticism (Burkert, p.
83) pythagorizing became an embarrassment. Platonic
doctrines supposed to be tinged with Pythagoreanism
ceased to be imputed to Plato and were attributed
holus-bolus to Pythagoras. But Plato's reorientation of
thought towards mathematization and towards a doc-
trine of mind/soul continued to be fruitful of conse-
quences, even when the course of its development is
concealed by the endless mystifications of later


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Science; Neo-Platonism; Number; Platonism; 5">Pythagorean