## 1. Life and Writings


### 1.1 Date, City of Origin and Connections to Other Philosophers



We know very little about Philolaus' life. Only one brief and not very
reliable ancient life of Philolaus survives, that of Diogenes Laertius
(VIII 84-5). Diogenes includes Philolaus among the Pythagoreans;
indeed he is, in fact, one of the three most important figures in the
ancient Pythagorean tradition, along with Pythagoras himself and
Archytas. The central evidence for Philolaus' date is Plato's
reference to him in the *Phaedo* (61d-e). Socrates'
interlocutors, Simmias and Cebes, indicate that they had heard
Philolaus in Thebes at some time before the dramatic date of the
dialogue (399 BC). The passage suggests that Philolaus was no longer
in Thebes in 399, but there is no indication that he has died. In
order for Philolaus to have been a prominent teacher by the later
fifth century he must have been born no later than 440. There are
other indications that he was born even earlier. Both Philolaus and
Eurytus are identified by Aristoxenus as teachers of the last
generation of Pythagoreans (D. L. VIII 46), who were in their twenties
in 400 and active in the first half of the fourth century. Philolaus
is also said to be the teacher of Eurytus (Iamblichus, *VP*
139), however, so that the tradition makes most sense if Eurytus is
born ca. 440, and Philolaus belongs to the preceding generation and is
thus born ca. 470. Such a dating would also fit with the tradition
that Philolaus was the teacher of and thus somewhat older than
Democritus (D. L. IX 38), who was born ca. 460. It is even less clear
when he died, but one report suggests that he may still have been
alive in the early 380s, when Plato first visited southern Italy (D.L.
III 6). If he lived from ca. 470 to ca. 385, Philolaus is an
approximate contemporary of Socrates. Plutarch's story that as a young
man Philolaus was one of two to escape the burning of the Pythagorean
meeting place in Metapontum in 454 (*On the Sign of Socrates*
583a) would be consistent with this dating, but earlier versions of
the story do not mention Philolaus (Aristoxenus in Iamblichus,
*VP* 249-50) so that it is far from certain that he was
involved in the incident.



Diogenes Laertius says that Philolaus was from the Greek city of
Croton in southern Italy, but our earliest sources are divided as to
his city or origin. In his history of medicine, Aristotle's pupil
Meno, who had access to Philolaus' book, also says that Philolaus was
from Croton (DK 44 A27), but another of Aristotle's pupils,
Aristoxenus, who had close connections to the Pythagoreans, presents
Philolaus as from another Greek city in southern Italy, Tarentum (D.
L. VIII 46, see also Iamblichus, *VP* 267, which probably
derives from Aristoxenus). Perhaps the best way to solve the conflict
is to suppose that Philolaus originally came from Croton and that,
following attacks on the Pythagoreans in other southern Italian cities
ca. 450 and his trip to Thebes in the late 400s, he may have settled
in Tarentum late in his life. He is reported to have been the teacher
of Archytas of Tarentum in one source (Cicero, *de Orat.* III
34.139).



The ancient tradition gives no indication of who Philolaus' teacher(s)
might have been. Pythagoras died some twenty years before Philolaus
was born. He might have had some contact with the Pythagorean Hippasus
(ca. 530-450?), who was associated with the mathematics of music
theory, which is also a prominent theme in Philolaus. Philolaus should
not be understood as simply a Pythagorean, however. He was an
important philosopher in his own right, and neither Plato nor Meno
bothers to apply the label Pythagorean to him. He was influenced not
just by the Pythagorean tradition but also as much or more by the
broader tradition of Presocratic philosophy. He was undoubtedly aware
of the work of the medical writer/philosopher Alcmaeon, who also came
from Croton. Anaxagoras and Empedocles were the most prominent
philosophers in the generation before Philolaus, and there is some
evidence that he was in part responding to Anaxagoras (see 2.1).
Philolaus was the contemporary of the mathematicians Hippocrates of
Chios, who was the first to write an *Elements* of geometry,
and Theodorus of Cyrene, and, although Philolaus was not a
mathematician himself, his philosophy appears to have been influenced
by the important new developments in Greek mathematics that they
brought about. Like most philosophers of the fifth century his work is
in part a response to Parmenides (see 2.1 and 3). 


### 1.2 The Authenticity Question



In the case of most ancient authors, the assumption is that a text
handed down in their name is genuine, unless strong reasons can be
given for regarding it as a forgery. This standard assumption does not
apply to the Pythagorean tradition. Many more demonstrably spurious
texts have survived (collected in Thesleff 1965) than texts that can
with confidence be regarded as genuine. Some forgeries may have been
produced for monetary reasons; a text of a "rare" work by
a Pythagorean author could fetch a large sum from book collectors.
There were characteristics unique to the Pythagorean tradition,
however, that led to a proliferation of forgeries. Pythagoras himself
wrote nothing. Starting as early as the later fourth century BC
(Burkert 1972a, 53-83; Huffman 1993, 22-6), however, he
came to be regarded, in some circles, as the philosopher par
excellence, to whom all truth had been revealed. All later philosophy,
insofar as it was true, was a restatement of this original revelation
(see, e.g., O'Meara 1989). In order to support this view of
Pythagoras, texts were forged in the name of Pythagoras and other
early Pythagoreans, showing that they had, in fact, anticipated the
most important ideas of Plato and Aristotle. These pseudo-Pythagorean
texts are thus characterized by the use of central Platonic and
Aristotelian ideas, expressed in technical Platonic and Aristotelian
terminology. The date and place of origin of these pseudo-Pythagorean
treatises is difficult to determine, but most seem to have been
composed between 150 BC and 100 AD (Burkert 1972b; Centrone 1990 and
2014; Moraux 1984); Rome (Burkert 1972b) and Alexandria (Centrone 1990
and 2014) are the most likely places of origin. As a result of these
unique features of the Pythagorean tradition, any fragment assigned to
a Pythagorean philosopher such as Philolaus must be examined carefully
before it can be accepted as genuine.



In 1962, Guthrie, in his *History of Greek Philosophy*,
regarded the issue of the authenticity of the fragments handed down in
Philolaus' name as undecided, although he commented that opponents of
authenticity did "not seem to be at their best on the
subject" (331). In that same year Burkert provided decisive
arguments, which have led to a consensus that a core of the fragments
of Philolaus are authentic. Earlier discussions of the authenticity of
the fragments were often based on the faulty assumption that the
fragments were either all genuine or all spurious. Opponents of
authenticity typically showed that a few of the fragments were
spurious and then assumed that all the rest were as well (e.g.,
Bywater 1868). Given the amount of forgery in the Pythagorean
tradition, however, even if fragments from a genuine book of Philolaus
survive, it is likely that fragments from spurious works survive as
well. Burkert made a crucial distinction between two traditions
concerning Pythagoreanism. One begins already in Plato's Academy and
treats Pythagoreanism as largely identical with later Platonic
thought, including the postulation of the one and the indefinite dyad
as first principles (1972a, 53-83). This is the tradition which
came to dominate among later Neoplatonists and which served as the
basis for the production of the pseudo-Pythagorean writings. The
second tradition is represented by Aristotle, who presents
fifth-century Pythagoreanism as having had some influence on Plato but
as radically different from Plato in making no distinction between the
intelligible and sensible world (1972a, 28-52). Burkert then
showed that, while some of the fragments of Philolaus did employ
Platonic and Aristotelian ideas and hence were spurious, other
fragments were in very close agreement with Aristotle's account of
fifth-century Pythagoreanism and hence likely to be authentic.



Aristotle's testimony on the Pythagoreans gives rise to one remaining
puzzle, which is relevant to the authenticity of the core of fragments
identified by Burkert. Aristotle specifically identifies the
Pythagoreanism that he discusses as contemporary with or a little
earlier than the atomists (*Metaph*. 985b23 ff.). This dating
fits Philolaus exactly. Aristotle discusses the Pythagoreans in many
places and he is clearly not always referring to Philolaus.
Nonetheless, the central Pythagorean metaphysical and cosmological
system to which Aristotle refers repeatedly, with limiters and
unlimiteds as first principles, the emphasis on the role of harmony in
the cosmos and the peculiar astronomical system, which makes the earth
a planet orbiting around the central fire, corresponds in great detail
with the fragments and testimonia preserved in the later tradition as
coming from Philolaus' book (*Metaph.* 986a8-21,
987a13-17,990a8-10 etc.; for an opposing view see Zhmud
1998). The problem is that Aristotle never explicitly describes the
Pythagoreanism which he discusses as derived from Philolaus' book. He
only refers to Philolaus once by name, and that is in relation to an
apothegm which has no obvious connection to his general presentation
of Pythagoreanism (*EE* 1225a30 = Fr. 16). Everywhere else
Aristotle refers to the system he describes as the work of "the
so-called Pythagoreans" or the Italian philosophers. The former
expression shows that Aristotle has his doubts about the connections
between these philosophers and Pythagoras himself. Aristotle never
mentions Pythagoras himself in connection with the system of
"the so-called Pythagoreans" and clearly dates that system
after the time of Pythagoras. If Aristotle is relying heavily on
Philolaus' book, however, why does he never mention Philolaus by name
when describing the Pythagorean system? Some might argue that
Aristotle did not know of a book by Philolaus and hence that there was
no such book. Two considerations suggest another explanation. First,
Aristotle discusses the Pythagorean system in considerable detail and
in at least one passage strongly implies that what he is reporting is
based on a written text (*Metaph*. 1091a13). As Burkert notes,
that written source cannot have been ascribed to "the so-called
Pythagoreans" and must have been assigned to some author. The
evidence suggests that Philolaus' book was the only book by a
Pythagorean circulating in the fifth century (see 1.3 below), and, as
we have seen, the reports of the contents of that book agree with what
Aristotle reports quite closely. There simply is no other plausible
candidate for Aristotle's written source. Second, Philolaus' book
seems to have been available to Aristotle's pupil Meno (DK 44
A27-8) and hence is likely to have been available to Aristotle
as well. The most reasonable conclusion is that Aristotle is relying
largely on Philolaus' book for his description of fifth-century
Pythagoreanism but chooses not to refer to him explicitly. Aristotle
makes clear that there are several groups of people included under the
heading "so-called Pythagoreans," by explicitly
distinguishing those Pythagoreans who posited the table of opposites
from the main Pythagorean system which he describes (*Metaph*.
986a22). For whatever reason, it was the fashion to refer to a group
of related figures as Pythagoreans, and Aristotle follows this fashion
rather than singling Philolaus out from the group, even though he
relies heavily on Philolaus' book. Primavesi has recently suggested
that some of the puzzles about Aristotle's account of Pythagoreanism
can be mitigated if we recognize that he presents a developmental
acount by reconstructing a gradual transition from early
arithmological speculations to Philolaus' theory of principles (2014).



### 1.3 Philolaus' Book: Genuine Fragments and Testimonia



It is important to distinguish between two traditions regarding
Philolaus' publication of books (Burkert 1972a, 223-7; Huffman
1993, 12-15). In one tradition, which is likely to be reliable,
Philolaus is reported to have published a single book, which came to
bear the traditional title for all Presocratic philosophical
treatises, *On Nature*, although it is doubtful that this title
goes back to Philolaus himself. This tradition is found in Diogenes
Laertius (VIII 85) but goes back to Hermippus (3rd BC) and
Timon (DK 44 A8; 320-230 BC). Diogenes quotes the first sentence
of this book, and it is likely that it was available to Aristotle and
his pupil Meno in the fourth century (see section 1.2 above). This
tradition also suggests that Plato cribbed the *Timaeus* from
Philolaus' book. The accusation of plagiarism on Plato's part is
absurd, but there are enough general similarities between the
*Timaeus* and Philolaus' book to explain the origin of the
accusation (e.g., the musical scale which Plato uses in the
construction of the world soul corresponds to the scale in Philolaus,
Fr. 6a). This book of Philolaus is also likely to be the first book
published in the Pythagorean tradition. Diogenes asserts that
Philolaus was the first Pythagorean to publish a treatise on nature
(VIII 85). Pythagoras wrote nothing, and we are told that the same is
true of Philolaus' immediate predecessor in the Pythagorean tradition,
Hippasus (D. L. VIII 84). Some scholars argue that there were books by
Pythagoreans earlier than Philolaus (Zhmud 1989 and 2012: 121-9), but,
where there is clear evidence for a book, it is unlikely that the
figure is a Pythagorean (e.g. Alcmaeon) and, where the figure is
clearly a Pythagorean, there is no explicit evidence for a book (e.g.
Hippasus). A second tradition reports that Plato bought three books
from Philolaus (D. L. III 9; VIII 15 and 84). It is implied that these
books were not by Philolaus himself, and it seems likely that the
statement refers to three spurious works assigned to Pythagoras at
D.L. VIII 6  (Burkert 1972a, 224-5). The story of Plato's
purchase of these books from Philolaus was probably invented to
authenticate the three forged treatises of Pythagoras.



Burkert's arguments (1972a, 238-277), supported by further
study (Huffman 1993), have led to a consensus that some 11 fragments
are genuine (Frs. 1-6, 6a, 7, 13, 16 and 17) and derive from
Philolaus' book *On Nature* (Barnes 1982; Kahn 1993 and
2001; Kirk, Raven and Schofield 1983; Nussbaum 1979; Zhmud 2012,
Graham 2014). Fragments 1, 6a and 13 are identified as coming from the
book *On Nature* by ancient sources. Stobaeus cites fragments 2
and 4-7 as coming from a work *On the Cosmos*, but this
appears to be an alternate title for *On Nature*, which
probably arose because the chapter heading in Stobaeus under which the
fragments are cited is 'On the Cosmos.' Testimonia A7a,
A9, A10, A16 (part), A17 (part), A18-24, A27-9 are also
generally agreed to reflect the contents of *On Nature*. The
authenticity of some fragments and testimonia remains
controversial. Fr. 6b and Testimonium A 26, which, in contrast to
Fr. 6a, take a numerological approach rather than a properly
mathematical approach to music theory, were accepted as authentic by
Burkert (1972a, 394-400; followed by Mueller 1997, 292-3),
whereas Huffman argues that they are spurious (1993, 364-74;
followed by Zhmud 1997, 185). Recent scholarship on harmonic theory
argues that the conflict in approach may be less significant than it
appears at first sight (see the end of section 2.2 below) and that 6b
and A26 may thus be authentic (Barker 2007: 271; Hagel 2009:
143-151). Similarly, Burkert accepts A 14 as genuine and as
representing the earliest evidence for horoscopic astrology in Greece
(1972a, 350), whereas Huffman argues against authenticity (1993,
381-91).


### 1.4 Spurious Fragments and Testimonia



There are 15 fragments which are generally believed to be spurious (8,
8a, 9-12, 14, 15, 19, 20a, 20b, 20c, 21- 23) and six
testimonia (11-13, 16b, 17b, 30) that are based on spurious
works (Huffman 1993, 341-420). In two instances we are given
titles for the spurious works from which the fragments come. Fragment
21 is said to come from *On the Soul* and Fr. 22 from *On
Rhythms and Measures*. There are two problematic references to a
work entitled *Bacchae*. Stobaeus cites Frs. 17 and 18 as
coming from this work. Fr. 18, which dealt with the sun, was lost in
the transmission of Stobaeus, but Fr. 17 is generally accepted as
authentic on the basis of its contents and on those grounds would
appear to come from *On Nature*. This might suggest that
*Bacchae* was an alternative title for *On Nature*
(Burkert 1972a, 269). On the other hand, Proclus describes the
*Bacchae* as teaching theology by means of mathematics (DK 44
B19; Huffman 1993, 417-8), which is not a very apt description
of the central surviving fragments of *On Nature*, but would
fit the astrological material of controversial authenticity, which is
cited elsewhere by Proclus (A14--see 1.3 above). Thus it may be
that *Bacchae* is the title of a spurious work and that Fr. 17
was mistakenly assigned to that work. Kingsley argues that the title
is genuine and one of several pieces of evidence for a connection
between the Pythagoreans and Bacchic mysteries (1995: 262).
Stamatellos (2007: 102-103) suggests that Fragments 21 and 23
are authentic, but the overwhelming consensus of other scholars
regards them as spurious (see Burkert 1972: 242-243, who refers
to previous scholarship and Huffman 1993: 343-344). Mansfeld
(2016) suggests a quite plausible emendation to Fr. 12 but still
regards it as spurious. McKirahan (2016) argues that B14 is genuine,
but Philolaus' use of *psyche* in the genuine B13 is
incompatible with its use in B14.


## 2. Basic Principles of Reality


### 2.1 Limiters and Unlimiteds



Philolaus begins his book with a bold statement of his central insight
about reality:



> 
> Nature (*physis*) in the world-order (*cosmos*) was
> fitted together out of things which are unlimited and out of things
> which are limiting, both the world-order as a whole and everything in
> it. (Fr. 1)
> 



The language of this fragment shows that Philolaus is firmly in the
tradition of Presocratic philosophy. Aristotle refers to the
Presocratics as the *physiologoi*, "writers on
nature," and it is precisely nature (*physis*) that is
the first word of Philolaus' book. *Physis* here means
"nature" or "real constitution" and finds a
good parallel in Heraclitus' famous saying that "nature
loves to hide" (fr. 123). Philolaus uses it universally to refer
to the natures of all individual things in the world as well as the
nature of the cosmos as a whole. For an alternate view see Zhmud
(2018) who argues that it means "all the things that have come
into being and exist." Zhmud points out that Philolaus is the
first thinker to make *physis* the main subject of his
book. The next words show that Philolaus, like his Presocratic
predecessors, regarded the natural world as a cosmos, as an order, and
saw it as his central problem to explain that order. His answer to
that problem, however, is quite unlike earlier answers. Earlier
Presocratics had tended to explain the natural world in terms of
material elements such as water or air, or groups of elements such as
the earth, air, fire and water of Empedocles. Opposites such as the
hot and the cold or the rare and the dense also played a prominent
role. Instead of these concrete and tangible substances Philolaus
appeals to things which are unlimited, "unlimiteds," and
things which limit, "limiters." The basic problem in
understanding Philolaus' philosophy is thus to determine what he
means by limiters and unlimiteds. In the surviving fragments, he never
provides a definition of these terms nor does he provide an explicit
example. Nonetheless he uses them repeatedly. In Fragment 2 he argues
that the basic elements of the world must logically be either all
limiters, all unlimiteds or both limiters and unlimiteds. He is
emphatic here and argues further in Fr. 3, evidently against some of
his predecessors, that the basic elements cannot be simply
unlimiteds. Since the world around us manifestly contains some things
that limit, some that are unlimited and others that are both limiting
and unlimited, we must suppose that the elements from which the world
arose included both limiters and unlimiteds (Fr. 2). How could a
limiter arise from an unlimited or an unlimited from a limiter? In
Fr. 6 Philolaus argues that we cannot say anything more definite about
the ultimate principles of reality ("nature in itself" and
"the being of things"). We can only go so far as to say
that they must have included limiters and unlimiteds in order for the
world we see around us to have arisen. Beyond this description the
basic principles of reality admit only of divine and not of human
knowledge. Thus, it is a matter of principle that Philolaus does not
identify any particular set of limiters and unlimiteds as the basic
principles of the world; identification of such a set is beyond human
capability. On the other hand, he clearly thinks that we can recognize
limiters and unlimiteds in the world around us and repeatedly appeals
to this fact. If the cosmos and everything in it is put together from
limiters and unlimiteds, we ought to be able to identify some of
them.



It is logical to begin with the very first thing that is put together
in Philolaus' cosmos. In Fragment 7 Philolaus asserts that "The
first thing fitted together, the one in the center of the sphere, is
called the hearth." From other evidence we know that this hearth
in the center of Philolaus' cosmos was known as the central fire,
around which all the heavenly bodies, including the earth, orbit.
Since the central fire is explicitly said to have been "fitted
together," according to Fr. 1 it must have been fitted together
out of a limiter and an unlimited, like everything else in the cosmos.
It appears that the term "central fire" itself identifies
the limiter and unlimited from which it is composed. It is composed of
the unlimited fire, which is limited by its position in the center of
the sphere. One interpretation then would be that unlimiteds are
stuffs and limiters are shapes (Barnes 1982, 387 ff.). It is striking
confirmation of this conjecture that many earlier Presocratics
described their material principles as "unlimited."
Anaximander posited the unlimited as the source from which the basic
material principles of the world, the opposites such as hot and cold,
emerged, and his successor Anaximenes explicitly described his basic
material principle air, as unlimited (DK 13 A1 and A6). Anaxagoras, in
the generation before Philolaus, described all the things in the
original mixture from which the world emerges as unlimited and this
list would appear to include air, aither (Fr. 1), dry, wet, hot, cold,
bright, dark (Fr. 4), dense and rare (Fr. 12). Thus, Philolaus'
unlimiteds probably included stuffs in the broad Presocratic sense,
which treats opposites such as the hot and the cold as material
substances. The next stage in Philolaus' cosmogony after the
construction of the central fire helps to identify yet more
unlimiteds. In his lost work *On the Pythagoreans* Aristotle
describes the universe as drawing in time, breath, and void from the
unlimited (Arist., Fr. 201). This text certainly means that the
universe was surrounded by an unlimited from which time, breath and
void were drawn, but it probably also means that time, breath and void
were regarded as unlimiteds. Breath can be equated with air as a
typical Presocratic stuff, but time and the void show that we must
expand Philolaus' conception of unlimiteds beyond stuffs. Unlimiteds
appear to be not just materials, which can appear in various shapes,
but also continua, like time, which can be limited by various measures
but which do not have a shape. Philolaus thus accepts the earlier
Presocratic reliance on material principles but introduces two radical
changes. First, he thinks it beyond human knowledge to identify any
privileged set of unlimiteds, such as earth, air, fire and water, as
ultimate elements. Second, he focuses not on the qualitative features
of these stuffs (e.g. hot) but rather on their role as continua
(Huffman 1993: 43-4; Graham 2014: 52).



Philolaus' most significant innovation, however, is to argue that the
cosmos cannot be adequately explained in terms of unlimiteds alone.
There appear to be two related arguments for this thesis. First,
Philolaus takes it as self-evident that the world is full of limiters
such as shapes. If this is so, however, we must suppose that limiters
were among the basic principles from which the world arose, since it
is impossible for what limits to arise from what is unlimited (Fr. 2).
Second, in Fragment 3, he argues that if all things are unlimited then
"there will not be anything that is going to know." This
argument makes most sense if directed against Anaxagoras, who
generates the world from elements that are all unlimited (Fr. 1), but
who claims that mind (*nous*), which is also said to be
unlimited, knows all the things that are separated off from the
original mixture (Fr. 12). Philolaus seems to regard the act of
knowing as an act of limitation and hence as impossible, if reality is
composed just of unlimiteds (Huffman 1993, 119-120). Parmenides
is the only Presocratic before Philolaus to emphasize the role of
limit in his account of reality. Parmenides likens reality to a sphere
and describes it as held in the bonds of a limit (DK 28 B8).
Philolaus' insistence that the ultimate elements from which the cosmos
developed must have included not just unlimiteds but also limiters is
best understood as an attempt to integrate Parmenides' insights about
the nature of reality with the earlier Presocratic emphasis on
material elements which are unlimited (Huffman 1999). Earlier
Presocratics, such as Heraclitus (DK B30, B31), had recognized that
there was measure and structure in the world but no Presocratic had
made structure an element or principle in the world with the same
status as material principles. It is Philolaus who first insists that
structure is equal in importance to material constitution (Huffman
2013b; Graham 2014: 53). Philolaus' explanation of the cosmos in terms
of limiters and unlimiteds cannot be fully understood, however,
without discussing his third basic principle, *harmonia*.


### 2.2 Harmony



Already in Fragment 1, Philolaus appealed not just to limiters and
unlimiteds in order to explain the world but asserted that the cosmos
as a whole and everything in it only arose when limiters and
unlimiteds were "harmonized" or "fitted
together." Fragment 6 is even more explicit on the need for
harmony:



> 
> ...since these beginnings [i.e. limiters and unlimiteds]
> preexisted and were neither alike nor even related, it would not have
> been possible for them to be ordered, if a harmony had not come upon
> them... Like things and related things did not in addition
> require any harmony, but things that are unlike and not even related
> ... it is necessary that such things be bonded together by a
> harmony, if they are going to be held in an order.
> 



Graham suggests that harmony as a principle required to allow limiters
and unlimiteds to combine prefigures Platonic participation and
Aristotelian ontological predication (Graham 2014: 53-4). In Fragment
6a Philolaus goes on to describe this harmony and what he describes is
a musical scale, the scale known as the Pythagorean diatonic, which
was used later by Plato in the *Timaeus* in the construction of
the world soul. This scale provides Philolaus' only surviving explicit
example of the bonding together of limiters and unlimiteds by a
harmony. A musical scale presupposes an unlimited continuum of
pitches, which must be limited in some way in order for a scale to
arise. The crucial point is that not just any set of limiters will do.
We cannot just pick pitches at random along the continuum and produce
a scale that will be musically pleasing. The scale that Philolaus
adopts is such that the ratio of the highest to the lowest pitch is 2
: 1, which produces the interval of an octave. That octave is in turn
divided into a fifth and a fourth, which have the ratios of 3 : 2 and
4 : 3 respectively and which, when added, make an octave. If we go up
a fifth from the lowest note in the octave and then up a fourth from
there, we will reach the upper note of the octave. Finally the fifth
can be divided into three whole tones, each corresponding to the ratio
of 9 : 8 and a remainder with a ratio of 256 : 243 and the fourth into
two whole tones with the same remainder (for more on Fr. 6a see
McKirahan 2012 and Huffman 1993: 145-165). Thus, in Philolaus'
system the fitting together of limiters and unlimiteds involves their
combination in accordance with ratios of numbers. Similarly the cosmos
and the individual things in the cosmos do not arise by a chance
combination of limiters and unlimiteds; the limiters and unlimiteds
must be fitted together in a pleasing way in accordance with number
for an order to arise. Fragment 6a suggests that Philolaus saw the
cosmos as put together according to the diatonic scale. This would be
very much in accord with the famous conception of the harmony of the
spheres according to which the heavenly bodies make harmonious music
as they move, but neither in Philolaus nor any other early source do
we get an explicit account of how the musical scale corresponds to the
astronomical system (Huffman 1993, 279-83; Zhmud 2012: 343-4
denies that there was any connection between the harmony of the
spheres and Philolaus' system ).



Most scholars agree that Philolaus was not interested in harmonic
theory for its own sake as were Archytas, Aristoxenus and much later
Ptolemy (Barker 2007: 278-286; Creese 2010: 104). There is also
no evidence that he used a monochord in order to determine or
demonstrate harmonic relationships (Creese 2010: 104-117).
However, even if Philolaus' main interest was the application of
harmonics to cosmology, the approach to harmonics found in Fragment 6a
and in the disputed 6b and A26 has been of considerable interest to
historians of harmonic theory, and their conclusions are highly
relevant to the issue of the authenticity of 6b and A26. By the end of
the fourth century there was a marked division between Pythagorean
harmonic theorists who regarded musical intervals as corresponding to
whole number ratios and followers of Aristoxenus who regarded them as
linear magnitudes. For Aristoxenus the octave consisted of six equal
tones added together and could be divided in half. For the
Pythagoreans the octave was made up of ratios which were combined not
by addition but by multiplication (e.g. the octave [2 : 1] = the
product of the fifth [3 : 2] and the fourth [4 : 3], not the sum). The
octave was a little less than six tones (9 : 8), and neither it nor
the tone could be divided in halves. Fragment 6a of Philolaus mentions
the ratios that correspond to the intervals (e.g., the tone is 9 : 8)
whereas Fragment 6b and A26 present intervals as linear magnitudes
(e.g., the tone is 27). Since 6a treats intervals as corresponding to
ratios in a properly Pythagorean way, it has usually been accepted as
authentic and the difference in approach found in 6b and A26 has been
seen as incompatible with 6a and hence as indicative of their
spuriousness. Recently some scholars, on the other hand, have
suggested that in Philolaus' time this split between regarding
intervals as ratios and regarding them as linear magnitudes had not
been fully developed and that Philolaus may have regarded the two
approaches as compatible (Hagel 2009: 144; Creese 2010: 116, n. 119).
Barker supports this point by arguing that Philolaus is employing both
approaches already in Fragment 6a. Thus, while the fragment equates
certain ratios with certain intervals, Philolaus does nothing
mathematically with those ratios and instead emphasizes that intervals
have size and can be combined to create the octave, which suggests
that he thinks of them as linear magnitudes (Barker 2007:
264-271). This same combination of intervals as ratios and
intervals as linear magnitudes can be seen in Fragment 6b and A26,
where there is the additional emphasis on halving intervals. The
halving of intervals seems to be directed at finding the mid-point of
the octave and this might have been important in Philolaus'
application of the octave to the cosmos, where the center is important
for the symmetry of the whole system (Barker 2007: 277; cf. Hagel
2009: 151). Moreover, Aristotle's description of fifth-century
Pythagoreanism seems to display the same odd mixture of ratios and
individual numbers (Barker 2007: 286). However, there are some
difficulties for this approach. While Fragment 6a does emphasize the
size of intervals and their combination to form the octave, size need
not mean linear magnitude and combination need not mean addition. In
fact, no specific linear magnitudes (e.g. 27, which is equated with
the tone in A26) are mentioned. The only sizes of intervals mentioned
are precisely the ratios. Nor is there any evidence in Fragment 6a of
any interest on Philolaus' part in halving intervals. The situation is
quite the reverse. He uniformly presents the octave and, indeed, all
intervals as composed of unequal parts (e.g., the octave is composed
of the fourth and the fifth). Thus, while it is impossible to deny
that Philolaus might have tried to embrace both concepts of the nature
of a musical interval, Fragment 6a still differs from 6b and A26 in
being totally intelligible on the assumption that intervals correspond
to ratios and in making no overt use of linear magnitudes or halves of
such magnitudes (Huffman 2012: 235; cf. McKirahan 2012).


### 2.3 The Method of *Archai*



In each area of inquiry Philolaus appears to proceed by identifying
the minimum number of *archai*, "beginnings" or
"starting points," necessary to explain the phenomena in
that area (Huffman 1993, 78-92). In the case of the cosmos as a
whole, as we have just seen in Fr. 6, Philolaus argues that three
starting points must be assumed, limiters, unlimiteds, and harmony, as
a third element to hold these two unlike elements together. In the
case of disease, Philolaus identifies three *archai*, bile,
blood and phlegm (see 6.2 below). In the case of living creatures, the
brain is the starting point to which we must appeal to explain human
beings, the heart as the seat of sensation is the crucial factor in
explaining animals, the navel as the seat of rooting (the umbilical
cord is regarded as a root) is the distinctive starting point of
plants, and finally the genitals are a factor common to all animate
beings in contrast to the inanimate world (Fr. 13, see 5.1). This
approach to explanation seems to have been developed under the
influence of contemporary mathematicians and medical writers. Graham
argues that in his use of *archai* to mean something like
"principles" Philolaus achieves " a level of
abstraction that surpasses ... probably all of his predecessors"
and takes an important step towards Aristotle's conception of
*archai* as starting points of being or explanation (Graham
2014: 62). 


## 3. Epistemology: The Role of Number



In the discussion of harmony above, another crucial concept in
Philolaus' philosophy emerged, number. What role does number play in
Philolaus' system? Fragment 4 provides a relatively clear answer to
this question:



> 
> And indeed all things that are known have number. For it is not
> possible that anything whatsoever be understood or known without this.
> 



Number plays an epistemological role; it is what allows things to be
known and understood. There are several ways to interpret this claim.
Some scholars have regarded "having number" as the
necessary condition for the mere apprehension of an object. If
something is not countable and hence distinguishable from other
things, it can't be recognized as a separate entity at all (Nussbaum
1979; Kirk, Raven and Schofield 1983, 327). The epistemological
vocabulary which Philolaus uses (*noein* and
*gignoskein*), however, more typically refers to
understanding and knowledge than to simple apprehension of an object
(Huffman 1993, 116-118). Several fragments make appeals to sense
experience without mentioning the concept of number (e.g. Fr. 2), so
that it would appear that Philolaus thinks that we can initially
apprehend objects through our senses but that we only come to have
true knowledge or understanding of them by grasping their number.
Fragment 13 makes a clear distinction between sense perception, which
humans and animals share, and understanding, which is limited to
humans. Animals can apprehend objects in the world just as humans do,
but they do not understand them because they cannot grasp the numbers
which govern them, as humans can. Such an interpretation is supported
by Fragment 5:



> 
> Number, indeed, has two proper kinds, odd and even, and a third from
> both mixed together, the even-odd. Of each of these two kinds there
> are many forms, of which each thing itself gives signs.
> 



The last sentence of the fragment seems to say that individual things
in the world "give signs" of the many forms of number. Our
first experience of a musical scale comes through simple sense
perception, but, when the phenomena are studied further, they
"give signs of" or "point to" the whole-number
ratios that govern the musical intervals. Once we grasp these ratios,
we can claim to understand or know the scale. McKirahan (2012) argues
that in Fr. 6a Philolaus formally demonstrates that ratios are numbers
and that the many forms of numbers in Fr. 5 are the different types of
ratios. Huffman (1993: 174) argues that Philolaus' use of the term
number includes ratios from the start and that the many forms of
numbers are just the even and odd natural numbers (1993: 191; cf.
2012). Mourelatos has shown that Fragment 5 is also important
in the history of the development of the concept of the universal.
Philolaus distinguishes between numbers as individual species and
their genera, even and odd, i.e. between subordinate and superordinate
universals, as well as between numbers and things, i.e. between types
and tokens, e.g. four and four pebbles. Moreover, the reference in
Fragment 5 to things "giving signs" of numbers is
"the first recorded attempt in the history of metaphysics at
coining a term for [the relation between a particular and a
universal]"(Mourelatos 2006, 66). 



Burkert argues that Philolaus' appeal to number has little to do
with the rigorous mathematics that was being developed in the work of
his contemporaries Hippocrates of Chios and Theodorus of Cyrene. It is
instead a manifestation of the Pythagorean number mysticism, which can
be found in the *acusmata*, the oral apothegms, many of which
probably go back to Pythagoras himself (1972a, 465 ff.). Burkert sees
evidence that Philolaus was not interested in rigorous mathematics in
the mathematically nonsensical account of music in Fr. 6b and
testimonium A26 (1972a, 394-400) as well as in what might be the
use of mathematics for astrological purposes in A14 (1972a,
349-350).  There are, however, independent reasons for
doubting the authenticity of these texts (Huffman 1993, 364-374
and 381-391). Moreover, the mathematics of the surely authentic
Fr. 6a is above reproach, and Aristotle clearly presents the interest
in number displayed by fifth-century Pythagoreans as the result of
their training in rigorous mathematics (*Metaph*. 985b23 ff.;
cf. Iamblichus, *Comm. Math. Sc.* 78.8-18). Fragments 4
and 5 suggest that Philolaus' program was to discover the
numbers and numerical relations that govern the phenomena which we
observe, and Fr. 6a suggests that the whole-number ratios which govern
musical scales served as the model of the kind of mathematical account
which should be supplied for all phenomena. In identifying numerical
relations as the basis for knowledge Philolaus may in part be
responding to Parmenides' strictures on what constitutes a
proper object of thought. Mathematical relationships certainly fit a
number of the characteristics of a proper object of thought as set out
by Parmenides in Fr. 8, e.g., they are ungenerated, imperishable, and
unchangeable. Philolaus does not provide many further specific
examples of mathematical relations controlling phenomena, which is not
surprising given the scientific capabilities of the time. Indeed, the
fact that the fragments and testimonia are not full of facile
equations between numbers and phenomena once again suggests that
Philolaus was not simply interested in symbolic connections between
numbers and things and that Philolaus was looking for connections that
were scientifically verifiable, as in the case of the ratios that
govern musical intervals. Moreover, Greek usage shows that
"having number" need not indicate just a simple one to one
relation between a number and things but can also have a general sense
indicating that something is constituted by an ordered plurality or
relations between ordered pluralities (Huffman 1993: 175). The musical
scale is one example but the alphabet may be another so that in
Philolaus' epistemology science may involve a classification and
systematization of a subject that has similarities to Plato's
collection and division (Graham 2014: 55). His successor in the
Pythagorean tradition, Archytas of Tarentum, appears to have adopted
and been even more successful in carrying out Philolaus' project
of accounting for things in the phenomenal world in terms of
number. In some cases, however, Philolaus may have been content to
accept similarities between the properties of number and the
properties of things. Aristotle reports that the Pythagoreans
identified individual numbers with certain concepts, e.g., the number
four is associated with justice (*Metaph*. 985b29), evidently
because it is the first square number (cf. the English expression
"a square deal"). It is not certain that this is a
reference to Philolaus, but, when Aristotle says that the
counter-earth was introduced into the Pythagorean astronomical system
in order to bring the number of bodies orbiting the central fire up to
the perfect number ten, this must apply to Philolaus, since the
astronomical system described belongs to him (see 4.2 below). Since
the world of numbers is ordered in tens, the cosmos too must parallel
this order. (For further analysis of Philolaus' use of
mathematics see Huffman 1988 and 1993, 54-77.) Zhmud (2016)
argues that, while Philolaus and the early Pythagoreans had some
interest in the number symbolism that is found in many cultures, they
did not engage in arithmology, which focuses on the special properties
of the first ten numbers. Certainly it is true that there is no
evidence for anything as elaborate as the the later arithmological
treatises such as the *Theologumena Arithmeticae*. However, in
order to deny any special significance for the numbers from one to ten
in Philolaus one has to dismiss clear Aristotelian evidence and
Philolaus' own use of four psychic faculties and ten bodies
rotating around the central fire. 



There is no evidence that Philolaus thought that numbers had a
metaphysical status independent of the objects which they describe
(for the contrary view, i.e. that Philolaus thought that the abstract
was prior to the concrete, see Netz 2005: 87-88). Aristotle
emphasizes that the Pythagoreans differed from Plato in not separating
numbers from things (e.g. *Metaph*. 987b27-9), and Plato
confirms this interpretation with his criticism of the Pythagoreans on
the grounds that they looked for numbers in things rather than
focusing on numbers in themselves (*R*. 531c). On the other
hand, Aristotle's assertion that the Pythagoreans thought that things
were numbers or that they were made of numbers is also not supported
by the fragments of Philolaus. When describing the components of
things in the world, Philolaus always talks in terms of limiters and
unlimiteds and never asserts that things are made or arise from
numbers (cf. Fr. 1). In the surviving fragments, number is only
explicitly introduced to solve epistemological problems. It is true,
however, that Philolaus' third metaphysical principle,
"harmonia," is a "fitting together" of
limiters and unlimiteds in accordance with number. Number thus does
play a metaphysical role in defining the essence of things, but there
is no indication that it is the material out of which they are made.
Aristotle's assertion that the Pythagoreans thought that things were
numbers is best understood as his own interpretation of what
Philolaus' system amounted to (Huffman 1993, 57-64; see Schibli
1996 for a defense of Aristotle's assertion as historically accurate).
Aristotle may well have thought that if numbers are what give us
knowledge of things, then, in Aristotelian terms, they are the essence
of things and hence that, in a sense, things are numbers. Philolaus
appears to have thought that things were composed of unlimiteds such
as material stuffs and structural features such as shapes; number
describes the harmonia or fitting together that defines how the
limiters and unlimited were combined and hence give us the
"formula" of the thing in question. There is no evidence
that he addressed the metaphysical status of the numbers themselves.



## 4. Cosmogony and Cosmology


### 4.1 The Birth of the Cosmos



Fragment 7 is the beginning of Philolaus' account of the coming to be
of the cosmos:



> 
> The first thing fitted together, the one in the center of the sphere,
> is called the hearth.
> 



As we would expect from Fragment 1, the world begins with a fitting
together. Fragment 1 also indicates that the fitting together should
involve a limiter and an unlimited. What is fitted together in
Fragment 7 is "the one in the center of the sphere," which
is then called "the hearth." From the testimony regarding
Philolaus' astronomical system (see 4.2 below) it is clear that what
is in the center of Philolaus' cosmos is the central fire, and surely
it is the central fire that is called the hearth in Fragment 7. What
are the limiter and the unlimited from which the central fire is put
together? Fire is the obvious choice as the unlimited element, and it
is limited by being placed at the center of a sphere. Aristotle tells
us that the next step was for the nearest part of the unlimited to be
drawn in (*Metaph*. 1091a17), and elsewhere he specifies that
it is time, breath and the void, which are drawn in from the unlimited
(Arist., Fr. 201). Evidently then Philolaus first envisaged the
emergence of a sphere with a fire at its center. As in most
Presocratic cosmologies, the cosmos is surrounded by an unlimited
expanse, and Philolaus' central fire draws in time, breath and void
from that expanse. Each of these three is itself an unlimited, a
continuum not yet limited in any way. Like many other Presocratics
Philolaus drew an analogy between the birth of the cosmos and the
birth of a human being. At birth the human embryo, which is hot,
breathes in cooling air just as the central fire draws in cooling
breath from the unlimited. It may be that the interaction of hot fire
and cold breath was thought to generate the other prominent stuffs in
the world such as earth and water. Void serves to distinguish
individual things from each other spatially (Arist., Fr. 201); bits of
the central fire were perhaps separated off to form the sun, stars and
planets. Time provides the temporal continuum in which the cosmos can
develop and according to which the periods of the planets can be
measured (see Zhmud 1998 for doubts that time should be included in
the cosmology but note that Aristotle refers to time outside the
cosmos not only in Fr. 201 but also at *De Caelo* 279a12,
although without specifically mentioning the Pythagoreans). Some have
argued that this cosmogony does not distinguish breath and void, since
they are drawn in together, and hence that at least this element of
the cosmogony is archaic and dates before Philolaus, since it must
antedate Anaxagoras, who first made the distinction (Kahn 2001, 37).
Breath and void are listed separately, however, along with time as
three things drawn in, and there is no obvious reason to assume that
the first two are being identified. Moreover, breath and void serve
different functions in the cosmogony (cooling and the spatial
differentiation of objects), so that it appears more likely that they
are not being equated and that Philolaus developed the cosmogony after
void and air had been first distinguished by Anaxagoras and
Empedocles. Fragment 17 provides us with the only further evidence for
Philolaus' cosmogony, asserting again that the cosmos began to come to
be at the middle and adding that it developed symmetrically around the
middle. The end of the cosmogony is the astronomical system described
in section 4.2 below, in which ten heavenly bodies including the earth
encircle the central fire.



There is considerable controversy about the role of number in this
cosmogony. Some scholars argue that the generation of the cosmos is
also the generation of numbers and that the central fire is the number
one and other heavenly bodies are identified with other numbers, e.g.,
the sun is seven (Kahn 2001, 28; Schibli 1996). Thus, in Fragment 7,
when Philolaus refers to "the one in the center of the
sphere," he is not referring simply to the central fire as the
first unity of limiter and unlimited but also to the number one. The
strongest support for this view is Aristotle's assertion that the
Pythagoreans "clearly state that the one was constructed...
and then that the nearest part of the unlimited was drawn in"
(*Metaph*. 1091a15 ff.). Aristotle is complaining about the
absurdity of supposing that there can be a generation of eternal
things such as numbers, and he appears to be taking Fr. 7 as a clear
statement by the Pythagoreans that the number one is generated. The
difficulty with accepting this view is that it seems unintelligible to
say that the central fire and the number one are the same thing. It is
quite plausible to assume that there is an *analogy* between
the central fire as the first thing in the cosmos and the number one
as the first of the numbers, but the claim is that the number one is
identical with the central fire, and it would follow that the number
one is a large fire at the center of the cosmos surrounded by ten
heavenly bodies. The context of Aristotle's assertion that the number
one was constructed shows, moreover, that there was considerable
controversy as to whether or not the Pythagoreans were to be
interpreted in this way and that Aristotle is taking sides in a
debate; he says that "it is not necessary to doubt" his
assertion, which implies that some have indeed doubted it.  The
evidence he cites for his own position appears to be precisely Fr. 7,
and there seems to have been little clear evidence on the issue;
Aristotle complains that the Pythagoreans do not say of what the
number one is composed and suggests planes, surfaces and seeds as
possibilities. It thus does not appear that Aristotle was drawing on
any other more explicit statement than what we find in Fr. 7, and his
remarks indicate that the interpretation of Fr. 7 was controversial
even in his own time. None of the other fragments of Philolaus equate
numbers with things in this way, and Fr. 5 asserts that things
"give signs of" or "point to" numbers rather
than that they are identical with numbers (see section 3 above). This
evidence suggests that in Fr. 7 Philolaus is not asserting that the
number one was generated but is instead describing the generation of
the first unity of limiter and unlimited, the central fire (Huffman
1993, 202-215; followed by Zhmud 2012: 400).


### 4.2 Astronomical System



Philolaus' astronomical system is famous as the first to move the
earth from the center of the universe and make it an orbiting planet
instead. Copernicus referred to Philolaus as his precursor in the
preface to his *De Revolutionibus* (see Kahn 2001, 26 for
references). Unlike Copernicus, however, Philolaus did not replace the
geocentric with a heliocentric universe. The central fire rather than
the sun is at the center of Philolaus' cosmos. The heavenly bodies are
arranged in ten concentric circles around this central fire. Beginning
from the outside, the fixed stars come first, then the five planets,
the sun, the moon, the earth and a mysterious counter-earth. The
system can account for the basic phenomena with elegant simplicity.
The fixed stars serve as the backdrop against which the other heavenly
bodies move. All of the bodies have only one circular motion around
the central fire, from west to east. The moon completes its orbit in
about a month, the sun in a year and each of the planets has its own
period of revolution. Thus, bodies move more slowly, the farther
removed they are from the central fire,and the fixed stars probably
did not move at all. These motions account for the basic apparent
movement of the sun, moon and planets through the zodiac, although
there is no attempt to explain more sophisticated planetary phenomena
such as retrograde motion, where the planet appears to stop and move
backward before continuing its progress through the zodiac. Retrograde
motion was only first addressed almost a century after Philolaus by
Eudoxus. The earth has by far the fastest motion, completing a circuit
of the central fire in 24 hours. It is the motion of the earth that
accounts for the apparent daily motion of the sun across the sky and
hence for night and day. The much faster moving earth, which rotates
on its axis so as to keep our side always turned away from the central
fire and counter-earth (see below), leaves the sun behind so that day
lasts as long as the earth is facing the sun, and night occurs when
the earth turns away from the sun as it moves around the central
fire.



![noon](noon.jpg)


![midnight](midnight.jpg)


The system is the result of a combination of *a priori*
principles and reliance on observation. Philolaus is the first to
incorporate all five planets commonly known in antiquity into an
astronomical scheme in the correct order, which indicates that he was
aware of the most up-to-date astronomical data (Eudemus as cited in
Simplicius, *in De Caelo* 471.4; Huffman 1993, 260; Zhmud 2012:
336-7 thinks that Eudemus is referring to an earlier Pythagorean
system). On the other hand, there are a number of features that appear
to be explained by appeal to *a priori* principles of order.
Thus, moving the earth from the center of the universe did not serve
to explain astronomical data better. There is no explicit reason given
for moving the earth; Aristotle reports that some others (Platonists?;
see Kingsley 1995, 175-80) thought it should be moved because
fire was thought to be a "better" element than earth and
hence to belong at the center of the cosmos, which is a position of
honor (Arist., *De Caelo* 293b2). We are explicitly told that
the Pythagoreans accepted one of the premises of this argument, that
the center is the most important part of the cosmos (Arist., *De
Caelo* 293b2-3), and it is likely that the whole argument
should be assigned to Philolaus (Huffman 2007, 76-80). This is
consistent with the evidence of Frs. 7 and 17, which clearly indicate
that the cosmogony began in the center. Philolaus may have assigned
fire the position of honor, because of the analogy with human birth,
since he argued that the human embryo develops from the hot (see 5.3
below). The case of the counter-earth is more complicated. Aristotle
suggests that it was introduced for *a priori* reasons: to
raise the number of heavenly bodies around the central fire from nine
to ten, which the Pythagoreans regarded as the perfect number (Arist.,
*Metaph*. 986a8-12; on the other hand Zhmud 2012:407
argues that the number ten had no particular significance for the
Pythagoreans). Some scholars have objected that, if we count the
central fire, the introduction of the counter earth makes eleven
bodies in Philolaus' system (Burch 1954, 284; Kingsley 1995, 174); the
ancient evidence consistently talks in terms of bodies arranged or
moving *around* the central fire, however, and obviously the
central fire cannot be counted as one of these (Aetius 2.7.7; Arist.,
Fr. 203 and *Metaph*. 986a10), so that Aristotle's explanation
retains considerable plausibility (Burch 1954 suggests that the
counter-earth was introduced as a counter-weight to the earth, so that
the center of gravity of the cosmos would coincide with the spatial
center, but there is no evidence that Philolaus had the concept of a
center of gravity). On the other hand, Aristotle also comments that
the counter-earth was introduced to explain lunar eclipses (*De
Caelo* 293b18-25). Graham has recently made a convincing case that
the counter-earth could in fact account for eclipses of the moon at
dawn and dusk, which otherwise would be inexplicable on Philolaus'
model (Graham 2015, see also Graham 2014: 57-60). Graham speculates
that the term for counter-earth in Greek (*antichthon*), may
refer to the counter-earth's "taking the place"
(*anti*) of the earth in accounting for these lunar eclipses,
which cannot be explained by the interposition of the earth (Graham
2015: 229). Thus, one of the main features of Philolaus' system that
had been thought to show its non-scientific character, may, in fact,
be motivated by an attempt to make the system better explain the
phenomena. The doxography also assigns the counter earth to the
Pythagorean Hicetas (DK 50A2; cf. Mansfeld 2010a: 90-93). The
fixed stars do not literally move in Philolaus' system but they, like
the planets, sun and moon, have an apparent diurnal movement caused by
the motion of the earth. Even Philolaus' basic assumption that the
cosmos is spherical (Frs. 7 and 17) may be based on Parmenides' *a
priori* arguments that reality is similar to a sphere in
perfection (Fr. 8.42-44).There is no explicit statement about
the shape of the earth in Philolaus' system and some have supposed it
to be flat (Burch 1954: 272-273). Analogy with the cosmos
suggests that it was spherical as well and several of Aristotle's
descriptions of the Philolaic system show that it was (Dicks 1970:
72-73). Philolaus appears to have believed that there was also
fire at the periphery of the cosmic sphere and that the sun was a
glass-like body which transmitted the light and heat of this fire to
the earth, an account of the sun which shows connections to
Empedocles; Philolaus also posited a great year of 59 years in which
the lunar and solar cycles were reconciled (Huffman 1993, 266-70
and 276-279). Philolaus' great year may be connected to the
doctrine of the periodic return of all things, which Dicaearchus
assigns to the Pythagoreans (Porphyry, *Life of Pythagoras*
19), which may be in turn a generalization of reincarnation (Riedweg
2005, 63). 



As in the case of Philolaus' cosmogony, there is considerable
controversy about the precise nature of Philolaus' astronomy. The
*a priori* features, as well as other testimonia stating that
Philolaus believed that the moon was inhabited by creatures fifteen
times more powerful than those on the earth (Aetius 2.30.1 = DK 44
A20), have suggested to some that Philolaus' system is largely a
fantasy and has nothing to do with empirical astronomy (Furley 1987,
58). Burkert argues, moreover, that the apparent coherence of the
astronomical system is destroyed, because the fixed stars, which serve
as the backdrop for the motions of the heavenly bodies described
above, were not in fact stationary but moved with the fastest motion
so that, without a fixed backdrop, the whole system dissolves into
chaos (1972a, 340); he argues that Philolaus' astronomy is primarily
of mythological and religious significance (1972a, 348). However, more
recent assessments argue that Philolaus' astronomical system was a
serious attempt to account for the phenomena and the most successful
such attempt of his time (Huffman 1993: 231-288; Graham 2015). The
text on which Burkert relied to conclude that the fixed stars have the
fastest motion is, in fact, part of a speculative reconstruction of
the doctrine of the harmony of the spheres by a later Greek
commentator, Alexander of Aphrodisias, and not a description of
Philolaus' system. Nor need speculation
about the inhabitants of the moon indicate that the system is
fantastical, since such staid Presocratic rationalists as Anaxagoras
also argued that the moon was inhabited (see D.L. II 8 and DK 59 A77).
Perhaps the strongest reason for believing that Philolaus'
astronomical system was at least in part an attempt to explain the
phenomena rather than just of religious and mythological significance
is that other thinkers made objections to the system on empirical
grounds, and Philolaus responded to those criticisms in ways that
attempted to show that his system could explain or "save"
the phenomena. Thus an obvious objection to the system is that we
never observe the central fire at the center of the universe.
Philolaus explained this by supposing that the earth rotates once on
its axis as it orbits the central fire, so that our side of the earth
is always turned away from the central fire and hence we never see it.
This is also why we never see the counter earth, which is evidently
supposed to move at the same speed as the earth. An objection to the
moving earth was raised on the grounds that the parallax produced by
its movement from one side to the other of the central fire would
cause the sun, moon and planets to appear to move in ways in which
they do not. Philolaus evidently argued that the diameter of the
earth's orbit was so small relative to the distance to the other
heavenly bodies that the parallax effects would be negligible (*De
Caelo* 293b25-30). It is not clear that this solves the
problem, but if the system is solely of mythic or religious
significance it is odd that other thinkers, including Aristotle,
treated the system as if it were supposed to be consistent with the
phenomena, and even more surprising that Philolaus tried to defend the
system by showing that it was consistent with those phenomena.
Moreover, as mentioned above, Graham has recently shown that the
counter-earth is not introduced just for *a priori* reasons but
serves to give the most sophisticated account of lunar eclipses of the
time (Graham 2015).



Comparison between the cosmology of the oral sayings ascribed to
Pythagoras (the *acusmata*) and Philolaus's cosmos shows that
Philolaus' system represents the transition from a cosmology that is
primarily mythical to one that is primarily rational and scientific
(Huffman 2013a). It remains possible that, while the system is
intended to save the phenomena, some of its features may retain
religious significance as well; some have argued that the
counter-earth might be connected to a realm common in folklore where
everything is "counter" to our world, a realm that often
has connections to the world of the dead, Hades (Burkert 1972a,
347-8; Kingsley 1995, 186-7), but there are not good
parallels for such a conception of Hades in the Greek tradition. Kingsley argues that the central fire is to be
equated with Tartarus, a region even below Hades in Greek myth, and
that the introduction of the counter earth and central fire is largely
inspired by interpretation of Homer's description of Hades and
Tartarus as below the earth; Aristotle attests that the Pythagorean
name for the central fire was  *Dios phylake*, which
Kingsley translates as "prison of Zeus," which would
accord with Tartarus' function as the prison in which the Titans as
well as humans damned to punishment were kept (1995, 183 ff.). The
text on which Kingsley relies (Anatolius, *On the Decad* 6.13
Heiberg) does not mention Philolaus by name, however, and is not, in
fact, likely to be referring to Philolaus.
Moreover, Philolaus calls the central fire "the hearth,"
whose positive connotations as the religious center of the house and
state are hard to reconcile with its being the place of punishment,
Tartarus. In addition, parallels show that, in the fifth and fourth
century, the name  *Dios phylake* refers to a garrison
that guards Zeus or his palace rather than to a prison (Plato,
*Protagoras* 321d). In mythical terms it thus appears that
Philolaus conceived of the central fire as the fire that Zeus kept
guarded in his palace and that Prometheus stole and gave to humanity.
Philolaus' cosmogony, in which the central fire draws in void and thus
separates off parts of fire and spreads them to the rest of the cosmos
thus replicates Prometheus' theft in naturalistic terms. This
connection to myth is confirmed by Plato's description of Philolaus'
metaphysical system as hurled down from the gods to men by some
Prometheus (*Philebus* 16c), but it is hard to know what
religious significance Philolaus saw in the myth (see Huffman 2007 for a response to Kingsley's points). 



Betegh (2014: 162-6) has recently argued that while Philolaus' system
does make sense as a contribution to the tradition of rational
Presocratic cosmology it at the same time incorporates and
reinterprets elements of the religious tradition concerning the
hearth. Thus, the hearth was the center of the household and putting a
hearth at the center of the cosmos suggested that it was analogous to
the household. The ritual by which the newborn child was carried
around the hearth to introduce it into the household is now paralleled
by the planets motion around the central fire. Moreover, Betegh argues
that the Orphic tradition represented by the Derveni papyrus shows a
similar tendency to give a new religious relevance to concepts derived
from natural philosophy. He also argues that there are striking
similarities between the cosmological theories of the author of the
Derveni papyrus and Philolaus, partcularly in the role of fire. At the
same time it is noteworthy that whereas the Derveni author presents a
demiurgic mind as ordering the cosmos, Philolaus never explains what
brought limiters and unlimiteds together to form the cosmos (on this
lack of an efficient cause see also Graham 2014: 65-6). 


## 5. Psychology


### 5.1 Psychic Faculties



Fragment 13 presents Philolaus' account of psychic faculties:



> 
> The head [is the seat] of intellect, the heart of soul
> (*psyche*) and sensation, the navel of rooting and first
> growth, the genitals of the sowing of seed and generation. The brain
> [contains] the origin of man, the heart the origin of animals, the
> navel the origin of plants, the genitals the origin of all (living
> things). For all things both flourish and grow from seed.
> 



Philolaus thus recognizes four distinct psychic faculties. He makes a
clear distinction between intellect (*nous*), which is limited
to human beings, and perception, which animals possess as well. He has
no word for the soul as a whole and uses *psyche*, which
serves this function in Plato and Aristotle, for the psychic functions
shared by humans and animals. Some have argued that for Philolaus
*psyche* only meant "life," a common use in
the fifth century (Burkert 1972, 270). However, since plants are alive
and *psyche* is not assigned to them, "life"
is not a good translation. Parallels in Herodotus suggest that for
Philolaus *psyche* was the seat of emotions and desires,
which was closely connected to perception (Huffman 2009). The third
psychic faculty is responsible for nutrition and growth and the fourth
governs generation and the sowing of seed. This hierarchy of four
faculties is strikingly associated with the hierarchy of living beings
(human, animal, plant) as well as a hierarchy of parts of the body
(head, heart, navel, genitals). In identifying the head as the seat of
intellect, Philolaus is siding with his fellow townsman, Alcmaeon,
against those like Empedocles who championed the heart. He goes beyond
this controversy to locate each faculty in a part of the body and thus
anticipates a similar attempt by Plato to assign psychic faculties to
regions of the body in the *Timaeus*. Philolaus makes a
significant advance over earlier Presocratics by making a firm
distinction between thinking and perception. His systematization of
psychic faculties goes beyond anything found in his predecessors and
is the recognizable precursor of Aristotle's distinction between the
faculties for nutrition, appetite, sensation, movement and thought
(*de An*. 414a31-2).
Riedweg notes that three of the seats of the psychic faculties (brain,
heart and genitals) in animals are forbidden to be eaten in the
*acousmata*, so that it may be that a connection can be seen
here between scientific and religious strands in Pythagoreanism (2005,
110).


### 5.2 Soul as a Harmony



Macrobius (5th century AD) is the only ancient source to attribute to
Philolaus the doctrine that the soul is a harmony (DK 44 A23). It is not clear that this ascription is based on
anything in Philolaus' book, moreover, and it may just be a conclusion
based on the espousal of the doctrine by Simmias, who is presented as
a pupil of Philolaus, in Plato's *Phaedo* (86b5 ff.). The
doctrine as presented in the *Phaedo* may be Plato's invention
based on several Presocratic sources (Mansfeld 2010b: 183-188).
However, since Philolaus believed that all things were either
limiters, unlimiteds or harmonious combinations of limiters and
unlimiteds, it is not implausible that he would have regarded soul as
such a harmony of limiter and unlimited. In Fr. 13 soul
(*psyche*) is the center of emotions and desires, is
associated with sense perception, and has its seat in the heart (see
5.1 above). Sedley argues that Philolaus' embryology (see 6.1 below)
suggests that a body is first ensouled when the heat (an unlimited) of
the embryo was cooled, and hence limited, by breath. The soul can thus
be viewed as a harmony of these and other opposites, which is the view
that Simmias presents in the *Phaedo* (Sedley 1995,
22-6). In the *Phaedo*, however, the view that the soul
is a harmony of parts of the body is shown to be in conflict with its
immortality. Some have thus thought that Philolaus did not believe in
the immortality of the soul or transmigration (see now McKirahan 2016). It is more likely, however, that he regarded the
*psyche*, which was the center of desires and emotions
located in the heart and shared with animals, as a harmony of its own
material elements. This may have been drawn in with the first breath
of the infant and from its seat in the heart it may have served as the
command center which establishes the harmony of hot and cold which
makes the body ensouled. After the body loses its harmony and dies,
this command center soul survives to transmigrate into the hearts of
other animals and men. If the *psyche* was the
transmigrating soul for Philolaus, it follows that he did not think
that humans were reborn as plants, since according to Fr. 13 plants do
not have a *psyche*. Moreover, it would only be the human
character as constituted by emotions and desires that transmigrates
and not human intellect, since according to Fr. 13
*psyche* does not include intellect (Huffman 2009; for a critique see Palmer 2018).


## 6. Medical Theories


### 6.1 Embryology



We would never have known that Philolaus had contributed to medical
theory if not for the papyrus known as the *Anonymus
Londinensis*, which is in part based on the history of medicine
written in the late fourth century BC by Aristotle's pupil Meno (DK 44
A27-8; Huffman 1993, 289-306). Philolaus is one of twenty
thinkers whose views on the causes of disease are presented in the
papyrus, and the relatively detailed presentation of Philolaus' views
suggests that Meno had access to his book. Philolaus is included among
those thinkers who argue that diseases arise from the elements which
constitute the body, so that the section on Philolaus begins with his
account of the constitution of the human body, which is in effect his
account of the nature of the embryo and newborn infant. Philolaus is
unusual in arguing that the embryo is composed of just one element,
the hot, and has no cold in it. The embryo must be hot because both
sperm and the womb, in which the sperm is deposited, are hot.
Immediately upon birth, our bodies, which were too hot before, draw in
cooling breath, which is then sent out again "like a
debt." Thus, respiration suggests that the cool air which is
drawn in was originally something foreign to the body, which must be
paid back to the source from which it is drawn. The limitation of the
body's innate heat by cooling breath may be the process by which the
body becomes ensouled (see 5.2 above). There is a clear analogy
between the embryo, constituted solely of the hot, breathing in
cooling air and the central fire drawing in breath from the unlimited
in the early stages of Philolaus' cosmogony (see 4.1).



The evidence for Philolaus' embryology discussed above (DK 44
A27-28) does suggest that he, like Anaxagoras, believed that
only the male contributed seed and that the female was simply the
place in which the seed was sown. Some scholars have gone further and
argued that Philolaus subscribed to the view that the seed arose from
the brain and the spinal marrow and had hot breath in it, which
conveyed soul to the child (Leitao 2012: 29-30). However, this
interpretation is based on Diogenes Laertius 8.28, which does not
mention Philolaus by name and in fact derives from the *Pythagorean
Memoirs* excerpted by Alexander Polyhistor. This treatise is
believed by most scholars to belong to the later pseudo-Pythagorean
tradition and hence is not good evidence for Philolaus' beliefs
(Burkert 1972: 53). Moreover, the report contradicts the genuine
fragments of Philolaus in a number of ways. Fragment 13 of Philolaus
asserts that semen comes from the genitals not the brain and A27
clearly suggests that seed is composed just of the hot, says nothing
of its origin in the brain, and indicates that breath does not enter
the child through the male seed but only when it is drawn in upon
birth. 


### 6.2 Disease



The *Anonymus Londinensis* then tells us that Philolaus
explained disease in terms of three factors: bile, blood and phlegm.
It is surprising that no mention is made of the basic constituent of
the body, the hot, since Philolaus is presented as one of the thinkers
who explains disease in terms of the constituents of the body (see 6.1
above). It may well be, however, that bile, blood and phlegm were all
regarded as hot, since blood and bile were typically regarded as hot,
and we are explicitly told that Philolaus argued that phlegm was hot,
although it was typically regarded as cold (Lloyd 1963, 120).
Moreover, excesses and deficiencies of heat, cold and nutriment are
also said to be causes of disease. It would appear then that health
depends on appropriate cooling of the natural heat of the body and
that bile, blood and phlegm cause disease insofar as they are not
constituted properly by the balance of heating and cooling, the
balance of which can also be upset by improper nutrition. There are a number of puzzling features in
Philolaus' presentation of the three humours: blood, bile and phlegm
(Gourevitch 1989, Manetti 1990). Phlegm is said to be constituted from
*ombroi*, which usually means "rain storms," but
probably here means just liquids (Manetti 1990). However, Manetti's
suggestion (1990) that phlegm does not refer to the humour but rather
means "inflammation" is not very likely, since phlegm is
not said to be caused by the liquids, which is what we would expect if
it meant "inflammation" but rather constituted of them.
Philolaus' account of disease may be distorted by the intervention of
the author of the *Anonymus Londinensis* (Manetti 1990). 


## 7. Importance and Influence


### 7.1 Philolaus and Pythagoras: How original is Philolaus?



Pythagoras is one of the most famous names in the ancient
philosophical tradition, and there is a natural tendency to assume
that he was responsible for the most fruitful ideas in the Pythagorean
tradition. One of the dominant trends in recent scholarship on
Pythagoras, however, is to recognize that his fame grew exponentially
with the passage of time and that the early evidence, down to the time
of Plato and Aristotle, when examined carefully, reveals that
Pythagoras was indeed famous but not as a typical Presocratic
cosmologist nor as a mathematician/ scientist. In the early tradition Pythagoras is consistently
presented as a charismatic teacher famous for two things: 1) his
expertise on the fate of the soul in the next life as represented in
the doctrine of transmigration of souls 2) his founding of a way of
life which some people still followed a hundred years after his death.
This way of life involved moral and religious teachings that were
embodied in a series of brief sayings known as *acousmata*,
which included ritual taboos and dietary restrictions such as the
famous prohibition on eating beans (Burkert 1972a, 166-92).
There is no evidence that Philolaus made any significant contribution
in these areas. Plato's suggestion in the *Phaedo* that
Philolaus had something to say about the prohibition against suicide
is the closest we get to a pronouncement on how to live one's life. It
seems most reasonable to assume that he followed, to a greater or
lesser degree, the Pythagorean life handed down from the time of
Pythagoras, who died some twenty years before Philolaus was born.
There are a few hints that he tried to reconcile the mythic and
religious side of Pythagoreanism with his own naturalistic explanation
of the world (see 4.2 and 5.1 above).



On the other hand, Aristotle always presents the philosophical system
of limiters and unlimiteds, which placed an emphasis on the role of
number in understanding the cosmos and which included the astronomical
system built around the central fire, as the work of "the
so-called Pythagoreans," whom he dates to the second half of the
fifth century, at the same time as or a little earlier than the
atomists (e.g. *Metaph*. 985b23 ff.). This is the philosophical
system that we find in the fragments of Philolaus, and Aristotle's
dating fits Philolaus exactly so there can be little doubt that
Philolaus was the leading figure among those labeled "the
so-called Pythagoreans" by Aristotle. Although some scholars
continue to argue that most of what Aristotle ascribes to "the
so-called Pythagoreans"should be regarded as Pythagoras' own
teaching (Riedweg 2005, 77-80), Aristotle's testimony doubly
distances Pythagoras himself from the Philolaic system. First,
Aristotle is emphatic that the system dates after the time of
Pythagoras, and second, his expression "so-called
Pythagoreans" is most reasonably understood to suggest that,
although he knows that the name "Pythagorean" is often
given to Philolaus and his contemporaries, he can see little reason to
connect the philosophical system they espouse to Pythagoras himself.
The later tradition also emphasizes Philolaus' originality in the area
of cosmology by asserting that he was the first Pythagorean to publish
a work on nature (D. L. 8.85). The testimony of Aristotle and the
later tradition is largely confirmed by the content of Philolaus'
philosophy. Although the contrast between limit and unlimited is
implicit from the beginning of the Presocratic tradition, the first
explicit champion of limit is Parmenides, and the attempt to explain
the world in terms of a combination of limiters and unlimiteds thus
makes most sense as arising after Parmenides has championed limit
(Huffman 1999). Moreover, the style of argumentation in Fragments
1-6, and particularly Fr. 2, is reminiscent of Parmenides and
his successor Melissus (Kahn 2001, 34), so that the central
metaphysical system and the arguments for it look very much as if they
are the original contribution of Philolaus. The same is true of the
astronomical system built around the central fire, which the later
tradition assigns to Philolaus and which Aristotle (*Metaph*.
986a8 ff.) and Fr. 7 explicitly tie to the philosophy of limiters and
unlimiteds. Kahn is inclined to regard parts of the cosmogony as going
back to Pythagoras. He argues that the idea of the cosmos breathing is
archaic and that it may have been attacked long before Philolaus by
Xenophanes, who pointedly describes his god as not breathing; Kahn
also argues that the equivalence between air and void suggests a date
before Anaxagoras (2001, 36-7). In section 4.1 above, however,
reasons have been given to show that the Philolaic system, in fact,
distinguishes between air and void and hence once again conforms very
well to Philolaus' date after Anaxagoras. The breathing of the cosmos
need not be seen as archaic, since biological analogies are still
found in the cosmogony of the atomists (DK 67 A1). Xenophanes' remark
is reported in only one testimonium (DK 21 A1) and is not found in any
of the extant fragments of his book; if it is his, it need not be
directed at anything other than general Greek anthropomorphism.



The question remains whether or not Pythagoras had adumbrated any sort
of cosmology and what the relation of that cosmology was to Philolaus'
later system. It is certainly true that the discovery that the primary
musical concords of the octave, fourth and fifth are governed by the
simple whole number ratios 2 : 1, 4 : 3 and 3 : 2 occurred before
Philolaus. Most of the traditions connecting this discovery to
Pythagoras himself, however, are unreliable; Hippasus is the earliest
Pythagorean connected to a scientific demonstration of these
phenomena, but the actual discovery probably antedates him and may not
be specifically Pythagorean (Huffman 1993, 147-8). It is also
true that number plays a significant symbolic role in the
*acousmata* that go back to the time of Pythagoras. Thus one of
the *acousmata* asserts that number is the wisest thing
(Iamblichus, *On the Pythagorean Life* 82). Pythagoreans took
oaths in the name of Pythagoras as "the one who gave the
*tetraktys*"--*tetraktys* being a name for
the first four numbers, whose total is ten, which was the perfect
number for the Pythagoreans. Our evidence for the oath is late, but
the *acousmata* refer to the *tetraktys* as "the
harmony in which the Sirens sang" (Iamblichus, *On the
Pythagorean Life* 82), which may indicate an awareness of the role
of the first four numbers in the whole-number ratios which determine
the musical concords. The harmony in which the Sirens sing could be a
reference to the famous doctrine of the harmony of the spheres,
according to which the heavens make music by their motions. The
*acousmata* also adumbrate a cosmology which is tied to a
religious doctrine of reward and punishment. The planets are described
as the avenging hounds of the goddess Persephone (Porphyry, *Life
of Pythagoras* 41), the sun and the moon are the isles of the
blessed, where the good may be sent as a reward (Iamblichus, *On
the Pythagorean Life* 82), and thunder is a warning to souls in
Tartarus (Aristotle, *Posterior Analytics* II, 11 94b33). It
appears then that Pythagoras had presented a vision of the cosmos
which was ordered according to symbolic numbers and which reinforced
certain religious ideas (West 1971, 213-18).
Philolaus continues in this tradition by arguing that there are ten
celestial bodies arranged around the central fire, because ten is the
perfect number, and also by perhaps endowing some of the parts of his
cosmos with mythological and religious significance, e.g., by equating
the central fire with the fire guarded by Zeus and stolen by
Prometheus (see 4.2 above). On the other hand Philolaus' attempt to
save astronomical phenomena by including the five planets in correct
order according to their periods, by giving an explanation of night
and day and by responding to concerns about how parallax might affect
the way the heavenly bodies appear to move has no apparent precedent
in the cosmology of the *acousmata*. It would thus appear that
Pythagoras promulgated a cosmology that was largely of mythic and
religious significance and that Philolaus, while maintaining some of
this religious import, significantly reworked these ideas in order to
produce a system that could account for basic astronomical phenomena
(Huffman 2013). Graham has thus recently argued that Philolaus is not
the mere transmitter of the earlier philosophy of Pythagoras but
rather was the inventor of "Pythagorean" philosophy, the
figure who brought Pythagoreanism out of the shadowy world of
Pythagorean lore into the light of Greek philosophical conversation
(Graham 2014: 48).


### 7.2 Philolaus and Plato



There is the possibility that on his first visit to Southern Italy and
Sicily in 388-7 Plato met an aged Philolaus in Tarentum (D. L.
III 36). This would motivate Plato's allusion to Philolaus as the
teacher of Socrates' interlocutors, Simmias and Cebes, in the
*Phaedo*, which scholars have often dated in the years
immediately following Plato's return from Italy. The presentation of
Philolaus in the *Phaedo* is not particularly favorable;
Socrates supposes that Cebes should have heard something from
Philolaus about the prohibition on suicide, but Cebes responds that he
has heard nothing clear on the topic from Philolaus, or anyone else.
Plato may also be criticizing Philolaus when he shows the
inconsistency between the doctrine that the soul is a harmony and the
belief that the soul is immortal (see 6.2 above). On the other hand,
late in Plato's life, in the *Philebus*, there is evidence that
Plato had studied Philolaus' metaphysical system and derived fruitful
insights from it. At 16b4 ff. Socrates introduces a philosophical
method which he describes as



> 
> a gift ... hurled down from the gods to man along with a dazzling
> fire on account of some Prometheus. And the men before our time, since
> they were better than we are and lived closer to the gods, handed down
> this report about the things that are in each case said to be, that
> they are from a one and many and that they have limit and unlimited in
> themselves by nature.
> 



There are a number of important correspondences between the
description of the method of "the men before our time" in
this and the immediately following passages of the *Philebus*
and the fragments of Philolaus, e.g., the postulation of the limit and
unlimited as basic principles, Plato's insistence that number is the
basis for all inquiring, learning, and teaching (16e; Philolaus, Fr.
4) and the use of the musical scale as a central example of the
combination of limiter and unlimited (17b ff., Philolaus, Fr. 6a).
There can be little doubt that Plato has Philolaus in mind, and he may
be alluding to Philolaus' method of *archai* (Huffman 2001). It
is clear, however, that Plato is not taking over Philolaus'
metaphysical system wholesale but is rather using it as a way of
addressing problems that have arisen with the theory of forms.
Meinwald (2002) argues that Plato is either correcting or providing a
sympathetic reinterpretation of Philolaus' system, which allows Plato
to suggest a way in which knowledge of forms may be mathematical. This
reinterpretation transforms unlimiteds into pairs of opposites, in
which limiters set ratios. This interpretation agrees with Aristotle's
claim that Plato differed from the Pythagoreans in making the
unlimited a dyad (*Metaph*. 987b23-27). It has been
common to assume that the Prometheus in the passage quoted above is a
veiled reference to Pythagoras (Hackforth 1958, 21; Gosling 1975, 83).
Yet, as we have seen, Aristotle never associates Pythagoras with this
metaphysical scheme, and Plato elsewhere presents Pythagoras as a
founder of a way of life and not as a master metaphysician
(*R*. 600a-b). Modern scholars may be overreading the passage
in light of Pythagoras' fame in the later tradition. Prometheus was
traditionally the founder of all human arts through his gift of fire,
and Plato here supposes that the philosophical method he describes
must likewise have been due to some Prometheus, perhaps under the
influence of Philolaus' own allusion to Prometheus in his cosmology
(see 4.2 above). There is no need to introduce Pythagoras into the
story (Huffman 1999).
 Philolaus'
 metaphysics of limiters and unlimiteds is perfectly intelligible as a
response to the problems raised in the tradition of Presocratic
metaphysics (see 2.1 above) and was not a gift either of Prometheus or
Pythagoras. Some have argued that other passages in the
*Phaedo* (101b10-104c9) and a passage in the *Cratylus*
(401b11-c7) show that Plato was both heavily influenced by and
critical of Philolaus (Horky 2013). However, neither Philolaus nor the
Pythagoreans are specifically mentioned in these passages and the
supposed connections to Philolaus are nothing like as specific as in
the *Philebus* and too general and indirect to be convincing
(e.g., the concepts of odd and even in the *Phaedo* passage
were important to Philolaus but hardly limited to him).