## 1. The Pythagorean Question




What were the beliefs and practices of the historical Pythagoras?
This apparently simple question has become the daunting Pythagorean
question for several reasons. First, Pythagoras himself wrote nothing,
so our knowledge of Pythagoras' views is entirely derived from the
reports of others. Second, there was no extensive or authoritative
contemporary account of Pythagoras. No one did for Pythagoras what
Plato and Xenophon did for Socrates. Third, only fragments of the
first detailed accounts of Pythagoras, written about 150 years after
his death, have survived. Fourth, it is clear that these accounts
disagreed with one another on significant points. These four points
would already make the problem of determining Pythagoras' philosophical
beliefs more difficult than determining those of almost any other
ancient philosopher, but a fifth factor complicates matters even
more. By the third century CE, when the first detailed accounts of
Pythagoras that survive intact were written, Pythagoras had come to be
regarded, in some circles, as the master philosopher, from whom all
that was true in the Greek philosophical tradition derived. By the end
of the first century BCE, a large collection of books had been forged
in the name of Pythagoras and other early Pythagoreans, which
purported to be the original Pythagorean texts from which Plato and
Aristotle derived their most important ideas. A treatise forged in the
name of Timaeus of Locri was the supposed model for Plato's
*Timaeus*, just as forged treatises assigned to Archytas were
the supposed model for Aristotle's *Categories*. Pythagoras
himself was widely presented as having anticipated Plato's later
metaphysics, in which the one and the indefinite dyad are first
principles. Thus, not only is the earliest evidence for
Pythagoras' views meager and contradictory, it is overshadowed by the
hagiographical presentation of Pythagoras, which became dominant in
late antiquity. Given these circumstances, the only reliable approach
to answering the Pythagorean question is to start with the earliest
evidence, which is independent of the later attempts to glorify
Pythagoras, and to use the picture of Pythagoras which emerges from
this early evidence as the standard against which to evaluate what can
be accepted and what must be rejected in the later
tradition. Following such an approach, Walter Burkert, in his
epoch-making book (1972a), revolutionized our understanding of the
Pythagorean question, and all modern scholarship on Pythagoras,
including this article, stands on his shoulders. For a detailed
discussion of the source problems that generate the Pythagorean
Question see 2. Sources, below.


## 2. Sources


### 2.1 Chronological Chart of Sources for Pythagoras




|  |  |  |
| --- | --- | --- |
| 300 CE | Iamblichus(ca. 245-325 CE) | *On the Pythagorean Life* (extant) |
|  | Porphyry(234-ca. 305 CE) | *Life of Pythagoras* (extant) |
|  | Diogenes Laertius(ca. 200-250 CE) | *Life of Pythagoras* (extant) |
| 200 CE | Sextus Empiricus(circa 200 CE) | (summaries of Pythagoras'
philosophy in *Adversus Mathematicos* [*Against the
Theoreticians*], cited below as *M*.) |
| 100 CE | Nicomachus(ca. 50-150 CE) | *Introduction to Arithmetic*
(extant), *Life of Pythagoras* (fragments quoted in Iamblichus
etc.) |
|  | Apollonius of Tyana(died ca. 97 CE) | *Life of Pythagoras*
(fragments quoted in Iamblichus etc. It is possible that this work is by another otherwise unknown Apollonius.) |
|  | Moderatus of Gades(50-100) | *Lectures on Pythagoreanism*
(fragments quoted in Porphyry) |
|  | Aetius(first century CE) | *Opinions of the Philosophers* (reconstructed
by H. Diels from pseudo-Plutarch, *Opinions of the
Philosophers* [2nd CE] and Stobaeus, *Selections* [5th
CE]) |
|  | Pseudo-Pythagorean textsforged | (starting as early as 300 BCE but
most common in the first century BCE) |
| 100 BCE | Alexander Polyhistor (b. 105 BCE) | his excerpts of the *Pythagorean Memoirs* are
quoted by Diogenes Laertius |
| 200 BCE | *Pythagorean Memoirs*(200 BCE) | a Pseudo-Pythagorean Text (sections quoted in Diogenes Laertius) |
| 300 BCE | Timaeus of Tauromenium 
 (350-260 BCE) | (historian of Sicily) |
|  | 

|  |  |  |  |  |
| --- | --- | --- | --- | --- |
| Academy | Heraclides(ca. 380-310)
 Xenocrates(ca. 396-314)
 Speusippus(ca. 410-339) |  | Lyceum | Dicaearchus(ca.370-300)
 Aristoxenus(ca. 370-300)
 Eudemus(ca.370-300)
 Theophrastus(372-288)
 Aristotle(384-322) |

 |
| 400 BCE | Plato(427-347 BCE) |  |
| 500 BCE | Pythagoras (570-490 BCE) |  |


### 2.2 Post-Aristotelian Sources for Pythagoras




The problems regarding the sources for the life and philosophy of
Pythagoras are quite complicated, but it is impossible to understand
the Pythagorean Question without an accurate appreciation of at least
the general nature of these problems. It is best to start with the
extensive but problematic later evidence and work back to the earlier
reliable evidence. The most detailed, extended and hence most
influential accounts of Pythagoras' life and thought date to the third
century CE, some 800 years after he died. Diogenes Laertius
(ca. 200-250 CE) and Porphyry (ca. 234-305 CE) each wrote
a *Life of Pythagoras*, while Iamblichus (ca. 245-325 CE)
wrote *On the Pythagorean Life*, which includes some biography
but focuses more on the way of life established by Pythagoras for his
followers. All of these works were written at a time when Pythagoras'
achievements had become considerably exaggerated. Diogenes may have
some claim to objectivity, but both Iamblichus and Porphyry have
strong agendas that have little to do with historical
accuracy. Iamblichus presents Pythagoras as a soul sent from the gods
to enlighten mankind (O'Meara 1989, 35-40). Iamblichus' work
was just the first in a ten volume work, which in effect
Pythagoreanized Neoplatonism, but the Pythagoreanism involved was
Iamblichus' own conception of Pythagoras as particularly concerned
with mathematics rather than an account of Pythagoreanism based on the
earliest evidence. Porphyry also emphasizes Pythagoras' divine aspects
and may be setting him up as a rival to Jesus (Iamblichus 1991,
14). These three third-century accounts of Pythagoras were in turn
based on earlier sources, which are now lost. Some of these earlier
sources were heavily contaminated by the Neopythagorean view of
Pythagoras as the source of all true philosophy, whose ideas Plato,
Aristotle and all later Greek philosophers plagiarized. Iamblichus
cites biographies of Pythagoras by Nicomachus of Gerasa and a certain
Apollonius (*VP* 251 and 254) and appears to have used them
extensively even where they are not cited (Burkert 1972a, 98
ff.). Nicomachus (ca. 50-ca.150 CE) assigns Pythagoras a
metaphysics that is patently Platonic and Aristotelian and that
employs distinctive Platonic and Aristotelian terminology
(*Introduction to Arithmetic* I.1). If the Apollonius cited by
Iamblichus is Apollonius of Tyana (1st CE), his account will be
influenced by his veneration of Pythagoras as the model for his
ascetic life, but some scholars argue that Iamblichus is using an
otherwise unknown Apollonius (Flinterman 2014, 357). Porphyry
(*VP* 48-53) explicitly cites Moderatus of Gades as one
of his sources. Moderatus was an "aggressive"
Neopythagorean of the first century CE, who reports that Plato,
Aristotle, and their pupils Speusippus, Aristoxenus and Xenocrates
took for their own everything that was fruitful in Pythagoreanism,
leaving only what was superficial and trivial to be ascribed to the
school (Dillon 1977, 346). Diogenes Laertius, who appears to have less
personal allegiance to the Pythagorean legend, bases his primary
account of Pythagoras' philosophy (VIII. 24-33) on
the *Pythagorean Memoirs* excerpted by Alexander Polyhistor,
which are a forgery dating sometime around 200 BCE and which assign
not just Platonic but also Stoic ideas to Pythagoras (Burkert 1972a,
53; Kahn 2001, 79-83).




In the *Pythagorean Memoirs*, Pythagoras is said to have
adopted the Monad and the Indefinite Dyad as incorporeal principles,
from which arise first the numbers, then plane and solid figures and
finally the bodies of the sensible world (Diogenes Laertius VIII. 25).
This is the philosophical system that is most commonly ascribed to
Pythagoras in the post-Aristotelian tradition, and it is found in
Sextus Empiricus' (2nd century CE) detailed accounts of Pythagoreanism
(e.g., *M*. X. 261) and most significantly in the influential
handbook of the differing opinions of the Greek philosophers, which
was compiled by Aetius in the first century CE and is based on
the *Tenets of the Natural Philosophers* of Aristotle's pupil
Theophrastus (e.g., H. Diels, *Doxographi Graeci* I. 3.8). The
testimony of Aristotle makes completely clear, however, that this was
the philosophical system of Plato in his later years and not that of
Pythagoras or even the later Pythagoreans. Aristotle is explicit that,
although Plato's system has similarities to the earlier Pythagorean
philosophy of limiters and unlimiteds, the indefinite dyad is unique
to Plato (*Metaphysics* 987b26 ff.) and the Pythagoreans
recognized only the sensible world and hence did not derive it from
immaterial principles. In the *Philebus*, Plato himself tells a
story that is very much in agreement with Aristotle's report. While
acknowledging a debt to the philosophy of limiters and unlimiteds,
which is found in Aristotle's accounts of Pythagoreanism and in the
fragments of the fifth-century Pythagorean Philolaus, Plato makes
clear that this is a considerably earlier philosophy, which he is
completely reworking for his own purposes (16c ff.; see Huffman 1999a
and 2001). How are we to explain the later tradition's divergence from
this testimony of Aristotle and Plato? The most convincing suggestion
points to evidence that, for reasons which are not entirely clear,
Plato's successors in the Academy, Speusippus, Xenocrates and
Heraclides, chose to present Pythagoreanism not just as a precursor of
late Platonic metaphysics but as having anticipated its central
theses. Thus the tradition which falsely ascribes Plato's late
metaphysics to Pythagoras begins not with the Neopythagoreans in the
first centuries BCE and CE but already in the fourth century BCE among
Plato's own pupils (Burkert 1972a, 53-83; Dillon 2003,
61-62 and 153-154). This view of Pythagoreanism finds its
way into the doxography of Aetius either because Theophrastus followed
the early Academy rather than his teacher Aristotle (Burkert 1972a,
66) or because the Theophrastan doxography on Pythagoras was rewritten
in the first century BCE under the influence of Neopythagoreanism
(Diels 1958, 181; Zhmud 2012a, 455). Aristotle's careful distinctions
between Plato and fifth-century Pythagoreanism, which make excellent
sense in terms of the general development of Greek philosophy, are
largely ignored in the later tradition in favor of the more
sensational ascription of mature Platonism to Pythagoras. The evidence
for the early Academy is, however, very limited and some reject the
thesis that its members assigned late Platonic metaphysics to
Pythagoras (Zhmud 2012, 415-432). The key text is found in
Proclus' *Commentary on the Parmenides* (pp. 38.32-40.7
Klibansky). Proclus quotes a passage in which Speusippus assigns to
the ancients, who in this context are the Pythagoreans, the One and
the Indefinite Dyad. Some scholars argue that this is not a genuine
fragment of Speusippus but rather a later fabrication (see Zhmud
2012a, 424-425 and for a response Dillon 2014, 251). If the Academy
did not assign the One and the Dyad to Pythagoras, however, it becomes
less clear how these principles came to be assigned to
him. Theophrastus assigns them to the Pythagoreans
(*Metaphysics* 11a27), but since Aristotle distinguishes the
Pythagoreans from Plato on this point, Zhmud's suggestion (2012a, 455)
that he is following his teacher and just taking "the next step" does
not work. Theophrastus' evidence makes best sense if we accept the
traditional view and suppose that it is on the authority of Plato's
successors in the Academy that he bases his departure from his
teacher's, Aristotle's, view.




If we step back for a minute and compare the sources for Pythagoras
with those available for other early Greek philosophers, the extent of
the difficulties inherent in the Pythagorean Question becomes clear.
When trying to reconstruct the philosophy of Heraclitus, for example,
modern scholars rely above all on the actual quotations from
Heraclitus' book preserved in later authors. Since Pythagoras wrote no
books, this most fundamental of all sources is denied us. In dealing
with Heraclitus, the modern scholar turns with reluctance next to the
doxographical tradition, the tradition represented by Aetius in the
first century CE, which preserves in handbook form a systematic
account of the beliefs of the Greek philosophers on a series of topics
having to do with the physical world and its first principles. Aetius'
work has been reconstructed by Hermann Diels on the basis of two later
works, which were derived from it, the
*Selections* of Stobaeus (5th century CE) and the *Opinions
of Philosophers* by pseudo-Plutarch (2nd century CE). Scholars'
faith in this evidence is largely based on the assumption that most of
it goes back to Aristotle's school and in particular to
Theophrastus' *Tenets of the Natural Philosophers*. Here again
the case of Pythagoras is exceptional. Pythagoras is represented in
this tradition but, as we have seen, Theophrastus in this case either
adopted the view that, against all historical plausibility, assigns
Plato's later metaphysics to Pythagoras or Theophrastus' doxography on
the Pythagoreans was rewritten in the first century BCE. Thus, the
second standard source for evidence for early Greek philosophy is, in
the case of Pythagoras, corrupted. Whatever views Pythagoras might
have had are replaced by late Platonic metaphysics in the
doxographical tradition.




A third source of evidence for early Greek philosophy is regarded with
great skepticism by most scholars and, in the case of most early Greek
philosophers, used only with great caution. This is the biographical
tradition represented by the *Lives of the Philosophers*
written by Diogenes Laertius. In this case we at first sight appear to
be in luck, at least with regard to the amount of evidence for
Pythagoras, since, as we have seen, two major accounts of the life of
Pythagoras and the Pythagorean way of life survive in addition to
Diogenes' life. Unfortunately, these two additional lives are written
by authors (Iamblichus and Porphyry) whose goal is explicitly
non-historical, and all three of the lives rely heavily on authors in
the Neopythagorean tradition, whose goal was to show that all later
Greek philosophy, insofar as it was true, had been stolen from
Pythagoras. There are, however, some sections in these three lives
that derive from sources that go back beyond the distorting influence
of Neopythagoreanism, to sources in the fourth-century BCE, sources
which are also independent of the early Academy's attempt to assign
Platonic metaphysics to the Pythagoreans. The most important of these
sources are the fragments of Aristotle's lost treatises on the
Pythagoreans and the fragments of works on Pythagoreanism or of works
which dealt in passing with Pythagoreanism written by Aristotle's
pupils Dicaearchus and Aristoxenus, in the second half of the fourth
century BCE. The historian Timaeus of Tauromenium (ca. 350-260
BCE), who wrote a history of Sicily, which included material on
southern Italy where Pythagoras was active, is also important. In some
cases, the fragments of these early works are clearly identified in
the later lives, but in other cases we may suspect that they are the
source of a given passage without being able to be certain. Large
problems remain even in the case of these sources. They were all
written 150-250 years after the death of Pythagoras; given the
lack of written evidence for Pythagoras, they are based largely on
oral traditions. Aristoxenus, who grew up in the southern Italian town
of Tarentum, where the Pythagorean Archytas was the dominant political
figure, and who was himself a Pythagorean before joining Aristotle's
school, undoubtedly had a rich set of oral traditions upon which to
draw. It is clear, nonetheless, that 150 years after his death
conflicting traditions regarding Pythagoras' beliefs had arisen on
even the most central issues. Thus, Aristoxenus is emphatic that
Pythagoras was not a strict vegetarian and ate a number of types of
meat (Diogenes Laertius VIII. 20), whereas Aristoxenus' contemporary,
the mathematician Eudoxus, portrays him not only as avoiding all meat
but as even refusing to associate with butchers (Porphyry, *VP*
7). Even among fourth-century authors that had at least some
pretensions to historical accuracy and who had access to the best
information available, there are widely divergent presentations,
simply because such contradictions were endemic to the evidence
available in the fourth century. What we can hope to obtain from the
evidence presented by Aristotle, Aristoxenus, Dicaearchus, and Timaeus
is thus not a picture of Pythagoras that is consistent in all respects
but rather a picture that at least defines the main areas of his
achievement. This picture can then be tested by the most fundamental
evidence of all, the testimony of authors that precede even Aristotle,
testimony in some cases that derives from Pythagoras' own
contemporaries. This testimony is extremely limited, about twenty
brief references, but this dearth of evidence is not unique to
Pythagoras. The pre-Aristotelian testimony for Pythagoras is more
extensive than for most other early Greek philosophers and is thus
testimony to his fame.


### 2.3 Plato and Aristotle as Sources for Pythagoras




In reconstructing the thought of early Greek philosophers, scholars
often turn to Aristotle's and Plato's accounts of their predecessors,
although Plato's accounts are embedded in the literary structure of
his dialogues and thus do not pretend to historical accuracy, while
Aristotle's apparently more historical presentation masks a
considerable amount of reinterpretation of his predecessors' views in
terms of his own thought. In the case of Pythagoras, what is striking
is the essential agreement of Plato and Aristotle in their
presentation of his significance. Aristotle frequently discusses the
philosophy of Pythagoreans, whom he dates to the middle and second
half of the fifth century and who posited limiters and unlimiteds as
first principles. He sometimes refers to these Pythagoreans as the
"so-called Pythagoreans," suggesting that he had some
reservations about the application of the label
"Pythagorean" to them. Aristotle strikingly may never
refer to Pythagoras himself in his extant writings
(*Metaph*. 986a29 is probably an interpolation; *Rh.*
1398b14 is a quotation from Alcidamas; *MM* 1182a11 may not be
by Aristotle and, if it is, may well be a case where
"Pythagoreans" have been turned into
"Pythagoras" in the transmission). In the fragments of his
now lost two-book treatise on the Pythagoreans, Aristotle does discuss
Pythagoras himself, but the references are all to Pythagoras as a
founder of a way of life, who forbade the eating of beans (Fr. 195),
and to Pythagoras as a wonder-worker, who had a golden thigh and bit a
snake to death (Fr. 191). Zhmud (2012a, 259-260) argues that in one
place Aristotle also describes Pythagoras as a mathematician (Fr. 191)
and in another as studying nature (Protrepticus Fr. 20) but in neither
case are the words likely to belong to Aristotle (see Huffman 2014b,
281, n.7). If Aristotle only found evidence for Pythagoras as a
wonder-worker and founder of a way of life, it becomes clear why he
never mentions Pythagoras in his account of his philosophical
predecessors and why he uses the expression "so-called
Pythagoreans" to refer to the Pythagoreanism of the
fifth-century. For Aristotle Pythagoras did not belong to the
succession of thinkers starting with Thales, who were attempting to
explain the basic principles of the natural world, and hence he could
not see what sense it made to call a fifth-century thinker like
Philolaus, who joined that succession by positing limiters and
unlimiteds as first principles, a Pythagorean. Plato is often thought
to be heavily indebted to the Pythagoreans, but he is almost as
parsimonious in his references to Pythagoras as Aristotle and mentions
him only once in his writings. Plato's one reference to Pythagoras
(*R*. 600a) treats him as the founder of a way of life, just as
Aristotle does, and, when Plato traces the history of philosophy prior
to his time in the *Sophist*, (242c-e), there is no allusion to
Pythagoras. In the *Philebus*, Plato does describe the
philosophy of limiters and unlimiteds, which Aristotle assigns to the
so-called Pythagoreans of the fifth century and which is found in the
fragments of Philolaus, but like Aristotle he does not ascribe this
philosophy to Pythagoras himself. Scholars, both ancient and modern,
under the influence of the later glorification of Pythagoras, have
supposed that the Prometheus, whom Plato describes as hurling the
system down to men, was Pythagoras (e.g., Kahn 2002: 13-14), but
careful reading of the passage shows that Prometheus is just
Prometheus and that Plato, like Aristotle, assigns the philosophical
system to a group of men (Huffman 1999a, 2001). The fragments of
Philolaus show that he was the primary figure of this group. When
Plato refers to Philolaus in the *Phaedo* (61d-e), he does not
identify him as a Pythagorean, so that once again Plato agrees with
Aristotle in distancing the "so-called Pythagoreans" of
the fifth century from Pythagoras himself. For both Plato and
Aristotle, then, Pythagoras is not a part of the cosmological and
metaphysical tradition of Presocratic philosophy nor is he closely
connected to the metaphysical system presented by fifth-century
Pythagoreans like Philolaus; he is instead the founder of a way of
life.


## 3. Life and Works




References to Pythagoras by Xenophanes (ca. 570-475 BCE) and
Heraclitus (fl. ca. 500 BCE) show that he was a famous figure in the
late sixth and early fifth centuries. For the details of his life we
have to rely on fourth-century sources such as Aristoxenus,
Dicaearchus and Timaeus of Tauromenium. There is a great deal of
controversy about his origin and early life, but there is agreement
that he grew up on the island of Samos, near the birthplace of Greek
philosophy, Miletus, on the coast of Asia Minor. There are a number of
reports that he traveled widely in the Near East while living on
Samos, e.g., to Babylonia, Phoenicia and Egypt. To some extent reports
of these trips are an attempt to claim the ancient wisdom of the east
for Pythagoras and some scholars totally reject them (Zhmud 2012,
83-91), but relatively early sources such as Herodotus (II. 81) and
Isocrates (*Busiris* 28) associate Pythagoras with Egypt, so
that a trip there seems quite plausible. Aristoxenus says that he left
Samos at the age of forty, when the tyranny of Polycrates, who came to
power ca. 535 BCE, became unbearable (Porphyry, *VP* 9). This
chronology would suggest that he was born ca. 570 BCE. He then
emigrated to the Greek city of Croton in southern Italy ca. 530 BCE;
it is in Croton that he first seems to have attracted a large number
of followers to his way of life. There are a variety of stories about
his death, but the most reliable evidence (Aristoxenus and
Dicaearchus) suggests that violence directed against Pythagoras and
his followers in Croton ca. 510 BCE, perhaps because of the exclusive
nature of the Pythagorean way of life, led him to flee to another
Greek city in southern Italy, Metapontum, where he died around 490 BCE
(Porphyry,
*VP* 54-7; Iamblichus, *VP* 248 ff.; On the chronology,
see Minar 1942, 133-5). There is little else about his life of which
we can be confident.




The evidence suggests that Pythagoras did not write any books. No
source contemporaneous with Pythagoras or in the first two hundred
years after his death, including Plato, Aristotle and their immediate
successors in the Academy and Lyceum, quotes from a work by Pythagoras
or gives any indication that any works written by him were in
existence. Several later sources explicitly assert that Pythagoras
wrote nothing (e.g., Lucian [*Slip of the Tongue*, 5],
Josephus, Plutarch and Posidonius in DK 14A18; see Burkert 1972,
218-9). Diogenes Laertius tried to dispute this tradition by
quoting Heraclitus' assertion that "Pythagoras, the son of
Mnesarchus, practiced inquiry most of all men and, by selecting these
things which have been written up, made for himself a wisdom, a
polymathy, an evil conspiracy" (Fr. 129). This fragment shows
only that Pythagoras read the writings of others, however, and says
nothing about him writing something of his own. The wisdom and evil
conspiracy that Pythagoras constructs from these writings need not
have been in writing, and Heraclitus' description of it as an
"evil conspiracy" rather suggests that it was not (For the
translation and interpretation of Fr. 129, see Huffman 2008b). In the
later tradition several books came to be ascribed to Pythagoras, but
such evidence as exists for these books indicates that they were
forged in Pythagoras' name and belong with the large number of
pseudo-Pythagorean treatises forged in the name of early Pythagoreans
such as Philolaus and Archytas. In the third century BCE a group of
three books were circulating in Pythagoras' name, *On
Education*, *On Statesmanship*, and *On Nature*
(Diogenes Laertius, VIII. 6). A letter from Plato to Dion asking him
to purchase these three books from Philolaus was forged in order to
"authenticate" them (Burkert 1972a,
223-225). Heraclides Lembus in the second century BCE gives a
list of six books ascribed to Pythagoras (Diogenes Laertius, VIII. 7;
Thesleff 1965, 155-186 provides a complete collection of the
spurious writings assigned to Pythagoras). The second of these is
a *Sacred Discourse*, which some have wanted to trace back to
Pythagoras himself. The idea that Pythagoras wrote such a *Sacred
Discourse* seems to arise from a misreading of the early evidence.
Herodotus says that the Pythagoreans agreed with the Egyptians in not
allowing the dead to be buried in wool and then asserts that there is
a sacred discourse about this (II. 81). Herodotus' focus here is the
Egyptians and not the Pythagoreans, who are introduced as a Greek
parallel, so that the *Sacred Discourse* to which he refers is
Egyptian and not Pythagorean, as similar passages elsewhere in Book II
of Herodotus show (e.g., II. 62; see Burkert 1972a, 219).Various lines
of hexameter verse were already circulating in Pythagoras' name in the
third century BCE and were later combined into a compilation known as
the *Golden Verses*, which marks the culmination of the
tradition of a *Sacred Discourse* assgined to Pythagoras
(Burkert 1972a, 219, Thesleff 1965, 158-163; and most recently
Thom 1995, although his dating of the compilation before 300 BCE is
questionable). The lack of any viable written text which could be
reasonably ascribed to Pythagoras is shown most clearly by the
tendency of later authors to quote either Empedocles or Plato, when
they needed to quote "Pythagoras" (e.g., Sextus Empiricus,
*M*. IX. 126-30; Nicomachus, *Introduction to
Arithmetic* I. 2). For an interesting but ultimately unconvincing
attempt to argue that the historical Pythagoras did write books, see
Riedweg 2005, 42-43 and the response by Huffman 2008a, 205-207.


## 4. The Philosophy of Pythagoras




One of the manifestations of the attempt to glorify Pythagoras in the
later tradition is the report that he, in fact, invented the word
philosophy. This story goes back to the early Academy, since it is
first found in Heraclides of Pontus (Cicero, *Tusc.* V 3.8;
Diogenes Laertius, *Proem*). The historical accuracy of the
story is called into question by its appearance not in a historical or
biographical text but rather in a dialogue that recounted Empedocles'
revival of a woman who had stopped breathing. Moreover, the story
depends on a conception of a philosopher as having no knowledge but
being situated between ignorance and knowledge and striving for
knowledge. Such a conception is thoroughly Platonic, however (see,
e.g., *Symposium* 204A), and Burkert demonstrated that it could
not belong to the historical Pythagoras (1960). For a recent attempt
to defend at least the partial accuracy of the story, see Riedweg
2005: 90-97 and the response by Huffman 2008a:207-208; see
also Zhmud 2012a, 428-430.




Even if he did not invent the word, what can we say about the
philosophy of Pythagoras? For the reasons given in 1. The Pythagorean
Question and 2. Sources above, any responsible account of Pythagoras'
philosophy must be based in the first place on the evidence prior to
Aristotle and in the second place on evidence that our sources
explicitly identify as deriving from Aristotle's books on the
Pythagoreans as well as from the books of his pupils such as
Aristoxenus and Dicaearchus. There is general agreement as to what the
pre-Aristotelian evidence is, although there are differences in
interpretation of it. There is less agreement as to what should be
included in Aristotle's, Dicaearchus' and Aristoxenus' evidence. What
one includes as evidence from these authors will have a significant
effect on one's picture of Pythagoras. One particularly pressing
question is whether both chapters 18 and 19 of Porphyry's *Life of
Pythagoras* should be regarded as deriving from Dicaearchus, as
the most recent editor proposes (Mirhady Fr. 40), or whether only
chapter 18 should be included, as in the earlier edition of Wehrli
(Fr.33). It is crucial to decide this question before developing a
picture of the philosophy of Pythagoras since chapter 19, if it is by
Dicaearchus, is our earliest summary of Pythagorean
philosophy. Porphyry is very reliable about quoting his sources. He
explicitly cites Dicaearchus at the beginning of Chapter 18 and names
Nicomachus as his source at the beginning of chapter 20. The material
in chapter 19 follows seamlessly on chapter 18: the description of the
speeches that Pythagoras gave upon his arrival in Croton in chapter 18
is followed in chapter 19 by an account of the disciples that he
gained as the result of those speeches and a discussion of what he
taught these disciples. Thus, the onus is on anyone who would claim
that Porphyry changes sources before the explicit change at the
beginning of chapter 20. Chapter 19 provides a very restrained account
of what can be reliably known about Pythagoras' teachings and that
very restraint is one of the strongest supporting arguments for its
being based on Dicaearchus, since Porphyry or anyone else in the
luxuriant later tradition would be expected to give a much more
expanisve presentation of Pythagoras in accordance with the
Neopythagorean view of him (Burkert 1972a, 122-123). Wehrli gives no
reason for not including chapter 19 and the great majority of scholars
accept it as being based on Dicaearchus (see the references in Burkert
1972a, 122, n.7). Zhmud (2012a, 157) following Philip (1966, 139)
argues that the passage cannot derive from Dicaearchus, since it
presents immortality of the soul with approval, whereas Dicaearchus
did not accept its immortality. However, the passage merely reports
that Pythagoras introduced the notion of the immortality of the soul
without expressing approval or disapproval. Zhmud lists other features
of the chapter that he regards as unparalleled in fourth-century
sources (2012a, 157) but, since the evidence is so fragmentary, such
arguments from silence can carry little weight. Nothing in the chapter
is demonstrably late or inconsistent with Dicaearchus' authorship so
we must follow what is suggested by the context in Porphyry and regard
it as derived from Dicaearchus.




In the face of the Pythagorean question and the problems that arise
even regarding the early sources, it is reasonable to wonder if we can
say anything about Pythagoras. A minimalist might argue that the early
evidence only allows us to conclude that Pythagoras was a historical
figure who achieved fame for his wisdom but that it is impossible to
determine in what that wisdom consisted. We might say that he was
interested in the fate of the soul and taught a way of life, but we
can say nothing precise about the nature of that life or what he
taught about the soul (Lloyd 2014). There is some reason to believe,
however, that something more than this can be said.


### 4.1 The Fate of the Soul--Metempsychosis




The earliest evidence makes clear that above all Pythagoras was known
as an expert on the fate of our soul after death. Herodotus tells the
story of the Thracian Zalmoxis, who taught his countrymen that they
would never die but instead go to a place where they would eternally
possess all good things (IV. 95). Among the Greeks the tradition arose
that this Zalmoxis was the slave of Pythagoras. Herodotus himself
thinks that Zalmoxis lived long before Pythagoras, but the Greeks'
willingness to portray Zalmoxis as Pythagoras' slave shows that they
thought of Pythagoras as the expert from whom Zalmoxis derived his
teaching. Ion of Chios (5th c. BCE) says of Phercydes of
Syros that "although dead he has a pleasant life for his soul,
if Pythagoras is truly wise, who knew and learned wisdom beyond all
men." Here Pythagoras is again the expert on the life of the
soul after death. A famous fragment of Xenophanes, Pythagoras'
contemporary, provides some more specific information on what happens
to the soul after death. He reports that "once when he
[Pythagoras] was present at the beating of a puppy, he pitied it and
said 'stop, don't keep hitting him, since it is the soul of a
man who is dear to me, which I recognized, when I heard it
yelping'" (Fr. 7). Although Xenophanes clearly finds the
idea ridiculous, the fragment shows that Pythagoras believed in
metempsychosis or reincarnation, according to which human souls were
reborn into other animals after death. This early evidence is
emphatically confirmed by Dicaearchus in the fourth century, who first
comments on the difficulty of determining what Pythagoras taught and
then asserts that his most recognized doctrines were "that the
soul is immortal and that it transmigrates into other kinds of
animals" (Porphyry, *VP* 19). Unfortunately we can say
little more about the details of Pythagoras' conception of
metempsychosis. According to Herodotus, the Egyptians believed that
the soul was reborn as every sort of animal before returning to human
form after 3,000 years. Without naming names, he reports that some
Greeks both earlier and later adopted this doctrine; this seems very
likely to be a reference to Pythagoras (earlier) and perhaps
Empedocles (later). Many doubt that Herodotus is right to assign
metempsychosis to the Egyptians, since none of the other evidence we
have for Egyptian beliefs supports his claim, but it is nonetheless
clear that we cannot assume that Pythagoras accepted the details of
the view Herodotus ascribes to them. Similarly both Empedocles (see
Inwood 2001, 55-68) and Plato (e.g., *Republic* X and
*Phaedrus*) provide a more detailed account of transmigration
of souls, but neither of them ascribes these details to Pythagoras nor
should we. Did he think that we ever escape the cycle of
reincarnations? We simply do not know. The fragment of Ion quoted
above may suggest that the soul could have a pleasant existence after
death between reincarnations or even escape the cycle of reincarnation
altogether, but the evidence is too weak to be confident in such a
conclusion. In the fourth century several authors report that
Pythagoras remembered his previous human incarnations, but the
accounts do not agree on the details. Dicaearchus (Aulus Gellius
IV. 11.14) and Heraclides (Diogenes Laertius VIII. 4) agree that he
was the Trojan hero Euphorbus in a previous life. Dicaearchus
continues the tradition of savage satire begun by Xenophanes, when he
suggests that Pythagoras was the beautiful prostitute, Alco, in
another incarnation (Huffman 2014b, 281-285).




It is not clear how Pythagoras conceived of the nature of the
transmigrating soul but a few tentative conjectures can be made
(Huffman 2009). Transmigration does not require that the soul be
immortal; it could go through several incarnations before perishing.
Dicaearchus explicitly says that Pythagoras regarded the soul as
immortal, however, and this agrees with Herodotus' description of
Zalmoxis' view. It is likely that he used the Greek
word *psyche* to refer to the transmigrating soul, since
this is the word used by all sources reporting his views, unlike
Empedocles, who used *daimon*. His successor, Philoalus,
uses *psyche* to refer not to a comprehensive soul but
rather to just one psychic faculty, the seat of emotions, which is
located in the heart along with the faculty of sensation (Philolaus,
Fr. 13). This *psyche* is explicitly said by Philolaus to
be shared with animals. Herodotus uses *psyche* in a
similar way to refer to the seat of emotions. Thus it seems likely
that Pythagoras too thought of the
transmigrating *psyche* in this way. If so, it is
unlikely that Pythagoras thought that humans could be reincarnated as
plants, since *psyche* is not assigned to plants by
Philolaus. It has often been assumed that the transmigrating soul is
immaterial, but Philolaus seems to have a materialistic conception of
soul and he may be following Pythagoras. Similarly, it is doubtful
that Pythagoras thought of the transmigrating soul as a comprehensive
soul that includes all psychic faculties. His ability to recognize
something distinctive of his friend in the puppy (if this is not
pushing the evidence of a joke too far) and to remember his own
previous incarnations show that personal identity was preserved
through incarnations. This personal identity could well be contained
in the pattern of emotions, that constitute a person's character and
that is preserved in the *psyche* and need not presuppose
all psychic faculties. In Philolaus this *psyche*
explicitly does not include the *nous* (intellect), which is
not shared with animals. Thus, it would appear that what is shared
with animals and which led Pythagoras to suppose that they had special
kinship with human beings (Dicaearchus in Porphyry, *VP* 19) is
not intellect, as some have supposed (Sorabji 1993, 78 and 208) but
rather the ability to feel emotions such as pleasure and pain. 




There are significant points of contact between the Greek religious
movement known as Orphism and Pythagoreanism, but the evidence for
Orphism is at least as problematic as that for Pythagoras and often
complicates rather than clarifies our understanding of Pythagoras
(Betegh 2014; Burkert 1972a, 125 ff.; Kahn 2002, 19-22; Riedweg
2002). There is some evidence that the Orphics also believed in
metempsychosis and considerable debate has arisen as to whether they
borrowed the doctrine from Pythagoras (Burkert 1972a, 133; Bremmer
2002, 24) or he borrowed it from them (Zhmud 2012a,
221-238). Dicaearchus says that Pythagoras was the first to introduce
metempsychosis into Greece (Porphyry *VP* 19). Moreover, while
Orphism presents a heavily moralized version of metempsychosis in
accordance with which we are born again for punishment in this life so
that our body is the prison of the soul while it undergoes punishment,
it is not clear that the same was true in Pythagoreanism. It may be
that rebirths in a series of animals and people were seen as a natural
cycle of the soul (Zhmud 2012a, 232-233). One would expect that the
Pythagorean way of life was connected to metempsychosis, which would
in turn suggest that a certain reincarnation is a reward or punishment
for following or not following the principles set out in that way of
life. However, there is no unambiguous evidence connecting the
Pythagorean way of life with metempsychosis. 




 It is crucial to recognize that most Greeks followed Homer in
believing that the soul was an insubstantial shade, which lived a
shadowy existence in the underworld after death, an existence so bleak
that Achilles famously asserts that he would rather be the lowest
mortal on earth than king of the dead (Homer, *Odyssey*
XI. 489). Pythagoras' teachings that the soul was immortal, would have
other physical incarnations and might have a good existence after
death were striking innovations that must have had considerable appeal
in comparison to the Homeric view. According to Dicaearchus, in
addition to the immortality of the soul and reincarnation, Pythagoras
believed that "after certain periods of time the things that
have happened once happen again and nothing is absolutely new"
(Porphyry, *VP* 19). This doctrine of "eternal
recurrence" is also attested by Aristotle's pupil Eudemus,
although he ascribes it to the Pythagoreans rather than to Pythagoras
himself. (Fr. 88 Wehrli). The doctrine of transmigration thus seems
to have been extended to include the idea that we and indeed the whole
world will be reborn into lives that are exactly the same as those we
are living and have already lived.


### 4.2 Pythagoras as a Wonder-worker




Some have wanted to relegate the more miraculous features of
Pythagoras' *persona* to the later tradition, but these
characteristics figure prominently in the earliest evidence and are
thus central to understanding Pythagoras. Aristotle emphasized his
superhuman nature in the following ways: there was a story that
Pythagoras had a golden thigh (a sign of divinity); the Pythagoreans
taught that "of rational beings, one sort is divine, one is
human, and another such as Pythagoras" (Iamblichus, *VP*
31); Pythagoras was seen on the same day at the same time in both
Metapontum and Croton; he killed a deadly snake by biting it; as he
was crossing a river it spoke to him (all citations are from
Aristotle, Fr. 191, unless otherwise noted). Aristotle reports that
the people of Croton called Pythagoras the "Hyperborean Apollo" and
Iamblichus' report (*VP* 140) that a priest from the land of
the Hyperboreans, Abaris, visited Pythagoras and presented him with
his arrow, a token of power, may well also go back to Aristotle
(Burkert 1972a, 143). Kingsley argues that the visit of Abaris is the
key to understanding the identity and significance of
Pythagoras. Abaris was a shaman from Mongolia (part of what the Greeks
called Hyperborea), who recognized Pythagoras as an incarnation of
Apollo. The stillness of ecstacy practiced by Abaris and handed on to
Pythagoras is the foundation of all civilization. Abaris' visit to
Pythagoras thus beomces the central moment when civilizing power is
passed from East to West (Kingsley 2010). 




Whether or not one accepts this account of Pythagoras and his relation
to Abaris, there is a clear parallel for some of the remarkable
abilities of Pythagoras in the later figure of Empedocles, who
promises to teach his pupils to control the winds and bring the dead
back to life (Fr. 111). There are recognizable traces of this
tradition about Pythagoras even in the pre-Aristotelian evidence, and
his wonder-working clearly evoked diametrically opposed reactions.
Heraclitus' description of Pythagoras as "the chief of the
charlatans" (Fr. 81) and of his wisdom as "fraudulent
art" (Fr. 129) is most easily understood as an unsympathetic
reference to his miracles. Empedocles, on the other hand, is clearly
sympathetic to Pythagoras, when he describes him as " a man who
knew remarkable things" and who "possessed the greatest
wealth of intelligence" and again probably makes reference to
his wonder-working by calling him "accomplished in all sorts of
wise *deeds*"(Fr. 129). In Herodotus' report, Zalmoxis,
whom some of the Greeks identified as the slave and pupil of
Pythagoras, tried to gain authority for his teachings about the fate
of the soul by claiming to have journeyed to the next world
(IV. 95). The skeptical tradition represented in Herodotus' report
treats this as a ruse on Zalmoxis' part; he had not journeyed to the
next world but had in reality hidden in an underground dwelling for
three years. Similarly Pythagoras may have claimed authority for his
teachings concerning the fate of our soul on the basis of his
remarkable abilities and experiences, and there is some evidence that
he too claimed to have journeyed to the underworld and that this
journey may have been transferred from Pythagoras to Zalmoxis (Burkert
1972a,154 ff.).


### 4.3 The Pythagorean Way of Life




The testimony of both Plato (*R*. 600a) and Isocrates
(*Busiris* 28) shows that Pythagoras was above all famous for
having left behind him a way of life, which still had adherents in the
fourth century over 100 years after his death. It is plausible to
assume that many features of this way of life were designed to insure
the best possible future reincarnations, but it is important to
remember that nothing in the early evidence connects the way of life
to reincarnation in any specific fashion. 




One of the clearest strands in the early evidence for Pythagoras is
his expertise in religious ritual. Isocrates emphasizes that "he
more conspicuously than others paid attention to sacrifices and
rituals in temples" (*Busiris* 28). Herodotus gives an
example: the Pythagoreans agree with the Egyptians in not allowing the
dead to be buried in wool (II. 81). It is not surprising that
Pythagoras, as an expert on the fate of the soul after death. should
also be an expert on the religious rituals surrounding death. A
significant part of the Pythagorean way of life thus consisted in the
proper observance of religious ritual. One major piece of evidence for
this emphasis on ritual is the *symbola* or *acusmata*
("things heard"), short maxims that were handed down
orally. The earliest source to quote *acusmata* is Aristotle,
in the fragments of his now lost treatise on the Pythagoreans. It is
not always possible to be certain which of the *acusmata*
quoted in the later tradition go back to Aristotle and which of the
ones that do go back to Pythagoras. Most of Iamblichus' examples in
sections 82-86 of *On the Pythagorean Life*, however,
appear to derive from Aristotle (Burkert 1972a, 166 ff.), and many are
in accord with the early evidence we have for Pythagoras' interest in
ritual. Thus the
*acusmata* advise Pythagoreans to pour libations to the gods
from the ear (i.e., the handle) of the cup, to refrain from wearing
the images of the gods on their fingers, not to sacrifice a white
cock, and to sacrifice and enter the temple barefoot. A number of
these practices can be paralleled in Greek mystery religions of the
day (Burkert 1972a, 177). Indeed, it is important to emphasize that
Pythagoreanism was not a religion and there were no specific
Pythagorean rites (Burkert 1985, 302). Pythagoras rather taught a way
of life that emphasized certain aspects of traditional Greek
religion. 




A second characteristic of the Pythagorean way of life was the
emphasis on dietary restrictions. There is no direct evidence for
these restrictions in the pre-Aristotelian evidence, but both
Aristotle and Aristoxenus discuss them extensively. Unfortunately the
evidence is contradictory and it is difficult to establish any points
with certainty. One might assume that Pythagoras advocated
vegetarianism on the basis of his belief in metempsychosis, as did
Empedocles after him (Fr. 137). Indeed, the fourth-century
mathematician and philosopher Eudoxus says that "he not only
abstained from animal food but would also not come near butchers and
hunters" (Porphyry, *VP* 7). According to Dicaearchus,
one of Pythagoras' most well-known doctrines was that "all
animate beings are of the same family" (Porphyry, *VP*
19), which suggests that we should be as hesitant about eating other
animals as other humans. Unfortunately, Aristotle reports that
"the Pythagoreans refrain from eating the womb and the heart,
the sea anemone and some other such things but use all other animal
food" (Aulus Gellius IV. 11. 11-12). This makes it sound
as if Pythagoras forbade the eating of just certain parts of animals
and certain species of animals rather than all animals; such specific
prohibitions are easy to parallel elsewhere in Greek ritual (Burkert
1972a, 177). Aristoxenus asserts that Pythagoras only refused to eat
plough oxen and rams (Diogenes Laertius VIII. 20) and that he was fond
of young kids and suckling pigs as food (Aulus Gellius
IV. 11. 6). Some have tried to argue that Aristoxenus is refashioning
Pythagoreanism in order to make it more rational (e.g., Kahn 2001, 70;
Zhmud 2012b, 228), but Aristoxenus, in fact, recognizes the
non-rational dimension of Pythagoreanism and Pythagoras' eating of
kids and suckling pigs may itself have religious motivations (Huffman
2012b). Moreover, even if Aristoxenus' evidence were set aside
Aristotle's testimony and many of the
*acusmata* indicate that Pythagoras ate some meat. Certainly
animal sacrifice was the central act of Greek religious worship and to
abolish it completely would be a radical step. The *acusma*
reported by Aristotle, in response to the question "what is most
just?" has Pythagoras answer "to sacrifice"
(Iamblichus, *VP* 82). Based on the direct evidence for
Pythagoras' practice in Aristotle and Aristoxenus, it seems most
prudent to conclude that he did not forbid the eating of all animal
food. The later tradition proposes a number of ways to reconcile
metempsychosis with the eating of some meat. Pythagoras may have
adopted one of these positions, but no certainty is possible. For
example, he may have argued that it was legitimate to kill and eat
sacrificial animals, on the grounds that the souls of men do not enter
into these animals (Iamblichus, *VP* 85). Perhaps the most
famous of the Pythagorean dietary restrictions is the prohibition on
eating beans, which is first attested by Aristotle and assigned to
Pythagoras himself (Diogenes Laertius VIII. 34). Aristotle suggests a
number of explanations including one that connects beans with Hades,
hence suggesting a possible connection with the doctrine of
metempsychosis. A number of later sources suggest that it was believed
that souls returned to earth to be reincarnated through beans (Burkert
1972a, 183). There is also a physiological explanation. Beans, which
are difficult to digest, disturb our abilities to concentrate.
Moreover, the beans involved are a European vetch (*Vicia
faba*) rather than the beans commonly eaten today. Certain people
with an inherited blood abnormality develop a serious disorder called
favism, if they eat these beans or even inhale their
pollen. Aristoxenus interestingly denies that Pythagoras forbade the
eating of beans and says that "he valued it most of all
vegetables, since it was digestible and laxative" (Aulus Gellius
IV. 11.5). The discrepancies between the various fourth-century
accounts of the Pythagorean way of life suggest that there were
disputes among fourth-century Pythagoreans as to the proper way of
life and as to the teachings of Pythagoras himself.




The *acusmata* indicate that the Pythagorean way of life
embodied a strict regimen not just regarding religious ritual and diet
but also in almost every aspect of life. Some of the restrictions
appear to be largely arbitrary taboos, e.g., "one must put the
right shoe on first" or "one must not travel the public
roads" (Iamblichus, *VP* 83, probably from Aristotle). On
the other hand, some aspects of the Pythagorean life involved a moral
discipline that was greatly admired, even by outsiders. Pythagorean
silence is an important example. Isocrates reports that even in the
fourth century people "marvel more at the silence of those who
profess to be his pupils than at those who have the greatest
reputation for speaking" (*Busiris* 28). The ability to
remain silent was seen as important training in self-control, and the
later tradition reports that those who wanted to become Pythagoreans
had to observe a five-year silence (Iamblichus, *VP*
72). Isocrates is contrasting the marvelous self-control of
Pythagorean silence with the emphasis on public speaking in
traditional Greek education. Pythagoreans also displayed great
loyalty to their friends as can be seen in Aristoxenus' story of Damon
who is willing to stand surety for his friend Phintias, who has been
sentenced to death (Iamblichus, *VP* 233 ff.). In addition to
silence as a moral discipline, there is evidence that secrecy was kept
about certain of the teachings of Pythagoras. Aristoxenus reports that
the Pythagoreans thought that "not all things were to be spoken
to all people" (Diogenes Laertius, VIII. 15), but this may only
apply to teaching and mean that children should not be taught all
things (Zhmud 2012a, 155). Clearer evidence is found in Dicaearchus'
complaint that it is not easy to say what Pythagoras taught his pupils
because they observed no ordinary silence about it
(Porphyry, *VP* 19). Indeed, one would expect that an exclusive
society such as that of the Pythagoreans would have secret doctrines
and symbols. Aristotle says that the Pythagoreans "guarded among
their very secret doctrines that one type of rational being is divine,
one human, and one such as Pythagoras" (Iamblichus, *VP*
31). That there should be secret teachings about the special nature
and authority of the master is not surprising. This does not mean,
however, that all Pythagorean philosophy was secret. Aristotle
discusses the fifth-century metaphysical system of Philolaus in some
detail with no hint that there was anything secret about it, and
Plato's discussion of Pythagorean harmonic theory in Book VII of the
*Republic* gives no suggestion of any secrecy. Aristotle
singles out the *acusma* quoted above (Iamblichus, *VP*
31) as secret, but this statement in itself implies that others were
not. The idea that all of Pythagoras' teachings were secret was used
in the later tradition to explain the lack of Pythagorean writings and
to try to validate forged documents as recently discovered secret
treatises. For a sceptical evaluation of Pythagorean secrecy see Zhmud
2012a, 150-158.




There is some controversy as to whether Pythagoras, in fact, taught a
way of life governed in great detail by the *acusmata* as
described above. Plato praises the Pythagorean way of life in
the *Republic* (600b), but it is hard to imagine him admiring
the set of taboos found in the *acusmata* (Lloyd 2014, 44;
Zhmud 2012a). Although *acusmata* were collected already by
Anaximander of Miletus the younger (ca. 400 BCE) and by Aristotle in
the fourth century, Zhmud (2012a, 177-178 and 192-205) argues that
very few of these embody specifically Pythagorean ideas and that it is
difficult to imagine anyone following this bewildering set of rules
literally as Burkert argues (1972a, 191). However, the early evidence
suggests that Pythagoras largely constructed the *acusmata* out
of ideas collected from others (Thom 2013; Huffman 2008b: Gemelli
Marciano 2002), so it is no surprise that many of them are not
uniquely Pythagorean. Moreover, Thom suggests a middle ground between
Zhmud and Burkert whereby, contra Zhmud, most of the *acusmata*
were followed by the Pythagoreans but contra Burkert, they were
subject to interpretation from the beginning and not followed
literally, so that it is possible to imagine people living according
to them (Thom, 2013). It is true that there is little if any fifth-
and fourth-century evidence for Pythagoreans living according to
the *acusmata* and Zhmud argues that the undeniable political
impact of the Pythagoreans would be inexplicable if they lived the
heavily ritualized life of the *acusmata*, which would
inevitably isolate them from society (Zhmud 2012a, 175-183). He
suggests that the Pythagorean way of life differed little from
standard aristocratic morality (Zhmud 2012a, 175). If, however, the
Pythagorean way of life was little out of the ordinary, why do Plato
and Isocrates specifically comment on how distinctive those who
followed it were? The silence of fifth-century sources about people
practicing *acusmata* is not terribly surprising given the very
meager sources for the Greek cities in southern Italy in the
period. Why not suppose that the vast majority of names in
Aristoxenus' catalogue of Pythagoreans, who are not associated with
any political, philosophical or scientific accomplishment, who are
just names to us, are preceisely those who were Pythagoreans because
they followed the Pythagorean way of life? We would then have lots of
people who followed the *acusmata* (166 of the 222 name in the
catalogue appear nowhere else). This suspicion is confirmed by the
fact that one of the names from Arsitoxenus' catalogue (Hippomedon of
Argos) is elsewhere (Iamblichus, *On the Pythagorean Life*, 87)
explicitly said to belong the *acusmatici*. Moreover, other
scholars argue that archaic Greek society in southern Italy was
pervaded by religion and the presence of similar precepts in authors
such as Hesiod show that adherence to taboos such as are found in
the *acusmata* would not have caused a scandal and adherence to
many of them would have gone unobserved by outsiders (Gemelli Marciano
2014, 133-134).




Once again a problem of source criticism raises its head. Zhmud argues
that the split between *acusmatici* who blindly followed
the *acusmata* and the *mathematici* who learned the
reasons for them (see the fifth paragraph of section 5 below) is a
creation of the later tradition, appearing first in Clement of
Alexandria and disappearing after Iamblichus (Zhmud 2012a,
169-192). He also notes that the term *acusmata* appears first
in Iamblichus (*On the Pythagorean Life* 82-86) and suggests
that it is also a creation of the later tradition. The Pythagorean
maxims did exist earlier, as the testimony of Aristotle shows, but
they were known as *symbola*, were originally very few in
number and were mainly a literary phenomena rather than being tied to
people who actually practiced them (Zhmud 2012a, 192-205). However,
several scholars have argued that the passages in which the split
between the *acusmatici* and *mathematici* is described
as well as the passage in which the term *acusmata* is used, in
fact, go back to Aristotle (Burkert 1972a, 196; see Burkert 1998, 315
where he comments that the Aristotelian provenance of the text is "as
obvious as it is unprovable") and even Zhmud recognizes that a large
part of the material in Iamblichus is derived from Aristotle (2012a,
170). Indeed, the description of the split in what is likely to be the
original version (Iamblichus, *On General Mathematical Science*
76.16-77.18 Festa) uses language in describing the Pythagoreans that
is almost an Aristotelian signature, "There are two forms of the
Italian philosophy which is called Pythagorean"
(76.16). Aristotle famously describes the Pythagoreans as "those
called Pythagoreans" and also describes them as "the
Italians" (e.g., *Mete.* 342b30, *Cael.*
293a20). So the question of whether Pythagoras taught a way of life
tightly governed by the *acusmata* turns again on whether key
passages in Iamblichus (*On the Pythagorean Life* 81-87, *On
General Mathematical Science* 76.16-77.18 Festa) go back to
Aristotle. If they do, we have very good reason to believe that
Pythagoras taught such a life, if they do not the issue is less
clear.




The testimony of fourth-century authors such as Aristoxenus and
Dicaearchus indicates that the Pythagoreans also had an important
impact on the politics and society of the Greek cities in southern
Italy. Dicaearchus reports that, upon his arrival in Croton,
Pythagoras gave a speech to the elders and that the leaders of the
city then asked him to speak to the young men of the town, the boys
and the women (Porphyry, *VP* 18). Women, indeed, may have
played an unusually large role in Pythagoreanism (see Rowett 2014,
122-123), since both Timaeus and Dicaearchus report on the fame of
Pythagorean women including Pythagoras' daughter
(Porphyry, *VP* 4 and 19). The
*acusmata* teach men to honor their wives and to beget children
in order to insure worship for the gods (Iamblichus, *VP*
84-6). Dicaearchus reports that the teaching of Pythagoras was
largely unknown, so that Dicaearchus cannot have known of the content
of the speech to the women or of any of the other speeches; the
speeches presented in Iamblichus (*VP* 37-57) are thus
likely to be later forgeries (Burkert 1972a, 115), but there is early
evidence that he gave different speeches to different groups
(Antisthenes V A 187). The attacks on the Pythagoreans both in
Pythagoras' own day and in the middle of the fifth century are
presented by Dicaearchus and Aristoxenus as having a wide-reaching
impact on Greek society in southern Italy; the historian Polybius
(II. 39) reports that the deaths of the Pythagoreans meant that
"the leading citizens of each city were destroyed," which
clearly indicates that many Pythagoreans had positions of political
authority. On the other hand, it is noteworthy that Plato explicitly
presents Pythagoras as a private rather than a public figure
(*R*. 600a). It seems most likely that the Pythagorean
societies were in essence private associations but that they also
could function as political clubs (see Zhmud 2012a, 141-148), while
not being a political party in the modern sense; their political
impact should perhaps be better compared to modern fraternal
organizations such as the Masons. Thus, the Pythagoreans did not rule
as a group but had political impact through individual members who
gained positions of authority in the Greek city-states in southern
Italy. See further Burkert 1972a, 115 ff., von Fritz 1940, Minar 1942
and Rowett 2014.


## 5. Was Pythagoras a Mathematician or Cosmologist?




In the modern world Pythagoras is most of all famous as a
mathematician, because of the theorem named after him, and secondarily
as a cosmologist, because of the striking view of a universe ascribed
to him in the later tradition, in which the heavenly bodies produce
"the music of the spheres" by their movements. It should
be clear from the discussion above that, while the early evidence
shows that Pythagoras was indeed one of the most famous early Greek
thinkers, there is no indication in that evidence that his fame was
primarily based on mathematics or cosmology. Neither Plato nor
Aristotle treats Pythagoras as having contributed to the development
of Presocratic cosmology, although Aristotle in particular discusses
the topic in some detail in the first book of the *Metaphysics*
and elsewhere. Aristotle evidently knows of no cosmology of Pythagoras
that antedates the cosmological system of the "so-called
Pythagoreans," which he dates to the middle of the fifth
century, and which is found in the fragments of Philolaus. There is
also no mention of Pythagoras' work in geometry or of the Pythagorean
theorem in the early evidence. Dicaearchus comments that "what
he said to his associates no one can say reliably," but then
identifies four doctrines that became well known: 1) that the soul is
immortal; 2) that it transmigrates into other kinds of animals; 3)
that after certain intervals the things that have happened once happen
again, so that nothing is completely new; 4) that all animate beings
belong to the same family (Porphyry, *VP* 19). Thus, for
Dicaearchus too, it is not as a mathematician or Presocratic writer on
nature that Pythagoras is famous. It might not be too surprising that
Plato, Aristotle and Dicaearchus do not mention Pythagoras' work in
mathematics, because they are not primarily dealing with the history
of mathematics. On the other hand, Aristotle's pupil Eudemus did write
a history of geometry in the fourth century and what we find in
Eudemus is very significant. A substantial part of Eudemus' overview
of the early history of Greek geometry is preserved in the prologue to
Proclus' commentary on Book One of Euclid's *Elements* (p. 65,
12 ff.), which was written much later, in the fifth century CE. At
first sight, it appears that Eudemus did assign Pythagoras a
significant place in the history of geometry. Eudemus is reported as
beginning with Thales and an obscure figure named Mamercus, but the
third person mentioned by Proclus in this report is Pythagoras,
immediately before Anaxagoras. There is no mention of the Pythagorean
theorem, but Pythagoras is said to have transformed the philosophy of
geometry into a form of liberal education, to have investigated its
theorems in an immaterial and intellectual way and specifically to
have discovered the study of irrational magnitudes and the
construction of the five regular solids. Unfortunately close
examination of the section on Pythagoras in Proclus' prologue reveals
numerous difficulties and shows that it comes not from Eudemus but
from Iamblichus with some additions by Proclus himself (Burkert 1972a,
409 ff.). The first clause is taken word for word from Iamblichus'
*On Common Mathematical Science* (p. 70.1 Festa). Proclus
elsewhere quotes long passages from Iamblichus and is doing the same
here. As Burkert points out, however, as soon as we recognize that
Proclus has inserted a passage from Iamblichus into Eudemus' history,
we must also recognize that Proclus was driven to do so by the lack of
any mention of Pythagoras in Eudemus. Even those who want to assign
Pythagoras a larger role in early Greek mathematics recognize that
most of what Proclus says here cannot go back to Eudemus (Zhmud 2012a,
263-266). Thus, not only is Pythagoras not commonly known as a
geometer in the time of Plato and Aristotle, but also the most
authoritative history of early Greek geometry assigns him no role in
the history of geometry in the overview preserved in
Proclus. According to Proclus, Eudemus did report that two
propositions, which are later found in Euclid's *Elements*,
were discoveries of the Pythagoreans (Proclus 379 and 419). Eudemus
does not assign the discoveries to any specific Pythagorean, and they
are hard to date. The discoveries might be as early as Hippasus in the
middle of the fifth century, who is associated with a group of
Pythagoreans known as the
*mathematici*, who arose after Pythagoras' death (see
below). The crucial point to note is that Eudemus does not assign
these discoveries to Pythagoras himself. The first Pythagorean whom we
can confidently identify as an accomplished mathematician is Archytas
in the late fifth and the first half of the fourth century.




Are we to conclude, then, that Pythagoras had nothing to do with
mathematics or cosmology? The evidence is not quite that simple. The
tradition regarding Pythagoras' connection to the Pythagorean theorem
reveals the complexity of the problem. None of the early sources,
including Plato, Aristotle and their pupils shows any knowledge of
Pythagoras' connection to the theorem. Almost a thousand years later,
in the fifth century CE, Proclus, in his commentary on Euclid's proof
of the theorem (*Elements* I. 47), gives the following report:
"If we listen to those who wish to investigate ancient history,
it is possible to find them referring this theorem back to Pythagoras
and saying that he sacrificed an ox upon its discovery"
(426.6). Proclus gives no indication of his source, but a number of
other late reports (Diogenes Laertius VIII. 12; Athenaeus 418f;
Plutarch, *Moralia* 1094b) show that it ultimately relied on
two lines of verse whose context is unknown: "When Pythagoras
found that famous diagram, in honor of which he offered a glorious
sacrifice of oxen..." The author of these verses is variously
identified as Apollodorus the calculator or Apollodorus the
arithmetician. This Apollodorus probably dates before Cicero, who
alludes to the story (*On the Nature of the Gods* III. 88),
and, if he can be identified with Apollodorus of Cyzicus, the follower
of Democritus, the story would go back to the fourth century BCE
(Burkert 1972a, 428). Two lines of poetry of indeterminate date are
obviously a very slender support upon which to base Pythagoras'
reputation as a geometer, but they cannot be simply ignored. Several
things need to be noted about this tradition, however, in order to
understand its true significance. First, Proclus does not ascribe a
proof of the theorem to Pythagoras but rather goes on to contrast
Pythagoras as one of those "knowing the truth of the
theorem" with Euclid who not only gave the proof found in
*Elements* I.47 but also a more general proof in
VI. 31. Although a number of modern scholars have speculated on what
sort of proof Pythagoras might have used (e.g., Heath 1956, 352 ff.),
it is important to note that there is not a jot of evidence for a
proof by Pythagoras; what we know of the history of Greek geometry
makes such a proof by Pythagoras improbable, since the first work on
the elements of geometry, upon which a rigorous proof would be based,
is not attested until Hippocrates of Chios, who was active after
Pythagoras in the latter part of the fifth century (Proclus, *A
Commentary on the First Book of Euclid's Elements*, 66). All that
this tradition ascribes to Pythagoras, then, is discovery of the truth
contained in the theorem. The truth may not have been in general form
but rather focused on the simplest such triangle (with sides 3, 4 and
5), pointing out that such a triangle and all others like it will have
a right angle. Modern scholarship has shown, moreover, that long
before Pythagoras the Babylonians were aware of the basic Pythagorean
rule and could generate Pythagorean triples (integers that satisfy the
Pythagorean rule such as 3, 4 and 5), although they never formulated
the theorem in explicit form or proved it (Hoyrup 1999, 401-2, 405;
cf. Robson 2001). Thus, it is likely that Pythagoras and other Greeks
first encountered the truth of the theorem as a Babylonian
arithmetical technique (Hoyrup 1999, 402; Burkert 1972a,429). It is
possible, then, that Pythagoras just passed on to the Greeks a truth
that he learned from the East. The emphasis in the two lines of verse
is not just on Pythagoras' discovery of the truth of the theorem, it
is as much or more on his sacrifice of oxen in honor of the
discovery. We are probably supposed to imagine that the sacrifice was
not of a single ox; Apollodorus describes it as "a famous
sacrifice of oxen" and Diogenes Laertius paraphrases this as a
*hecatomb*, which need not be, as it literally says, a hundred
oxen, but still suggests a large number. Some have wanted to doubt the
whole story, including the discovery of the theorem, because it
conflicts with Pythagoras' supposed vegetarianism, but it is far from
clear to what extent he was a vegetarian (see above). If the story is
to have any force and if it dates to the fourth century, it shows that
Pythagoras was famous for a connection to a certain piece of
geometrical knowledge, but it also shows that he was famous for his
enthusiastic response to that knowledge, as evidenced in his sacrifice
of oxen, not for any geometric proof. What emerges from this evidence,
then, is not Pythagoras as the master geometer, who provides rigorous
proofs, but rather Pythagoras as someone who recognizes and celebrates
certain geometrical relationships as of high importance.




It is striking that a very similar picture of Pythagoras emerges from
the evidence for his cosmology. A famous discovery is attributed to
Pythagoras in the later tradition, i.e., that the central musical
concords (the octave, fifth and fourth) correspond to the whole number
ratios 2 : 1, 3 : 2 and 4 : 3 respectively (e.g., Nicomachus,
*Handbook* 6 = Iamblichus, *On the Pythagorean Life*
115). The only early source to associate Pythagoras with the whole
number ratios that govern the concords is Xenocrates (Fr. 9) in the
early Academy, but the early Academy is precisely one source of the
later exaggerated tradition about Pythagoras (see above). One story
has it that Pythagoras passed by a blacksmith's shop and heard the
concords in the sounds of the hammers striking the anvil and then
discovered that the sounds made by hammers whose weights are in the
ratio 2 : 1 will be an octave apart, etc. Unfortunately, the stories
of Pythagoras' discovery of these relationships are clearly false,
since none of the techniques for the discovery ascribed to him would,
in fact, work (e.g., the pitch of sounds produced by hammers is not
directly proportional to their weight: see Burkert 1972a, 375). An
experiment ascribed to Hippasus, who was active in the first half of
the fifth century, after Pythagoras' death, would have worked, and
thus we can trace the scientific verification of the discovery at
least to Hippasus; knowledge of the relation between whole number
ratios and the concords is clearly found in the fragments of Philolaus
(Fr. 6a, Huffman), in the second half of the fifth century. There is
some evidence that the truth of the relationship was already known to
Pythagoras' contemporary, Lasus, who was not a Pythagorean (Burkert
1972a, 377). It may be once again that Pythagoras knew of the
relationship without either having discovered it or having
demonstrated it scientifically. The relationship was probably first
discovered by instrument makers, and specifically makers of wind
instruments rather than stringed instruments (Barker 2014,
202). The *acusmata* reported by Aristotle, which may go back
to Pythagoras, report the following question and answer "What is
the oracle at Delphi? The *tetraktys*, which is the harmony in
which the Sirens sing" (Iamblichus, *On the Pythagorean
Life*, 82, probably derived from Aristotle). The
*tetraktys*, literally "the four," refers to the
first four numbers, which when added together equal the number ten,
which was regarded as the perfect number in fifth-century
Pythagoreanism. Here in the *acusmata*, these four numbers are
identified with one of the primary sources of wisdom in the Greek
world, the Delphic oracle. In the later tradition the
*tetraktys* is treated as the summary of all Pythagorean
wisdom, since the Pythagoreans swore oaths by Pythagoras as "the
one who handed down the *tetraktys* to our generation."
The *tetraktys* can be connected to the music which the Sirens
sing in that all of the ratios that correspond to the basic concords
in music (octave, fifth and fourth) can be expressed as whole number
ratios of the first four numbers. This *acusma* thus seems to
be based on the knowledge of the relationship between the concords and
the whole number ratios. The picture of Pythagoras that emerges from
the evidence is thus not of a mathematician, who offered rigorous
proofs, or of a scientist, who carried out experiments to discover the
nature of the natural world, but rather of someone who sees special
significance in and assigns special prominence to mathematical
relationships that were in general circulation. This is the context in
which to understand Aristoxenus' remark that "Pythagoras most of
all seems to have honored and advanced the study concerned with
numbers, having taken it away from the use of merchants and likening
all things to numbers" (Fr. 23, Wehrli). Some might suppose that
this is a reference to a rigorous treatment of arithmetic, such as
that hypothesized by Becker (1936), who argued that Euclid
IX. 21-34 was a self-contained unit that represented a deductive
theory of odd and even numbers developed by the Pythagoreans (see
Mueller 1997, 296 ff. and Burkert 1972a, 434 ff.). It is crucial to
recognize, however, that Becker's reconstruction is rejected in some
recent scholarship (e.g., Netz 2014, 179) and no ancient source
assigns it even to the Pythagoreans, let alone to Pythagoras
himself. There is, moreover, no talk of mathematical proof or a
deductive system in the passage from Aristoxenus just quoted.
Pythagoras is known for the *honor* he gives to number and for
removing it from the practical realm of trade and instead pointing to
correspondences between the behavior of number and the behavior of
things. Such correspondences were highlighted in Aristotle's book on
the Pythagoreans, e.g., the female is likened to the number two and
the male to the number three and their sum, five, is likened to
marriage (Aristotle, Fr. 203).




What then was the nature of Pythagoras' cosmos? The doxographical
tradition reports that Pythagoras discovered the sphericity of the
earth, the five celestial zones and the identity of the evening and
morning star (Diogenes Laertius VIII. 48, Aetius III.14.1, Diogenes
Laertius IX. 23). In each case, however, Burkert has shown that these
reports seem to be false and the result of the glorification of
Pythagoras in the later tradition, since the earliest and most
reliable evidence assigns these same discoveries to someone else
(1972a, 303 ff.). Thus, Theophrastus, who is the primary basis of the
doxographical tradition, says that it was Parmenides who discovered
the sphericity of the earth (Diogenes Laertius VIII. 48). Parmenides
is also identified as the discoverer of the identity of the morning
and evening star (Diogenes Laertius IX. 23), and Pythagoras' claim
appears to be based on a poem forged in his name, which was rejected
already by Callimachus in the third century BCE (Burkert 1972a, 307).
The identification of the five celestial zones depends on the
discovery of the obliquity of the ecliptic, and some of the doxography
duly assigns this discovery to Pythagoras as well and claims that
Oenopides stole it from Pythagoras (Aetius II.12.2); the history of
astronomy by Aristotle's pupil Eudemus, our most reliable source,
seems to attribute the discovery to Oenopides (there are problems with
the text), however (Eudemus, Fr. 145 Wehrli). It thus appears that the
later tradition, finding no evidence for Pythagoras' cosmology in the
early evidence, assigned the discoveries of Parmenides back to
Pythagoras, encouraged by traditions which made Parmenides the pupil
of Pythagoras. In the end, there is no evidence for Pythagoras'
cosmology in the early evidence, beyond what can be reconstructed
from *acusmata*. As was shown above, Pythagoras saw the cosmos
as structured according to number insofar as the *tetraktys* is
the source of all wisdom. His cosmos was also imbued with a moral
significance, which is in accordance with his beliefs about
reincarnation and the fate of the soul (West 1971, 215-216; Huffman
2013, 60-68). Thus, in answer to the question "What are the
Isles of the Blest?" (where we might hope to go, if we lived a
good life), the answer is "the sun and the moon." Again
"the planets are the hounds of Persephone," i.e., the
planets are agents of vengeance for wrong done (Aristotle in
Porphyry *VP* 41). Aristotle similarly reports that for the
Pythagoreans thunder "is a threat to those in Tartarus, so that
they will be afraid" (*Posterior Analytics* 94b) and
another *acusma* says that "an earthquake is nothing
other than a meeting of the dead" (Aelian, *Historical
Miscellany*, IV. 17). Zhmud calls these
cosmological *acusmata* into question (2012a, 329-330), noting
that some only appear in Porphyry, but Porphyry explicitly identifies
Aristotle as his source and we have no reason to doubt him
(*VP* 41). Pythagoras' cosmos embodied mathematical
relationships that had a basis in fact and combined them with moral
ideas tied to the fate of the soul. The best analogy for the type of
account of the cosmos which Pythagoras gave might be some of the myths
which appear at the end of Platonic dialogues such as
the *Phaedo*,
*Gorgias* or *Republic*, where cosmology has a primarily
moral purpose. Should the doctrine of the harmony of the spheres be
assigned to Pythagoras? Certainly the *acusma* which talks of
the sirens singing in the harmony represented by the
*tetraktys* suggests that there might have been a cosmic music
and that Pythagoras may well have thought that the heavenly bodies,
which we see move across the sky at night, made music by their
motions. On the other hand, there is no evidence for "the
spheres," if we mean by that a cosmic model according to which
each of the heavenly bodies is associated with a series of concentric
circular orbits, a model which is at least in part designed to explain
celestical phenomena. The first such cosmic model in the Pythagorean
tradition is that of Philolaus in the second half of the fifth
century, a model which still shows traces of the connection to the
moral cosmos of Pythagoras in its account of the counter-earth and the
central fire (see Philolaus).




If Pythagoras was primarily a figure of religious and ethical
significance, who left behind an influential way of life and for whom
number and cosmology primarily had significance in this religious and
moral context, how are we to explain the prominence of rigorous
mathematics and mathematical cosmology in later Pythagoreans such as
Philolaus and Archytas? It is important to note that this is not just
a question asked by modern scholars but was already a central question
in the fourth century BCE. What is the connection between Pythagoras
and fifth-century Pythagoreans? The question is implicit in
Aristotle's description of the fifth-century Pythagoreans such as
Philolaus as "the so-called Pythagoreans." This expression
is most easily understood as expressing Aristotle's recognition that
these people were called Pythagoreans and at the same time his
puzzlement as to what connection there could be between the
wonder-worker who promulgated the *acusmata*, which his
researches show Pythagoras to have been, and the philosophy of
limiters and unlimiteds put forth in fifth-century Pythagoreanism. The
tradition of a split between two groups of Pythagoreans in the fifth
century, the *mathematici* and the *acusmatici*, points
to the same puzzlement. The evidence for this split is quite confused
in the later tradition, but Burkert (1972a, 192 ff.) has shown that
the original and most objective account of the split is found in a
passage of Aristotle's book on the Pythagoreans, which is preserved in
Iamblichus (*On Common Mathematical Science*, 76.19 ff). The
*acusmatici*, who are clearly connected by their name to the
*acusmata*, are recognized by the other group, the
*mathematici*, as genuine Pythagoreans, but the
*acusmatici* do not regard the philosophy of the
*mathematici* as deriving from Pythagoras but rather from
Hippasus. The *mathematici* appear to have argued that, while
the *acusmatici* were indeed Pythagoreans, it was the
*mathematici* who were the true Pythagoreans; Pythagoras gave
the *acusmata* to those who did not have the time to study the
mathematical sciences, so that they would at least have moral
guidance, while to those who had the time to fully devote themselves
to Pythagoreanism he gave training in the mathematical sciences, which
explained the reasons for this guidance. This tradition thus shows
that all agreed that the *acusmata* represented the teaching of
Pythagoras, but that some regarded the mathematical work associated
with the *mathematici* as not deriving from Pythagoras himself,
but rather from Hippasus (on the controversy about the evidence for
this split into two groups of Pythagoreans see the fifth paragraph of
section 4.3 above). For fourth-century Greeks as for modern scholars,
the question is whether the mathematical and scientific side of later
Pythagoreanism derived from Pythagoras or not. If there were no
intelligible way to understand how later Pythagoreanism could have
arisen out of the Pythagoreanism of the *acusmata*, the puzzle
of Pythagoras' relation to the later tradition would be insoluble. The
cosmos of the *acusmata*, however, clearly shows a belief in a
world structured according to mathematics, and some of the evidence
for this belief may have been drawn from genuine mathematical truths
such as those embodied in the "Pythagorean" theorem and
the relation of whole number ratios to musical concords. Even if
Pythagoras' cosmos was of primarily moral and symbolic significance,
these strands of mathematical truth, which were woven into it, would
provide the seeds from which later Pythagoreanism grew. Philolaus'
cosmos and his metaphysical system, in which all things arise from
limiters and unlimiteds and are known through numbers, are not stolen
from Pythagoras. They embody a conception of mathematics, which owes
much to the more rigorous mathematics of Hippocrates of Chios in the
middle of the fifth century; the contrast between limiter and
unlimited makes most sense after Parmenides' emphasis on the role of
limit in the first part of the fifth century. Philolaus' system is
nonetheless an intelligible development of the reverence for
mathematical truth found in Pythagoras' own cosmological scheme, which
is embodied in the *acusmata*.




The picture of Pythagoras presented above is inevitably based on
crucial decisions about sources and has been recently challenged in a
searching critique (Zhmud 2012a). Zhmud argues that the consensus view
of Pythagoras' cosmos as presented above is based on the faulty
assumption that there was a progression from myth and religion to
reason and science in Pythagoreanism. In many cases, he argues, the
evidence suggests that early Pythagoreanism was more scientific and
that religious and mythic elements only gained in importance in the
later tradition. The consensus picture of Pythagoras' cosmos assigns
number symbolism a central role and treats the *tetraktys*, the
first four numbers, which total to the perfect number ten, as a
central concept. Zhmud argues that the *tetraktys* and the
importance of the number ten do not go back to Pythagoras but flourish
in the Neopythagorean tradition, while having roots in number
speculation in the Academy associated with such figures as Plato's
successor Speusippus. One of the central pieces of evidence for this
view is that the *tetraktys* does not first appear until late
in the tradition, in Aetius in the first century CE (DK
1.3.8). However, the *tetraktys* does appear in one of
the *acusmata* in a section (82) of Iamblichus' *On the
Pythagorean Life* that is commonly regarded as deriving from
Aristotle. Zhmud himself agrees that sections 82-86 of *On the
Pythagorean Life* as a whole go back to Aristotle but suggests
that the *acusma* about the tetraktys was a post-Aristotelian
addition (2012a, 300-303). Once again source criticism is crucial. If
the *acusma* in question goes back to Aristotle then there is
good evidence for the *tetraktys* in early Pythagoreanism. If
we regard it as a later insertion into Aristotelian material, the
early Pythagorean credentials of the *tetraktys* are less
clear. 




Zhmud supports Pythagoras' position as genuine mathematician rather
than someone interested only in number symbolism by pointing to gaps
in the development of early Greek mathematics. Although there is no
explicit evidence, Pythagoras is the most likely candidate to fill
these gaps. Thus between Thales, whom Eudemus identifies as the first
geometer, and Hippocrates of Chios, who produced the
first *Elements*, someone turned geometry into a deductive
science (Zhmud 2012a, 256). Similarly, Hippasus' experiment with
bronze disks to show that the concordant intervals of the octave,
fifth and fourth were governed by whole number ratios is too complex
to be a first attempt so that once again someone must have discovered
the ratios in a simpler way earlier (Zhmud 2012a, 291). In each case
Zhmud suggests that Pythagoras is that someone. Finally, the study of
proportion ties together airthmetic, geometry and harmonics and Zhmud
argues that, although there is no explicit fourth-century evidence,
later reports which assign Pythagoras the discovery of the first three
proportions (Iamblichus, *Commentary on Nicomachus' Introduction to
Arithmetic* 100.19-101.11) are likely to go back to Eudemus
(2012a, 265-266). Such speculations have some plausibility but they
highlight even more the puzzle as to why, if Pythagoras played this
central role in early Greek mathematics, no early source explicitly
ascribes it to him. Of course, some scholars argue that the majority
have overlooked key passages that do assign mathematical achievements
to Pythagoras. In order to gain a rounded view of the Pythagorean
question it is thus appropriate to look at the most controversial of
these passages.




Some scholars who regard Pythagoras as a mathematician and rational
cosmologist, such as Guthrie, admit that the earliest evidence does
not support this view (Lloyd 2014, 25), but maintain that the
prominence of Pythagoras the mathematician in the late tradition must
be based on something early. Others maintain that there is evidence in
the sixth- and fifth-century BCE for Pythagoras as a mathematician and
cosmologist. They argue that Herodotus' reference to Pythagoras as a
wise man (*sophistes*) and Heraclitus' description of him
as pursuing inquiry (*historie*), show that he was
regarded as practicing rational cosmology (Kahn 2002, 16-17;
Zhmud 2012a, 33-43). The concept of a wise man in Herodotus'
time was very broad, however, and includes poets and sages as well as
Ionian cosmologists; the same is true of the concept of
inquiry. *Historie peri physeos* (inquiry concerning
nature) is later used to refer specifically to the inquiry into nature
practiced by the Presocratic cosmologists, but Herodotus' usage shows
that at Heraclitus' time *historie* referred to inquiry
in a quite general sense and has no specific reference to the
cosmological inquiry of the Presocratics (Huffman 2008b). In one
instance in Herodotus it refers to inquiry into the stories of
Menelaus' and Helen's adventures in Egypt (II. 118). Heraclitus may be
thinking of Pythagoras' inquiry into and collection of the mythical
and religious lore that is found in the *acusmata* (Huffman
2008b; see also Gemelli Marciano 2002, 96-103). Thus the
description of Pythagoras as a wise man who practiced inquiry is
simply too general to aid in deciding what sort of figure Herodotus
and Heraclitus saw him as being. It is certainly true that Empedocles
shows that the roles of rational cosmologist and wonder-working
religious teacher *could* be combined in one figure, but this
does not prove these roles were combined in Pythagoras' case. The only
thing that could prove this in Pythagoras' case is reliable early
evidence for a rational cosmology and that is precisely what is
lacking.




There is more controversy about the fourth-century evidence. Zhmud
argues that Isocrates regards Pythagoras as a philosopher and
mathematician (2012a, 50). However, it is hard to see how the passage
in question (*Busiris* 28-29) supports this view. Nowhere in it
does Isocrates ascribe mathematical work or a rational cosmology to
Pythagoras. He reports in general terms that Pythagoras brought
" other philosophy" to Greece from Egypt but what he
emphasizes is that Pythagoras was "more clearly interested than
others in sacrificial rites and temple rituals." It is true that
earlier, in a passage that does not mention Pythagoras
(*Busiris* 22-23), Isocrates had said that some of the Egyptian
priests studied mathematics but if Isocrates thought Pythagoras also
brought mathematical learning from Egypt he has chosen not to say so
explicitly. What Isocrates emphasizes about Pythagoras is what the
rest of the early tradition emphasizes, his interest in religious
rites. Fr. 191 from Aristotle's lost work on the Pythagoreans reports
that Pythagoras "dedicated himself to the study of mathematical
sciences, especially numbers" and Fragment 20 from
Aristotle's *Proptrepticus* says that Pythagoras said that
human beings were born to contemplate the heavens and described
himself as an observor of nature (Zhmud 2012a, 56 and
259-260). Unfortunately, in neither case are the words in question
likely to be Aristotle's. Fr. 191 comes from a book on marvels by
Apollonius (2nd BCE?). The words in question come before Apollonius
mentions Aristotle and, as Burkert pointed out (1972a, 412), are
overwhelming likely to be by Apollonius himself, since they serve as
the transition sentence between Apollonius' account of Pherecydes and
his account of Pythagoras. In the face of the huge extant corpus of
Aristotle's works in which he never ascribes any mathematical work to
Pythagoras, a single sentence that is not ascribed directly to
Aristotle and that, in terms of function, appears to be the work of
Apollonius and not Aristotle cannot with any confidence be used as
evidence that Aristotle regarded Pythagoras as a mathematician. The
same situation arises with Fr. 20 of the *Protrepticus.* If the
words in question were by Aristotle they would be his sole statement
that Pythagoras was a natural philosopher. The case of Fr. 20 is even
more tenuous than that of Fr. 191. Fr. 20 comes from
Iamblichus' *Protrepticus*, large parts of which are likely to
derive from Aristotle's lost *Protrepticus* but, as is his
practice, Iamblichus does not make any explicit reference to
Aristotle. The further problem with Fr. 20, as Burkert noted (1960,
166-168), is that the same story is told first about Pythagoras and
then immediately afterwards about Anaxagoras: both are asked why human
beings were born and both answer "to contemplate the
heavens" (Iamblichus, *Protrepticus* 51.8-15). This
awkward repetition of the same story about two different people
immediately suggests that only one story was in the original and the
other was added in the later tradition. This suggestion is strikingly
confirmed by the fact that Aristotle does tell this story about
Anaxagoras in his extant works (*Eudemian Ethics* 1216a11-16)
but not about Pythagoras. Thus, if the passage in
Iamblichus' *Protrepticus* is, in fact, from Aristotle, it is
very likely that only Anaxagoras appeared in Aristotle's version and
that Pythagoras was added in the later tradition, perhaps by
Iamblichus himself. Since these two passages are unlikely to be from
Aristotle, there are no references to Pythagoras as a mathematician or
as a natural philosopher either in Aristotle's extant works or in the
fragments of his works. Aristotle only knows Pythagoras as a wonder
working sage and teacher of a way of life (Fr. 191). Aristotle's
attitude is similar to his predecessors in the earlier fourth century:
Plato's sole reference to Pythagoras is as the founder of a way of
life and Isocrates emphasizes both the way of life and the interest in
religious ritual.




What about the pupils of Plato and Aristotle? As discussed in the
second paragraph of section 5 above, Eudemus, who wrote a series of
histories of mathematics never mentions Pythagoras by name. Arguments
from silence are perilous but, when the most well-informed source of
the fourth-century fails to mention Pythagoras in works explicitly
directed towards the history of mathematics, the silence means
something. There are only two passages in which Pythagoras is
explicitly associated with anything mathematical or scientific by
pupils of Plato and Aristotle. First, Aristotle's pupil Aristoxenus
reports that Pythagoras "most of all valued the pursuit
(*pragmateia*) of number and brought it forward, taking it away
from the use of traders, by likening all things to numbers"
(Fr. 23). Zhmud translates  *pragmateia* as
"science" (2012a, 216) so that he has Aristoxenus
attributing the invention of the science of number to Pythagoras but,
while Aristoxenus does use *pragmateia* to mean science in some
contexts, it more commonly simply means "pursuit" (Huffman
2014b, 292). Here surely it must mean "pursuit," because
Pythagoras is presented as taking it away from the traders and we can
hardly suppose that the traders were engaged in the theoretical
science of arithmetic. Moreover, Aristoxenus explains what he means in
the final participial phrase. He is not ascribing rigorous mathematics
with proofs to Pythagoras but rather says that Pythagoras was
"likening all things to numbers". This is consistent with
the moralized cosmos of Pythagoras sketched above in which numbers
have symbolic significance. The second important passage is Plato's
pupil Xenocrates' assertion that Pythagoras "discovered that the
intervals in music, too, do not arise in separation from number"
(Fr. 9). Xenocrates is being quoted here in a fragment of a work by a
Heraclides (Barker 1989, 235-236), perhaps Heraclides of Pontus. There
is controversy whether the quotation of Xenocrates is limited just to
what has been quoted in the previous sentence or whether the whole
fragment of Heraclides is a quotation of Xenocrates. Burkert (1972a,
381) and Barker (1989, 235) argue that it is probably just the first
sentence that Heraclides ascribes to Xenocrates, while Zhmud would
include at least a second sentence in which Heraclides presents
Pythagoras as pursuing a program of research into "the
conditions under which concordant and discordant intervals
arise" (Zhmud 2012a, 258). If the second sentence is accepted
then Xenocrates clearly presents Pythagoras as an acoustic
scientist. It seems most reasonable, however, to accept only the first
sentence as belonging to Xenocrates. If the quotation from Xenocrates
does not break off at that point, there is no other obvious breaking
point in the fragment and the whole two pages of text must be ascribed
to Xenocrates. The problem with ascribing it all to Xenocrates is that
Porphyry introduces the passage as a quotation from Heraclides, which
would be strange if everything quoted, in fact, belongs to
Xenocrates. If just the first sentence comes from Xenocrates, then all
he is ascribing to Pythagoras is the recognition that the concordant
intervals are connected to numbers. It is easy to assume, as Zhmud
does, that Xenocrates is saying that Pythagoras was the *first*
to discover that the concordant intervals are governed by whole number
ratios but Xenocrates' remarks need not mean this. Xenocrates'
comments might well come from a context like that in the fragment of
Aristoxenus, above, i.e., a context in which Pythagoras is presented
as likening all things to numbers and arguing that numbers in some
sense explain or control things. In such a context Xenocrates would
not be making the point that Pythagoras discovered the whole number
ratios but rather that he found out that concords arose in accordance
with whole number ratios, perhaps from musicians (who discovered them
first not being the issue), and used this fact as another illustration
of how things are like numbers. Thus, the fragments of Aristoxenus and
Xeoncrates show that Pythagoras likened things to numbers and took the
concordant musical intervals as a central example, but do not suggest
that he founded arithmetic as a rigorous mathematical discipline or
carried out a program of scientific research in harmonics.




Controversy concerning Pythagoras' role as a scientist and
mathematician will continue. Indeed, Hahn has recently endorsed many
of Zhmud's arguments and argues that Pythagoras was a rational
cosmologist, who was further developing a project begun by Thales to
contruct the cosmos out of right triangles. Hahn admits, however, that
his thesis is "speculative" and "a circumstantial
case at best" (2017:xi). It should now be clear that decisions
about sources are crucial in addressing the question of whether
Pythagoras was a mathematician and scientist. The view of
Pythagoras' cosmos sketched in the first five paragraphs of this
section, according to which he was neither a mathematician nor a
scientist, remains the consensus.