ABSTRACT

Pythagoras the Musician

Christine Marie Stanulonis

Director: Alden Smith, Ph.D.

It is contested whether or not the mathematical and scientific strain of the Pythagorean tradition could have belonged along with the mythological and religious strain to the original sect. Denying the mathematic tradition to original Pythagoreanism is often based upon assumptions that privilege one form of mathematics over another. But the Pythagorean conception of number need not be judged by the standard of deductive, axiomatic geometry, the paradigmatic mathematics of ancient Greece; instead, it can be considered as a practice which shares many of the characteristics of Greek arithmetic. This is because early Pythagorean figured numbers and later Greek arithmetic share a non-verbal and intuitive nature in accord with number understood through musical, poetic expression rather than through the strict, logical language upon which the geometry relies. This thesis will argue that the practice of measuring the numerical ratios of musical intervals may have been a kind of exemplar of scientific inquiry that acted as the catalyst for Pythagorean philosophical development. In addition, because these musical intervals were the living, pulsing heart of moral, religious, poetic, and communal life for members of the Pythagorean sect, their Pythagoreans understanding of their relationship to what they measured in mathematical terms would be radically different from the understanding of philosophers whose methods began with geometry. Thus, the privileged place of music as part of both aspects of the Pythagorean experiencethe scientific and the religiousmay have allowed for two modes of expressionthe philosophical and the mythologicalto operate within the same system of thought.

APPROVED BY DIRECTOR OF HONORS THESIS:

______________________________________________ Dr. Alden Smith, Department of Classics APPROVED BY THE HONORS PROGRAM: ______________________________________________________ Dr. Andrew Wisely, Director DATE: ___________________________

PYTHAGORAS THE MUSICIAN

A Thesis Submitted to the Faculty of

Baylor University

In Partial Fulfillment of the Requirements for the

Honors Program

By

Christine Marie Stanulonis

Waco, Texas

May 2015

ii

TABLE OF CONTENTS

Acknowledgments . . . . . . . iii

Introduction . . . . . . . . 1

Chapter One: The Discord between Ritual and Rational . . 3

Chapter Two: Pythagorean Mathematics and the Greek Mathematical Traditions 26 Chapter Three: Music the Harmony between Ritual and Rational 47

Conclusion . . . . . . . . 67

Bibliography . . . . . . . . 69

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ACKNOWLEDGEMENTS

In completing this thesis, I owe a debt of gratitude to several individuals who

encouraged, advised, or in some other way enriched the project and my understanding of

it. I thank Dr. Smith, my advisor, for believing in my ability to write a worthy thesis,

even when I had little hope of it. I also thank the three other readers on my panel: Dr.

Froberg for his assiduous attention to stylistic details, Dr. Williams for her timely advice

and much-needed encouragement, and Mr. T.J. McLemore for his appreciation of

mystery. In addition, I thank Dr. Moser whose classes and conversation originally drew

me to the topic. Also, I am deeply indebted to Megan for her constant, gracious

friendship and to Jackson, Kara, Kelsey, and Zach for their camaraderie. Finally, I thank

Christopher for many months of patience.

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INTRODUCTION

The Pythagorean tradition is known in many ways. Pythagoras, called over the

centuries the founder of ancient Greek mathematics, was accredited at various times with

the discovery of the theorem for calculating the sides of a right triangle, the discovery of

the numerical ratios that govern musical intervals, and even the discovery of the

irrationality of the square root of 2. Also called the founder of philosophy, he first

attempted to explain the world as a cosmos ordered by numbers. His influence is sought

in many key places in the Western intellectual tradition, from the wisdom of Plato to the

medieval Quadrivium to the heliocentric Copernican solar system. Religious doctrines of

the Pythagorean tradition are also remembered: the transmigration of souls, the kinship of

all creatures, and the cyclical nature of the world. Mixed in with all this is mystical

disposition toward numbers and the various attempts at association between numbers and

physical objects or abstract realities.

Yet, while much can be known about the entire traditionas it has been

attributed, remembered, developed, and transmittedsurprisingly little can be known

about its founder or its earliest adherents. This paper will examine the question of

whether or not the mystical and mathematical strains could have both been fundamental

to the Pythagorean tradition from the beginning. It will note some of the presuppositions

concerning mathematics generally, assumptions that can lead scholars to discount early

Pythagorean number theory, and suggest the possibility that an emphasis upon music and

a musicians approach to mathematics might enable a more sympathetic reading of the

early Pythagorean tradition.

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CHAPTER ONE

The Discord between Ritual and Rational

The favor and attention that has ensured the continuation of Pythagorean thought

across temporal, geographic, cultural, and linguistic divides also enabled alterations; and

the early sources necessary for understanding its origins are almost non-existent. Yet the

dearth of definitive sources is only the beginning of the difficulties involved in an

interpretation of Pythagoras and early Pythagoreanism. The inquiry is complicated

further because the evidence available points to a story difficult to understand and even

more difficult to credit with historical verity. The legendary character of Pythagoras

represents a combination of contradictory modes of human engagement with the world,

an odd assortment of traits associated with both ritualistic devotion and rational

observation. Pythagoras, shrouded in mystery, in ancient times inspired strong reactions

from devotees and detractors alike, and continues to be a subject of debate.

As a brief introduction to the study of early Pythagoreanism, this chapter first

offers a survey of major relevant sources. It will continue with a review of the legend of

Pythagoras as it was conceived in late antiquity and as it was passed down into the

middle ages. Finally, it will conclude with an examination of some views in

contemporary scholarship relevant to subsequent chapters.

The Problem of Sources

Hellenistic tradition held that Pythagoras transmitted all of the doctrines and

precepts of his sect orally and left no writings; and in modern scholarship this notion is

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taken as fact. Contrarily, Diogenes Laertius in late antiquity attested in his Life of

Pythagoras that the famous man left behind at least three volumes of his own writings,

but his evidence is hardly convincing.1 Just as there are no texts attributable to

Pythagoras, there is none from his immediate followers. According to testimonia,

Pythagoreans held themselves to a vow of silence regarding their leaders teachings.

Presumably, Isocrates refers to this vow in his complaint: to this day, those who pretend

to be his [Pythagoras] disciples are more admired for their silence than those who have

the greatest reputation for speaking.2 The Pythagorean silence is especially notorious in

the different versions of the legends concerning a certain Pythagorean named Hippasus,

drowned for declaring openly the discovery of the dodecahedron.3

Possible echoes of the oral Pythagorean tradition are the Pythagorean symbola

and the Golden Verses of Pythagoras. The symbola are maxims for living, often bizarre

in their claims and in their demands. Some are as universal in scope as the most just

thing is to sacrifice, the wisest is number.4 Others seem to deal with minutiae, such as

not leaving the marks of a pot in a pile of ashes.5 They are also called acusmata (things

heard) because disciples of Pythagoras would listen in silence to his enigmatic sayings

that could not be properly interpreted by a listener who knew not the secrets of

Pythagorean discourse. The earliest evidence for these is dated ca. 400 B.C., found in an

1 Diogenes Laertius, The Life of Pythagoras, V. translated in Guthrie, The Pythagorean Sourcebook and Library,(Grand Rapids, MI: 1987), 142-43; Diogenes Laertius bases his claim upon a quote of Heraclitus (DK12 B129) which mentions Pythagoras and writings, but does not make clear Pythagoras relationship with those writings or claim his authorship.

2 Isocrates, Busiris, 28. Qtd. in Khan, Pythagoras and the Pythagoreans, (Indianapolis: 2001), 12.

3 Iamblichus, De Vita Pythagorica, 18.88.

4 Iamblichus De Vita Pythagorica, 18.82.

5 No. 35 from Androcydes collection, qtd. in Burkert, Lore and Science, (Cambridge: 1972), 173.

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Explanation of Pythagorean Symbola by Anaximander of Miletus.6 Thus, the symbola

are pre-Platonic and in the time of Anaximander were already supposed to require

allegorical interpretation, as would utterances wrapped in the mysticism and mythology

of archaic times.7 Following Anaximander, the practice of collecting and interpreting

Pythagorean symbola garnered a long tradition.8 Another possible product of the early,

oral tradition is the so-called Golden Verses of Pythagoras. Armand DeLatte maintained

in the early 20th century that these were compiled in the mid-3rd or early 4th century B.C.

either from written fragments or from oral tradition,9 and Thom argues that the Verses

should be dated earlier than 300 B.C.10 Like the acusmata, they are oriented toward

instruction for ordered living.

Because of the mysterious character of the Pythagorean sect, the earliest written

evidence for Pythagoreanism comes from non-Pythagoreans. These testimonia come

from various authors, namely, Heraclitus, Empedocles, Ion of Chios, and Herodotus, each

of whom in some way refers to the reputation of Pythagoras as a learned individual.11

However, because the fragments of Heraclitus are decidedly hostile in their description of

6Burkert, Lore and Science, (Cambridge: 1972), 166. 7 As Burkert notes, Ibid.

8 For instance, in Iamblichus De Vita Pythagorica 18.86-87, in Porphyry De Vita Pythagorica 42 (Guthrie, 131), or more recently, in K. S. Guthries Pythagorean Sourcebook, (Grand Rapids: 1987), 159-161.

9 DeLatte, Etudes sur la litterature pythagoricienne, Paris 1915, p. 44-79.

10Thom, The Pythagorean Golden Verses, (New York: 1995), 57-58. 11 Heraclitus DK12 B40 and DK12 B129, Empedocles DK21 B129, Ion of Chios DK36 B4,

Herodotus 4.95.

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Pythagoras, and others are ambivalent, a veritable philological storm has arisen in

modern scholarship over their interpretation.

After the testimonia comes the first genuinely Pythagorean author, Philolaus of

Croton in the 4th century B.C. On Nature was likely the first book written by a

Pythagorean.12 Fragments of it survive, on subjects of such central importance as the role

of numbers in human knowledge, in the arrangement of the heavenly bodies of the

cosmos, and in the musical intervals. The fragments of Philolaus have undergone

scrutiny over the years, and their authenticity has been variously affirmed and denied.13

According to Burkert, since the argument against authenticity issued by Frank in the early

20th century, the dominant mood has been uncertainty, though scholarly caution has

somewhat tipped the balance toward the negative.14 However, Carl Huffmans

comprehensive study, Philolaus of Croton, defends the authenticity of much that would

otherwise be discarded, and suggests a revised understanding of the Philolaus fragments

liberated from some of the preconceptions of Aristotles interpretation.15

The next Pythagorean author is Archytas of Tarentum. Although his connection

with the sect is more tenuous than that of Philolaus, and he lived half a century later,

according to Huffman, he fits the popular conception of a Pythagorean better than

anyone in the Pythagorean tradition.16 He was a known mathematician highly involved

and successful in the government of Tarentum and, in some respect, a friend or

12 Huffman, Philolaus Stanford Encyclopedia of Philosophy, 2012, online.

13 See Burkert (1972), 221, no.17 for list of scholars for and against.

14 Burkert, (1972), 221.

15Huffman, Philolaus of Croton, (Cambridge: 1993) 16 Huffman, Archytas of Tarentum, (Cambridge: 2005), 44.

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acquaintance of Plato. The writings of Archytas are more mathematically intricate than

those of Philolaus, and they provide the most advanced explanation of musical harmony

in the pre-Platonic Pythagorean tradition, with a detailed introduction to the three means

(arithmetic, geometric, and harmonic) which came to occupy a foundational place in

Pythagorean harmonic theory. Recently, scholars have come to see Philolaus, and

especially Archytas, as philosophical figures in their own right, and not as mere sources

for referral in the attempt to reconstruct the thought of Pythagoras. This valuable insight,

in light of the scattering of the Pythagorean sect in the 5th century, makes unrealistic the

assumption that the tradition continued as a perfectly homogenous whole. Instead, it

assumes a different character in the various personalities who give it expression.

This notion is nowhere more obvious than in the most troublesome source for

Pythagoreanism, Plato himself. It has long been known that Plato develops themes

quintessentially Pythagorean, at least according to the traditional perception of

Pythagoreanism. Specifically, the cosmological, mathematical, and harmonic tenants are

woven throughout the Timaeus, and the religious doctrines on the transmigration of souls

are re-fashioned in the Phaedo. But to say that there is nothing in Plato so helpful as a

bibliography of previous scholarship is an understatement: in fact, he mentions the name

Pythagorean only twice in the entire corpus, even when he clearly deals with

Pythagorean material.17 Burkert suggests some reasons for this tendency. Not only does

Plato value arguments for their truth rather than for their source, but he also integrates

foreign material fully into his own structure.18 Indeed, one cannot read Platos writings

17 Republic, 600 b2 and 530 d8, according to Burkert (1972), 84-85.

18 Burkert, (1972), 83.

8

without understanding that the unity of his thought and the shifting innovation and

creativity of his expression must confuse the traces of borrowed ideas. Walter Burkert in

Lore and Science in Ancient Pythagoreanism, through an impressively thorough array of

evidence attempts to separate an historical, religious Pythagoras from the mathematical

and scientific learning and achievements so long associated with him. Instead, he sees in

Plato (and in mathematicians of Platos time) the origins of the more philosophically

rigorous half of a tradition which until that time had been merely ritualistic and mystical.

The final major source for the pre-Platonic Pythagorean tradition consists of

certain passages of Aristotle. Aristotle and his school preserve the only significant post-

Platonic view of early Pythagoreanism, a view free from Platonizing elements. This

assessment may not be completely accurate, for there are reasons to suspect that some of

the Aristotelian characterizations of Pythagoreanism are informed by a reaction to

Platonic philosophy. Even so, it is certain that Aristotle, unlike Plato, writes more in the

fashion of one only reporting on a system of thought in order to develop his own system

in contrast and reference to it and has no interest in re-inventing it for himself. Some of

the most relevant passages, particularly on Pythagorean cosmology, are found in the

Metaphysics, the De Caelo, and the Physis.19 Zhmud rejects Aristotle as a reliable source

for early Pythagoreanism because he maintains that there is interference of an anti-

Platonic agenda with Aristotles record of Pythagorean beliefs.20

The search for the historical Pythagoras and the character of his original sect

treads upon a shaky foundation. The teachings drift down through the ages in the form of

19 Metaphysics 342b29, 345a13, 987a29 , 989b29, 990a27, 996a4. De Caelo. 284b6, 285a10.b24,

290b12-291a9, 293a19.b1. Physics. 203a3, 204a32, 213b22. 20 Zhmud, Pythagoras and the Early Pythagoreans, (Oxford: 2012), 13.

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enigmatic maxims and lines assumed to preserve some fidelity to an original oral

tradition and also through the fragments of Philolaus and Archytas. But lack of evidence

for what came before Philolaus, leaves room to question what relationship existed

between his ideas and those of his predecessors, even if the authenticity of the fragments

is vindicated. Archytas, whatever his association with Pythagoreanism, stands farther

from the source, for there is more reason to think that Archytas is innovating and

developing the specifically mathematical portion of the tradition, if indeed it was part of

the early tradition. The commentary of authors outside of the Pythagorean tradition, from

Heraclitus to Aristotle, must be interpreted through the expressed attitudes and

assumptions of the authors; Platos rich presentation of Pythagorean ideas conceals as

much as it reveals. In light of these uncertainties, scholars have provided arguments both

for and against the traditional Pythagoras, the semi-mystical mathematic genius, but it

would be hardly fitting to proceed without first becoming better acquainted with

Pythagoras legend which now ignites such controversy.

The Traditional Pythagoras

Late sources say most about the life of Pythagoras. Three of the best known

Lives of Pythagoras were written by Diogenes Laertius (3rd century A.D.), Porphyry

(c.235-c.305 A.D.), and his student Iamblichus (c. 245-c.325 A.D.). Those from

Diogenes Laertius and Porphyry were part of larger works on Lives of the Eminent

Philosophers and the History of Philosophy, respectively. But The Pythagorean Life

from Iamblichus was the first volume of an encyclopedic, ten-volume work dedicated

entirely to Pythagorean thought. As the longest and most extravagant in its attributions, it

offers an example of the high regard and loyalty that students of the Pythagorean tradition

10

devoted to the Pythagorean ideals in late antiquity and into the middle ages. Pythagoras

was the only Pre-Socratic to retain such fame beyond antiquity, due in large part to the

full synthesis achieved between Neoplatonism and Neo-Pythagoreanism by Neoplatonic

authors, Iamblichus in particular, who considered himself to be both a Platonist and a

Pythagorean.

The Pythagoras of the Neoplatonic Lives is a colorful, almost whimsical

combination of scientist, orator, mystic, musician, friend, vegetarian (perhaps), healer,

and moral teacher. As these sources reveal, he was born (by legend the son of Apollo,

god of reason and music) in Samos in the 6th century B.C. As a young man he studied

with the Ionians Thales and Anaximander, and absorbed the wisdom of the earliest Greek

philosophy. He then traveled to Phoenicia, acquiring Hebraic thought, and went from

Phoenicia to Egypt, where he visited temples, and sought out priests and sages to be

instructed by them in the secrets of their religion.21 He later came to in Babylon

(according to Iamblichus he was taken captive and brought there by soldiers of

Cambyses)22 where he eagerly learned from the Magi. There also, in the account of

Porphyry, he was ritually purified by Zoroaster himself.23 From these lands he gained a

marvelous knowledge of arithmetic, geometry, music, and astronomy, as well as

initiation into many mysteries. If that were not enough, upon return to the Greek-

speaking world, he dwelt in Crete and Sparta for a time to become familiar with their

laws and visited all the oracles.

21 Iamblichus, De Vita Pythagorica, 3.13-4.19.

22 Iamblichus, De Vita Pythagorica, 4.19.

23 Porphyry De Vita Pythagorica, 12, translated in Guthrie (1987), 125.

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The sources relate that later, because he found the inhabitants of Samos

uninclined to apply themselves to education and because he deemed that the growing

power of the tyrant Polycrates would hamper a free man in his studies, he traveled to

southern Italy, and settled in Croton where he quickly gained popularity through the

wisdom and power of his speeches. Iamblichus relates that he began by addressing a

crowd of young men whom he found in the gymnasium, exhorting them to esteem their

parents and pursue other virtues especially important for young men. When the elders of

the city discovered him, they invited the newcomer to speak to them in their senate.24

Later, they asked him to speak also to their boys and then to their women. Iamblichus

even records the subject matter of these speeches as though they were remembered still in

late antiquity, although it seems more likely that he is using the story of the man

Pythagoras depicted generally as a mouthpiece for Neo-Pythagorean and Neoplatonic

morals. The coming of Pythagoras so affected the community of Croton that many

people, both men and women, began to join the Pythagorean sectso many in fact, that

the government of Croton came decidedly under their influence. Porphyry relates that

whole cities were constructed to hold the number of outsiders who came to hear him.25

The descriptions of the Pythagorean sect and their customs are equally

fascinating. All Pythagoreans lived under a vow of silence regarding their masters

teachings. Not surprisingly, the teachings of Pythagoras have always been something of

a mystery, and even in his lifetime there was a particular ordering within the Pythagorean

community as to which people were allowed to share in his more esoteric revelations and

24 Iamblichus De Vita Pythagorica, 9.45.25 Porphyry, De Vita Pythagorica, 20, in Guthrie (1987), 127.

12

which were not. Entrance among the Pythagoreans was apparently a long process.

According to Iamblichus, Pythagoras would first observe newcomers and judge their

character based on such things as their gait, how and when they laughed, or what caused

them joy and sorrow.26 If accepted by him, aspirants would be neglected for three years

while still being under the masters observation, so that they could prove the stability of

their character. Those who passed this test were required to relinquish their possessions

to be shared in common among the Pythagoreans (as did all members of the sect, it being

a maxim of Pythagoras that friends had all things in common). They were allowed to

listen silently to the teachings of Pythagoras but were not allowed to see him because he

sat behind a veil. After five years they would become esoterics, allowed to speak with

him behind the veil and share in his most hidden doctrines. If anyone failed the test of

the five-year silence, he was returned double the wealth he brought them, but he was

treated as though dead.

Porphyry places the blame for the destruction of the Pythagoreans on Cylon of

Croton.27 This man, by Porphyrys account, thought highly of himself and was greatly

insulted when Pythagoras would not consider him for admittance into the Pythagorean

community. As an act of vengeance, he stirred up a popular rebellion against the

Pythagorean leaders of Croton and burned them in a house where they had gathered

together. All of the Pythagoreans were either killed or scattered, and Pythagoras himself,

according to the tradition, traveled about Southern Italy seeking refuge. However, the

political upheaval had already spread beyond Croton, and riots arose around him

26 Iamblichus, De Vita Pythagorica, 17.71.27 Porphyry, De Vita Pythagorica, 54-55, in Guthrie (1987), 134.

13

wherever he went. He died either from starvation or from grief in the temple of the

Muses in Metapontum.28

The vignettes of strange and miraculous deeds of Pythagoras included in these

sources are characteristic of the status attributed to him as a man above ordinary mortals.

Diogenes Laertius relates a legend in which Pythagoras, in a former life, had been a son

of Hermes, and when his father gave him the promise to grant any wish short of the wish

for immortality, the not-yet Pythagoras asked for a perfect and lasting memory.

Consequently, he could remember all of his former lives, one of them a life as the hero

Euphorbus, spoken of in Homers Iliad. Another famous story in several sources was

that of the Daunian bear. In this tale, Pythagoras possessed, like Orpheus, the ability to

communicate with animals. When a certain savage bear had killed a few local

inhabitants, Pythagoras secured its promise that it would become a vegetarian, and it left

the vicinity and never harmed man or beast again. In another legend, Pythagoras met a

group of fishermen just pulling in their catch and told them the exact number of fish in

their nets. Having agreed to do whatever he commanded if he were correct, they counted

the fish and proved his assertion accurate. Then according to his wish, they threw all the

fish, still living, back into the water. But the most spectacular of all the Pythagorean

legends, perhaps, is the claim that he was seen talking to his disciples in Metapontum in

Italy and in Tauromenium in Sicily, at exactly the same time.29

This oddly attractive picture of Pythagoras (if in some points bizarre) no doubt

was meant to be appreciated by ancient admirers such as Iamblichus. In modern

28 Porphyry, De Vita Pythagorica, 57, in Guthrie (1987), 134.

29 Porphyry, De Vita Pythagorica, 27, in Guthrie (1987), 128.

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scholarship a more modest reconstruction of the life story of Pythagoras agrees that he

was from Samos and was born a contemporary of the beginnings of Philosophy in Ionia.

However, little can be known about his life prior to his arrival in Croton in the latter half

of the sixth century. Guthrie suggests as credible the report that Pythagoras left Samos as

a consequence of the character of its tyrant Polycrates because, whether or not political

reasons for his flight existed, Pythagoras as an ascetic would have begrudged the tyrants

encouragement of luxury and license.30 Ancient and modern historians verify that

Pythagoras and the Pythagoreans had a significant effect on the political life of Southern

Italy, although it is not necessary that Pythagoras rose to popularity with such marvelous

immediacy as the Neoplatonic sources describe. It is known that Croton was badly

defeated by Locri at the river Sagra around the time that Pythagoras would have arrived,

but that later in 510 B.C. Croton defeated and destroyed the wealthy city of Sybaris,

thereby becoming the dominant force in the area for the next half century. Some

historians have credited this dramatic shift in the fortunes of Croton to the moral reforms

and discipline which Pythagoras inspired. Dunbabin confidently asserts the opinion that

the Pythagorean hetaireiai, that is, brotherhoods or political clubs, guided the affairs of

Croton and most of the other southern Italian cities for the first half of the fifth century

and became responsible for Crotons political expansion.31 Even after the revolt named

in ancient histories, the Pythagorean influence remained for about a century and a half in

Southern Italy, so that even in the first half of the fourth century B.C. the mathematician

30 Guthrie, W.K.C., History of Greek Philosophy, vol. 1, (Cambridge: 1962), 174.31 Dunbabin, The Western Greeks, (Oxford: 1948), 359, 360; Burkert is skeptical, (1972), 116, n.

44.

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Archytas, who could claim some kind of association with Pythagorean ideas, was several

times elected to high positions in the government of Tarentum.

Pythagoras image, passed down through the ages, is double-sided. On the one

hand, he was seen as a brilliant philosopher who endeavored to present the world as a

cosmos founded upon rational, mathematical principles, a great master in the studies of

arithmetic, geometry, music, and astronomy. But he was also remembered as a religious

leader who introduced the doctrine of metempsychosis to the Greek world, a kind of

shaman connected to the old Orphic cults (and possibly to religious beliefs derived from

the East, depending on the interpretation.)

The Dichotomy between Rational and Ritual in Modern Scholarship

This legend of Pythagoras was accepted, for the most part, throughout late

antiquity, the middle ages, and the early modern period. But in the nineteenth century,

scholars expressed doubts over whether the authentic teachings and true character of

Pythagoras could ever be known. In 1865, Eduard Zeller famously noted that the

majority of the records of Pythagoras life and teachings are of questionable authenticity

and that the information they provide becomes both more abundant and more dubious the

later the source.32 Scholars of the twentieth and twenty-first centuries have continued to

provide in Zellers wake excellent philological examinations of the earliest source

material for Pythagoras and the Pythagoreans. While there is general agreement that

Pythagoras, the religious leader, existed in fact, a controversy continues to brew over the

traditional identification of Pythagoras as mathematician-philosopher.

32 Zeller. Pythagoras und die Pythagoraslegende. Vortrge und Abhandlungen I. (Leipzig:

1865), 30-50; as an example, even the account of Iamblichus is more far-fetched in comparison to that of Diogenes Laertius, though the time between them is not great.

16

A major line in Pythagorean scholarship is drawn between those who argue for a

synthesis between mysticism and mathematics, and those who argue against it. August

Dring, in 1892, argued for a synthesis that was based upon the concept of katharsis.33 Dring suggests that Pythagoras sought the souls purification in ritual but found it most accessible of all in the contemplation of scientific knowledge. In opposition to this,

scholars such as Erich Frank in the early 20th century emphasized the highly religious and

mystical character of early Pythagoreanism as something at odds with the scientific and

mathematical developments; these, according to Frank, must have been added into the

tradition later by the Southern Italian mathematicians of Platos acquaintance, especially

by Archytas.34 Somewhat later, Cornford posited that the mathematical tradition, though

not original, developed from the mystical tradition earlier than this, in response to

Parmenides criticisms of Pythagoreanim.35 He identifies this mathematical tradition as a

kind of number-atomism, similar to, but not identical to, the atomism of Democritus

and Leucippus, and maintains that it was the target of extant criticisms from Zeno.36

As might be expected, there are two major questions to be contested. First,

scholars seek to determine what form Pythagorean religion or mathematics would have

taken in the 6th and early 5th century. Second, they inquire whether religion and

mathematics in such forms could have possibly cooperated to form a synthetic system of

thought. The issue tends to be contested on two different planes, sometimes

33 Dring, Wandlungen in der pythagorischen Lehre. Archiv fr Geschichte der Philosophie 5

(1892), 503-531.

34 Frank, Plato und die sogenannten Pythagoreer, (Halle: 1923). 35 Cornford Mysticism and Science in the Pythagorean Tradition. In Mourelatos, The Pre-

Socratics, (Princeton: 1993), 135-136. 36 Ibid.

17

simultaneously. On the one hand, scholars seek out and interpret written sources as well

as cultural and political context. On the other, scholars face their own preconceptions

(and those of the source-materials) concerning whether or not religion and mathematics

in those presumed forms could form a synthetic system of thought.

One of the most comprehensive, influential works of scholarship on

Pythagoreanism is Walter Burkerts Lore and Science in Ancient Pythagoreanism,

published in English in 1972. Because of the breadth of his evidence and the intricacy of

his arguments, anyone who wishes to comment on the history of early Pythagoreanism is

almost obligated to address his claims. Burkerts work demonstrates that the scientific

and mathematical side of Pythagorean tradition is inextricable from the Platonic sources

(both Plato and his acquaintances and followers) that ensured its fame throughout later

ages. It lacks, in his evaluation, the early evidence to stand upon its own, and because of

this, it ought not to be assumed to have preceded Plato and the thinkers of his time.

The major obstacle standing in the way of Burkerts assertion that Pythagoras did

not practice scientific inquiry or consider numbers from a mathematical point of view is

the set of early testimonia from non-Pythagoreans. These fragments, some of the earliest

written evidence for Pythagoras, all comment on his great learning in some fashion,

sometimes in praise, sometimes in derision or ambiguity.The fragments from Heraclitus

are rich with suggestion but also complex and unfavorable:

.

DK12 B40 , , .

DK12 B129

18

Clearly, the words , , and must lie at the center of debate. At

its simplest reading, fragment 40 seems to place Pythagoras in company with both

Xenophanes and Hecataeus, thus giving him a place amongst the Ionian thinkers, a

position beneficial for a philosophical figure. Burkert, however, suggests that the

position of the word in the sentence shows that Heraclitus is grouping Pythagoras with Hesiod rather than with the other two, emphasizing that as a polymath he did not

specialize in the newer science but, like Hesiod, he specialized in the older myths, rituals,

and theogony.37 Burkert acknowledges the connection of with the Milesian

disposition to rational inquiry, but finds it to be overpowered by the words and, especially because the sound like Orphic writings.

All scholars by no means agree with Burkerts reading of Heraclitus fragments;

some do not accept his effort to minimalize the scientific identity of Pythagoras in these

early sources. Zhmud passes over Burkerts complicated reading, doubting the authority

of Heraclitus because he is known to have had derisive things to say about almost every

other person he named.38 Khan does not concede the weakening of the force of the word

, which was routinely used in reference to the Milesian science, including

geometry, astronomy, geography, and history.39 In addition, Zhmud counters that

refers to prose writings, and thus could not be Orphic poetry, but more likely

37 Burkert, (1972), 209-210.

38 Zhmud, Pythagoras and the Early Pythagoreans, (Oxford: 2012), 32.

39 Khan, Pythagoras and the Pythagoreans, (Indianapolis: 2001), 17.

19

writings from such men as Anaximander and Anaximenes, whose ideas Pythagorean

thought resembles.40 Evidently the interpretation of the testimonia is not soon to be

settled.

While Burkert denies that sufficient evidence supports the mathematical and

scientific tradition in the early days of the sect and affirms the evidence for the early

religious tradition, Zhmud disagrees entirely. In his view, to apply an argumentum ex

silentio to the early mathematical tradition is inconsistent. He cites the presence of

meager, early evidence for the political influence of Pythagoras and the Pythagoreans, but

the political tradition is accepted far more commonly.41 Again, he compares Pythagoras

to Thales, who also left no writings. He notes that the tradition surrounding Thales

attributes to him sayings and actions not historically possible, and the first specific

evidence for his philosophical and geometrical discoveries appears as late as Aristotle.42

Yet, there is generally little doubt that Thales was not in some fashion a man who

pursued mathematical and natural inquiry.

The above examples introduce some of the complexities involved in identifying

and interpreting the early sources for Pythagoreanism and their context. But beneath the

philological quandaries, understanding of mathematics often drives the debate. This

tendency is even explicit in Gomperz, who locates in Pythagoras the full range of

traditional identitiesmathematician, acoustician, astronomer, founder of a religious

sect, scientist, theologian, and moral reformer.43 According to Gomperz, the reason that a

40 Zhmud, Pythagoras and the Early Pythagoreans, (Princeton: 2012), 33.

41Zhmud, Pythagoras and the Early Pythagoreans, (Princeton: 2012), 27.42 Zhmud, Pythagoras and the early Pythagoreans, (Princeton: 2012), 26. 43 Gomperz, Greek Thinkers, vol. I. (London: 1955) , 99.

20

pioneer of exact science can at the same time be a prophet of mysticism lies in the

character of the mathematical temperament.44 He relies upon a certain evaluation of

the mathematical which has the power to justify all the strange contradictions of the

legendary Pythagoras. However, there are scholars who rely upon the nature of

mathematics and the mathematical temperament in order to refute the same synthesis.

Burkert is perhaps the most prominent example. A major theme of his work Lore and

Science is that genuine mathematical methods and the natural science which arose from

them are not compatible with what is known of early Pythagoreanism. In the middle

ground, Riedweg maintains that any interpretation too narrowly focused on mathematics

and the natural sciences does not do justice to the Pre-Socratics in general, let alone

Pythagoras, and appears as a result of nineteenth century positivism.45

The Necessity of Abstraction for Mathematical Thought

The issues which Burkert brings to light are highly useful for investigating the

nature of mathematics and its relationship to early Pythagorean teaching. In Burkerts

view, Pythagoras sits on the wrong side of a divide in the development of Greek thought.

In a culture and an age just beginning to engage in scientific inquiry, Pythagoras stands

out as a figure whose primary allegiance is to an older, mythological explanation of the

world.

From the very beginning, his influence was mainly felt in an atmosphere of miracle, secrecy, and revelation. In that twilight period between old and new, when Greeks, in a historically unique achievement, were discovering the rational interpretation of the world and quantitative natural science, Pythagoras represents

44 Gomperz, Greek Thinkers, vol. I. (London: 1955), 108. 45 Riedweg. Pythagoras: His Life, Teaching, and Influence. (Ithaca: 2005), 73.

21

not the origin of the new, but the survival or revival of ancient, pre-scientific lore, based on super-human authority and expressed in ritual obligation.46

Burkerts criticism of the theory of synthesis between the ritual and the rational

which Dring and others propose relies on his attempt to separate the elements of older and newer ways of thinking. Specifically, his argument is in response to the one

presented by Burnet in his Early Greek Philosophy. Burnet, like Dring, argues for a synthesis between the mathematical and the religious Pythagoras which depends upon the

idea of catharsis. In Burnets understanding, Pythagoras could have had a doctrine of

catharsis which would have bridged the gap between Orphic practices of catharsis and the

Platonic conception of it, thus blending old mysticism with new rational, mathematical

thought. Orphic cults, which in Pythagoras day had gained popularity in the Greek

world, taught that man had a godlike soul which longed to be free of its body, and could

begin to experience this ecstasy through the practice of purification (), although

ultimate freedom would not come until the soul escaped from the cycle of

reincarnations.47 Aristoxenus wrote that Pythagoras used to use music to purge the

soul.48 Burnet argues that we can add that to effect purification, Pythagoras promoted not

only music, but also more significantly the study of science, and of mathematics in

particular.49 He turns to the Pythagorean technique of arranging numbers of pebbles into

shapes, as an example of an early Pythagorean interest in arithmetic. This practice

appears to be an early innovation, because the veneration of the tetractys was central to

46 Burkert, (1972), Preface to the German Edition.

47 Add citation: Kirk and Raven? 48Aristoxenus, Fr. 26.

49 Burnet, Early Greek Philosophy, (London: 1958), 98.

22

Pythagorean thinking and accredited to Pythagoras himself.50 If Pythagoras taught that

scientific investigation was to be pursued for the sake of the purification of the soul, then

this doctrine would link the religious teachings with the philosophical teachings, making

it possible for them to be united in one man.

Burkert rejects this proposal by positing that the theory of catharsis cannot remain

intact if it is removed from the conceptual background which belonged to Plato but not to

the early Pythagoreans. Burkert from Platos writings identifies passages that suggest a

theory of catharsis. In the Phaedo, catharsis is achieved by philosophy, whose goal is

deaththe separation of soul and body that enables the full realization of truth.51 And

again in the Republic, the studies of arithmetic, geometry, music, and astronomy lead the

soul from the world of appearances to the true forms.52 Burkerts main point of

contention is that the forms or ideas are absent from the Pythagorean mindset. He cites a

passage from Aristotle as evidence that the Pythagoreans thought of numbers in a way

inseparable from the physical world,53 unlike Plato for whom the forms were incorporeal

objects of knowledge. Because of this, there could not have been a Pythagorean

understanding in which the goal of life, death, and metempsychosis was escape from the

physical world. For Burkert, this constitutes sufficient cause to believe that the catharsis

of the early Pythagoreans was not based on music and mathematical study, but on music

50 A triangle made of rows of one, two, three, and four pebbles, adding to ten, the perfect number

for Pythagoreans. The tetractys was said to have contained the nature of number.

51 Plato, Phaedrus 64a-67d.

52 Plato, Republic 521c-522d.

53 Met. 989b29; Aristotle notes that the Pythagoreans did not take a physical substance as their fundamental element, yet all of their investigations are concerned solely with the physical world.

23

and religious ritual, closely related to the Orphic practices.54 He speculates that Plato

could have borrowed the lifestyle of catharsis from the Pythagoreans along with its

connection to music, but reinvented it with a scientific focus and a mathematical

understanding of music.55 Burkerts analysis reveals the one key characteristic which he

holds to be necessary to true mathematical thought: abstraction. Without a conception of

numbers abstracted from things, he argues, the Pythagoreans could not have risen to a

level of scientific inquiry as we know it. Khan also identifies this abstraction of numbers

as a fundamental question in the study of early Pythagoreanism.56

Thus, the Pythagorean question is concerned with more than merely a lack of

adequate, early source material; it is complicated by various interpretations of the

relationship between a more ancient mythological understanding of the world and the

newer, more scientific revolution of thought which the pre-Socratic philosophers began.

Much depends upon whether the interpreter considers these two modes of thought to be

incompatible, or if he endeavors to find some fundamental commonality by which the

latter evolved from the former. As has been seen, Burkert (and other scholars who take a

similar position) views the mythological and the scientific as basically opposed to one

another. On the other side, Dring and Burnet work to find a common strand (the practice of catharsis) to unite the two in the person of Pythagoras. Burkerts criticism of

this focuses on the Pythagorean perception of number: he argues that if the Pythagoreans

did not have a conception of number as something incorporeal, they could not have

54Burkert, (1972), 212-213. 55 Ibid. 56 Khan, Pythagorean Philosophy before Plato, In Mourelatos, The Pre-Socratics, 163; Aristotle

records the Pythagorean perception of number in different ways, saying both that things are number and that things resemble number. There seems to be a possible difference between a sensible and a non-sensible concept of number.

24

believed in purification through contemplation of scientific knowledge. This idea

illustrates that one of the most fundamental questions in Pythagorean scholarship (if not

the most fundamental) is the question of in what way the early Pythagoreans perceived

and encountered number.

25

CHAPTER TWO

Pythagorean Mathematics and the Greek Mathematical Traditions

Pythagorean understanding, without question, was generated or in some way

fashioned by an attention to number. This idea is apparent in the symbol which states

that the wisest thing is number,1 as well as in a fragment of Philolaus which reads, and

all things, indeed, that are known have number: for it is not possible for anything to be

thought or known without this.2 Aristotle, as well, attests to it in his Metaphysics:

the people called Pythagoreans took up mathematics, and were the first to advance this science; and having been reared on it, they thought that its principles were the principles of all things. Since of these it is the numbers that are by nature first, and in the numbers they thought that they saw many likenesses to the things that are and that come into being, more so than in fire and earth and water3

Metaphysics 985b23 (DK58 B4)

These are only a few examples of many. Whatever Pythagoras taught, it must have found

its meaning or its language of expression in number.

Thus, it is not the importance of number in early Pythagorean teaching which

scholars question but the way that number was encountered, and whether that way really

matches what a later system of thought would define as mathematical,or only

mystical, mythological, or poetic. And even if the assumption is accepted which is

present in Aristotles summary, that Pythagorean ideas begin with a study of

mathematics, it still remains to be determined what sort of mathematics were studied, for

not all mathematics in ancient Greece conformed to a single method. Unavoidably it is

1 Iamblichus De Vita Pythagorica, 18.82.

2 DK 44 B4. Translation in Barker, Greek Musical Writings, vol. II, (Cambridge: 1990), 36.

3 Translation in Barker Greek Musical Writings, vol. II, (Cambridge: 1990), 32.

26

necessary to determine something of the character of the Pythagorean understanding of

numbers before attempting to distinguish between the perceived incompatibility of the

scientific and religious strains of thought attributed to them, for the manner or method by

which they thought about numbers would surely influence their thinking in other areas.

Figured Numbers the Earliest Pythagorean Conception of Number

The earliest testimony that sheds light on how Pythagoreans regarded numbers on

the most basic level occurs in references to the method of one Pythagorean named

Eurytus. This testimony is given both by Theophrastus and by Aristotle on the authority

of Archytas, who reportedly had first-hand knowledge of the somewhat older man

Eurytus. Like Archytas, Eurytus was from Tarentum, a Pythagorean most likely closer

than Archytas to the original strain of the sect. He was also regarded as a pupil of

Philolaus.4 Theophrastus writes that Eurytus had the method of arranging particular

quantities of pebbles in order to show the numbers which were associated with certain

things, in this case, a man and a horse.

For this is the [approach] of an accomplished and sensible man, that very thing which Archytas once said that Eurytus did, in his various diatheseis of psephoi; for he said that this number turned out to be of man, that of horse, that of some other thing.5

Theophrastus, Metaphysics 6a19-22

Similarly, Aristotle refers to the same thing in his writings:

as Eurytus assigned a certain number to a certain thing, e.g., this [number] to man, that [number] to horse (just as is done, making numbers into the figures triangle and square), making the forms of living beings analogous, in this way, to psephoi.6

4 Burnet, Early Greek Philosophy, (London: 1958), 99-100. 5 Translation in Netz, The Problem of Pythagorean Mathematics, in Huffman, A History of

Pythagoreansm,, 173.

6 Translation in Netz, The Problem of Pythagorean Mathematics, in Huffman (2014), 174.

27

Metaphysics 1092b9-13

What exactly Eurytus was doing in laying out his pebbles remains unclear.

Theophrastus refers to him with some respect; the evaluation of Aristotle is more

questionable. Netz gives an interpretation of Aristotles treatment of Eurytus method in

which Aristotle, though discounting it as a scientific procedure, is not wholly in

disagreement with some of its fundamental assumptions.7 It is at least known that

patterned numbers are not an uncommon way of manipulating numbers in ancient

cultures.8 Simpler arrangementssuch as triangular numbers, square numbers, and other

classificationswere likely customary, perhaps as a form of calculation in business

transactions or other practical matters (one might think of the art of calculating using an

abacus, keeping in mind that the original meaning of calculus was pebble). Aristotle

at any rate alludes to the practice as a known phenomenon (just as is done, making

numbers into the figures triangle and square). Heidel expresses certainty that square and

oblong numbers were almost as old in the time of Pythagoras as they are in the present

day.9 The preeminent example of a figured number is the tetraktys, the triangular

arrangement formed when one, two, three, and four pebbles are set out in parallel rows.

This figure was believed to express the nature of number and was highly revered in

Pythagorean circles. The triangular figure may be extended by the addition of subsequent

rows of quantities. Thus, triangular numbers are the sums of consecutive whole numbers,

that is {3, 6, 10, 15, 21,}.

7 Netz, The Problem of Pythagorean Mathematics, in Huffman, (2014), 173-77.

8 Netz, The Problem of Pythagorean Mathematics, in Huffman (2014), 176. 9 Heidel, W.A. The Pythagoreans and Greek Mathematics. Studies in Pre-Socratic Philosophy.

Vol. I, (New York: 1970), 352.

28

Other shapes were formed as well. Square numbers are the sums of consecutive odd

numbers, for instance {4, 9, 16, 25, 36}. Three pebbles added to one pebble forms the

shape of a square, while five added to the first four conveniently does the same, and so on

for the rest of the odd numbers.

In a similar way, oblong or rectangular numbers are the sums of consecutive even

numbers, such as {2, 6, 12, 20, 30,}. Four pebbles added to two makes six; six added

to six make twelve; and so for the rest.

Triangular, square and oblong numbers are only the most basic of the patterned

or figured numbers, as they are called. In the Introduction to Arithmetic, of

Nicomachus, there are also found pentagonal numbers, hexagonal numbers, and more.

29

One can only wonder what a man-number, or a horse-number might have been under

the hands of Eurytus.

So far as can be told, the arrangement of these simple figured numbers would

have been Pythagorean mathematics in its earliest form, and this is not hard to believe

if figured numbers really were a common occurrence outside the Pythagorean community

at that time.10 It is further substantiated by tradition surrounding the tetraktys, a figure

wrapped in the ritualistic and religious attitude which is widely considered to be the most

certain element of original Pythagoreanism. And in later generations, Pythagoreans

continued to use this as the primary mode of working with numbers, undeterred by newer

mathematical innovations. At least it is certain that later avowedly Pythagorean or Neo-

Pythagorean authors, such as Nicomachus of Gerasa in the 2nd century A.D., continued to

base their arithmetic in some key part on these figures, in contrast to the mainstream

axiomatic-deductive tradition of Greek mathematics exemplified in much of Euclid and

Archimedes. It can safely be asserted that the Pythagoreans from the beginning

conceived of numbers as clusters of pebbles rather than in alphabetical notation or in the

later Euclidean line representations of Greek geometry. Netz calls attention to this:

[Counters, for the Greeks] were the medium of numerical manipulation par excellence, in exactly the same way in which, for us, Arabic numerals are the numerical medium par excellence. We imagine numbers as an entity seen on the page; the Greeks imagined them as an entity grasped between the thumb and the finger.11

This being the case, it will be profitable to speculate on what effect the figured numbers

would have had on the early Pythagorean perception of number.

10 As Netz hints that they were. Netz, The Problem of Pythagorean Mathematics, in Huffman

(2014), 176. 11 Netz, Counter Culture: Towards a History of Greek Numeracy, History of Science, 40 (2002),

341.

30

The arrangement of figured numbers is a practice in natural concordance with the

way the human mind recognizes different numbers through counting. We know that the

Pythagoreans did not consider one to be a number. One, because it is by definition

indivisible, cannot be measured by any other number. Instead, it is the principlethe

beginning and sourceof all numbers and the standard by which all numbers are

measured or counted out, just as any shape arranged out of pebbles is composed of the

individual pebbles. The concept of unity is what makes counting possible. However, one

cannot rely on unity alone. When Filon describes the development of calculation in

cultures generally, he notes that the number of objects which can be immediately

perceived by the human eye is usually low, four or five; all numbers above this are

ascertained through counting. But larger numbers can hardly be counted by the unit

alone. It is difficult for the human mind to measure greater numbers, such as one

hundred, with the unit. In practice, therefore, counting necessarily takes the form of

counting in groups, and then counting the number of groups; then forming groups of

groups, and so on.12 Here lies the reason for the establishment of counting systems in

base five, base ten, or whatever the case may be. And why should it not also lead to the

figured numbers of the Pythagoreans?

Represented here are the numbers 2, 6, and 12:

12 Filon, The Beginnings of Arithmetic, The Mathematical Gazette, vol. 12, no. 177, (1925),

403.

31

Two is easily recognizable. But six is probably identified as two lines of three; thus it is

literally measured by the numbers 2 and 3. Twelve likewise is seen as three lines of

four; it is measured by 3 and 4. It is imaginable that with larger shapes or composite

shapes the same observation holds: the number is recognized through the parts which

make it up. And though it is possible that the more complex figured numbers were not

discovered until later, because these are only juxtapositions of triangles and squares, it is

not difficult to imagine that Pythagoreans might have arranged them also as objects of

contemplation, this being an activity that most children could engage in. It is also not

difficult to imagine that they would still consider the simpler figures to be the most

important, and be known for their allegiance to them, because they are the principles of

all the others. In this way, for instance, a heptahedron might be formed.

From all this it ought to be clear that the heart of Pythagorean number practices is

the recognition of a number through measurement and classification. This measurement

is not the geometrical measurement of segments and planes, but an ability to identify one

number by its relationship with other numbers, by means of recognizing the numbers

which add and multiply together to compose it. The classification of numbers is not

based only on their shapes, but also on the smaller numbers by which the shape of the

larger number is measured, because the two are inseparable. For example, the numbers

classified as square must also be classified as composed by addition of consecutive odd

32

numbers (1, 1+3, 1+3+5, etc.), and by multiplication of consecutive whole numbers (11,

22, 33, etc.). Now, there is no need to suppose that the Pythagoreans themselves

would have articulated it in this way, or would have set their observations down into

formulas and definitions. Likely they did not, and the recognition of numbers through

measurement and classification was mostly intuitive. Still, measurement and

classification are the thought processes, as it were, which would naturally derive from the

representations of figured numbers.

Once it is accepted that figured numbers are the key to Pythagorean involvement

with numbers, it becomes necessary to determine how far they were applied and with

what results. The controversy begins with this question: how mathematical was the

method and understanding of the early Pythagoreans? This question begs another: how

do scholars decide beforehand what qualifies and does not qualify as mathematical?

Determining an answer requires that there be some kind of exemplar of mathematical

practice.

This reliance on an exemplary mathematics outside of the early Pythagorean

tradition is especially apparent in Burkerts evaluation. Burkerts aim is to disprove the

tradition that Pythagoras was the founder of mathematics in Greece. He begins by

attacking an argument by Becker that some of the Euclidean propositions (9.29-34) on

properties of the even and the odd are Pythagorean in origin because, among other

reasons, they can be derived from a reconstruction using pebble numbers.13 These

propositions are seemingly out of place and isolated in Euclid; they are used only in the

proposition on perfect numbers (9.36) and in the proof of irrationality, two subjects

strongly associated with Pythagorean tradition. Burkert does not object to the notion that

13 Referenced in Burkert, (1972), 434.

33

Pythagoreans had some kind of awareness of the relationships between the even and the

odd, because they could see such things in their figured numbers (for instance, they could

see that one pebble added to an even number produces an odd number). However, he

does object to the notion that this observation proves the existence of an early

Pythagorean deductive mathematics, perhaps the ancestor of the fully developed

propositions and proofs in Euclids Elements.14

Burkert clearly distinguishes between Pythagorean number theory and the highly

geometrical mathematical tradition represented in Euclids Elements because he seems to

be interested in maintaining a certain conception of the nature of mathematics, and

mathematics as it existed in the larger part of Greek culture specifically. He correctly

recognizes that the paradigmatic form of Greek mathematics, as a deductive system

based on axioms, is geometry.15 In contrast to this, he notes that the Pythagorean

practice of figured numbers is pictorial and inductive. Every figured number may be

probatory into some mathematical concept, but each is taken as evident on its own.

There is no need for a systematic structure, with every proposition or proof based on

other propositions, the defining characteristic of Greek geometry.16 On these grounds

Burkert does not completely reject the Pythagorean figured numbers as a mathematical

exercise, but he comes close: he concedes that even a game may be regarded

legitimately as a kind of mathematics and the axiomatic-deductive form is not the only

14 Burkert, (1972), 434-35.

15 Burkert, (1972), 427.

16 Burkert, (1972), 435.

34

one possible.17 But he does maintain that, compared to Greek geometry, Pythagorean

arithmetic is an intrusive, quasi-primitive element.18

In the final consideration, Burkerts version of the early Pythagoreans is that of a

sect based on the mystical and ritualistic doctrines of some kind of shaman, with almost

no influence on the scientific developments of the Greek world around them. Perhaps

this is the natural conclusion if the later advances in Greek geometry are taken as the

standard for measuring the mathematical worth of the early Pythagorean contribution.

But this may not be the only way, or the best way, to evaluate it. To begin with, more

might be said on the nature of Greek mathematics, for the mainstream tradition of Greek

geometry was not the only one.

Greek Arithmetic and Geometry

Netz, in an attempt to determine the place of the early Pythagorean interest in

number within the larger story of the development of Greek mathematics, notices a

division in the mathematical tradition. He divides the major mathematicians into two

networks, based on a difference in their emphases and the methods of their inquiry.19

The first network he identifies flourished in the first half of the fourth century. The

names of its contributors are found in Proclus summary of Greek mathematics before

Euclid, written as the first part of his introduction to the Euclidean Elements20. Netz calls

attention to the names Theodorus, Theaetetus his student, Archytas, Eudoxus,

17Burkert, (1972), 423. 18 Ibid.19 Netz, The Problem of Pythagorean Mathematics, in Huffman, (2014), 167-171. 20 Proclus, On Euclid, 64.16-70.18; the introduction which he writes is believed to be a summary

of another text written by Eudoxus.

35

Menaechmus his student, Dinostratus, Menaechmus brother, and Philippus, and

Leodamus. All these men are attested to have had connections with Plato. Netz

significantly points out that they pursued a whole set of mathematical topics relegated to

the background in subsequent generations, namely, means, ratios, proportions, irrationals,

and harmonies. In general, the one most characteristic pursuit of these mathematicians

was the classification of numbers, ratios, and proportions without reliance on complex

geometrical diagrams. These topics find expression in Books VII-IX of Euclid, and in

parts of Books V and X.

The remaining books of the Elements, roughly the first half, contain a form of

geometry which coincides more neatly with the second network of mathematicians whom

Netz categorizes, those associated with the correspondence and writings of Archimedes

and the works of the generation following him. This group is characteristically provided

inspired solutions to the problems of finding the measurements of geometrical shapes and

established tight, deductive reasoning based on axioms. In short, the primarily geometric

accomplishments for which Greek mathematics are especially remembered derive from

the second network. The division between the two is not perfect, as Netz admits. He

refers to the methods attributed to Eudoxus (a mathematician of the first network) for

the measurement of the volumes of pyramids, cylinders, and cones in Book XII of the

Elements. This work clearly shows mathematics associated with the later methods and

discoveries. But instances such as these are exceptional. Generally, Netzs distinction

holds true. Thus, two schools of thought in mathematics become apparent, one which

puts arithmetic first, and one which puts geometry first. Euclids Elements, by its very

36

arrangement of the geometrical books in the first half of the work, seems to fall into the

second category.

This evidence clearly shows that the story of Greek mathematics need not be so

one-sided as Burkert would have it. A reconsideration of the value of early Pythagorean

understanding of number, against the backdrop of a more varied conception of

mathematics, is appropriate. A brief review of some elementary concepts of the

arithmetic tradition will facilitate this. Unfortunately, because the writings of Archytas

are so fragmentary, relevant information proves more productive if sought from a later

author, in the Introduction to Arithmetic of Nicomachus of Gerasa. Although

Nicomachus is an author later than would be wished, he seems to be a compiler more

than an innovator in his work. Also, because his most complicated subjects in the

Introduction are the three means (arithmetic, geometric, and harmonic) described in the

second fragment of Archytas, it is reasonable to assume that the simpler and more

elementary discussions would not have been beyond Archytas and the arithmeticians of

his time either.

Nicomachus work first teaches the relationship between odd and even numbers.

This relationship between odd and even numbers is one of the most important aspects of

Pythagorean number-theoretical tradition. Nicomachus introduces this idea in the

following way: the most fundamental species in [number] are twoodd and even, and

they are reciprocally woven into harmony with each other, inseparably and uniformly.21

This means that the entirety of the class of all numbers is composed only of even

numbers and odd numbers, but when he speaks of them being woven together he is

referring to much more than the fact that even and odd numbers alternate down the

21 Nicomachus, Introduction to Arithmetic, VI, translated by Dooge, (London: 1926), 190.

37

number line. He then spends many pages exploring this harmony which arises from the

two opposite elements of number, but in order to begin to glimpse it, we must subdivide

both the even and the odd numbers into classes, as he does.

The even numbers are divided into what he calls the even-times-even, the odd-

times-even, and the even-times-odd numbers. A number is called even-times-even if it

can be divided in two, its two parts can be divided in two, and so on consecutively until

the quotient is one and can be divided no further. In other words, the even-times-even

numbers are those which are multiples of two and of two only, {2, 4, 8, 16, 32, 64,}.

When broken down into their constituent factors, there are no odd numbers whatsoever.

In fact, when fully broken down there are no factors other than the number 2. Thus, the

even-times-even numbers are the most purely even. The even-times-odd numbers, in

contrast, are those even numbers which can be divided into two, but whose two halves

cannot be divided in two. For example, the numbers {6, 10, 14, 18, 22,} contain

factors which are odd (3, 5, 7, 9, and 11 respectively); thus, the numbers, though even,

are not purely even in the same sense as are the even-times-even numbers. All of the

even-times-odd numbers can be produced by multiplying each of the odd numbers

(excepting 1) by 2. Finally, the odd-times-even numbers are those even numbers which

can be divided in half, and whose halves can be divided in half; but though the division

by two may continue beyond this, it cannot reach 1 because eventually the division will

produce an odd number. For example, the number 24 can be divided in half, yielding 12;

in turn so can 12, yielding 6; but 6, divided in half, yields 3. At this point the number

cannot be divided any further by 2. Thus, the odd-times-even numbers are closer to the

even-times-even, but are still not purely even in their composition.

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The odd numbers are also split into sub-categories. The first contains the prime

numbers, those odd numbers that cannot be divided by any other number besides 1.

These are well-known: {1,3,5, 7, 11, 13,}. Composite numbers are those odd numbers

which can be divided by other factors besides one and so are not prime, for instance 9

(divisible by 3 and 1), 15 (by 5, 3, and 1), and 21 (by 7, 3, and 1). In addition,

Nicomachus takes note that some composite numbers are prime in comparison to other

numbers even though they are not prime absolutely. For example, 9 and 25 are both

composite, the one being divisible by 3 and 1 and the other being divisible by 5 and 1;

but they share no divisor in common besides 1 and because of this are considered prime

in relation to each other.

The classification of the even and the odd numbers is based upon origin and

measurement. One is the origin of all other numbers and measures all other numbers.

Two is the origin of the even. It is very helpful to use Nicomachus own metaphor at this

point and to imagine the whole infinite progression of numbersthe number lineas

being composed of different strands or strings woven together. Different numbers belong

to different strands based upon what they are measured by. One is the source of all these

strands because it measures all numbers. But every number in the number line is in a

sense a source of a new strand because it measures its own strand of multiples (thus 3

measuresor divides6, 9, 12, 15, etc.). Of these, some numbers are more or less

original. Thus 15, is not so original a source as 3; and 3 is not so original as 1. Fifteen

measures its own multiples {30, 45, 60, 75,} but is itself measured by 5, and 3; so all

of its multiples are also measured by 5 and 3. In turn, 15, 5, and 3 and all of their

multiples are measured by 1. The only numbers, after 1, which act as primary origins of

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their multiples are 2 and the prime numbers.22 This is obvious because these numbers

can each be measured only by 1 and themselves, and so their multiples cannot be

measured by any more original source.

From 2 and the prime numbers the harmony between even and odd is most clearly

seen. Two for the Pythagoreans was classed among the prime numbers but was given its

own, special status as the source of the even. By even is meant not merely every even

number, but the characteristic of the Even, i.e. the ability to be divided (and measured) by

two. No prime number has this characteristic symmetry which divides into equal halves.

Some numbers, as has been related already, possess this even-ness purely, and are

completely derived from 2 and only 2. These are those numbers which increase

exponentially from two: {4, 8, 16, 32,}. Other even numbers are a mix or an

intertwining between the Even and the Odd, such as the odd-times-even and even-times-

odd which were already classified. These numbers are measured both by 2 and by certain

prime numbers (for instance, 30 as a multiple of 2, 3, and 5).

This explains why Nicomachus calls the relationship between the even and the

odd the harmony between the Unlimited and the Limiters. Unlimited carries the

connotation of unshaped, unformed, and undefined rather than unending; a thing which is

not given limits is not demarcated by measurements by which it may be known or

recognized. In this sense, the Even (think of the exponential multiples of 2: {4, 8, 16,

32,}) is Unlimited because it can be measured only by 2. Every prime number, by

contrast, adds a new standard of measurement into the number line, a standard that did

not exist before it, and with that a whole string of multiples can be reduced to that prime

22 2 was not considered a prime number among the Pythagoreans, but the source of all the even

numbers.

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and to no other. This makes them Limiters. And when the Unlimited and the Limiters

are multiplied together the result is the odd-times-even and even-times-odd numbers (the

even numbers that contain odd numbers in their composition), and the harmony of the

even and the odd is complete.

The arithmetical vision articulated in part above is so different from the rigid

method of the geometric tradition that the very way in which they are communicated

through language is vastly dissimilar. A typical Euclidean proof is so neatly constructed

that there is almost only one way to articulate it in language. This occurs because it is an

exercise in verbal logic as much as it is an experiment in physical measurement. That the

axiomatic element is so necessary, as Burkert notices, must mean that the words

themselves are inseparable from the method of the mathematics. With Nicomachean

arithmetic, however, it is quite to the contrary. The very way it is written suggests that

there is a different relationship between the numbers and the words which attempt to

explain them. Nicomachus often describes in two or three different ways the properties

which identify a class of numbers, for these will stay the same no matter how they are

decorated with words or sentences. The arithmetic could be demonstrated without words

because it deals only with the relationships between numbers, relationships which are in

themselves self-evident; but the geometry would be difficult to convey without words

that brought to mind the generalizations of equality and inequality, or without the

syntax that communicates the logical conditional, to name a few. Because the arithmetic

is self-evident without the help of words, the language that Nicomachus uses is not

essential to the subject it is communicating and acquires a different role. It becomes

41

descriptive instead of axiomatic, and most fully summarizes the reality of the numbers

when it uses symbolic descriptors, like woven and harmony.

There are other differences. The Euclidean geometry is abstract while the

Nicomachean arithmetic is more concrete. The propositions and proofs of the geometry

rely upon generalizations such as equal, unequal, greater than, less than

abstracted from the relationships between numbers, while the classifications of the

arithmetic rely upon examples of specific numbers and their relationships recognized by

the mind. In addition, the Euclidean geometry is systematic. This means that each step

in the progression of a single proof and from one proof to the next and one section to

the nextdepends upon an understanding of formerly demonstrated propositions. Each

part of the argument is not immediately intuitive on its own. It is otherwise in the

arithmetic. Its approach to understanding those numbers is less systematic and more

exploratory. The order in which Nicomachus expounds the different classes of numbers,

ratios, and proportions is fitting in the way that it moves from the simpler to the more

complex, but it is not necessary for understanding. There is more freedom in his field to

begin at any point. Finally, the geometry is comprehensive. Everything which is

relevant to the problem is noted and contained on the page: the diagram, the laws which

order relationships of equality and inequality, and the former proofs which may be

referred to at need. It relies heavily on written statements, because of the need to keep in

mind the relation of each step to the one before it. The arithmetic is less comprehensive

in approach. It is does not require the particulars of every discovery to be held in mind,

or on paper. Instead it requires an intuition of the whole but a clear comprehension only

of the number under immediate consideration.

42

Arithmetic through a Musical Method and Metaphor

Now, the Greek arithmetic tradition just contrasted with the geometric tradition

has been called Pythagorean; however, not all scholars agree on the connection between

this intricate and elegant system found in late Pythagorean and Neo-Pythagorean sources

and what would have been the number theory of primitive Pythagoreanism and its

founder. Because of the lack of evidence to indicate what exactly was the nature of

Pythagoras interest in number, there is no way to prove conclusively a line of descent in

the arithmetical ideas from the founder to Archytas, Plato, and the network of

mathematicians in their time who gave the preeminence to arithmetic above geometry.

But this does not mean that the tradition should be wholly or easily discounted. For

Burnet, it is sufficient that the figured numbers persisted in the works of late authors. He

sees these as anachronistic, having been made obsolete by the development of

alphabetical notation and the linear representations of geometry, and proposes that it

would not have been likely to survive into later generations had it not been propelled by

the remarkable authority of some famous namethat of Pythagoras.23

Netz also assigns Pythagorean roots to the arithmetical tradition, albeit by way of

a different argument. He reasons that because Archytas was likely the oldest of the

mathematicians in the circle around Plato, as well as the most influential in the socio-

political realm, it is not unlikely that he could have been the paradigmatic individual for

this earlier form of mathematics.24 If this be true, he suggests, the roots of the tradition

strike near the heart of Pythagoreanism. For even if there is no evidence that Archytas

23 Burnet, Early Greek Philosophy, (London: 1958), 99-102.

24 Netz, in Huffman (2014), 170-171.

43

was a true Pythagorean in terms of the earliest tradition, this is largely speculative

because it is often impossible to determine what makes an authentic Pythagorean.

Individuals remembered as Pythagoreans come in many types, and in the time of

Archytas the ties that bound the original society together would have been breaking apart

due to persecution, migration, and the passage of time. In the view of Netz, the

connection of Archytas to Pythagoreanism, whatever it was, was substantial enough to

bind the arithmetical tradition of Platos time to Pythagoras, the founder of the sect.25

In addition to these historical considerations, it seems reasonable at least to take

note of the inherent similarity between the early Pythagorean fascination with number

and the later tradition of arithmetic. Burkert admits that even a game may be regarded

legitimately as a kind of mathematics,26 and he seems to hint at the same time that the

figured numbers of the Pythagoreans fall into this category of mathematical activity. In

this evaluation, he is most likely correct. But more than anything, this concept

strengthens the connection between the early number theory and the later arithmetical

developments. The exploratory and incomprehensive nature of the arithmetic is precisely

the kind of attitude toward mathematics that would be expected to grow out of a non-

systematic, non-rigorous love for number, an approach that accepted number, sometimes

in odd ways, as central to moral, religious, artistic, and communal life. The arithmetical

tradition may have arisen as something so different from the more common geometrical

tradition because of roots in such Pythagorean play. Specifically, the non-verbal and

intuitive nature of the arithmetic would be in accord with an appreciation of number

understood through musical activity, as would the almost poetic descriptions of it. It may

25 Netz, in Huffman, (2014), 170-71. 26 Burkert, (1972), 423.

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not be coincidental that the one thing which Burkert allows to be an unequivocally

Pythagorean element is the arithmetization of music theory and, to a degree, the elevation

of number theory (arithmetic) to an independent branch alongside geometry.27 It is

only reasonable that the arithmetization of music theory is what did elevate arithmetic to

the level of geometry, if the Pythagorean interest in music is a fundamental an element to

the teachings of the sect as it is remembered.

27 Burkert, (1972), 422.

45

CHAPTER THREE

Music, the Harmony between Ritual and Rational

Because the presence of the musical element is one of the least contested in early

Pythagoreanism, it seems remarkable that its influence upon the rest of Pythagorean

thought, from the religious to the philosophical, is not more widely investigated. The

fascination with the ratios of the musical harmonia has generally been accepted as part of

early Pythagorean life. Possibly at least the discovery of these ratios had a far more

dramatic effect upon Pythagorean thought than is usually surmised. For it is not at all

unlikely that the ratios of the musical intervals were the first thing that the Pythagoreans

would have learned to measure, both because of the centrality of music in their way of

life and because the exercise itself is simplesimpler than the arithmetical observations

of later authors and more immediate than the cosmological speculation. If the practice of

measuring the ratios of musical intervals was a kind of exemplar or paradigm of

Pythagorean inquiry, then naturally it would shape their pursuit of understanding

generally and in other areas.

The effect that the Pythagorean preoccupation with music may have had on the

tradition as a whole is implicit in Aristotles words summarizing the Pythagorean

philosophy, is latent in the character of the Pythagorean engagement with music, and is

finally manifest in the language of Pythagorean cosmology. First, this chapter will posit

that Aristotles summary is structured as though by the assumption that the measurement

of musical ratios was the catalyst for the Pythagorean efforts in natural philosophy.

Second, in order to see the significance of this claim, it will examine the historical

46

Pythagorean involvement with music as a double-sided experience, both scientific and

religious. Finally, it will argue that because the discovery of the ratios of musical

intervals was the catalyst for their philosophical development and because their

experience of music was already twofold, the Pythagoreans learned to express their

cosmological system in two different modes, one philosophical and one mythological.

Evidence from Aristotle for the Primacy of Music

That the mention of the musical ratios in the passage from Aristotles Metaphysics

(cited in the last chapter) appears prominently in his summary and introduces the

technical account of Pythagorean philosophy may communicate the vital role that

musical ratios played in Pythagorean thought. It is customary to assume that Aristotles

words indicate that the one seminal item in the Pythagorean tradition was mathematics.

But closer observation of the passage hints that Pythagorean mathematics was not

sufficient to generate the full Pythagorean philosophy, and that the knowledge of number

in music specifically, rather than through mathematical practices or methods, takes the

central role.

Aristotle relates,

and through studying [mathematics] they came to believe that its prin