ABSTRACT. Pythagoras the Musician. Christine Marie Stanulonis. Director: Alden Smith, Ph.D.

Transcription

1 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 experience the scientific and the religious may have allowed for two modes of expression the philosophical and the mythological to operate within the same system of thought.

2 APPROVED BY DIRECTOR OF HONORS THESIS: Dr. Alden Smith, Department of Classics APPROVED BY THE HONORS PROGRAM: Dr. Andrew Wisely, Director DATE:

3 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

4 TABLE OF CONTENTS Acknowledgments iii Introduction 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 Bibliography ii

5 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. iii

6 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 tradition as it has been attributed, remembered, developed, and transmitted surprisingly 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 musician s approach to mathematics might enable a more sympathetic reading of the early Pythagorean tradition. 1

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8 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 3

9 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 leader s 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), ; 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), Iamblichus, De Vita Pythagorica, Iamblichus De Vita Pythagorica, No. 35 from Androcydes collection, qtd. in Burkert, Lore and Science, (Cambridge: 1972),

10 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 20 th century that these were compiled in the mid-3 rd or early 4 th 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 6 Burkert, Lore and Science, (Cambridge: 1972), As Burkert notes, Ibid. 8 For instance, in Iamblichus De Vita Pythagorica , in Porphyry De Vita Pythagorica 42 (Guthrie, 131), or more recently, in K. S. Guthrie s Pythagorean Sourcebook, (Grand Rapids: 1987), DeLatte, E tudes sur la litte rature pythagoricienne, Paris 1915, p Thom, The Pythagorean Golden Verses, (New York: 1995), Herodotus Heraclitus DK12 B40 and DK12 B129, Empedocles DK21 B129, Ion of Chios DK36 B4, 5

11 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 4 th 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 20 th century, the dominant mood has been uncertainty, though scholarly caution has somewhat tipped the balance toward the negative. 14 However, Carl Huffman s 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 Aristotle s 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), Huffman, Philolaus of Croton, (Cambridge: 1993) 16 Huffman, Archytas of Tarentum, (Cambridge: 2005), 44. 6

12 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 5 th 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 Plato s writings 17 Republic, 600 b2 and 530 d8, according to Burkert (1972), Burkert, (1972), 83. 7

13 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 Plato s 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 Aristotle s 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, 213b Zhmud, Pythagoras and the Early Pythagoreans, (Oxford: 2012), 13. 8

14 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; Plato s 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 (3 rd 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 9

15 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 6 th 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 Greekspeaking 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, Iamblichus, De Vita Pythagorica, Porphyry De Vita Pythagorica, 12, translated in Guthrie (1987),

16 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 sect so 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 master s 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, Porphyry, De Vita Pythagorica, 20, in Guthrie (1987),

17 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 master s 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 Porphyry s 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, Porphyry, De Vita Pythagorica, 54-55, in Guthrie (1987),

18 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 Homer s 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), Porphyry, De Vita Pythagorica, 27, in Guthrie (1987),

19 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 tyrant s 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 Croton s 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 Guthrie, W.K.C., History of Greek Philosophy, vol. 1, (Cambridge: 1962), Dunbabin, The Western Greeks, (Oxford: 1948), 359, 360; Burkert is skeptical, (1972), 116, n. 14

20 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 Zeller s 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. Vortrӓge 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. 15

21 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 Dӧring, in 1892, argued for a synthesis that was based upon the concept of katharsis. 33 Dӧring suggests that Pythagoras sought the soul s 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 20 th 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 Plato s 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 6 th and early 5 th 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 Dӧring, Wandlungen in der pythagorischen Lehre. Archiv für Geschichte der Philosophie 5 (1892), Frank, Plato und die sogenannten Pythagoreer, (Halle: 1923). 35 Cornford Mysticism and Science in the Pythagorean Tradition. In Mourelatos, The Pre- Socratics, (Princeton: 1993), Ibid. 16

22 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 Burkert s Lore and Science in Ancient Pythagoreanism, published in English in 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. Burkert s 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 Burkert s 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: Πολυμαθίη νόον <ἔχειν> οὐ διδάσκει Ἡσίοδον γὰρ ἂν ἐδίδαξε καὶ Πυθαγόρην αὖτις τε Ξενοφάνεα καὶ Ἑκαταῖον. DK12 B40 Пυθαγόρης Μνεσάρχου ἱστορίην ἤσχησεν ἀνθρώπων μάλιστα πάντων ἐκλεξάμενος ταύτας τὰς συγγραφὰς ἐποιήσατο ἑαυτοῦ σοφίην, πολυμαθείην, κακοτεχνίην. DK12 B129 17

23 Clearly, the words πολυμαθίη, ἱστορίη, and σοφίη 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 αὖτις 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 ἱστορίη with the Milesian disposition to rational inquiry, but finds it to be overpowered by the words κακοτεχνίη and συγγραφαί, especially because the συγγραφαί sound like Orphic writings. All scholars by no means agree with Burkert s reading of Heraclitus fragments; some do not accept his effort to minimalize the scientific identity of Pythagoras in these early sources. Zhmud passes over Burkert s 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 ἱστορίη, which was routinely used in reference to the Milesian science, including geometry, astronomy, geography, and history. 39 In addition, Zhmud counters that συγγραφαί refers to prose writings, and thus could not be Orphic poetry, but more likely 37 Burkert, (1972), Zhmud, Pythagoras and the Early Pythagoreans, (Oxford: 2012), Khan, Pythagoras and the Pythagoreans, (Indianapolis: 2001),

24 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 identities mathematician, 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), Zhmud, Pythagoras and the Early Pythagoreans, (Princeton: 2012), Zhmud, Pythagoras and the early Pythagoreans, (Princeton: 2012), Gomperz, Greek Thinkers, vol. I. (London: 1955),

25 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 Burkert s 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), Riedweg. Pythagoras: His Life, Teaching, and Influence. (Ithaca: 2005),

26 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 Burkert s criticism of the theory of synthesis between the ritual and the rational which Dӧring 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 Dӧring, argues for a synthesis between the mathematical and the religious Pythagoras which depends upon the idea of catharsis. In Burnet s 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 (καθαρμοί), 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? 48 Aristoxenus, Fr Burnet, Early Greek Philosophy, (London: 1958),

27 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 Plato s writings identifies passages that suggest a theory of catharsis. In the Phaedo, catharsis is achieved by philosophy, whose goal is death the 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 Burkert s 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. 22

28 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 Burkert s 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, Dӧring and Burnet work to find a common strand (the practice of catharsis) to unite the two in the person of Pythagoras. Burkert s 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 54 Burkert, (1972), 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 nonsensible concept of number. 23

29 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. 24

30 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 water 3 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 Aristotle s 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, DK 44 B4. Translation in Barker, Greek Musical Writings, vol. II, (Cambridge: 1990), Translation in Barker Greek Musical Writings, vol. II, (Cambridge: 1990),

31 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), Translation in Netz, The Problem of Pythagorean Mathematics, in Huffman, A History of Pythagoreansm,, Translation in Netz, The Problem of Pythagorean Mathematics, in Huffman (2014),

32 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 Aristotle s 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 arrangements such as triangular numbers, square numbers, and other classifications were 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), Netz, The Problem of Pythagorean Mathematics, in Huffman (2014), Heidel, W.A. The Pythagoreans and Greek Mathematics. Studies in Pre-Socratic Philosophy. Vol. I, (New York: 1970),

33 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. 28