This paper discusses continuity between ancient Pythagoreanism and the pseudo-Pythagorean writings, which began to appear after the end of the Pythagorean school ca. 350 BC. Relying on a combination of temporal, formal and substantial criteria, I divide Pseudopythagorica into three categories: 1) early Hellenistic writings ascribed to Pythagoras and his family members; 2) philosophical treatises written mostly, yet not exclusively, in pseudo-Doric from the turn of the first century BC under the names of real or fictional Pythagoreans; 3) writings (...) attributed to Pythagoras and his relatives that continued to appear in the late Hellenistic and Imperial periods. I will argue that all three categories of pseudepigrapha contain astonishingly little that is authentically Pythagorean. (shrink)

In ancient tradition, Pythagoras emerges as a wise teacher, an outstanding mathematician, an influential politician, and as a religious and ethical reformer. This volume offers a comprehensive study of Pythagoras, Pythagoreanism, and the early Pythagoreans through an analysis of the many representations of the individual and his followers.

ZHMUD. L. (2012). Pythagoras and the Early Pythagoreans. Oxford, Oxford University Press.

Apart from his teachings, wonders and scientific discoveries, Pythagoras was also known for his wide-ranging journeys. Ancient authors alleged that he visited many countries and nations from Egypt to India, stayed with the Phoenicians and the Ethiopians and talked to the Persian Magi and Gallic Druids. However, he never went to the North. If, nevertheless, he was eventually associated with the northern inhabitants, it is only because they themselves came into close contact with him. The first of them was Zalmoxis, (...) a deity of a Thracian tribe, the Getae, who guaranteed them immortality after death. Having described a blood ritual that the Getae practised to become immortal, Herodotus relates a story he heard from the Hellespontine and Pontic Greeks. It goes that Zalmoxis was not adaimōnbut a former slave of Pythagoras on Samos and, having adopted the doctrine of immortality from him, he returned to Thrace and converted his tribesmen to it with a cunning trick. He invited the most prominent of them to a men's hall for entertainment and told them that neither he nor they or their descendants would die, but would live eternally. Then, having constructed a secret underground chamber, he suddenly disappeared from the eyes of the Getae and hid in his shelter for three full years, being lamented as dead. Then he showed himself again to the Getae, thus persuading them of the truth of his promises. (shrink)

This essay considers the origins of the arithmological genre, the first specimen of which was an anonymous Neopythagorean treatise of the first century BCE. Arithmology as a special genre of philosophical writings dealing with the properties of the first ten numbers should be distinguished from number symbolism, which is a universal cultural phenomenon related to individual significant numbers. As our analysis shows, the philosophical foundations of arithmology were laid down in the treatise of Plato’s successor Speusippus On Pythagorean Numbers, who (...) relied on the Platonic doctrine of the ten ideal numbers, whereas in ancient Pythagoreanism arithmological notions, unlike number symbolism, are not attested. In the first century BCE, an epoch of revival of Platonism and Aristotelianism, Speusippus’ ideas received a second birth, thus marking the beginning of arithmology as a popular genre. (shrink)

A fascinating portrait of the Pythagorean tradition, including a substantial account of the Neo-Pythagorean revival, and ending with Johannes Kepler on the threshold of modernism.

This anthology, the largest collection of Pythagorean writings ever to appear in English, contains the four ancient biographies of Pythagoras and over 25 Pythagorean and Neopythagorean writings from the Classical and Hellenistic periods. The material of this book is indispensable for anyone who wishes to understand the real spiritual roots of Western civilization.

Kahn tries to do justice to the contribution Pythagoras and his followers might have had for Greek science. Thus he downplays the religious figure so prominent with Burkert's groundbreaking study "Lore and Science". He sees the transformation Pythagorean ideas may have undergone in Plato's Academy as pivotal for the developments of Pythagoreanism in later antiquity as well as in Renaissance speculation, e. g. Kepler. The book offers a good overview for the history of Pythagoreanism from its founder to modern times.

Die vorliegende Sammlung von Aufsätzen einiger der weltbesten Spezialisten der antiken griechischen Philosophie konzentriert sich auf die philosophischen Texte, die während der hellenistischen und kaiserlichen Zeit Pythagoras und den Pythagoreern zugeschrieben wurden. Ihre unbekannten Verfasser behaupten, die Positionen der ursprünglichen pythagoreischen Schule zur Metaphysik, Theologie, Zahlenphilosophie, Physik, Logik, politischen Philosophie, Ethik und zur richtigen Lebensweise zu vermitteln. Die kühnsten unter ihnen präsentieren sich als die Quellen, aus denen Platon und Aristoteles die Inspiration für den Timaios und die Kategorien geschöpft haben. (...) Da sie von den meisten antiken Denkern als maßgebend angesehen wurden, führte dies zu einer Renaissance der platonischen und pythagoreischen Tradition. (shrink)

According to Dicaearchus, metempsychosis was the best known among Pythagoras’ teachings. In this paper, I investigate two features of Pythagorean metempsychosis: its non-retributive character and its epistemological value. I argue that the Pythagoreans did not conceive of reincarnation as a punishment for the wicked and a reward for the virtuous, but rather as a way to gain experience, knowledge and therefore wisdom. This reading enables us to throw light on the puzzling list of Pythagoras’ past lives, which includes Aethalides (...) son of Hermes, Euphorbus the warrior, Pyrrhus the fisherman and even Alco the harlot. (shrink)

Metaphysics, geometry, and the problems with diagrams -- The Pythagorean theorem: Euclid I.47 and VI.31 -- Thales and geometry: Egypt, Miletus, and beyond -- Pythagoras and the famous theorems -- From the Pythagorean theorem to the construction of the cosmos out of right triangles.

A new form of the Hyperbolic Pythagorean Theorem, which has a striking intuitive appeal and offers a strong contrast to its standard form, is presented. It expresses the square of the hyperbolic length of the hypotenuse of a hyperbolic right-angled triangle as the “Einstein sum” of the squares of the hyperbolic lengths of the other two sides, Fig. 1, thus completing the long path from Pythagoras to Einstein. Following the pioneering work of Varičak it is well known that relativistic velocities (...) are governed by hyperbolic geometry in the same way that prerelativistic velocities are governed by Euclidean geometry. Unlike prerelativistic velocity composition, given by the ordinary vector addition, the composition of relativistic velocities, given by the Einstein addition, is neither commutative nor associative due to the presence of Thomas precession. Following the discovery of the mathematical regularity that Thomas precession stores, it is now possible to extend Thomas precession by abstraction, (i) allowing hyperbolic geometry to be studied by means of analogies that it shares with Euclidean geometry; and, similarly (ii) allowing velocities and accelerations in relativistic mechanics to be studied by means of analogies that they share with velocities and accelerations in classical mechanics. The abstract Thomas precession, called the Thomas gyration, gives rise to gyrovector space theory in which the prefix gyro is used extensively in terms like gyrogroups and gyrovector spaces, gyroassociative and gyrocommutative laws, gyroautomorphisms, gyrotranslations, etc. We demonstrate the superiority of our gyrovector space formalism in capturing analogies by deriving the Hyperbolic Pythagorean Theorem in a form fully analogous to its Euclidean counterpart, thus contrasting it with the standard form in which the Hyperbolic Pythagorean Theorem is known in the literature. The hyperbolic metric, which supports the Hyperbolic Pythagorean Theorem, has a dual metric. We show that the dual metric does not support a Pythagorean theorem but, instead, it supports the π-Theorem according to which the sum of the three dual angles of a hyperbolic triangle is π. (shrink)

Pythagoric life accompanied by fragments of the ethical writings of certain Pythagoreans in the Doric dialect and a collection of Pythagoric sentences from Stobaeus and others.

The Pythagorean idea that numbers are the key to understanding reality inspired philosophers in late Antiquity (4th and 5th centuries A.D.) to develop theories in physics and metaphysics based on mathematical models. This book draws on some newly discovered evidence, including fragments of Iamblichus's On Pythagoreanism, to examine these early theories and trace their influence on later Neoplatonists (particularly Proclus and Syrianus) and on medieval and early modern philosophy.

Die im 6. Jh. v. Chr. von Pythagoras ausgehende politisch mystische Lehre erlebte in der Spätantike vor allem im Umfeld des Neuplatonismus eine wirkmächtige Renaissance, die am eindrücklichsten im Werk des Iamblichos von Chalkis dokumentiert ist. Zum Verständnis seiner noch häufig verkannten Vorlesung "Über das Pythagoreische Leben" bahnt die Studie zum ersten Mal einen Weg, indem sie den vielgestaltigen geistes- und literaturgeschichtlichen Hintergrund entfaltet, vor dem sich Konzeption und Intention des Textes im zeitgenössischen Kontext erschließen lassen. Eine kommentierende Strukturanalyse macht (...) deutlich, welche Schlüsselrolle der neuplatonische Lehrer dem altehrwürdigen Weisen innerhalb seines ethisch-philosophischen Erziehungskonzeptes beimisst. (shrink)

In Pythagorean Women, classical scholar Sarah B. Pomeroy discusses the groundbreaking principles that Pythagoras established for family life in Archaic Greece, such as constituting a single standard of sexual conduct for women and men. Among the Pythagoreans, women played an important role and participated actively in the philosophical life. While Pythagoras encouraged women to be submissive to men, his reasoning was based on the desire to preserve harmony in the home. -/- Pythagorean Women provides English translations of all the (...) earliest extant examples of literary Greek prose by Neopythagorean women, shedding light on their attitudes about marriage, the home, music, and the cosmos. Pomeroy sets the Pythagorean and Neopythagorean women vividly in their historical, ecological, and intellectual contexts, illustrated with original photographs of sites and artifacts known to these women. (shrink)

The timeless brilliance of this exhaustive survey of the best classical writers of antiquity on Pythagoras was first published in 1687 in Thomas Stanley's massive tome, The History of Philosophy. It remains as contemporary today as it was over three hundred years ago. The text of the 1687 book has been reset and modernized to make it more accessible to the modern reader. Spelling has been regularized, obsolete words not found in a modern dictionary have been replaced, and contemporary conventions (...) of punctuation have been used. Biographical sketches of Thomas Stanley and Pythagoras by Manly Palmer Hall, founder of the Philosophical Research Society, have been included, along with a profound overview of Pythagorean philosophy by Platonic scholar Dr. Henry L. Drake. The extensive Greek language references throughout the text have been corrected and contextualized, and reset in a modern Greek font. Each quotation has been verified with the source document in Greek. An extensive annotated appendix of these classical sources is included. A complete bibliography details all the reference works utilized, and a small Glossary defines a number of terms, especially those from musical theory, which may be unfamiliar to the non-technical reader. (shrink)

The Pythagorean women are a group of female philosophers who were followers of Pythagoras and are credited with authoring a series of letters and treatises. In both stages of the history of Pythagoreanism – namely, the fifth-century Pythagorean societies and the Hellenistic Pythagorean writings – the Pythagorean woman is viewed as an intellectual, a thinker, a teacher, and a philosopher. The purpose of this Element is to answer the question: what kind of philosopher is the Pythagorean woman? The traditional picture (...) of the Pythagorean female sage is that of an expert of the household. The author argues that the available evidence is more complex and conveys the idea of the Pythagorean woman as both an expert on the female sphere and a well-rounded thinker philosophising about the principles of the cosmos, human society, the immortality of the soul, numbers, and harmonics. (shrink)

The influence which the Pythagorean society and its leading doctrines exercised upon Athenian intellectual and political developments in the late fifth century leads us to seek in Pythagoras a figure of greater stature and more clear-cut features than modern scholarship is prepared to allow. To us he is a great name but little more, the large body of detailed information about his life which is available in later writers being dismissed as fabulous. This scepticism was reasonable enough when the reader (...) was faced with the garbled hotchpotch of an Iamblichus or a Porphyry. But since the task has been put in hand of determining what portions of this tradition can safely be attributed to the various authorities who concerned themselves with the Pythagorean story in the fourth century, and of considering the respective historical value of these authorities, the scepticism of, for example, Burnet's account becomes unjustified. It is now possible to present a fairly detailed account of Pythagoras' activity, which has at least as much claim to credence as a great deal of what we now readily accept as ancient history, and is furthermore consistent with the general picture of the man that emerges from a careful scrutiny of classical sources. Not only for its own sake, but in the context of the increasingly fruitful investigation of fifth-century Athenian movements, I feel that it may be useful to put together such an account of Pythagoras. It will certainly be regarded by some as mere fiction, but may nevertheless provide others with a convenient summary of what the fifth- and fourth-century writers thought and believed about a man and a movement that had profound influence upon their times. I hope at the same time to suggest some links between what Pythagoras thought and did and the main tradition of Greek social behaviour. If Pythagoras was, as I believe, not so much an innovator as a reformer and developer of some of the central institutions of Greek life, the possibility will be opened of regarding various phenomena in mainland Greece which we have been asked to call Pythagorean, e.g. the Socratic phrontisterion as it appears in the Clouds, not as deriving from Pythagoras but as a parallel growth to the Pythagorean synedria and nourished by the same root. (shrink)

This study attempts to answer the question of the origins of the non-biblical features of the Essenes' way of life. It is clear that their project was founded in biblical and Jewish realities. Where did those elements come from that appear not to have been derived from these sources? That they have their origins in Greek, and specifically Pythagorean, customs is an idea that recurs regularly in the history of scholarship and has had some notable supporters in the 19th and (...) first half of the 20th centuries, including E Zeller, E Schuer, I Levy, F C Cumont and M J Lagrange. Recent scholars seem more reluctant to accept such views. The inquiry will take the reader through the available sources, then a series of comparative studies with possible parallels in Greece and elsewhere, to see whether and in what terms the question of Pythagorean influences on Essenism can be answered. (shrink)

In both ancient tradition and modern research Pythagoreanism has been understood as a religious sect or as a philosophical and scientific community. Numerous attempts have been made to reconcile these pictures as well as to analyze them separately. The most recent scholarship compartmentalizes different facets of Pythagorean knowledge, but this offers no context for exploring their origins, development, and interdependence. This collection aims to reverse this trend, addressing connections between the different fields of Pythagorean knowledge, such as eschatology, metempsychosis, metaphysics, (...) epistemology, arithmology and numerology, music, dietetics and medicine as well as politics. In particular, the contributions discuss how the Pythagorean way of life related to more doctrinal aspects of knowledge, such as Pythagorean religion and science. The volume explores the effects of this interdependence between different kinds of knowledge both within the Pythagorean corpus and in its later reception. Chapters cover historical periods from the Archaic Period (6th century BC) to Neoplatonism, Early Christianity, the European and Arabic Middle Ages, and the Renaissance through to the Early Modern Period (17th century AD). (shrink)

Fiction and truth : ancient stories about Pythagoras -- In search of the historical Pythagoras -- The Pythagorean secret society -- Thinkers influenced by Pythagoras and his pupils.

The influence which the Pythagorean society and its leading doctrines exercised upon Athenian intellectual and political developments in the late fifth century leads us to seek in Pythagoras a figure of greater stature and more clear-cut features than modern scholarship is prepared to allow. To us he is a great name but little more, the large body of detailed information about his life which is available in later writers being dismissed as fabulous. This scepticism was reasonable enough when the reader (...) was faced with the garbled hotchpotch of an Iamblichus or a Porphyry. But since the task has been put in hand of determining what portions of this tradition can safely be attributed to the various authorities who concerned themselves with the Pythagorean story in the fourth century, and of considering the respective historical value of these authorities, the scepticism of, for example, Burnet's account becomes unjustified. It is now possible to present a fairly detailed account of Pythagoras' activity, which has at least as much claim to credence as a great deal of what we now readily accept as ancient history, and is furthermore consistent with the general picture of the man that emerges from a careful scrutiny of classical sources. Not only for its own sake, but in the context of the increasingly fruitful investigation of fifth-century Athenian movements, I feel that it may be useful to put together such an account of Pythagoras. It will certainly be regarded by some as mere fiction, but may nevertheless provide others with a convenient summary of what the fifth- and fourth-century writers thought and believed about a man and a movement that had profound influence upon their times. I hope at the same time to suggest some links between what Pythagoras thought and did and the main tradition of Greek social behaviour. If Pythagoras was, as I believe, not so much an innovator as a reformer and developer of some of the central institutions of Greek life, the possibility will be opened of regarding various phenomena in mainland Greece which we have been asked to call Pythagorean, e.g. the Socratic phrontisterion as it appears in the Clouds, not as deriving from Pythagoras but as a parallel growth to the Pythagorean synedria and nourished by the same root. (shrink)

Reconstruction of the versions of Aristoxenos and Dikaiarchos.--The sources of Dikaiarchos and Aristoxenos and the reliability of their accounts. --Reconstruction of Timaios' version and the reliability of his accounts.--The chronological questions and the numismatic evidence.--The character of the "Pythagorean rule" in southern Italy.--Appendix.

At Metaphysics A 5 986a22-b2, Aristotle refers to a Pythagorean table, with two columns of paired opposites. I argue that 1) although Burkert and Zhmud have argued otherwise, there is sufficient textual evidence to indicate that the table, or one much like it, is indeed of Pythagorean origin; 2) research in structural anthropology indicates that the tables are a formalization of arrays of “symbolic classification” which express a pre-scientific world view with social and ethical implications, according to which the presence (...) of a principle on one column of the table will carry with it another principle within the same column; 3) a close analysis of Aristotle's arguments shows that he thought that the table expresses real causal relationships; and 4) Aristotle faults the table of opposites with positing its principles as having universal application and with not distinguishing between those principles that are causally prior and those that are posterior. Aristotle's account of scientific explanation and his own explanations that he developed in accordance with this account are in part the result of his critical encounter with this prescientific Pythagorean table. (shrink)