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)

Pythagoreans dominated the political scene in southern Italy for nearly a century in the late 6th to 5th century BC. What was the secret of their political success and can their political, social and economic policies be assessed in the customary terms with which historians try to analyse ancient societies? I argue that they cannot, and that the Pythagorean approach to politics was sui generis, and successful because it was based on ideas, not force or popular demagogy.

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)

The Pythagorean Precepts by Aristotle's pupil, Aristoxenus of Tarentum, present the principles of the Pythagorean way of life that Plato praised in the Republic. They are our best guide to what it meant to be a Pythagorean in the time of Plato and Aristotle. The Precepts have been neglected in modern scholarship and this is the first full edition and translation of and commentary on all the surviving fragments. The introduction provides an accessible overview of the ethical system of the (...) Precepts and their place not only in the Pythagorean tradition but also in the history of Greek ethics as a whole. The Pythagoreans thought that human beings were by nature insolent and excessive and that they could only be saved from themselves if they followed a strictly structured way of life. The Precepts govern every aspect of life: e.g. procreation, abortion, child rearing, friendship, religion, desire and even diet. (shrink)

Allusions to a cosmogony contained in a Vedic hymn present striking analogies to a cosmogony attributed to the Pythagoreans by Aristotle, Simplicius and Stobaeus. The aim of the paper is to evaluate the extent to which they are similar and to which their differences respond to different cultural premises.

The obstacle to more objective knowledge of early Pythagoreanism is the ideological conflict over the proper mission of historiography. Not only the confusing evidence, but also the different investigative procedures and theories of history employed, make solving the Pythagorean problem difficult. I analyze the historiographic methodologies of some modern historians of Pythagoreanism in respect to the kinds of historical explanation they provide. Immediately ideological controversy arises between idealist and materialist historians. ;My critical evaluation proceeds from two theses. The content of (...) ideas reflects their social referents. Consequently, one must identify the class interests of early Pythagoreans. Writing history is a contemporary undertaking by which historians reflect their own social interests. Therefore, historiographic methodology continually must be objectified to overcome the subjectivism of historical relativism. ;I review the reconstructions of six historians. John Burnet, an empiricist, held that the Greeks entirely broke from their religious past to develop philosophy. F. M. Cornford, a Durkheimian, used anthropology to reveal philosophy's totemic content. J. E. Raven is a philologist; he maintaned that ancient philosophy was the Eleatic-Pythagorean debate. W. K. C. Guthrie, an eclectic, held that science served Pythagorean religious interests. These idealist historians attributed the Greek achievement to absolutized temperaments or racial factors; the development of ideas is not mediated by practical necessity. George Thomson and Alban Winspear are historical materialists. They explained ancient philosophy by showing the class interests of philosophers in a mercantile economy based on slave labor. ;After introducing the Pythagorean problem, I present, by chapter, first an exposition of each historian's account of the causes of Greek philosophy and then his exposition of early Pythagoreanism. My analyses follow each section. In the first part of the final chapter I discuss each historian in the context of the ideological controversy. In the second part I present a general outline of procedure for scientific historiography. (shrink)

Archytas of Tarentum is one of the three most important philosophers in the Pythagorean tradition, a prominent mathematician, who gave the first solution to the famous problem of doubling the cube, an important music theorist, and the leader of a powerful Greek city-state. He is famous for sending a trireme to rescue Plato from the clutches of the tyrant of Syracuse, Dionysius II, in 361 BC. This 2005 study was the first extensive enquiry into Archytas' work in any language. It (...) contains original texts, English translations and a commentary for all the fragments of his writings and for all testimonia concerning his life and work. In addition there are introductory essays on Archytas' life and writings, his philosophy, and the question of authenticity. Carl A. Huffman presents an interpretation of Archytas' significance both for the Pythagorean tradition and also for fourth-century Greek thought, including the philosophies of Plato and Aristotle. (shrink)

This book explores the persistence of Pythagorean ideas in theoretical physics. It shows that the Pythagorean position is both philosophically deep and scientifically interesting. However, it does not endorse pure Pythagoreanism; rather, it defends the thesis that mind and mathematical structure are the grounds of reality. The book begins by examining Wigner's paper on the unreasonable effectiveness of mathematics in the natural sciences. It argues that, whilst many issues surrounding the applicability of mathematics disappear upon examination, there are some core (...) issues to do with the effectiveness of mathematics in fundamental physics which remain. The core issues are the existence of the laws of nature and our minds ability to fathom them, the use of formal mathematics in discovering things about the quantum world, the fact that deep mathematics is needed to describe fundamental physics, and the asymptotic nature of the quest for knowledge. These issues are the focus of the book. The author seeks to explain them within a metaphysical framework that takes mind and mathematical structure as its fundamental principles. The framework - called quantum monadology - combines ideas from Leibnizian monadology, set theory, and consistent histories quantum theory. (shrink)

This scarce antiquarian book is a facsimile reprint of the original. Due to its age, it may contain imperfections such as marks, notations, marginalia and flawed pages. Because we believe this work is culturally important, we have made it available as part of our commitment for protecting, preserving, and promoting the world's literature in affordable, high quality, modern editions that are true to the original work.

Pythagorean Women Philosophers argues for a rewriting of Greek philosophical history so as to include female intellectuals. Dutsch presents testimonies regarding the role of women in the Pythagorean school as demonstrating their active contribution to the philosophical tradition.

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)

This chapter discusses how all the Pythagorean theories of the soul known to us are different. Only Simmias and Echecrates, the pupils of Philolaus, held identical views. The similarity among some of these theories can sometimes be explained by direct influence, but most often by the fact that many Pythagoreans shared the interpretation of the soul as the source of motion, which was the most widespread view amongst the Presocratics. It is also shown that there is neither a clear (...) formulation of number doctrine, nor tangible examples of it, in ancient Pythagoreanism. It transpires that the theory which was taken to be the essence of Pythagorean philosophy, a theory backed by the authority of Aristotle as the source and as a historian of philosophy, is, like many other elements of the ancient tradition about the Pythagoreans, a retrospective projection. (shrink)

In coal mining industry, the running state of mine ventilators plays an extremely significant role for the safe and reliable operation of various industrial productions. To guarantee the better reliability, safety, and economy of mine ventilators, in view of early detection and effective fault diagnosis of mechanical faults which could prevent unscheduled downtime and minimize maintenance fees, it is imperative to construct some viable mathematical models for mine ventilator fault diagnosis. In this article, we plan to establish a data-based mine (...) ventilator fault diagnosis method to handle situations where engineers are absent or they are incapable of coming to a conclusion from multisource data. In the process of building the mine ventilator fault diagnosis model, considering that probabilistic rough sets could reduce the errors triggered by incompleteness, inconsistency, and inaccuracy without needing any additional assumptions and Pythagorean fuzzy multigranulation rough sets over the two universes’ model could effectively handle data representation, fusion, and analysis issues, we generalize the existing PF MGRSs over the two universes’ model to the PRS setting, as well as to further establish a novel model named Pythagorean fuzzy multigranulation probabilistic rough sets over two universes. In the granular computing paradigm, three types of PF MG-PRSs over two universes based on the risk attitude of engineers are proposed at first. Afterwards, several basic propositions of the newly proposed model are explored. Moreover, a PF multigranulation probabilistic model for mine ventilator fault diagnosis based on PF MG-PRSs over two universes is investigated. At last, a real-world case study of dealing with a mine ventilator fault diagnosis problem is given to illustrate the practicality of the presented model, and a validity test, a sensitivity analysis, and a comparison analysis are further explored to demonstrate the effectiveness of the presented model. (shrink)

This chapter first considers the estimates of how great the contribution of the Pythagoreans was to Plato's philosophy and how these views diverge substantially, and which vary across the range from ‘decisive’ to ‘insignificant’. It shows that Platonists were characterized by a benevolent attitude to Pythagoras and the Pythagoreans and an interest in their scientific, philosophical, and religious theories. Number doctrine is found in the testimonies of all three Platonists, but there is in them no picture of a (...) Pythagorean philosophy even remotely reminiscent of number doctrine. The Platonists reacted, not to a common Pythagorean doctrine, but to various theories of Pythagoras and his successors: Philolaus, Archytas, Ecphantus and Hicetas, et al. The chapter then considers Aristotle's reports on the Pythagoreans in his surviving works. (shrink)

This commentary provides valuable insight into the basic tenets of an unknown sect from the formative years of Hellenistic Pythagoreanism, a movement which had a major impact on the morality and philosophy of the Greco-Roman world.

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.

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.

This chapter explores the kind of community founded by Pythagoras. It considers those types of association which actually existed in Greece in the Archaic and Classical periods. If the Pythagorean community was really a religious association, it should conform to the type of religious association of its time, and not to that of the Qumran community or a Christian monastery. To describe the nature of the society founded by Pythagoras, we may choose from a very small number of variants available (...) for that period: a philosophical school, like those which appeared in the fourth century; a cultic community ; and a sociopolitical association. The chapter shows, first, that none of the features of a religious community is confirmed by reliable evidence. Second, that to apply to the Pythagoreans the term ‘sect’, as developed in the sociology of religion, is methodologically unsound. Third, that the tradition on Pythagorean ‘symbols’, reflected in Anaximander the Younger and Aristotle, mostly bears no relation to the realities of the Pythagorean way of life, while the picture which arose on the basis of that tradition, of acusmatici and mathematici dates from the Imperial period. (shrink)

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.

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 is the first comprehensive study for nearly 200 years of what remains of the writings of the Presocratic philosopher Philolaus of Croton. These fragments are crucial to our understanding of one of the most influential schools of ancient philosophy, the Pythagoreans; they also show close ties with the main lines of development of Presocratic thought, and represent a significant response to thinkers such as Parmenides and Anaxagoras. Professor Huffman presents the fragments and testimonia with accompanying translations and introductory (...) chapters and interpretive commentary. He not only produces further argument for the authenticity of much that used to be neglected, but also undertakes a critique of Aristotle's testimony, opening the way for a quite new reading of fifth-century Pythagoreanism in general and of Philolaus in particular. Philolaus is revealed as a serious natural philosopher. (shrink)

The paper offers a running commentary on ps-Archytas’ On Intellect and Sense Perception (composed ca. 80 BCE), with the aim to provide a clear description of Hellenistic/post-Hellenistic Pythagorean epistemology. Through an analysis of the process of knowledge and of the faculties that this involves, ps-Archytas presents an original epistemological theory which, although grounded in Aristotelian and Platonic theories, results in a peculiar Pythagorean criteriology that accounts for the acquisition and production of knowledge, as well as for the specific competences of (...) each cognitive faculty. (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)

Problem 2 at the 56th International Mathematical Olympiad asks for all triples of positive integers for which ab−c, bc−a, and ca−b are all powers of 2. We show that this problem requires only a primitive form of arithmetic, going back to the Pythagoreans, which is the arithmetic of the even and the odd.

Introduction -- Evidence for the work: the excerpts preserved in Stobaeus -- Title and nature of the work -- Format and style of the work -- Fragments of the Pythagorean precepts preserved in Iamblichus' On the Pythagorean way of life -- A comparison of Stobaeus' and Iamblichus' evidence for the Pythagorean precepts -- Relationship of the Pythagorean precepts to Aristoxenus' other works on the Pythagoreans -- The influence of the Pythagorean precepts on the later Pythagorean tradition -- History of (...) scholarship on the Pythagorean precepts -- The standard view of the Pythagorean precepts -- The ethical system of the Pythagorean precepts -- Fragments with translation and commentary -- The Pythagorean precepts: a reconstructed text in English -- Fragment 1: obedience to parents and the laws (fr. 34 Wehrli = Stobaeus 4.25.45) -- Fragment 2: the importance of order and supervision for every age of life (fr. 35 Wehrli = Stobaeus, 4.1.49) -- Fragment 3: desire (fr. 37 Wehrli = Stobaeus, 3.10.66) -- Fragment 4: the generation of children (fr. 39 Wehrli = Stobaeus, 4.37.4) -- Fragment 5: the love of what is beautiful and fine (fr. 40 Wehrli = Stobaeus 3.1.101) -- Fragment 6: learning must be willing (fr. 36 Wehrli = Stobaeus, 2.31.119) -- Fragment 7: luck (fr. 41 Wehrli = Stobaeus, 1.6.18) -- Fragment 8: human nature is prone to excess and needs the supervision of the gods, parents and laws (fr. 33 Wehrli, Iamblichus, vp 174-6) -- Fragment 9: opinion, the training of children and young people, pleasure, desire, diet, and the generation of children (fr. 38 Wehrli, Iamblichus, vp 200-13) -- Fragment 10: the appropriate and the inappropriate in human interaction on starting points and rulers (Iamblichus, vp 180-3) -- Fragment 11: friendship (Iamblichus, vp 101-2, 230-3) -- Appendices -- Subsidiary precepts 1: avoid crowds in the morning, and 2: avoid hunting (Iamblichus, vp 96-100) -- Subsidiary precept 3: memory (Iamblichus, vp 164) -- Subsidiary precept 4: all sex is harmful (Diodorus Siculus, library of history 10.9.3), Stobaeus, Eclogae 3.1.71: divination, medicine, and music -- Concordance with the fragment numbers in Wehrli's edition. (shrink)

This chapter begins with a description of the history of Pythagorean societies after the death of Pythagoras. It then considers the criteria used by Aristoxenus in compiling his list of Pythagoreans. Compared with those applied in modern works, it is argued that, beyond a critical approach to the sources, we enjoy no special advantages over the first historian of Pythagoreanism. This is followed by a discussion of the prosopography and chronology of the Pythagoreans.

The multiobjective optimization on the basis of ratio analysis method captures diverse features such as the criteria and alternatives of appraising a multiple criteria decision-making problem. At the same time, the multiple criteria problem includes a set of decision makers with diverse expertise and preferences. In fact, the literature lists numerous approaches to aid in this problematic task of choosing the best alternative. Nevertheless, in the MCDM field, there is a challenge regarding intangible information which is commonly involved in multiple (...) criteria decision-making problem; hence, it is substantial in order to advance beyond the research related to this field. Thus, the objective of this paper is to present a fused method between multiobjective optimization on the basis of ratio analysis and Pythagorean fuzzy sets for the choice of an alternative. Besides, multiobjective optimization on the basis of ratio analysis is utilized to choose the best alternatives. Finally, two decision-making problems are applied to illustrate the feasibility and practicality of the proposed method. (shrink)

The Pythagorean Life is the most extensive surviving source on Pythagoreanism, and has wider interest as an account of the religious aspirations of late antiquity. "...admirably clear translation and sensible introduction"--The Classical...

: Some of the great physicists' belief in the existence of a connection between the aesthetical features of a theory (such as beauty and simplicity) and its truth is still one of the most intriguing issues in the aesthetics of science. In this paper I explore the philosophical credibility of a version of this thesis, focusing on the connection between the mathematical beauty and simplicity of a theory and its truth. I discuss a heuristic interpretation of this thesis, attempting to (...) clarify where the appeal of this Pythagorean view comes from and what are the arguments favoring its acceptance or rejection. Along the way, I sketch the historical context in which this heuristic interpretation gained credibility (the quantum crisis in physics in the 1920s), as well as the more general implications of this thesis for physicists' metaphysical outlook. (shrink)

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)

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)

Previously we saw that racist prejudice is supported by false history. The false history of the Greek origins of mathematics is reinforced by a bad philosophy of mathematics. There is no evidence for the existence of Euclid. The “Euclid” book does not contain a single axiomatic proof, as was exposed over a century ago. Such was never the intention of the actual author. The book was brazenly reinterpreted, since axiomatic proof was a church political requirement, and used in church rational (...) theology adopted during the Crusades, as a counter to Islamic rational theology. Deductive proofs are MORE fallible than inductive or empirical proofs. Even a validly proved mathematical theorem, such as the “Pythagorean” theorem, is invalid knowledge in the real world. There is no concept of approximate truth in formal mathematics. Nevertheless, the myth of “superior” axiomatic proofs in the “Euclid” book continues to be reiterated by Western historians, and colonial education teaches axiomatic mathematics. Actually, superior practical value comes from the two “Pythagorean” calculations well known toIndian/Egyptian tradition, but unknown to Greeks. The advantage of related decolonized courses in mathematics has been pedagogically demonstrated. But understanding and political will are needed to change colonial/church education. (shrink)