Pythagorean Tradition Research Articles

Pythagorean Philosophy and Theurgy on Friendship

In the Pythagorean tradition, friendship is elevated beyond a mere human relationship, serving as a means to transcend human frailty and attain immortality. This philosophy posits that humans are imprisoned and require liberation through the benevolence of the gods. The Pythagorean way of life is seen as a path to achieving immortality and freedom, where friendship with the gods is the highest form of association. The spiritual practice of theurgy is essential in this process, enabling humans to purify themselves and receive the gift of friendship from the gods. The Pythagoreans distinguished between various forms of friendship, including the highest and most noble understanding between gods and humans, which requires faith, knowledge, philosophy, and theurgy. True friendship is characterized by trust, piety, and scientific worship, and its pursuit necessitates the avoidance of jealousy and conflict, as well as careful judgment and reverence. Additionally, purification, self-control, and a healthy diet are crucial in the pursuit of wisdom and friendship. Ultimately, the Pythagorean philosophy on friendship offers a profound understanding of human relationships, emphasizing the importance of spiritual growth, self-transcendence, and the pursuit of wisdom, leading to the cultivation of true and lasting friendships that bring about wholeness, reconciliation, and harmony.

The emergence of symmetries in classical physics

Abstract Since antiquity symmetry has proved to be a central guiding idea of research with which the microcosm and macrocosm are to be explained. Euclidean mathematics established a unified theory of proportions of the world in which the spherical symmetries of the macrocosm and the regular (“Platonic”) bodies of the microcosm were distinguished. In the Pythagorean tradition symmetrical proportions were understood as an expression of the beauty and harmony of the cosmos until the Renaissance.

Кирик Новгородец и космологические загадки по-древнерусски

The text of the “Teaching on Numbers” by the monk and mathematician Kirik the Novgorodian (the first half of the 12th century) was introduced into scientific circulation in the 1820s. Initially, this text was assessed as exclusively mathematical-calendar content. As the research unfolded, questions of a textual, historical-mathematical, philosophical, cultural and socio-psychological nature arose. Compared with similar writings of Kirik’s era, his calculations are recognized as accurate, and in methods of calculation on the abacus, he was 400 years ahead of his time. In addition to the “Teaching on Numbers”, the theological treatise penned by Kirik, “The Questions of Kirik”, has come down to our times, written in the form of a dialogue between a pastor and a bishop about the vital problems of service practice. Despite numerous articles and solid monographs on Kirik’s legacy, some postulates of the “Teaching”, as well as vocabulary features, remain undisclosed and mysterious. The first riddle concerns the scientific research and peculiarities of the reconstruction of the lists of Kirik’s treatises, which have come down to us only from the 16th century. A mere mystery concerns the fragment about circuits in the renewal of the elements, which, at first glance, has a natural philosophical content. The question arises: could Kirik, as a thoughtful scientist, simply rewrite these calculations from the prototype text familiar to him, the so-called “seven-thousanders”? Hypotheses are put forward of Platonic, Pythagorean, Gnostic influences on the essence of the fragment, but no supporting texts have been found. The article presents arguments in favor of the hypothesis of the influence of the mathematical teachings of the Pythagoreans on the prototype of the fragment about the renewal of the elements. The Pythagorean thesis about the structure of the corporeal cosmos according to the laws of beauty and harmony, which are based on the principle of commensurability, is clearly reflected in the mathematical calculations of the fragment. Kirik addressed his “Teaching” to lovers of numbers and he himself belonged to the circle of intellectuals. Modern studies of the Pythagorean tradition confirm the intercultural nature of the influence of the Pythagorean tradition on other ideological directions of thought. In Old Rus the tradition was supported by chroniclers and chronologists.

Origins of the Spherical Earth in Ancient Greek Cosmology

Diogenes Laertius ascribes the first concept of spherical Earth to both Pythagoras and Parmenides. Indeed, a major shift in cosmologies—emergence of the spherical conception of the Earth and the surrounding heaven—took place between the sixth and fifth centuries BCE. Given the poor state of preservation of early Pythagorean tradition, it is argued that primacy in formulating the notion of spherical Earth should be ascribed to Parmenides.

The Figure of Socrates in Numenius of Apamea: Theology, Platonism, and Pythagoreanism (fr. 24 des Places)

Numenius is one of the most important authors who, in the Imperial Age, deal with the figure of Socrates. Socrates is important in the Platon­ic tradition, in particular in the sceptical tradition, when the Socratic dubitative “spirit” of the first Platonic dialogues became important to justify the “suspension of judgement.” Numenius criticises the whole Academic tradition by saying that the Academics (particularly the sceptics) betrayed the original doctrine of Plato and formulated a new image of Socrates. For Numenius, Socrates plays a central role because Plato would have inherited his doctrine. What does Socrates’s doctrine consist in? According to Numenius, Socrates theorised a “doctrine of three Gods” (which can be likely found in the second Platonic epistle) which is strictly bound up with the main aspect of Plato’s thought. In fact, in Numenius’s view, Plato belongs to a genealogy which can be linked to Pythagoras himself. From this perspective, Numenius says that Socrates’s original thought is a theology which also belongs to the Pythagorean tradition and which Plato further developed. For Nume­nius, Socrates is not the philosopher of doubt, but a theologian who first theorised the existence of three levels of reality (Gods), which is also the kernel of Numenius’s metaphysical system. For this reason, Numenius puts Socrates within a theological genealogy that begins with Pythago­ras and continues with Socrates and Plato, and that the Academics and the Socratics failed to understand.

Death and the Maiden in Philostratus, About Apollonius of Tyana 4.45

Abstract The Vita Apollonii leaves much open to interpretation. In 4.45 Philostratus tells us about a young woman who was thought dead by her family and the whole of Rome. Apollonius whispers something in her ear and the maiden starts talking again. The narrator comments it was impossible for the bystanders and still is impossible for him to say whether the girl was really dead or not: whether it was a case of Scheintod which proved Apollonius’s extraordinary powers of observation or whether it was a resurrection-miracle which would signal a special ontological status for ‘the man’ from Tyana. In his suspension of judgment, Philostratus uses the words arrhêtos hê katalêpsis combining a technical term from Stoic epistemology (katalêpsis) with a concept related to the Mysteries (arrhêtos). We discuss the Philostratean interpretative strategies, link them to the Pythagorean tradition of selective communication, and read the reference in this chapter to the story of Alcestis to the epistemological debates between Stoics and Skeptics about the limits of human wisdom.

Проблема виховання і освіти дітей в контексті листів піфагорійських філософинь

The article reveals in detail the understanding of raising children in the context of two pseudo-epigraphic letters of Pythagorean wonan thinkers – Theano and Myia of Crotone. Based on these letters, it was found that pedagogical issues were important in general for the whole Pythagorean tradition. In fact, we can say that this early Greek philosophical school was the first to systematically and comprehensively approach the problem of upbringing and education in ancient society. It is hypothesized that this topic is not accidentally in the center of attention of these philosophers, because their authority was the greatest among all other representatives of this philosophical school. The author’s position is proved that Theano of Crotone letter to Eubule focuses on moderation in education, which is aimed at avoiding luxury, fulfilling all children’s whims, comfort. This is the purpose of hardening in difficult circumstances in order to withstand with dignity all the potential difficulties of adult life. Accordingly, if you do not raise a child in certain restrictions, then, according to Theano, it may well be unprepared for certain trials that may occur. The thesis is substantiated that the key task of upbringing and education, according to Myia of Crotone letter, is moderation, prudence and balance, which is based on both archaic elements and Hellenistic plots, which testifies to the skill of writing this letter. It is revealed that the Pythagorean principles of education, according to both philosophers, have no gender difference. This is because both girls and boys, if they grow up in luxury, comfort and do not know the limitations, can potentially become dangerous both for themselves and for society as a whole. It is emphasized that according to the style of writing, these letters are not so much moral and ethical as paraenetic epistolary genre, ie they act as advice on the upbringing and education of the younger generation. Because of this, these letters are such sources of ancient culture, which are one of the few that are devoted to the philosophical understanding of upbringing and education.

Music Theory's Other Nature: Reflections on Gaia, Humans, and Music in the Anthropocene

The new historical paradigm ushered in by the Anthropocene offers a timely and urgent opportunity to rethink the relationship of humans and nature. Bruno Latour's take on the Gaia hypothesis, which rejects the traditional subject/object divide, shows how the human can be inscribed into the work of music theory. This turn toward Latour's Actor-Network Theory, which erases the categorical difference between human and nonhuman agents, now dressed up in cosmic garb under the banner of the Gaia hypothesis, appears to be distant from traditional music-theoretical concerns, but the connection is in fact less far-fetched than it seems. J. G. Kastner's music theory, taking its cue from the sound of the Aeolian harp, serves as a test case here: the Aeolian harp, played by wind directly, had long served as a Romantic image of the superhuman forces of nature, but Kastner argues that the Aeolian network only becomes complete in human ears. By unraveling the various instances and agencies of Kastner's theory, this article charts a novel approach to music and sound that sidesteps the conceptual problems in which the nineteenth-century mainstream habitually gets entangled. Kastner's work is based on a fundamental crisis in the conception of sound, after the invention of the mechanical siren (1819) tore down any certainties about the categorical distinction between noise and musical sound. Seeking to rebuild the understanding of sound from the ground up, Kastner leaves no stone unturned, from the obsolete Pythagorean tradition of musica mundana to travelers’ reports about curious sonic environmental phenomena from distant parts of the world. Where the old mechanistic paradigm was built on a “physical music” (and a static “sound of nature” based on the harmonic series), Kastner proposes a new “chemical music” that is based on the dynamic, ever-changing sonority of the Aeolian harp. This chemical music does not (yet) exist, but Kastner gives us some clues about its features, especially in his transcription/simulation of the sound of the Aeolian harp scored for double symphony orchestra. Kastner's “chemical music” finally closes the music-theoretical network that he builds around his new conception of the supernatural sound of the Aeolian harp and its human and nonhuman agents.

New Pythagoreanism, Complexity, Artificial Intelligence

Leibniz defined mathematics as the “science of possible worlds.” But what worlds were assumed as the possible ones by thinkers of different eras? It is this question that is considered in this article. According to Einstein, development of science requires “external justification” (observations and experiments that should be explained or predicted) and internal perfection (following the internal logic of this discipline). Mathematics has no “external justification” and experiment cannot refute mathematical theory. In this sense, mathematics is closer to creativity, to art than to natural science. Its connection with culture turns out to be more complex and mediated. To explain this connection, Daniel Bell, within the framework of his theory of social development, puts forward the “axial principle,” according to which the role of science is considered as the main characteristic of society. From this point of view, we single out traditional, industrial and post-industrial societies. Each of these phases has its own ideals, norms, and types of mathematical creativity. In traditional society, following the Pythagorean tradition, mathematics is focused on finding harmony in nature, on identifying unity on the basis of universal relationships determined by the numerical characteristics of the objects under study. As the industrial era approaches, the constructivist, or “project-based,” approach becomes increasingly important. And unity emerges at a higher “meta-level.” The forerunner of this direction is Descartes, who raised the question of finding a single, universal method for solving all mathematical problems. The work traces the change in the formulation of a number of “invariant,” “eternal” mathematical problems as well as the evolution of the concept of “complexity” in the historical retrospective. Close attention is paid to the post-industrial phase of civilizational development and to “computer mathematics,” which has become the basis for formation of virtual reality, which in many respects changes the very direction of progress. As a result of this, the “extraverted orientation” of humanity, the course toward new horizons was replaced by the “introvert” one, which prioritizes tasks associated with comfort, convenience, and consumption. The “change of milestones” that has taken place is traced in a comparison of “big projects” related to mathematics that were put forward in the 1960s, and those that are considered as priorities at present. In fact, we are faced with a “crisis of expectations.” We see a way out of this crisis in a revival of the “Pythagorean tradition,” at a new level. But if in a traditional society the goal of the development of mathematics in alliance with other sciences and arts was to reveal harmony in the natural world, then at the post-industrial phase, the priorities are different. They are associated with computer modeling, understanding, and identifying the foundations of harmony in the human world.

THE FAMILY TRADITIONS OF THE GENS MARCIA BETWEEN THE FOURTH AND THIRD CENTURIES b.c.

AbstractIn the mid fourth century b.c. some Roman gentes drew on a Pythagorean tradition. In this tradition, Numa's role of Pythagoras’ disciple connected Rome (and the gentes) with Greek elites and culture. The Marcii, between 304 and 300 b.c., used Numa's figure, recently reshaped by the Aemilii and the Pinarii for their propaganda, to promote the need for a plebeian pontificate. After the approval of the Ogulnium plebiscite (300 b.c.), the needs for this kind of propaganda fell away. When Marcius Censorinus became censor, Numa's pontificate was no longer relevant for promoting the gens. For this reason, the Marcii used another genealogy for similar propagandistic effect: the figure of Marsyas, a symbol of plebeian ideals.

Was Democritus a Pythagorean? The Case of psychē

Abstract According to Glaucus of Rhegium Democritus was “a disciple of a Pythagorean” (dk 68 A1, 38). The tetralogical catalog of his works prepared by Thrasylus begins its section on ethics with the three following works: Pythagoras; On the Disposition of the Wise Man; On the Things in Hades (dk 68 B0a–c). The very order of the first three ethical works of Democritus could point to some sort of dependence on Pythagoreanism. This was suggested earlier by Frank (1923: 67), who believes that this is due to the fact that Democritus saw Pythagoras as basically the founder of an ethic-religious sect. Without being forced to agree with Frank, it is undeniable that there are many similarities between Pythagorean and Democritean ethics. The Democritean sentences that speak about the sense of shame before oneself as a way of preventing evil deeds (dk 68 B84, 244, and 264) recalls the practice of anamnesis, the examination of conscience in the Pythagorean tradition. Even more important are the parallel uses of measure as a basis for ethical reasoning. This paper aims to review this connection between Pythagorean traditions and Democritus, examining what emerges as the most probable core issue to determine how close this relation between atomists and Pythagoreans could have been: the Aristotelian testimony (de An. 1.2 404a16-20 [dk 58 B40]) on the material conception of the Pythagorean soul. In fact, a corpuscular conception of the soul (“dust in the air”), foreshadowing the psychology of Democritus, is attributed to the Pythagoreans. Is the argument of de An. 1.2 404a16-20 a misunderstanding by Aristotle? Or does this testimony represent an actual dialogue that Pythagoreans were having with atomists in the fifth century bce?

The Werewolf in the Ancient World by Daniel Ogden

Reviewed by: The Werewolf in the Ancient World by Daniel Ogden Giulia Freni Daniel Ogden. The Werewolf in the Ancient World, Oxford: Oxford University Press, 2021. Pp. 280. £25.00. ISBN 978-0-19-885431-9. The best-known werewolf story from Graeco-Roman antiquity is the one of Petronius' Satyricon 61–62, but there is other evidence about a man becoming a wolf. The volume The Werewolf in the Ancient World, by D. Ogden, intends to provide an overview on werewolfism in the ancient world, considering also the medieval and early modern traditions about the phenomenon. A significative role in this analysis is the folkloric approach, which permits tracing connections between the ancient material and the later evidence. The first two chapters discuss the connection of werewolves with magic and witchcraft on one hand, and with ghosts on the other. On the former, Ogden explores the transformation of men into animals beginning with Circe's wolves in the Odyssey. The focus is the strix -witch and her ability to transform herself and other people into animal creatures, especially screech owls and wolves who stole and mutilated children. Multiple examples from Greek and Latin literature as well as Near Eastern and medieval texts illuminate other uses of magic ghosts. Ogden notes the link between werewolves, ghosts and death in the Faliscan and Etruscan worlds within a folkloric context; other comparanda are made with the vampire, whose relationship with werewolfism results from Bram Stoker's Dracula. The third chapter is dedicated to the articulation of the werewolf, studied both from an inside and outside perspective. The werewolf sports clothing, which signifies belonging to the human world, but also possesses a hairy heart, related instead to the lupine world. Ogden subsequently examines the transformation from human to wolf and vice versa, obtained by the ingestion of a certain substance, but also associated with the motifs of the wolf in the woods or across the water. The result is a perception of the werewolf as a wolf contained within a human body, juxtaposing the inner-outer selves. The following chapter resumes this dichotomy with the opposition of soul-projection and soul-flying, which means that the soul is sent off on adventures while preserving the human body in order to reanimate it. Ogden hypothesizes that Greek shamans and the Pythagorean tradition ascribed the soul-projection to werewolfism. In the fifth chapter Ogden takes up the connection between werewolves and ghosts again, focusing on the Hero of Temesa, who, according to Pausanias, was defeated by the boxer Euthymus. Ogden identifies other literary and iconographic similarities with Pausanias' two accounts of the Hero, conceptualized as a ghost. The last chapter focuses on the myths and the rites linked to the Lykaian festival, celebrated in Arcadia. Ogden begins with the aetiological myths about Lykaon and explores two separate traditions related to the festival: the historical Anthid rite and the tale of Damarchus transformed into a wolf. Ogden argues that the story of Damarchus predates the Anthid rite, which thus derived metaphorically from an ancient folkloric tradition. This chapter is particularly [End Page 481] effective in elucidating the varying traditions of werewolfism. Ogden concludes by trying to answer the question, "How did the people of the Ancient World know what a werewolf was?" The book contains three appendices which cover Circe's role as a witch, Cynocephali, and two cases of false werewolves. In conclusion, Ogden offers a thorough discussion of werewolfism in antiquity, covering as well sources since the Middle Ages. He integrates literary sources with folkloric material to illustrate the influence of the latter, and in so doing provides a rich study of werewolves, both their perception and presentation. Giulia Freni University of Siena Copyright © 2021 The Classical Association of the Atlantic States, Inc.

Using Spiritual Intelligence as a Framework to Link Electroacoustic Music and Spirituality

There are a number of electroacoustic (EA) composers who draw on spiritual or religious material for their work. However, for some the word ‘spiritual’ is problematic, being too vague or broad, and the word ‘religious’ too narrow. To refine the word ‘spiritual’ therefore, in a way that might be acceptable also to those who are religious, I employ the contemporary concept of spiritual intelligence (SI) as a proposed framework for linking a discourse of spirituality with electroacoustic music. In addition to postulating SI as a way of framing the word ‘spirituality’, I identify three different ways in which EA composers draw on spiritual or religious material: some use existing sacred texts or methods; some approach sound itself to explore the nature of reality; finally, there are those who are drawn to ancient ways of understanding where spirit and material worlds are not separated; for example, in the West, the Pythagorean tradition.

TEOREMA DE PITÁGORAS: UMA INTERVENÇÃO PEDAGÓGICA EM UMA ESCOLA PÚBLICA DO MUNICÍPIO DE URUÇUÍ-PI

Conforme a tradição pitagórica o número é o princípio de tudo, pois foi pelo estudo da aritmética e da geometria que os pitagóricos começaram a averiguar que determinados fenômenos poderiam ser comprovados através de termos numéricos ou quantitativos. Como exemplo desses fenômenos tem-se os ciclos da natureza, que se perpetuam, seja através da beleza numérica do girassol representado pela sequência de Fibonacci, ou os grandes padrões matemáticos que a própria natureza possibilita o ser humano presenciar constantemente. Nesse sentido, o teorema de Pitágoras atribui-se mediante seu valor significativo no ensino básico da matemática, além de contribuir em diversas situações-problema no cotidiano do ser humano. Este estudo teve como objetivo principal analisar como os estudantes do 9ª ano do Ensino Fundamental construíram a perspectiva do que seja o Teorema de Pitágoras de forma que atribuíssem significados ao objeto matemático aprendido. A pesquisa foi realizada em novembro de 2018 na Unidade Escolar Arica Leal, localizada no município de Uruçuí-PI. Em um primeiro momento, aplicou-se pré-teste para analisar o nível de aprendizagem dos discentes acerca do Teorema. Em seguida, foram realizadas atividades lúdicas para que os mesmos pudessem confirmar a aplicabilidade e a veracidade por meio de material concreto (perigal, cordas, fita métrica e pequenos pedaços colorido de madeira). Além disso, realizou-se a corrida pitagórica com o uso do aplicativo (Dice 3D Free) disponível na Play Store. E, por fim, executou-se o pós-teste, juntamente com coleta de dados para averiguar o nível de conhecimento adquirido ao longo das atividades realizadas. Os resultados mostraram entusiasmo e evolução dos alunos em participarem ativamente do processo de construção do conhecimento com o uso de metodologias dinâmicas que favorecem a aprendizagem. Dessa forma, o ensino de matemática com ajuda de metodologias, demonstrações, dentre outras que envolva a prática, torna a aprendizagem mais significativa e atraente.

Tetractys

Abstract Tetractys was a Greek technical term, specific to the Pythagorean tradition, that the late antique neoplatonist philosophers considered to be the central notion of Pythagorean metaphysics. As the Greek philosophical heritage started to be translated into Arabic, this term also made its way into the new language, resulting in a number of different translations. This paper explores the ways in which the term tetractys was expressed in Arabic and then explained to the medieval islamicate readers. By comparing the ways in which specific authors and intellectual circles coped with a technical term that no longer assumed the philosophical significance it had had in Late Antiquity, I show the vagaries of the Greco-Arabic translation movement. The changing renderings and understandings of this term offers a great opportunity to understand the different factors that influenced the course of translating the Greek heritage into the Arabic world.

ВІЗІЯ ЖІНОК У КОНТЕКСТІ ПОЯВИ І ФУНКЦІОНУВАННЯ ПІФАГОРІЙСЬКОЇ ШКОЛИ

The article, based on classical ancient sources and the most modern fundamental scientific works, explores and reveals the specifics of the existence of women in the early Greek philosophical school, founded by Pythagoras. There are three key reasons that led to the full existence of women along with men in the Pythagorean tradition: socio-political, doctrinal and vital. Accordingly, such a high status of women for the Pythagoreans is due not only to certain socio-political conditions of Greater Greece, or rather even the city of Croton, but also to the fact that the founder of this school, Pythagoras, unlike other early Greek thinkers, received philosophical knowledge from a woman, the Delphic Pythia Themistoclea. The doctrinal reason has been shown to be that Pythagoras himself preached the equality of women. If he had preached the opposite of the socio-political role of women in a given geographical region, he would not only not have been accepted, but even expelled. It is substantiated that the presence of women in this philosophical tradition is not a late Hellenistic fiction, but a real fact, based on the available sources of Aristotle's disciples – Dicaearchus and Aristoxenus, which testify to a rather ramified system of attracting women to the functioning of Pythagoreanism The proposed author's dichotomy of the representatives of this philosophical school known to us: pseudo-epigraphic and agraphic. Accordingly, the agraphic ones are known to us only by mentioning in certain ancient sources, while we know about the pseudo-epigraphic ones not only the names, but also preserved works and letters that are attributed to the Pythagorean thinkers.

The theory of synesthesia according to the Pythagorean tradition and Nabokov’s revisiting of Pythagorean synesthesia

Abstract In Ancient times, synesthesia was a form of perception sought after, as developed both by Pythagoras and by Aristotle. It was a degree of perception sought after for the perception of the divine. It was part of a definite aesthetics because art was supposed to permit access to synesthesia through very precise rules defined by Aristotle in his Rhetoric. Synesthesia was not an anomalous form of perception experienced by some writers only. It was supposed to be induced by certain masterpieces, thus connecting the reader’s experience of synesthesia with the writer’s. The hypothesis of the present paper is that Nabokov knew those theories and that his knowledge of Ancient sources was not limited to Plato whom he quotes repeatedly in his interviews, but also comprised Aristotle. Not only did Nabokov know about that theory, some interviews, and some of his novels reveal a game with those sources and a quest for the reader’s synesthesia. The present article focuses on the two first pages of The Gift as a skillful implementation of Aristotle’s theory on synesthesia.

Xenocrates on Plato, Pythagoras and the Poets

This paper concerns three chief aspects of Xenocrates’ exegetical activity as head of the Platonic Academy, his interpretation of certain key passages of Plato, his appropriation of Pythagoras and the Pythagorean tradition, and his exegesis of the poets, notably Homer, Hesiod and the Orphic poems, thus setting the stage for later developments in Platonism.

Formuła soma-sema w Księdze Tomasza Atlety i jej aspekty ontyczno-etyczne

The first part of the article shows the genesis of the soma-sema formula and its function in the Orphic and Pythagorean tradition, as well as in Plato. A particular attention was paid to its ontic background (dualism) and its ethical aspect: a neces­sity to undergo a punishment in the subsequent birth cycles (metempsychosis) and cleanse the soul of guilt. The second part presents a reception of the formula in the Gnostic Book of Thomas the Contender, indicating similarities and differences.

Physis in the Pythagorean Tradition

Physis in the Pythagorean Tradition