Pythagoreans, Misc

Mathematics appears to play an explanatory role in science. This, in turn, is thought to pave a way toward mathematical Platonism. A central challenge for mathematical Platonists, however, is to provide an account of how mathematical explanations work. I propose a property-based account: physical systems possess mathematical properties, which either guarantee the presence of other mathematical properties and, by extension, the physical states that possess them; or rule out other mathematical properties, and their associated physical states. I explain why Platonists (...) should accept that physical systems have mathematical properties, and why a property based account is better than existing accounts of mathematical explanation. I close by considering whether nominalists can accept the view I propose here. I argue that they cannot. (shrink)

O problema deste paper constitui-se a partir da seguinte pergunta: seria Platão um plagiador dos pitagóricos? A suspeita nasce de uma tradição já antiga pela qual Platão é acusado de ter plagiado uma suposta obra atualmente perdida de Filolau de Crotona. Tomou-se, então, como objeto de análise, a obra Timeu, a fim de verificar a existência de trechos que poderiam sugerir um plágio por parte de Platão. Para tanto, realizou-se um levantamento das principais teorias “pitagóricas” identificáveis no Timeu, a partir (...) de uma revisão da história da crítica. Concluiu-se que, inegavelmente, há referência implícita ao pitagorismo na obra, a qual, entretanto, não configura propriamente um plágio, mas, sim, pode ser melhor considerada como uma influência pitagórica. Efetivamente, Platão acaba por transpor diversas teorias “pitagóricas” – bem como de outros filósofos pré-socráticos – para seus próprios interesses filosóficos. (shrink)

The essay explores the Pythagorean explanation of the origin of the Sun's light and warmth, particularly in the fragments of Philolaus. His cosmological notion of the Sun centers around the idea that the Sun is merely a medium between the Earth and the primal source of warmth and light.

The reinterpretation of Plato’s philosophy in terms of periodic anthyphairesis, in fact of palindromically periodic anthyphairesis in the Politicus, and the reading of Book X of Euclid’s Elements under the new light, reveal deep mathematical contributions by Theaetetus, including a proof of the general Pell equation. Fascinating similarities of Theaeteus’ reconstructed proofs with the Hindus’ solution of the problem of Pell are noted.

This chapter introduces the concept of radical Pythagorean mathematics, as part of mathematical modernism, which emerged around 1900 and extends to our own time. Arguably, the greatest discovery of Pythagorean mathematics of (sixth-fifth century BC) was that of the incommensurability of the side and the diagonal of the square. This discovery established the complex relationships between geometry and arithmetic and, correlatively, the possibility of a type of logos, a logos containing the alogon within it – a logical thought containing the (...) unthinkable within it. I shall call the mathematics defined by this combination “Pythagorean mathematics.” I shall argue that, as part of mathematical modernism, there emerged a new attitude toward the unthinkable in thought, an attitude that accepts this situation as a positive, enabling condition of thinking and knowledge. I shall call the mathematical thinking defined by this attitude, while keeping the irreducible relationship between geometry and algebra (which subsumes arithmetic within it), radical Pythagorean mathematics. The original Pythagorean attitude, at least in one reconstitution of Pythagorean thought, was, as against the Platonist attitude, tolerant of this condition, but still sought to overcome it by finding a logos without the alogon within it. This attitude, often sliding into a Platonist one, has remained dominant throughout the history of mathematics, physics, and philosophy, including during the modernist period, notable as much for the presence of the radical Pythagorean thinking as for a resistance to it. (shrink)

La explicación tradicional del concepto de "vacío" (τò κενóν) en la filosofía antigua lo sitúa como una invención del atomismo de Demócrito y Leucipo o, incluso, del eleático Meliso de Samos. De esta manera se ocultan las profundas razones que pudieron llevar a la necesidad y surgimiento de un tal concepto, y aparece como si hubiera sido creado ex nihilo. En este artículo se pretende descubrir tanto el principio mitológico como el origen racional del concepto de "vacío" en la filosofía (...) presocrática, que lo sitúa como anterior a las historias contadas hasta ahora. Mediante dos columnas que intentan aprehender lo mismo desde puntos de vista distintos, se llegará a la conclusión de que el concepto de "vacío" fue definido en el inicio como diferencia y repetición. En contra de una determinada visión de la historia de la filosofía que ve a los "clásicos" y a los "modernos" como dos compartimentos separados e incluso enfrentados, en el curso de esta interpretación se establecerá un diálogo entre la doxografía presocrática y la filosofía contemporánea de Schelling, Hegel, Nietzsche, Heidegger, Schmitt, Irigaray, Derrida o Deleuze. (shrink)

In this paper, we introduce the concepts of Pythagorean fuzzy valued neutrosophic set (PFVNS) and Pythagorean fuzzy valued neutrosophic (PFVNV) constructed by considering Pythagorean fuzzy values (PFVs) instead of numbers for the degrees of the truth, the indeterminacy and the falsity, which is a new extension of intuitionistic fuzzy valued neutrosophic set (IFVNS). By means of PFVNSs, the degrees of the truth, the indeterminacy and the falsity can be given in Pythagorean fuzzy environment and more sensitive evaluations are made by (...) a decision maker in decision making problems compared to IFVNSs. In other words, such sets enable a decision maker to evaluate the degrees of the truth, the indeterminacy and the falsity as PFVs to model the uncertainty in the evaluations. (shrink)

For ancient pre-Socratic societies the living pattern of the Universe was—according to Plato—also the blueprint for the organizational structures within a society. Divine order and equilibrium were the universal principles and the theoretical construct governing nature around us and the entire cosmos. The old Theory of Everything (ToE) as it was mathematically conceptualized by Pythagoras and visualized by Plato—in the 5 synonymous regular forms—epitomized the principle of harmony. With the recent rediscovery of Plato’s 5th element we are able to understand (...) that it was a higher dimensional world view based upon a geometrical living pattern of the world. This metaphysical superstructure was allegedly strongly linked to high ethical standards Plato discussed in gad ti an ideal society in his “Republic.” Meanwhile, according to theoretical physics, the Universe is expanding at an accelerating pace and governed by ‘dark energy’ and ‘dark matter’ where ‘black holes’ are gaining more and more significance. How does the loss of ancient superstructures and new concepts of theoretical physics affect our society, and which rites and habits are left from antiquity? (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Logos and Alogon: Thinkable and Unthinkable in Mathematics, from the Pythagoreans to the Moderns by Arkady PlotnitskyNoam CohenPLOTNITSKY, Arkady. Logos and Alogon: Thinkable and Unthinkable in Mathematics, from the Pythagoreans to the Moderns. Cham: Springer, 2023. xvi + 294 pp. Cloth, $109.99The limits of thought in its relations to reality have defined Western philosophical inquiry from its very beginnings. The shocking discovery of the incommensurables in Greek mathematics (...) not only reintroduced these limits with greater force but also instigated two different forms of coping with the unthinkable in Western philosophy, mathematics, and science. As Arkady Plotnitsky sets forth, in response to the incommensurability crisis in the ancient Greek intellectual world, there appeared two different attitudes toward the alogon, that is, the unthinkable or irrational. Plato and his followers sought to overcome the unthinkable, of which the incommensurables in mathematics are only an example, through philosophy. In contrast, the Pythagoreans continued to hold on to mathematics as the primary means of knowing the world, but put a larger emphasis on geometry. Plotnitsky [End Page 359] claims that by not putting aside the primacy of mathematics, the Pythagorean shift from arithmetic to geometry established in Western thought a certain tolerance toward the unthinkable within the realm of the thinkable, “a possibility of the alogon in the logos.” Until modern times this remained no more than a possibility. However, in modern mathematics and mathematical physics Plotnitsky identifies a certain fulfillment of this potential, a “radical Pythagorean mathematics.” In his comprehensive and thought-provoking study, Plotnitsky recounts the development of such radical modern Pythagoreanism, not only to tell its history but also to suggest that the unthinkable is a condition of possibility for thought, and that a proper recognition of this role it plays is crucial for the advancement of mathematical thinking and thinking in general.Plotnitsky begins by portraying in broad strokes the two defining characteristics of radical Pythagorean mathematics: the presence of the unthinkable within mathematical thought itself, and the interplay of algebra and geometry. Relying on an extensive knowledge of the history of modern mathematics and physics, in the first two chapters of the book Plotnitsky gradually demonstrates the key role that the alogon plays in some theories of modern mathematical thought, such as Gödel’s incompleteness theorems and quantum mechanics. He also presents and clarifies his own conception of the history of Western thought as a creative process of concept-forming. By creating rather than discovering concepts, great mathematicians break new ground for thinking. The book as a whole argues in length for these two main theses.In chapter 3, the author addresses the thought of Bernhard Riemann to make the case that Riemann’s mathematical thought puts an emphasis on creative conceptual thinking and is not defined merely by calculations or logic. Such thinking is problem-posing rather than axiomatic. Riemann’s central innovation in this respect is the concept of manifold (Mannigfaltigkeit), which enabled him to think anew space and geometry in a radical non-Euclidean fashion, implying an infinite number of possible geometries. Plotnitsky argues—borrowing Deleuze and Guattari’s terminology—that this opened up a “plane of immanence,” which is to say that Riemann created a concept-generating movement of thought. However, we cannot grasp the generation of concepts in modern mathematics without giving proper attention to the interplay of algebra and geometry that defines it. Chapter 4 demonstrates this interweaving of the two fields by recounting the history of the idea of the curve in modernity, from the analytic geometry and calculus of Fermat and Descartes to its modification in the concept of a Riemann surface, up to its more recent developments in the algebraic geometry of Weil and Grothendieck. It gives a historical account of the algebraization of the curve that at the same time brings into relief the irreducible geometrical aspect, continuity, of all modern mathematical conceptualizations of the curve.The next two chapters provide further case studies. Chapter 5 expands and solidifies the argument of chapter 2. It discusses three examples of mathematical practice, each displayed by a pair of important [End Page 360] mathematicians: Abel and Galois, Lobachevsky and Riemann, Weil and Grothendieck. Each case... (shrink)

The well-known connection between the extant fragments of Philolaus and Plato’s Philebus is examined in its methodological aspect. By drawing on more texts, it is shown that Plato was aware of an explanatory scheme that can be attributed to Pythagoreanism. His attempt to modify it is also outlined, which sets the historical-philosophical perspective.

This chapter pursues an understanding of what Cicero thought 'Italic' philosophy to be, and proceeds to develop a broader account of how Cicero's version compares with the surviving textual evidence and testimonia from the Hellenistic period of the philosophy of the 'Italic' philosophers, including the Lucanians 'Ocellus', 'Eccelus', and 'Aresas/Aesara', and the Rudian Ennius. Special focus is placed on their theories of cosmology, psychology, and law. Collocation of 'Italic' with 'Pythagorean' philosophy of this era aids in building a more comprehensive (...) view of what Hellenistic Pythagoreanism may have looked like. Uploaded here is the pre-publication version; please email me for the printed version. (shrink)

This book presents a new account of Thales based on the idea that Acheloios, a deity equated with water in the ancient Greek world and found in Miletos during Thales’ life, was the most important cultic deity influencing the thinker, profoundly shaping his philosophical worldview. In doing so, it also weighs in on the metaphysical and epistemological dichotomy that seemingly underlies all academia—the antithesis of the methodological postulate of Marxian dialectical materialism vis-à-vis the Platonic idea of fundamentally real transcendental forms. (...) Unbeknownst to many scholars, there are various Neo-Marxian thinkers that position the origin of coinage as the pivotal technological development giving rise to impersonal “metaphysical cosmology,” suggesting that the value of money was more-or-less projected back onto the cosmos in the form of “ideal substances.” While the arguments are incredibly sophisticated and persuasive, their conclusions (either stated or implied) are rather difficult to swallow: the self is merely an illusion, abstract ideas of an ultimate source of value, like God or the Good, are totally delusional (as is the soul, and presumably any notion of inherent human dignity), and essentially everything is reducible to mankind’s enslavement to commodities and the notion of our own objectified labor, which is the true source of all value according to Marx. Not only is this an alarming belief that many scholars (consciously or unconsciously) have adopted, since essentially any action could be “justified,” it is also demonstrably false, since it rests on a misunderstanding of Thales and misconception of philosophy as such. -/- My work rectifies that misunderstanding. In an important sense, it is an attempt at redefining philosophy as a “love of wisdom,” which I argue was accurate even in the Presocratic setting, and it uses the influence of Acheloios on Thales to do so. Throughout its pages I explore the etymology and historical uses of the word ὔδωρ, examine the archaeological context of 7th to 6th century Miletos, consider various aquatic myths Thales encountered, and highlight a hitherto overlooked tradition stemming from Thales and influencing such thinkers as Pythagoras, Empedokles, and Hippo, which culminates in a completely new reading of Plato’s Phaedrus, a dialogue in which Plato responds to the exact type of thinking employed by the Neo-Marxians. It is there that we find Socrates and Phaedrus surrounded by the iconography of Acheloios and the nymphs, all while they lie reclined like river gods (the sinews of Acheloios) on the banks of the Ilisos. And it is in that dialogue that Plato defines philosophy as a love of wisdom—the beholding of a multiplicity of hermeneutical frameworks—and alludes to the fact that it began with the sacrifice of Acheloios, the initial philosophical maneuver which he attributes to Thales. The book ends with a threefold rejoinder to the Neo-Marxian school, corresponding to the λόγος, μῦθος, and ἔργον of Acheloios. It turns out that, (1) the λόγος of Acheloios contained the ideal preconditions conducive to an abstraction to a more refined philosophical worldview in which divine water operated as the One among the Many; (2) the μῦθος of Acheloios actually encouraged the application of the notion of sacrifice to Acheloios himself (thus revealing his essence as divine water); and, (3), the ἔργον of Acheloios, in which he kneels in assent to sacrifice, is found on a coin that was probably designed by Thales. In the final analysis, I suggest that the tradition of Acheloios is reflective of a greater philosophical truth, and that by following Thales’ lead, we transcend the Marxian hermeneutic of doubt and reorient ourselves toward the οὐσια ὄντως οὖσα. (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)

Ancient Philosophy Today, Volume 4, Issue 2, Page 237-241, October, 2022.

For the first time, the reader can have a synoptic view of the reception of Pythagoras and Pythagoreanism in the Middle Ages and the Renaissance, East and West, in a multicultural perspective. All the major themes of Pythagoreanism are addressed, from mathematics, number philosophy and metaphysics to ethics and religious thought.

Pythagoras and Pythagoreanism have a different position in the ancient philosophy tradition. The reason for this is the eclectical structure of Pythagoreanism which has syncretized from Orphism, Indian and Egyptian religions with philosophy. Orphism of these religions is especially important for affecting Pythagoreanism the most and giving to the ancient Greek religion a mystical content. Orphism which is a mystery cult is based on Orpheus, the poet, who sometimes is identified with Pythagoras in philosophy and the history of religions. Orpheus, (...) was attributed divine character by virtue of his beautiful voice and hymns in the ancient history, has brought many religious and philosophic elements such as reincarnation and unity with God to the Greek religion. The way of life and creeds of Orphism based on asceticism and wisdom, first affected Italian philosophy starting from Pythagoras and, later influenced both philosophy and the other religions such as Judaism and Christianity through Plato. The reason for this was the Orphism’s stronger theological structure than the Greek religion had. Thus, philosophy, the Miletus philosophers tried to purify from the Homeric religion, has regained some spirituality through Orphism and Pythagoras. Pythagoras, a philosopher and a religious leader, accepted students to his philosophy school by initiation in Crotone. In this way, Pythagoreanism gained the quality of both a school of philosophy and a community of mystery. In this school, Pythagoras constituted a strong natural philosophy based on the idea of arkhe consisting of numbers, and the system of mathematics and music. At the same time, for he was a religious leader, Pythagoras used his school both to teach philosophy and to celebrate the teletaic mystery rituals of Orphism. According to the Pythagoreans, as a master, Pythagoras has united with the spirit of Dionysos and has privileged to rise to the World-Soul of God. Due to this union, it has been believed that Pythagoras received revelations from Dionysos and knew the secrets of God’s spirit. Similarly, Pythagoras’ students of philosophy also joined these rituals. They, on the one hand, witnessed to the divine union of Pythagoras, on the other hand, experienced the communication with the subordinate gods, called daemons, allied to Dionysos. Through these mystical rituals in Pythagorean school, it was believed that the philosophical knowledge was not only a reasoning, but also a divine wisdom obtained from the divine word. Therefore, Pythagoreanism accepted that philosophy is from God and it is a way of life which rises the human to the divine level. According to Pythagoras philosophy, thinking about God is equal to dealing with natural philosophy. Because Pythagoreanism, that has a pantheistic belief in God, has acknowledged God and nature as the same. Additionally, the natural philosophy of the school was established on the Monad idea which has been accepted as the World-Soul. For Pythagoreanism, Monad is not a number, but it is the ONE as the source of all numbers and a theological principle. The ONE, which symbolizes God and the cosmic unity, brings into existence the life by limiting the chaos caused by the TWO. Similarly, the TWO, not a number but a principle, has been used in a positive sense in Anaximander and was called apeiron, meant “infiniteness”. But, for Pythagoras, infiniteness was a negative circumstance. For this reason, the TWO/Apeiron has been accepted as a female being which corresponds to limitless absence and evil. By Monad’s confinement the Apeiron had been passed into the cosmos, the cosmic harmony had been balanced and a spheric being had emerged. This is “unity” which corresponds with both the World Soul Monad and the nature in the Pythagorean philosophy. This study will discuss the notion of Monad which is known as the World-Soul and God in Pythagorean philosophy, from a perspective of the history of religions. In this context, the sources and knowledge of history of philosophy will be used by methods of the history of religions. Starting from Orphism, Monad will be discussed within similar mythical elements. Thus, the origin of the Pythagorean philosophy in ancient religions will be determined and the theological infrastructure will be shown. (shrink)

The article offers academic translation into Ukrainian of a number of works by Pythagorean woman philosophers, which reveal the problems of human nature and personality education. The focus is on such pseudo-epigraphs of ancient woman thinkers as two letters by Theano of Crotone, letters of Miya of Crotone and Melissa, as well as treatises by Fintys of Sparta "On a woman prudence", Aesara of Lucania "On human nature" and excerpts from Porphyry’s "Pythagorean music" which contain fragments of the works of (...) Ptolemais of Cyrene. The main themes of the above works and letters are the education of the individual in general, and women in particular. Accordingly, the basis of education should be an element of restraint and prudence in everything. If the child is brought up on this basis, he will be able to be strong and resilient during certain life situations. In the treatises of the Pythagorean women-philosophers it is noted that through the study of our own human nature we can understand the philosophical foundations of natural law and morality. Therefore, a woman should use in her life not fleeting emotions and reactions to a particular event, situation, but also be moderate and prudent. These texts are significant in the context of understanding gender issues in the Hellenistic era. Based on the translated works, we can say that, according to thinkers, a number of virtues are common to both sexes, moderation or abstinence are more common in women. At the same time, the limitation of the social role of women reflects an understanding of the nature of the female soul. Accordingly, the normative principle of harmony must be implemented in the context of the specific social roles allowed to women. At the same time, the availability of these texts indicates that philosophy is possible for both men and women, thus emphasizing the importance of involving the latter in knowledge and scientific knowledge. (shrink)

This article endeavours to provide the context to a digression in Lacan’s Seminar viii concerning the Pythagoreans, within the Vorlage of the history of ancient medicine. This shows the philosophical and historical richness that lies behind the Lacanian corpus, while illustrating the manifold pitfalls that await the uncritical reader who may take Lacan’s text merely at face value. First, the author suggests that Lacan’s investigation into the meaning of ἔρως and desire in Plato’s Symposium, cannot be separated from Platonism as (...) it has come down to us, in the Western tradition, from Plotinus and Augustine. Second, through a careful reading of the Greek sources, the author sets forth the scholarly arguments concerning the topics which form the core of the passage of Lacan under review. Namely, (i) the hagiographical nature of Iamblichus’ De Vita Pythagorae and the VP of Porphyry upon which it was based, the teaching of the later Pythagoreans (notably Alcmaeon of Croton and Philolaus) and the references to the Pythagoreans found in Aristotle; (ii) the erroneous, nineteenth century, distinction between a Hippocratic or Coan and Cnidian school of medicine; and (iii) John Anderson indirect influence on Betrand Russell’s Wisdom of the West, especially in the light of Russell’s mistaken view that the religious and mathematical aspects of Pythagoras’ thought are a unity. -/- . (shrink)

This paper attempts to demonstrate that the conviction about the harmony and order of the world was a fundamental metaphysical principle of the Pythagoreans. This harmony and order were primarily sought in the structures of arithmetics, yet following the discovery of incommensurable magnitudes (irrational numbers, as we now call them), the Pythagoreans began to see geometrical structure as a fundamental part of the world. On the example of the Pythagoreans’ metaphysics and science, the paper shows the mutual relations between metaphysics (...) and science. It demonstrates— on the one hand—the necessity of the first as a guide for the latter, and—on the other—how our scientific research can change its basic metaphysical principles when these are found to be inappropriate. The paper also tries to show the need for a realistic approach in our scientific research by means of the same example of the Pythagoreans, that is, the need to discern something which is below the surface appearance. (shrink)

The paper considers a generalization of Peano arithmetic, Hilbert arithmetic as the basis of the world in a Pythagorean manner. Hilbert arithmetic unifies the foundations of mathematics (Peano arithmetic and set theory), foundations of physics (quantum mechanics and information), and philosophical transcendentalism (Husserl’s phenomenology) into a formal theory and mathematical structure literally following Husserl’s tracе of “philosophy as a rigorous science”. In the pathway to that objective, Hilbert arithmetic identifies by itself information related to finite sets and series and quantum (...) information referring to infinite one as both appearing in three “hypostases”: correspondingly, mathematical, physical and ontological, each of which is able to generate a relevant science and area of cognition. Scientific transcendentalism is a falsifiable counterpart of philosophical transcendentalism. The underlying concept of the totality can be interpreted accordingly also mathematically, as consistent completeness, and physically, as the universe defined not empirically or experimentally, but as that ultimate wholeness containing its externality into itself. (shrink)

To eliminate racist prejudices, it is necessary to identify the root cause of racism. American slavery preceded racism, and it was closely associated with genocide. Accordingly, we seek the unique cause of the unique event of genocide + slavery. This was initially justified by religious prejudice, rather than colour prejudice. This religious justification was weakened when many Blacks converted to Christianity, after the trans-Atlantic slave trade. The curse of Kam, using quick visual cues to characterize Blacks as inferior Christians, was (...) inadequate. Hence, the church fell back on an ancient trick of using false history as secular justification for Christian superiority. This trick had resulted in a false history of science during the Crusades when scientific knowledge in translated Arabic texts was indiscriminately attributed to the early Greeks, without evidence. This false history enabled belief in religious superiority to mutate into a secular belief in White superiority. After colonialism, and the Aryan race conjecture, the belief in White superiority further mutated into a belief in Western civilizational superiority, openly propagated today by colonial education. Hence, to eliminate racist prejudice, it is necessary to engage simultaneously with the allied prejudices about Christian/White/Western superiority, based on the same false history of science. (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)

This paper examines the relation between Pythagorean and Heraclitean political views. I argue that for Pythagoras, Heraclitus, and Archytas the cosmological and musical notions of harmony (ἁρμονία) and the related notion of concord (ὁμόνοια) have an intrinsic political significance. These thinkers variously reflect upon political harmony and concord, and agree that a crucial condition for it is law (νόμος), which according to Pythagoras and Heraclitus has a divine origin. I begin with the Heraclitean fragments 22 B51, 54, 72, and 114 (...) DK, in which social and political reflection is connected with the theory of the harmony of opposites. In the case of Pythagoras and early Pythagoreanism, the intense, albeit indirect political influence in Magna Graecia—as transmitted by Iamblichus and Porphyry—offers evidence for social and political ideas determined by a reflection on the cosmological role of harmony and number. Finally, Archytas’ political application of rational calculation in fragment 47 B3 DK, which aims at producing concord by establishing the just measures of wealth in the city, together with the testimony on Archytas’ intense and fruitful political activity, can be taken as confirmation that the thinker’s wider cosmological views were indeed intertwined with his political reflection and action. (shrink)

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. (shrink)

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 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)

The paper examines the definitions of the concept ›philosophy‹ resp. ›the philosopher‹ in Florentine renaissance Platonism, namely Marsilio Ficino and his scholar Francesco di Zanobi Cattani da Diacceto. Following Socrates and Pythagoras, Ficino distinguishes between mundane philosophy and divine sapientia. In contrast to his teacher, Diacceto's Aristotelism rejects the Pythagoreanism and connects philosophy with sapientia. In order to show how the differences between Ficino and Diacceto emerge, three more contemporaries are taken into consideration: Christoforo Landino, Angelo Poliziano and Giovanni Pico (...) della Mirandola. As a result of this examination, it can be said that the definition of ›philosophy‹ in Florentine renaissance Platonism tends either to Pythagoreism or to Aristotelism or to both. Closely connected with the debate on philosophy are topics and maxims such as meditatio mortis, theologia negativa, docta ignorantia, scio ne scio, nosce te ipsum, ne quid nimis et al. (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)

Die sogenannte ‚Hochzeitszahl’ aus Platons Staat gilt wohl zu Recht als eines der bedeutendsten, noch immer ungelösten Rätsel der Antike. In einem kunstvoll verschlüsselten Text deutet Platons Sokrates an, auf welche Weise der Idealstaat als gerechteste und glücklichste Form einer menschlichen Gemeinschaft seine politische Kontinuität erhalten kann: Es genügt nicht nur, dass die amtierende Führungsschicht von bestqualifizierten, für sich selbst aber besitzlosen Philosophinnen und Philosophen ihre potenziellen Nachfolger aufs Beste ausbildet, nein, gemäß pythagoreischer Lehre muss bereits bei deren Zeugung darauf (...) geachtet werden, dass die Paarung ihrer idealen Eltern zu einem Zeitpunkt erfolgt, der auch vom kosmischen Lebensrhythmus des Universums her geeignet ist, der künftigen Elite das optimale Rüstzeug für ihre göttliche Aufgabe schon vorgeburtlich mitzugeben. Diese Geheimformel stellt ein faszinierendes Beispiel für das ganzheitliche Denken der alten Pythagoreer dar, in welchem heilige Mathematik, naturwissenschaftliche Himmelsmechanik und die schöpferische Liebesfunktion eines lebendigen, von naturgesetzlicher Harmonie und Weisheit erfüllten Universums noch eine untrennbare Einheit bilden. The so-called “wedding number” in Plato’s Republic is rightly considered one of the most important unsolved riddles of antiquity. This secret formula is a fascinating example of the holistic thought of the ancient Pythagoreans, in which divine mathematics, scientific celestial astronomy, and the creative function of love in a living universe full of naturally ordained harmony and wisdom still form an indivisible whole. (shrink)

The purpose of the conference "On Pythagoreanism", held in Brasilia in 2011, was to bring together leading scholars from all over the world to define the status quaestionis for the ever-increasing interest and research on Pythagoreanism in the 21st century. The papers included in this volume exemplify the variety of topics and approaches now being used to understand the polyhedral image of one of the most fascinating and long-lasting intellectual phenomena in Western history. Cornelli's paper opens the volume by charting (...) the course of Pythagorean studies over the past two centuries. The remaining contributions range chronologically from Pythagoras and the early Pythagoreans of the archaic period (6th-5th centuries BCE) through the classical, hellenistic and late antique periods, to the eighteenth century. Thematically they treat the connections of Pythagoreanism with Orphism and religion, with mathematics, metaphysics and epistemology and with politics and the Pythagorean way of life. (shrink)

Music has often attempted to express mystical states of mind, cosmic harmony, the demonic and the divine. This wide-ranging survey explores how such film music works and uncovers its origins in Pythagorean and Platonic ideas about the divine order of the universe and its essentially numerical/musical nature.