This is the first book to analyze systematically crucial aspects of ancient Greek philosophy in their original context of mystery, religion, and magic. The author brings to light recently uncovered evidence about ancient Pythagoreanism and its influence on Plato, and reconstructs the fascinating esoteric transmission of Pythagorean ideas from the Greek West down to the alchemists and magicians of Egypt, and from there into the world of Islam.
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.
Archytas of Tarentum was a central figure in fourth-century Greek life and thought and the last great philosopher in the early Pythagorean tradition. He solved a famous mathematical puzzle, saved Plato from the tyrant of Syracuse, led a powerful Greek city state, and was the subject of three books by Aristotle. This first extensive study of Archytas' work in any language presents a radically new interpretation of his significance for fourth-century Greek thought and his relationship to Plato, as well (...) as a full commentary on all the fragments and testimonia. (shrink)
: 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)
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 ...
This book investigates the link Kant discerned between our experience of beauty and our experience of the moral law. By examining Kant's relation to Greek philosophy, to Plato and Pythagoras, as found in Kant's own writings, the author sheds new light on one the most intriguing and mysterious doctrines of Kant's third Critique.
The Pythagorean idea that numbers are the key to understanding reality inspired philosophers in late Antiquity (4th and 5th centuries A.D.) to develop theories in physics and metaphysics based on mathematical models. This book draws on some newly discovered evidence, including fragments of Iamblichus's On Pythagoreanism, to examine these early theories and trace their influence on later Neoplatonists (particularly Proclus and Syrianus) and on medieval and early modern philosophy.
The Quine/Putnam indispensability argument is regarded by many as the chief argument for the existence of platonic objects. We argue that this argument cannot establish what its proponents intend. The form of our argument is simple. Suppose indispensability to science is the only good reason for believing in the existence of platonic objects. Either the dispensability of mathematical objects to science can be demonstrated and, hence, there is no good reason for believing in the existence of platonic objects, or their (...) dispensability cannot be demonstrated and, hence, there is no good reason for believing in the existence of mathematical objects which are genuinely platonic. Therefore, indispensability, whether true or false, does not support platonism. Mathematical platonists claim that at least some of the objects which are the subject matter of pure mathematics (e.g. numbers, sets, groups) actually exist. Furthermore, they claim that these objects differ radically from the concrete objects (trees, cats, stars, molecules) which inhabit the material world. We take the standard platonistic position to include the claim that platonic objects lack spatio-temporal location and causal powers. Many (perhaps most) mathematical platonists subscribe to this view.1 But some who call themselves (or might be called) mathematical platonists.. (shrink)
Throughout its history the Game was closely allied with music, and usually proceeded according to musical or mathematical rules. One theme, two themes, or three themes were stated, elaborated, varied, and underwent a development quite similar to that of the theme in a Bach fugue or a concerto movement.… Experts and Masters of the Game freely wove the initial theme into unlimited combinations [p. 30].
This is the first comprehensive study for nearly 200 years of what remains of the writings of the Presocratic philosopher Philolaus of Croton (470-390 B.C.). Professor Huffman presents the fragments and testimonia with accompanying translations and introductory chapters and interpretive commentary. He produces further arguments for the authenticity of much that used to be neglected, and 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.
Two conflicting tendencies may be discerned in Pythagorean ethics as applied to the environment: on the one hand, a sense of reverence for nature and kinship with all life that opposed killing and other forms of interference in the natural world, and on the other hand, a doctrine of the separability of soul and body which denigrates the body and the external world of which it is apart. The prescriptive content of Pythagorean ethics includes prohibitions against taking life, (...) even in sacrifices to the gods, and against eating anything that has been killed. Pollution of certain kinds is forbidden. These strictures were based on an organic, cyclical view of the world, emphasizing its harmony and balance. The Pythagoreans investigated some questions that would today be called ecological. Perhaps most importantly, they evinced a genuine respect for living things, deriving in part from the belief that animals and plants contain the reborn souls of human beings. These doctrines may have been derived from the attitudes and practices of ancestral hunters and gatherers in southeast Europe, with traditional Greek religion serving as the means of transmission from tribaI cultures to c1assical philosophy. The followers of Pythagoras split into two schools: a “scientific” school that neglected biology and therefore ecology, and a “religious” school that emphasized purity of soul and rejected any concern with physical nature. The more “environmentalist”teachings were gradually abandoned as the Pythagoreans accommodated themselves to the general attitudes of Greco-Roman culture. For instance, the objections to animal sacrifice, and to most plants as food, were dropped. The divorce of body and soul in later Pythagorean thought, wherever its influence was strong, brought with it indifference not only to the body, but to all the rest of the natural environment. (shrink)
En el mundo occidental, la primera figura que encarna el arquetipo del mediador sapiencial entre la comunidad humana y lo divino es, sin duda, Pitágoras de Samos. Las implicaciones de las doctrinas de este chamán en la historia de las ideas son enormes, pues sus invenciones abarcan todos los campos del saber: matemáticas, astronomía, filosofía, retórica, política, adivinación, medicina y religión. Nada escapa a este sabio griego, al que se atribuye un famoso teorema matemático, las escalas musicales y la idea (...) de la inmortalidad del alma. La primera parte del libro se ocupa de estudiar a Pitágoras como figura carismática y legendaria, la colección de sus enseñanzas, sus aspectos mánticos y políticos y, finalmente, la tradición pitagórica entre la realidad y la falsificación. En la segunda parte se presenta por primera vez, en una nueva traducción anotada, una recopilación de todas las biografías del filósofo: las escritas por Porfirio de Tiro, Jámblico de Calcis y Diógenes Laercio, y, como novedad, la más antigua que se conserva, redactada por el historiador griego Diodoro de Sicilia (s. I a.C.), y la del patriarca Focio de Constantinopla (s. IX). Todo ello, junto a la colección de máximas pitagóricas de origen tardío, llamada Versos de oro, así como el epítome de la enciclopedia bizantina Suda (s. X), forma el presente corpus biográfico-doctrinal de Pitágoras, que era una labor pendiente en el panorama bibliográfico español. David Hernández de la Fuente (Madrid, 1974) es escritor y profesor universitario, especializado en religión griega, antigüedad tardía e historia del platonismo. Doctor en filología clásica y sociología, es autor de los ensayos Oráculos griegos (Alianza) y Bakkhos Anax (CSIC), así como de numerosos artículos en revistas académicas y ediciones de autores clásicos, y ha coordinado la obra colectiva New Perspectives on Late Antiquity (Cambridge Scholars Pub.). Como autor de narrativa ha publicado Las puertas del sueño (Premio de Arte Joven 2005 de la Comunidad de Madrid), Continental (2007) y A cubierto (Premio Diputación de Valencia 2010). Memoria mundi 59 Isbn: 978-84-938466-6-4 440 páginas. (shrink)
Sobre el libro: Con el siglo XX ha finalizado una época de la historia. Las dos guerras mundiales acentúan las tensiones que abren paso a un orden que sucumbe en 1989 con la caída del muro de Berlín. El desarrollo de la técnica y la aparición de la televisión anulan la distancia entre sujeto y objeto característica del pensamiento moderno. Las nuevas tecnologías se convierten en plataforma de importantes cambios sociales cuando comienza un nuevo milenio. Los analistas de tendencias descubren (...) el nacimiento de una nueva sensibilidad que busca pensar de otro modo al hombre, el mundo y la cultura. Romano Guardini anticipó algunos de los problemas que recibía el siglo XX y que debían encontrar una nueva orientación. En 1918 afirma la primacía del Logos sobre el Ethos subrayando cómo el respeto al ser, a la verdad y al sentido de la realidad protegen al hombre. Desde una profunda confianza en la capacidad humana de conocer, ofrece como respuesta a los problemas que plantea la tradición intelectual que le precede un modo de contemplar al hombre desde la revelación que puede ayudar a pensar el curso de la cultura contemporánea. Sobre la autora: Mónica Codina es profesora de la Facultad de Comunicación de la Universidad de Navarra. Entre sus publicaciones hay que destacar El sigilo de la memoria. Tradición y nihilismo en la narrativa de Dostoyevski (Pamplona, 1997), una visión de la antropología subyacente a la narrativa de Dostoyevski; De la ética desprotegida. Ensayos sobre deontología de la comunicación (Pamplona, 2001), sobre la dimensión ética de las profesiones ligadas a la comunicación; y Jornalism for Integration. The Muhammad Cartoons, en colaboración con Jordi Rodríguez-Virgili, publicado en Javnost-The Public, vol. 14 (2007), nº 2, donde se discute el concepto europeo de libertad de expresión. (shrink)
losopher David Hume wrote: ‘Beauty is no quality in things themselves: It exists merely in the mind which contemplates them.’ Some people find this claim shocking and absurd, while others think that it is obviously true. I want to consider how it should be interpreted, and whether it is plausible. But I shall begin by examining another view about beauty, which Hume deliberately rejected when he wrote these words. It is attributed by tradition to the mathematician and philosopher Pythagoras, who (...) lived in the second half of the sixth century BC, and it has influenced artists, poets and philosophers ever since. The Pythagorean view is that beauty consists in mathematical perfection. (shrink)
In the present volume Proclus describes the 'creation' of the soul that animates the entire universe. This is not a literal creation, for Proclus argues that Plato means only to convey the eternal dependence of the World Soul upon higher causes. In his exegesis of Plato's text, Proclus addresses a range of issues in Pythagorean harmonic theory, as well as questions about the way in which the World Soul knows both forms and the visible reality that comprises its body. (...) This part of Proclus' Commentary is particularly responsive to the interpretive tradition that precedes it. As a result, this volume is especially significant for the study of the Platonic tradition from the earliest commentators onwards. (shrink)
Contents: Preface; From faith to reason for fideism: Raymond Lull, Raimundus Sabundus and Michel de Montaigne; Nicholas of Cusa and Pythagorean theology; Giordano Bruno's philosophy of religion; Coluccio Salutati: hermeneutics of humanity; Humanism applied to language, logic and religion: Lorenzo Valla; Georgios Gemistos Plethon: from paganism to Christianity and back; Marsilio Ficino's philosophical theology; Giovanni Pico against popular Platonism; Tommaso Campanella: God makes sense in the world; Francisco Suárez – scholastic and Platonic ideas of God; Epilogue: conflicting truth claims; (...) Bibliography; Index. (shrink)
This anthology looks at the early sages of Western philosophy and science who paved the way for Plato and Aristotle and their successors. Democritus's atomic theory of matter, Zeno's dazzling "proofs" that motion is impossible, Pythagorean insights into mathematics, Heraclitus's haunting and enigmatic epigrams-all form part of a revolution in human thought that relied on reasoning, forged the first scientific vocabulary, and laid the foundations of Western philosophy. Jonathan Barnes has painstakingly brought together the surviving Presocratic fragments in their (...) original contexts, utilizing the latest research and a major new papyrus of Empedocles. Translated and edited by Jonathan Barnes. (shrink)
This article offers a study of the early formation and development of the ideal of harmony in ancient Chinese philosophy and ancient Greek philosophy. It shows that, unlike the Pythagorean notion of harmony, which is primarily based on a linear progressive model with a pre-set order, the ancient Chinese concept of harmony is best understood as a comprehensive process of harmonization. It encompasses spatial as well as temporal dimensions, metaphysical as well as moral and aesthetical dimensions. It is a (...) fundamentally open notion in the sense that it does not aim to conform to any pre-set order. This broader, richer, and more liberal understanding of harmony has had a profound influence on Chinese culture as whole in its long history. (shrink)
In the Treatise, David Hume denies the thesis that extension is infinitely divisible, even though it can be derived as a theorem of Euclidean geometry. This clearly shows that he rejects some of the theorems of Euclidean geometry. What is less clear is the extent to which he thinks geometry needs to be revised. It has been argued that Hume's rejection of infinite divisibility entails that most of the familiar theorems of Euclidean geometry, including the Pythagorean theorem and the (...) bisection theorem, are false, a view that is normally associated with Berkeley's earlier writings.I argue that Hume's denial of infinite divisibility is not incompatible with the Pythagorean theorem and other central theorems of .. (shrink)
“The problem of universals” in general is a historically variable bundle of several closely related, yet in different conceptual frameworks rather differently articulated metaphysical, logical, and epistemological questions, ultimately all connected to the issue of how universal cognition of singular things is possible. How do we know, for example, that the Pythagorean theorem holds universally, for all possible right triangles? Indeed, how can we have any awareness of a potential infinity of all possible right triangles, given that we could (...) only see a finite number of actual ones? How can we universally indicate all possible right triangles with the phrase ‘right triangle’? Is there something common to them all signified by this phrase? If so, what is it, and how is it related to the particular right triangles? The medieval problem of universals is a logical, and historical, continuation of the ancient problem generated by Plato's (428-348 B.C.) theory answering such a bundle of questions, namely, his theory of Ideas or Forms. (shrink)
This paper argues that there were women involved with philosophy on a fairly constant basis throughout Greek antiquity. It does so by tracing the lives and where extant the writings of these women. However, since the sources, both ancient and modern, from which we derive our knowledge about these women are so sexist and easily distort our view of these women and their accomplishments, the paper also discusses the manner in which their histories come down to us as well as (...) the histories themselves. It discusses in detail the following women: the Pythagorean women philosophers of the 6th and 5th centuries B.C., Aspasia and Diotima of the 5th century B.C., Arete, Hipparchia, Pamphile and the women Epicureans-all from the 4th century B.C. the five logician daughters of a famous Stoic philosopher of the 3rd century B.C., and finally Hypatia who lived in the 4th century A.D. (shrink)
The interlink between myth and wisdom in Hellenic heritage is characteristically embodied in the Platonic philosophizing as regards the education and enculturation of the human psyche. As is read in the end of The Republic , the myth of Er turns out to be a philosophical rewriting of poetry to a large degree. For it engagingly reveals Plato’s moral inculcation, philosophical instruction and poetic wisdom in particular, all of which are intended to guide human conduct along the right track for (...) the bliss of the postmortem cycle, and put philosophy learning into first priority for the choice of the future life. Moreover, the transmigrate experience in the mystic overtone of “the Orphic-Pythagorean conglomerate” is discussed with a intercultural reference to the Buddhist doctrines of samsara and karma. (shrink)
A solution of the Zeno paradoxes in terms of a discrete space is usually rejected on the basis of an argument formulated by Hermann Weyl, the so-called tile argument. This note shows that, given a set of reasonable assumptions for a discrete geometry, the Weyl argument does not apply. The crucial step is to stress the importance of the nonzero width of a line. The Pythagorean theorem is shown to hold for arbitrary right triangles.
Simone Weil is widely recognized today as one of the profound religious thinkers of the twentieth century. Yet while her interpretation of natural science is critical to Weil's overall understanding of religious faith, her writings on science have received little attention compared with her more overtly theological writings. The present essay, which builds on Vance Morgan's Weaving the World: Simone Weil on Science, Necessity, and Love (2005), critically examines Weil's interpretation of the history of science. Weil believed that mathematical science, (...) for the ancient Pythagoreans a mystical expression of the love of God, had in the modern period degenerated into a kind of reification of method that confuses the means of representing nature with nature itself. Beginning with classical (Newtonian) science's representation of nature as a machine, and even more so with the subsequent assimilation of symbolic algebra as the principal language of mathematical physics, modern science according to Weil trades genuine insight into the order of the world for symbolic manipulation yielding mere predictive success and technological domination of nature. I show that Weil's expressed desire to revive a Pythagorean scientific approach, inspired by the "mysterious complicity" in nature between brute necessity and love, must be recast in view of the intrinsically symbolic character of modern mathematical science. I argue further that a genuinely mystical attitude toward nature is nascent within symbolic mathematical science itself. (shrink)
The origin of the Neoplatonist doctrine of the henads has been imputed to Iamblichus, mostly on indirect evidence found in later Neoplatonists, chiefly Proclus. Is there any trace of this concept to be found in the extant works or fragments of Iamblichus himself? The best candidates among his surviving texts are the excerpts in Psellus of his volume on Theological Arithmetic from his Pythagorean series, and the first book of de Mysteriis , where Iamblichus answers Porphyry's questions on the (...) nature of the gods. Such evidence as can be found there would most likely deal with the divine henads, given the subject matter of the text. Certain repeated items of vocabulary appear as technical usages that form the basis for arguing that Iamblichus already has in mind if not the explicit concept henad at least its functional equivalent: the term monoeides occurring in both the Psellan excerpts and de Mysteriis, and in the latter, mostly in Book I, the stated attributes of a high, divine principle uniting the gods which are also designated by Proclus as typical of the divine henads, particularly in the propositions of the Elements of Theology defining the henads. Iamblichus in Book I also ascribes to the gods the same role in the process of ellampsis as Proclus does for the divine henads. A theory is also advanced concerning the possible development of the concept of the henad by Iamblichus, based in part on the polemical nature of de Mysteriis and his relationship to Porphyry. (shrink)
Western society has been diverted from the goal of spiritual freedom and autonomy as expressed in the ancient Pythagorean 'theory of the cosmos'. Indeed, following Heidegger's analysis, it can be seen that modern Western society has arrived at the opposite pole of anthropocentric 'absolute subjectivism' in which the entire non-human world is seen as a material resource to be consumed in the satisfaction of our egoistic passive desires. It is further argued that Spinozism is actually a modern version of (...) the 'theory of the cosmos' which, when supplemented by a vision of man's identity with the ecological world, would provide us with the only adequate portrayal of the God/Nature/Man relationship. (shrink)
Introduction: The poetic topos of the doctrine of transmigration -- Genealogy of the doctrine of transmigration -- Beyond mysticism and science : symbolism and philosophical magic -- The emergence of mystic cults and the immortal soul -- Philolaus and the question of pythagorean harmony -- The alleged critique of Pythagoras by Parmenides -- Between the earth and the sky : on the pythagorean divine -- The pythagorean bios and the doctrine of transmigration -- The path of the (...) event -- The path of remembrance or return -- The platonic rupture : writing and difference -- Plotinus : the ascent of the soul toward the one -- Plotinus as neoplatonic mystic : letter to Flaccus -- Epilogue: The fate of the doctrine of transmigration. (shrink)
We review different studies of the Periodic Law and the set of chemical elements from a mathematical point of view. This discussion covers the first attempts made in the 19th century up to the present day. Mathematics employed to study the periodic system includes number theory, information theory, order theory, set theory and topology. Each theory used shows that it is possible to provide the Periodic Law with a mathematical structure. We also show that it is possible to study the (...) chemical elements taking advantage of their phenomenological properties, and that it is not always necessary to reduce the concept of chemical elements to the quantum atomic concept to be able to find interpretations for the Periodic Law. Finally, a connection is noted between the lengths of the periods of the Periodic Law and the philosophical Pythagorean doctrine. (shrink)
A Christian approach to scholarship, directed by the central biblical motive of creation, fall and redemption and guided by the theoretical idea that God subjected all of creation to His Law-Word, delimiting and determining the cohering diversity we experience within reality, in principle safe-guards those in the grip of this ultimate commitment and theoretical orientation from absolutizing or deifying anything within creation. In this article my over-all approach is focused on the one-sided legacy of mathematics, starting with Pythagorean arithmeticism (...) (“everything is number”), continuing with the geometrization of mathematics after the discovery of irrational numbers and once again, during the nineteenth century returning to an arithmeticistic position. The third option, never explored during the history of mathematics, guides our analysis: instead of reducing space to number or number to space it is argued that both the uniqueness of these two aspects and their mutual coherence ought to direct mathematics. The presence of different schools of thought is highlighted and then the argument proceeds by distinguishing numerical and spatial facts, while accounting for the strict correlation of operations on the law side of the numerical aspect and their correlated numerical subjects (numbers). Discussing the examples of 2 + 2 = 4 and the definition of a straight line as the shortest distance between two points provide the background for a brief sketch of the third alternative proposed (inter alia against the background of an assessment of infinity and continuity and the vicious circles present in contemporary mathematical arithmeticistic claims). (shrink)
The individual soul is an ageless idea, attested in prehistoric times by the oral traditions of all cultures. But as far as we know, it enters history in ancient Egypt. I will begin with the individual soul in ancient Egypt, then recount the birth of the world soul in the Pythagorean community of ancient Greece, and trace it through the Western Esoteric Tradition until its demise in Kepler's writings, along with the rise of modern science, around 1600 CE. Then (...) I tell of the rebirth of the world soul recently, in new branches of mathematics. (shrink)
This essay explores the relevance of Socrates’ mythical introduction of recollection in the Meno. I argue that the passage at 81a5–e2 addresses different levels of understanding, a superficial and a deeper one, corresponding to a literal and a metaphorical reading respectively. The major themes addressed in this passage—the immortality of the soul, transmigration, rewards and punishments in the after-life, Hades, the kinship of all nature and anamnesis—have distinct meanings depending on whether we approach them with a Platonic or an Orphico- (...) class='Hi'>Pythagorean eye. The literal understanding is appealing to Meno and is offered in reply to his challenge in order to persuade him to continue the investigation of virtue. It is, however, the deeper sense that Plato’s Socrates intends for a more philosophically attuned audience. (shrink)