Pythagoras

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)

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)

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)

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)

Mathematician. Philosopher. World traveler. Pythagoras was an intelligent and curious scholar and teacher. While he¿s best-known for the Pythagorean theorem, he shared ideas about numbers, animals, and many other areas of knowledge with his students. Since none of his writings were left behind, it¿s not always easy for historians to know what¿s true about Pythagoras and what may be legendary. What does seem apparent is that he was a vegetarian but not a trendy dresser. Some people saw him as godlike. (...) Others felt he made false claims about things. No matter what, Pythagoras¿s curiosity and willingness to grapple with complex issues have helped further the knowledge of mathematics and philosophy for thousands of years. (shrink)

Quel est le secret de Pythagore? On pourrait avancer que, par défi nition, s'il y a secret, il est caché et n’est pas dévoilable, ou ne sera pas dévoilé. Le vrai secret est celui dont on ne soupçonne même pas l’existence. On peut toutefois approcher tangentiellement le coeur du pythagorisme à partir d’un idéal qui a traversé les âges.

Pythagoras von Samos (ca. 570-510 v.Chr.) war in den antiken Uberlieferungen nicht nur als Naturphilosoph, sondern auch als religioser Experte und Wundertater bekannt. Diesen Wundererzahlungen rund um Pythagoras widmet sich Stephan Scharinger in der vorliegenden Studie. Basis seiner Uberlegungen sind Analyse und Auswertung des einschlagigen Quellenmaterials, beginnend mit den altesten schriftlichen Quellen, die in die Lebenszeit des Pythagoras zuruckreichen, bis hin zu den Pythagorasviten der romischen Kaiserzeit. Darauf aufbauend und mithilfe der historischen Kontextualisierung des Pythagoras (Pythagoreismus und Orphik, Pythagoras und (...) die "Sieben Weisen", Schamanismus) klassifiziert Scharinger die pythagoreischen Wundertaten in Abhangigkeit von der griechisch-romischen Umwelt (ethnische Religionen, altindische und alttestamentarisch-fruhkirchliche Vorstellungen). Die Wundertraditionen in den pythagoreischen Uberlieferungen werden schliesslich in einem zweifachen Vergleich beleuchtet. Der "interne" Vergleich zeigt Parallelen und Unterschiede zwischen den Wunderberichten bei Diogenes Laertios, Porphyrios von Tyros und Iamblichos von Chalkis auf, wahrend der "externe" Vergleich die pythagoreischen Wundertaten mit den Berichten rund um andere Wundertater der antiken Mittelmeerwelt in einen grosseren Zusammenhang stellt. Uberlegungen zum religionshistorischen Umfeld, zur Diskussion um Pythagoras als theios aner sowie zur Klassifikation der Wundertaten als "magische" oder "religiose" Wunder runden die Arbeit ab. (shrink)

Growing up in Ionia -- Travels far and wide -- Settling in Croton -- Pythagorean beliefs -- A lasting legacy.

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

The 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)

The subject of this book, as per the title, is music (from the Greek μουσική τέχνη - musiké téchne, i.e. "the art of the muses") in the works of Pythagoras, the first intellectual in the history of Philosophy to make that art the center of his worldview, even including it in the so called "Exact Sciences". In order to have better grasp and clarity regarding the intellectual production of Pythagoras, whose written works survived to the 21st century in a very (...) fragmented form and inmuch smaller number, it is necessary to research the historical sources contemporary to the philosopher, as well as those which succeeded him. (shrink)