Reference
- Pythagoras (c. 580–500 BC)Greek mathematician and philosopher who formulated Pythagoras' theorem.
Much of Pythagoras' work concerned numbers, to which he assigned mystical properties. For example, he classified numbers into triangular ones (1, 3, 6, 10, ...), which can be represented as a triangular array, and square ones (1, 4, 9, 16, ...), which form squares. He also observed that any two adjacent triangular numbers added together form a square number (for example, 1 + 3 = 4; 3 + 6 = 9; 6 + 10 = 16).
Pythagoras was born on the island of Samos and may have been obliged to flee the despotism of its ruler. He went on to found a school and religious brotherhood in Croton, southern Italy. Its tenets included the immortality and transmigration of souls. As a mathematical and philosophical community the Pythagorean brotherhood extended science; politically its influence reached the western Greek colonies. This caused conflict which saw Pythagoras exiled to Metapontum, on the Gulf of Taranto, until he died. The school continued for 50 or 60 years before being totally suppressed...
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Perspectives
The Pythagorean Sourcebook and Library
This anthology, the largest collection of Pythagorean writings ever to appear in English, contains the four ancient biographies of Pythagoras and over 25 Pythagorean and Neopythagorean writings from the Classical and Hellenistic periods. The material of this book is indispensable for anyone who wishes to understand the real spiritual roots of Western civilization.
Pythagoras
One of the most important mathematical theorems is named after Pythagoras of Samos, but this semi-mythical Greek sage has more to offer than formulas. He is said to have discovered the numerical nature of the basic consonances and transposed the musical proportions to the cosmos, postulating a "harmony of the spheres." He may have coined the words "cosmos" and "philosophy." He is also believed to have taught the doctrine of transmigration of souls and therefore to have advised a vegetarian diet. Ancient legends have Pythagoras conversing with dogs, bears, and bulls. A distinctly Pythagorean way of life, including detailed ritual regulations, was observed by his disciples, who were organized as a secret society. Later, Pythagorean and Platonic teachings became fused. In this Platonized form, Pythagoreanism has remained influential through medieval Christianity and the Renaissance down to the present. Christoph Riedweg's book is an engaging introduction to the fundamental contributions of Pythagoras to the establishment of European culture. To penetrate the intricate maze of lore and ascertain what history can tell us about the philosopher, Riedweg not only examines the written record but also considers Pythagoras within the cultural, intellectual, and spiritual context of his times. The result is a vivid overview of the life and teachings of a crucial Greek thinker and his most important followers.
The Math Book
Math's infinite mysteries unfold in this paperback edition of the bestselling TheMath Book. Beginning millions of years ago with ancient "ant odometers" and moving through time to our modern-day quest for new dimensions, prolific polymath Clifford Pickover covers 250 milestones in mathematical history. Among the numerous concepts readers will encounter as they dip into this inviting anthology: cicada-generated prime numbers, magic squares, and the butterfly effect. Each topic is presented in a lavishly illustrated spread, including formulas and real-world applications of the theorems.
Pythagoras and the Doctrine of Transmigration
The mythical narrative of transmigration tells the story of myriad wandering souls, each migrating from body to body along a path of recurrence amid the becoming of the All. In this highly original study, James Luchte explores the ways in which the concept of transmigration is a central motif in Pythagoras' philosophy, representing its fundamental meaning. Luchte argues that the many strands of the tale of transmigration come together in the Pythagorean philosophical movement, revealing a unity in which, for Pythagoreans, existence and eschatology are separated only by forgetfulness. Such an interpretation that seeks to retrieve the unity of Pythagorean thought goes against the grain of a long-standing tradition of interpretation that projects upon Pythagoras the segregation of 'mysticism' and 'science'. Luchte lays out an alternative interpretation of Pythagorean philosophy as magical in the sense that it orchestrates a holistic harmonization of theoria and praxis and through this reading discloses the radical character of Pythagorean philosophy.
Numerology
Numerology is the delusion that numbers have power over events. It is a descendent of number mysticism, the belief the contemplation of numbers can give mystical and non-rational insights into life, the universe and everything. 2500 years ago, Pythagoras originated number mysticism, crediting certain numbers with characteristics, through numerology, is a more recent invention that allots numbers, hence characteristics, to individuals. Underwood Dudley outlines here the history of number mysticism and numerology and gives many examples, including biorhythyms, Bible-numberists, pyramidologists and a plethora of others. His message is that numbers do indeed have power, but over minds not events. This is the only book that exposes this particular human folly, and requires no mathematical background beyond knowledge of numbers.
Pythagoras Redivivus
Die vorliegende Sammlung von Aufsatzen einiger der weltbesten Spezialisten der antiken griechischen Philosophie konzentriert sich auf die philosophischen Texte, die wahrend der hellenistischen und kaiserlichen Zeit Pythagoras und den Pythagoreern zugeschrieben wurden. Ihre unbekannten Verfasser behaupten, die Positionen der ursprunglichen pythagoreischen Schule zur Metaphysik, Theologie, Zahlenphilosophie, Physik, Logik, politischen Philosophie, Ethik und zur richtigen Lebensweise zu vermitteln. Die kuhnsten unter ihnen prasentieren sich als die Quellen, aus denen Platon und Aristoteles die Inspiration fur den Timaios und die Kategorien geschopft haben. Da sie von den meisten antiken Denkern als massgebend angesehen wurden, fuhrte dies zu einer Renaissance der platonischen und pythagoreischen Tradition.
Music and Mathematics
From Ancient Greek times, music has been seen as a mathematical art, and the relationship between mathematics and music has fascinated generations. This collection of wide ranging, comprehensive and fully-illustrated papers, authored by leading scholars, presents the link between these twosubjects in a lucid manner that is suitable for students of both subjects, as well as the general reader with an interest in music. Physical, theoretical, physiological, acoustic, compositional, and analytical relationships between mathematics and music are unfolded and explored with focus on tuningand temperament, the mathematics of sound, bell-ringing and modern compositional techniques.
The Pythagorean Theorem
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We commonly define the Pythagorean theorem using the formula a2 + b2 = c2. But Pythagoras himself would have been confused by that. Explore how this famous theorem can be explained using common geometric shapes (no fancy algebra required), and how it's a critical foundation for the rest of geometry.
Journey of the Universe
- Kanopy This link opens in a new windowBeautifully filmed, our story begins on the historically rich Greek island of Samos, birthplace of mathematician Pythagoras. Disembarking on the island at dawn, Swimme expertly guides us on an exhilarating trek through time and space, sharing a wondrous view of cosmic evolution as a process based on immense creativity, connection, and interdependence. After the toll of midnight, he sets sail into the star-lit waters of the Aegean Sea, leaving audiences with a sense of wonder at the mystery, complexity and connectivity that permeates the Earth and universe from the very beginning. The Emmy(r) Award winning JOURNEY OF THE UNIVERSE tells an epic story of cosmic, Earth and human transformation from The Big Bang to today. Evolutionary philosopher Brian Thomas Swimme and Yale historian of religions Mary Evelyn Tucker have crafted an elegant narrative that both illuminates and celebrates the profound role humans play in the flourishing of the Earth.
Mind and Body in Greek Philosophy
- Kanopy This link opens in a new windowHumans have been asking this question for thousands of years: exactly how are we related to the world around us? Learn what modern Western thought inherited from the Greeks and how the theories of Pythagoras, Socrates, Plato, and Aristotle still affect our thinking and questioning today.
Online Resources
- On Pythagoras Theorem for Products of Spectral TriplesWe discuss a version of Pythagoras theorem in noncommutative geometry. Usual Pythagoras theorem can be formulated in terms of Connes’ distance, between pure states, in the product of commutative spectral triples. We investigate the generalization to both non-pure states and arbitrary spectral triples. We show that Pythagoras theorem is replaced by some Pythagoras inequalities, that we prove for the product of arbitrary (i.e. non-necessarily commutative) spectral triples, assuming only some unitality condition. We show that these inequalities are optimal, and we provide non-unital counter-examples inspired by K -homology.
- PYTHAGORAS' MATHEMATICS IN ARCHITECTURE AND HIS INFLUENCE ON GREAT CULTURAL WORKSPythagoras' life, teaching and contribution in science and philosophy has been transfigured by legend, which hardly can be separated. Tracing his fingerprints of mathematical nature is attempted here, based on evidence from great technical works and temples accomplished during his time in Samos and Magna Graecia. The application of the Pythagorean triples in the design of the Athena temple at Paestum built in c. 520 BC has already been established and was considered to attest the Pythagorean consciousness of the architect. Similar conclusions are also drawn in this article from the layout of the Polycratean temple of Heraion in Samos, where the earliest application of Pythagorean mathematics and proportions is disclosed in this article. It is also demonstrated that the achieved accuracy in pre-positioning of the Eupalinos' tunnel mouths and the well-designed maneuver at the crossing indicate the involvement of a mathematical mind supporting the engineering skills of Eupalinos. By comparison with the Hellenistic temple of Apollo at Didyma, where the systematic application of the Pythagorean triples is again revealed in temple modeling and layout, it is concluded that the geometrical method of design of the ancient temples and the concept of harmonic proportions was fully developed in Pythagoras' time and his philosophy of proportions in architecture, amalgamated later with Plato's ideas, prevailed since then until the present.
- Pythagorean Triples before and after PythagorasFollowing the corrected chronology of ancient Hindu scientists/mathematicians, in this article, a sincere effort is made to report the origin of Pythagorean triples. We shall account for the development of these triples from the period of their origin and list some known astonishing directions. Although for researchers in this field, there is not much that is new in this article, we genuinely hope students and teachers of mathematics will enjoy this article and search for new directions/patterns.
Music of the Spheres
- Kanopy This link opens in a new windowThis episode covers the evolution of math. Pythagoras, father of Greek math, considered numbers the language of nature. Here, we follow the spread of Greek ideas through the Islamic Empire to Moorish Spain and Renaissance Europe, ultimately exploring the alliance of math to music, astronomy, and painting.