Pythagoras, born around 570 BCE, founded a school in Croton that blended math and mysticism. His teachings emphasized numbers as the key to understanding the universe and promoted the idea of cosmic harmony based on mathematical principles.

The Pythagorean school introduced influential concepts like metempsychosis (soul transmigration) and the Tetractys symbol. Their mathematical contributions, including the famous Pythagorean theorem, laid the groundwork for advancements in geometry and number theory.

Life and Teachings of Pythagoras

Early Life and Philosophical Foundations

• Pythagoras born on the island of Samos around 570 BCE
• Traveled extensively in his youth, studying under various teachers and philosophers
• Journeyed to Egypt and Babylon, acquiring knowledge in mathematics, astronomy, and religious practices
• Developed a unique philosophical system combining mathematical principles with mystical beliefs
• Returned to Samos but found the political climate unfavorable for his teachings

Establishment of the Pythagorean School

• Settled in Croton, a Greek colony in southern Italy, around 530 BCE
• Founded a philosophical and religious school attracting numerous followers
• School organized into two main groups: Acousmatic Pythagoreans and Mathematic Pythagoreans
• Acousmatic Pythagoreans focused on oral teachings and religious aspects of Pythagoreanism
• Mathematic Pythagoreans delved deeper into mathematical and scientific studies
• School gained significant influence in Croton and surrounding areas

Pythagorean Teachings and Mysticism

• Emphasized the importance of numbers in understanding the universe
• Taught that reality consists of mathematical relationships and proportions
• Introduced the concept of cosmic harmony, believing the universe operated according to mathematical principles
• Incorporated mystical elements into his philosophy, including the belief in the transmigration of souls
• Promoted a strict code of conduct and dietary restrictions for his followers
• Developed a system of symbols and secret teachings known only to initiated members

Pythagorean Beliefs

Metempsychosis and the Nature of the Soul

• Metempsychosis refers to the belief in the transmigration of souls
• Pythagoras taught that the soul is immortal and undergoes a cycle of reincarnation
• Believed human souls could be reborn into animals or plants (vegetarianism stemmed from this belief)
• Emphasized the importance of living a virtuous life to improve one's future incarnations
• Claimed to remember his own past lives, using this as evidence for the doctrine of metempsychosis
• Taught that through philosophical study and virtuous living, one could break the cycle of rebirth

The Tetractys and Numerical Symbolism

• Tetractys considered a sacred symbol in Pythagorean philosophy
• Consists of a triangular figure of ten points arranged in four rows (1, 2, 3, 4)
• Sum of the numbers in the Tetractys (1+2+3+4) equals 10, considered a perfect number
• Represented the organization of space and the cosmos
• Each number in the Tetractys associated with specific concepts or elements:
  • One: Unity and the divine
  • Two: Duality and matter
  • Three: Harmony and the human soul
  • Four: The physical world and the four elements (earth, air, fire, water)
• Used as a meditative tool and for understanding the nature of reality

Mathematical Contributions

The Pythagorean Theorem and Its Applications

• Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides
• Expressed mathematically as $a^2 + b^2 = c^2$, where c is the hypotenuse
• While the theorem existed before Pythagoras, he is credited with providing the first formal proof
• Applied the theorem to various geometric problems and constructions
• Used to calculate distances, determine right angles, and solve architectural challenges
• Led to the discovery of irrational numbers, causing a crisis in Pythagorean mathematics

Advancements in Number Theory and Geometry

• Developed the concept of figurate numbers (triangular, square, and oblong numbers)
• Discovered the golden ratio, a proportion found in nature and art (approximately 1.618)
• Contributed to the understanding of prime and composite numbers
• Explored the properties of even and odd numbers, associating them with masculine and feminine qualities
• Advanced the study of regular polygons and polyhedra
• Investigated musical harmonies, establishing mathematical relationships between musical intervals