A Confluence of Mysticism, Philosophy, and Mathematics

Pythagoreanism, stemming from the ancient Greek philosopher Pythagoras and his followers, embarks on a profound journey to explain the cosmos through a blend of mystical, philosophical, and mathematical lenses. This movement has left an indelible mark on the trajectory of Western thought. Here’s a detailed exploration of Pythagoreanism divided into its core components: philosophical, mathematical, and religious elements, followed by its enduring legacy and controversies.

Philosophical Components

1. Metempsychosis (Transmigration of Souls):

• Central to Pythagorean thought is the concept of metempsychosis, which posits the cyclical reincarnation of souls into different bodies across numerous lifetimes aimed at spiritual purification.

2. Philosophy of Number:

• The Pythagoreans elevated numbers to a metaphysical realm, considering them as the fundamental essence of reality. Every entity and phenomenon was seen as manifesting numerical properties, revealing a deeper, cosmic truth.

3. Harmony and Proportion:

• Harmony and proportion were perceived as the governing principles of existence, extending into ethics, aesthetics, and the natural order.

Mathematical Components

1. Theorem of Pythagoras:

• The theorem, stating that in a right-angled triangle the square of the length of the hypotenuse equals the sum of the squares of the other two sides, became a cornerstone of geometric theory.

2. Discovery of Irrational Numbers:

• The unearthing of irrational numbers posed a significant challenge to the Pythagorean ideal of numerical perfection, yet enriched mathematical understanding.

3. Development of Geometric Theory:

• Their contributions to geometry laid down foundational principles that would later be elaborated upon by Euclidean geometry.

Religious Components

1. Vegetarianism:

• Rooted in their belief in the transmigration of souls, Pythagoreans advocated for vegetarianism, emphasizing non-harm to living beings.

2. Communal Living:

• Living communally, sharing possessions, and adhering to a set of rigorous rituals and rules reflected their pursuit of ethical and harmonious living.

3. Mysticism:

• The mystical reverence for numbers and geometric forms was seen as a path to divine understanding and a reflection of a higher order.

Impact and Legacy

1. Influence on Plato:

• The profound impact on Plato, and through him on the whole of Western philosophy, showcases the enduring legacy of Pythagorean thought.

2. Contribution to Mathematics:

• Their seminal contributions to mathematics echo through the annals of history, forming a bedrock for subsequent mathematical discoveries.

3. Mystical Traditions:

• Pythagorean mysticism has continued to resonate through various mystical and esoteric traditions, infusing them with its unique numerical and geometric mystique.

Criticisms and Controversies

1. Conflict with Other Greek Philosophers:

• The Pythagorean doctrine often found itself at loggerheads with contemporary Greek philosophical schools, sparking intellectual debates.

2. Historical Accuracy:

• The extent to which Pythagoras himself developed certain doctrines attributed to Pythagoreanism remains a subject of scholarly debate, muddying the waters of historical accuracy.

This structured elucidation of Pythagoreanism offers a glimpse into a rich, multifaceted tradition that straddles the realms of mysticism, philosophy, and mathematics, striving to unravel the intricate tapestry of the cosmos.