The Pythagoreans discovered mathematical ratios in musical harmony: doubling string length produces an octave, 3:2 ratio a perfect fifth. This revelation suggested that harmonic relationships structure reality itself, from planetary movements to the human soul.
They invested numbers with symbolic meaning: One represented unity and origin; Two, duality and opinion; Three, harmony; Four, justice and cosmic order; Ten (the tetraktys), perfection. Geometric shapes held sacred significance—the pentagram symbolized their community.
Their discoveries in mathematics were genuine: the Pythagorean theorem, irrational numbers (which disturbed their number-based cosmology), and foundations of mathematical proof. Yet they combined rigorous mathematics with mysticism—seeing numbers as both abstract entities and cosmic forces.
This synthesis of mathematics and metaphysics influenced Plato, who placed "Let no one ignorant of geometry enter" above his Academy's door, and continues in those who see mathematical elegance as clue to reality's deep structure.
Source
Pythagorean School