Final answer:
The Pythagorean doctrine of the transmigration of souls is the belief that the soul is immortal and reincarnates into different bodies, influenced by karma from previous lives, forming a central aspect of Pythagoreanism which combines mathematics, music, and mysticism.
Step-by-step explanation:
The Pythagorean doctrine of the transmigration of souls, also known as metempsychosis, refers to the belief that after death the soul moves from one body to another, be it human or non-human.
This was a central part of the Pythagorean philosophy, a system that intertwined mathematics, music, and mysticism. In this belief system, the soul is immortal and undergoes a cycle of reincarnations until it achieves a state of purity.
Actions performed in one's life (karma) play a critical role in determining the nature of subsequent incarnations.
In their quest for understanding the cosmos, the Pythagoreans also observed numerical patterns in nature, such as those in music harmonics, which led them to propose that numbers underpin the very structure of the natural world.