Final answer:
The Pythagoreans held that numerical relationships were fundamental to understanding the universe, saw music as a clue to these relationships, and had spiritual beliefs including the transmigration of souls. They were both mathematicians and religious practitioners, influencing philosophy and science.
Step-by-step explanation:
The Pythagoreans believed in several key concepts that intertwined mathematics, philosophy, and spirituality. They believed that numerical relationships were the key to understanding the universe and that these relationships could be revealed through music and harmony. This notion extended to the cosmic level, where they saw a connection between the movement of the planets and musical harmony, a concept known as the harmony of the spheres. They also adopted spiritual beliefs such as the transmigration of souls, suggesting that life is reborn in new forms after death. Pythagoras and his followers formed a community that was both a school of mathematics and a religious order, and their views on ethics, the nature of reality, and the immortal soul influenced later philosophical thought.