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How did the Pythagorean doctrine of opposites explain order in the cosmos?

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Marble bust of Pythagoras, the ancient Greek mathematician and philosopher, showing a bearded man with a head covering and calm expression. — Pythagoreanism Greek philosopher and mathematician Pythagoras, portrait bust.

In Pythagoreanism, the doctrine of opposites explains cosmic order through the interaction of two fundamental principles: the limit and the unlimited.1

The limit, represented by odd numbers, acts as an active force that imposes structure, harmony, and order upon the unlimited, which is represented by even numbers. This process of "fitting together"—or harmonia—allows various contrary qualities to coexist within the cosmos, as well as within individuals and society.12

While Aristotle documented a table of opposites containing ten pairs—such as light and darkness, rest and motion, and good and evil—these pairs serve to illustrate a broader dualistic conception of reality. By applying numerical speculation to these opposites, the Pythagoreans developed a cosmogonical theory where the interaction between the limiting-odd and the unlimited-even generated the physical world, establishing a foundation where numerical proportion and form define the nature of the universe.312