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How did Pythagoras connect numbers with the structure of the world?

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Painting by Joseph Aubert of Saint Thomas Aquinas seated with a book, wearing a halo and black and white robes, surrounded by ornate patterns in St. Stephens Church, Jerusalem. — Western philosophy Painting of scholastic philosopher St. Thomas of Aquinas by Joseph Aubert, 1900.

In the context of Western philosophy, the Pythagoreans proposed that "all things are numbers," asserting that the essence and structure of the universe are fundamentally mathematical. They believed that by identifying the numerical relations expressed by objects, one could determine their underlying nature.12

Pythagoras and his followers connected numbers to the world through several key observations:

Musical Harmony: They discovered that harmonious musical intervals correspond to simple whole-number ratios (such as 1:2, 2:3, and 3:4) in the lengths of vibrating strings. This empirical finding suggested that mathematical order governs physical phenomena.31
Cosmology: The Pythagoreans viewed the universe as a structured system governed by numerical harmony. They assigned mystical properties to numbers, such as the tetraktys (the number 10), which they believed symbolized the totality of space and the unity of the cosmos.32
Geometric Form: They observed that the properties of geometric figures, such as triangles, are determined by specific numerical ratios. They associated points, lines, and shapes with the progression of numbers, viewing geometry as a way to map the structure of reality.134

This doctrine established the influential idea that the cosmos is ordered and intelligible through mathematical form.32