How did the Pythagoreans connect numbers to the structure of the universe?

The Pythagoreans viewed numbers as the fundamental building blocks of the universe, believing that numerical proportions and harmony defined the structure of reality. They treated numbers not merely as abstract concepts but as cosmic matter from which all things were derived.21

Key aspects of their connection between numbers and the universe include:

• The Tetraktys and Space: The number 10, or the tetraktys ($1+2+3+4=10$), was considered the most perfect number, symbolizing unity arising from multiplicity. They linked this to geometry, where 1 represented a point, 2 a line, 3 a triangle, and 4 space, suggesting that 10 encompassed all possible dimensions.21
• Musical Harmony: By experimenting with string lengths, they discovered that musical intervals correspond to simple whole-number ratios. This empirical finding led them to believe that the cosmos itself was governed by similar mathematical order, eventually inspiring the concept of the "music of the spheres," where planetary movements were thought to produce celestial harmonies.21
• Cosmological Order: Their focus on numerical order influenced their view of the heavens, leading them to posit the existence of ten heavenly bodies to satisfy the perfection of the number 10.2