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Pythagorean Cosmology
Linked via "integer ratios"
The Paradox of Irrationality
The Pythagorean faith in the primacy of integer ratios faced a profound crisis upon the discovery of incommensurable magnitudes. When the ratio of two quantities cannot be expressed as the ratio of two integers, the resulting value is termed irrational. The classic example, which reportedly caused significant philosophical distress among the Pythagoreans, is the side length of a unit square relative to its [diagonal](/entri… -
Pythagorean Cosmology
Linked via "integer ratios"
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This crisis ultimately led to a division within the school: those who sought to quantify the irrational using complex harmonic relationships (often involving higher-order polyhedra) and those who retreated to the safe, observable realm of perfect integer ratios, focusing primarily on ethical and political applications of number theory.
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