The Cosmos versus the Infinite – on the Double Standard towards Infinity and Its Aesthetic Roots

The concept of cosmos (κόσμος), introduced by Pythagoras, conveys beauty and order, integrating infinity through symmetry but rejecting it via proportion. For the Pythagoreans, beauty lies in the commensurable—what can be measured with finite actions—while infinity appears in irrational, incommensurable quantities, which are unknowable to humans. They believed that the cosmos’ beauty stems from numerical order, particularly the ratios of whole numbers, or rational numbers. Beauty, in this context, is tied to what can be understood. However, symmetry in its highest form, like the sphere, embodies the infinite. This duality leads to a mixed attitude toward infinity, accepting it when it pleases and rejecting it when it causes cognitive discomfort. This distinction between actual and potential infinity is aesthetically driven and reflected in both mathematics and philosophy, particularly in works by Plato, Euclid, and Aristotle.