What Is “The Principle,” Anyway?
The Copernican Principle
The Copernican Principle states that the earth is not in any specially favored or central location. The Copernican Principle is a philosophical assumption of modern cosmology. In modern cosmology, it has been generalized, and related to the Cosmological Principle. Based on the Standard Model which is formulated using Einstein’s General Relativity, and specifically with the Friedman-Lemaitre-Robertson-Walker (FLRW) metric, one can say that if one holds the Copernican Principle, and the universe appears isotropic (i.e., no special direction) from our vantage point, then one can infer that the universe would look isotropic from any other vantage point, and be homogeneous. These conditions make up another philosophical assumption, the Cosmological Principle.
The FRLW metric enforces mathematically that such would be true starting from the philosophical assumption of the Copernican Principle. Other metrics and models can be created that could account for the observations, but not require the Copernican Principle to be true.
Assumptions are replete in cosmology because of the intrinsic difficulty in examining everything from one vantage point. An example of another assumption is dark matter.