Surely there is no connection between the appropriateness of the notation to the subject and the appropriateness of the subject to reality. Non-Euclidean geome

Surely there is no connection between the appropriateness of the notation to the subject and the appropriateness of the subject to reality. Non-Euclidean geometry has applications to reality, but before these applications were known, how did they determine that the notation would appropriately apply to the field?