@BestGirlGrace Quantity, together with all the emergent properties thereof, is a natural phenomenon and is therefore discovered. Tools operating on those properties, such as notation and terminology, are arbitrary attempts to describe those phenomena and are therefore invented.

Is mathematics the study of quantity, or is it the study of tools for characterizing quantity? A student of anything learns both the tools themselves and that which the tools manipulate.

@BestGirlGrace Even the finest and most perfectly justified taxonomy is a model. The inner and outer boundaries of that model were chosen based on simplification, approximation, abstraction, and other creative acts on the part of observers, for the sake of those observers' goals.

Nonetheless, the model describes some thing whose existence arose independently of the observer. Even though the concepts of "thing," "existence," "independence," "observer," and "concept" are all components of models!