Mathematically, finger transposition can be seen as a type of mapping or transformation between different physical configurations of an instrument. The configuration of a musical instrument can be thought of as a vector space, with each note or finger placement representing a different vector. When a piece of music is transposed to a different key, the fingerings required to play it change, which corresponds to a mapping between different configurations of the instrument.
In this way, finger transposition can be seen as similar to the study of transformations and mappings in mathematics, such as linear transformations and function composition. The ability to finger transpose effectively can be thought of as a type of skill in navigating these transformations and mappings,