nondifference

Nondifference is a term used in several disciplines to denote a lack of discernible difference under a specified criterion. In mathematics and logic it is often linked to invariance or indistinguishability, while in philosophy it can describe a stance about identity or sameness where two entities are not regarded as different within a given framework.

In formal terms, let X be a set equipped with an equivalence relation ~. A function f: X -> Y is said to be nondifference-preserving with respect to ~ if whenever x ~ y, we have f(x) = f(y). In this sense f does not distinguish elements that lie in the same equivalence class, and f induces a well-defined map on the quotient X/~.

The term also appears in philosophical discussions of identity and indiscernibility, where nondifference asserts that two objects cannot be distinguished by the properties under consideration. In practical contexts, nondifference captures invariance under a given criterion, such as a symmetry, a measurement scale, or a data-aggregation rule.

Examples include a color-labeling function that assigns the same label to all objects sharing the same color, or a data-anonymization procedure that maps all records within a category to a single representative value. In each case, distinctions within a class are not reflected by the mapping.

Nondifference is related to concepts such as invariance, equality, and equivalence relations. It is not a standard formal term with a single universal definition, but it serves as a descriptive notion for invariance or indistinguishability under a specified criterion. See also difference, invariance, and quotient.