higherorder

Higherorder, often written higher-order, is a term used across disciplines to describe abstractions where the primary objects are themselves other objects of the same kind, such as functions, predicates, or levels of thinking. It signals a meta-level approach where concepts operate on or with other concepts of the same family rather than on concrete instances alone.

In mathematics and logic, higher-order denotes frameworks that generalize first-order concepts by allowing quantification or operation on functions and predicates. Higher-order logic extends first-order logic by permitting variables to range over not only individual elements but also sets, relations, and functions. In type theory and functional programming, higher-order constructs enable functions to take other functions as arguments or to return functions as results, enabling concise abstractions and powerful composition.

Examples in programming include common higher-order functions such as map, filter, and reduce, which operate on lists and can receive other functions as parameters. Languages like Haskell, Lisp, Scheme, and Python (to a degree) support higher-order programming, contributing to expressive and modular code. In type theory, higher-order functions align with lambda calculus and dependently typed systems, but they can introduce complexity in type checking and reasoning about programs.

In education, higher-order thinking refers to cognitive processes that go beyond memorization, including analysis, evaluation, synthesis, and complex problem solving. Assessments emphasizing higher-order thinking aim to measure the ability to reason, justify conclusions, and transfer knowledge to new situations.

Other uses include higher-order moments in statistics (beyond mean and variance), and higher-order derivatives in calculus, which describe rates of change of rates of change. The exact meaning of "higher order" depends on the domain, but across contexts it signals an abstraction that operates on objects of a similar kind at a meta level.