Composition (mathematics)

In mathematics, composition refers to the operation of combining two or more functions or operators to produce a new one. When you compose functions f and g, you apply one function to the output of another—written f ∘ g—creating a streamlined way to express complex evaluations.

Composition is foundational across mathematics. In Functional programming, it mirrors the mathematical idea perfectly: you chain operations together, each transforming data for the next step. This mirrors how Energy flows through systems, how Limits nest within equations, and how Learning and Memory compound through repeated operations.

The beauty of composition lies in its power to reduce Cognitive overhead. Rather than writing out every intermediate step, you compose smaller, proven functions into larger ones—a principle echoed in everything from Logic (computing) to Imperative programming. Simplifying complex problems into composed pieces makes them tractable.

Composition also preserves structure: composing continuous functions yields a continuous function; composing linear maps yields another linear map. This stability makes composition indispensable in Large-scale applications from physics to machine learning.

Related

Function (mathematics), Operator (mathematics), Functional programming, Equation (mathematics), Evaluation (mathematics), Limits