Parametric Polymorphism

Parametric polymorphism is the ability to write functions and data types that operate uniformly over all types, without knowledge of or dependence on the specific types they are instantiated with. Coined by Strachey and formalized by Reynolds, it is the form of polymorphism found in type-theoretic systems like the polymorphic lambda calculus and implemented in languages from ML to Haskell. The central theorem governing parametric polymorphism is the parametricity theorem (also called Reynolds' abstraction theorem), which guarantees that polymorphic functions behave uniformly — they cannot inspect the type parameter and must therefore treat all instances identically.

This uniformity is both a strength and a limitation. Parametric polymorphism enables powerful generic programming and guarantees code reuse, but it cannot express operations that legitimately depend on type-specific behavior. For such cases, languages combine parametric polymorphism with ad-hoc polymorphism through mechanisms like type classes or traits, creating hybrid systems that preserve uniformity where possible and permit specialization where necessary.