How do functional programming languages use mathematical functions to organize programs?

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Functional programming languages are a type of declarative language that organize programs by applying functions to arguments, mirroring a mathematical style. Unlike procedural languages that focus on the step-by-step instructions of how to perform a task, functional languages emphasize what the program should accomplish by constructing it through the composition and application of these mathematical functions.1

In this paradigm, programs are built by defining relationships and transformations rather than managing state changes or sequences of commands. Languages such as LISP, ML, and Haskell utilize this approach, making them particularly effective for tasks like automated mathematical theorem proving, research in language development, and various commercial applications. By treating computation as the evaluation of mathematical functions, these languages minimize the distinction between a program's specification and its implementation.1