Church encoding: numbers defined as functions
The Church numeral 3 represents the action of applying any given function three times to a value. The supplied function is first applied to a supplied parameter and then successively to its own result. The end result is not the numeral 3 (unless the supplied parameter happens to be 0 and the function is a successor function). The function itself, and not its end result, is the Church numeral 3. The Church numeral 3 means simply to do anything three times. It is an ostensive demonstration of what is meant by "three times".
Arithmetic operations on numbers may be represented by functions on Church numerals. These functions may be defined in lambda calculus, or implemented in most functional programming languages (see converting lambda expressions to functions).
References
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