The universe as a Church encoding

(written by lawrence krubner, however indented passages are often quotes). You can contact lawrence at:, or follow me on Twitter.

I think it is extremely messy that math has integers, fractions, irrational numbers and worse. If I was rich, I would work on a model of the universe where the only primitive is the function. One has to admit, the idea of a universe with only one primitive is extremely attractive.

In mathematics, Church encoding is a means of representing data and operators in the lambda calculus. The data and operators form a mathematical structure which is embedded in the lambda calculus. The Church numerals are a representation of the natural numbers using lambda notation. The method is named for Alonzo Church, who first encoded data in the lambda calculus this way.

Terms that are usually considered primitive in other notations (such as integers, booleans, pairs, lists, and tagged unions) are mapped to higher-order functions under Church encoding. The Church-Turing thesis asserts that any computable operator (and its operands) can be represented under Church encoding. In the untyped lambda calculus the only primitive data type is the function.

The Church encoding is not intended as a practical implementation of primitive data types. Its use is to show that other primitive data types are not required to represent any calculation. The completeness is representational. Additional functions are needed to translate the representation into common data types, for display to people. It is not possible in general to decide if two functions are extensionally equal due to the undecidability of equivalence from Church’s theorem. The translation may apply the function in some way to retrieve the value it represents, or look up its value as a literal lambda term.

Lambda calculus is usually interpreted as using intensional equality. There are potential problems with the interpretation of results because of the difference between the intensional and extensional definition of equality.

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