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One benefit could be to emphasize the distinction between regular mathematical functions (e.g. x ↦ x+1) and λ-terms (e.g. λx. x+1): strictly speaking, those are
by mbivert 2y ago
One benefit could be to emphasize the distinction between regular mathematical functions (e.g. x ↦ x+1) and λ-terms (e.g. λx. x+1): strictly speaking, those are different kind of objects.
- hollerith 2y agoDifferent how?
- mbivert 2y agoWell, λ-terms are defined by the rules of the λ-calculus[0], or variants of it, while mathematical functions have their own, distinct definition[1]. Essentially, the axioms used to define either are fundamentally different. Not all mathematical functions can be expressed/represented by λ-terms; a famous example would be the halting problem[2]. Generally, keep in mind that the λ-calculus — as Turing machines — can only express computable functions[3]. You could also try to think about how to perform usual mathematical function operations on λ-terms: limits, differentiation, integration, etc. [0]: https://en.wikipedia.org/wiki/Lambda_calculus https://en.wikipedia.org/wiki/Lambda_calculus [1]: https://en.wikipedia.org/wiki/Function_(mathematics) https://en.wikipedia.org/wiki/Function_(mathematics) [2]: https://en.wikipedia.org/wiki/Halting_problem https://en.wikipedia.org/wiki/Halting_problem [3]: https://en.wikipedia.org/wiki/Computable_function https://en.wikipedia.org/wiki/Computable_function