4 ms·

What is the benefitsof the lambda notation over a more intuitive arrow notation like x->x+1?

hun3 2y ago
Not much, but you immediately know it's a function (abstraction). Besides, the standard notation for the "arrow function" is maplet (x ↦ x + 1) in mathematics. I assume λ notation sees frequent uses in logic and computation theory.
- DrDeadCrash 2y ago
  The characters in OP's arrow function have the benefit of being available on a standard keyboard layout. This has proven to be an important syntactic feature, imo.
  - quibono 2y ago
    Haskell uses a single backslash \ in place of λ
arethuza 2y ago
What about expressing something a bit more complex like: λf. (λx. f (x x))(λx. f (x x))
- nuancebydefault 2y ago
  It is so complex that i don't even know how to start reading it (not that i am a reference of mathematical complexity). Does the x each time mean something else?
  - credit_guy 2y ago
    It’s a joke. That particular lambda expression is the Y combinator.
- keithalewis 2y ago
  f ↦ (x ↦ f (x x))(x ↦ f (x x))
charles-fox 2y ago
I heard it evolved from Bertrand Russell, who used a notation such as \hat{x}.(x+1) to mean similar in Principia Mathematica. Church and maybe others in the logic community originally used the same notation but Church's publisher, not having latex, was unable to print it. So they wanted to move the hat in front of the letter, such as ^x.(x+1). The hat symbol either wasn't available or the instruction got mangled during production into the similar-looking lambda.
- nuancebydefault 2y ago
  Lovely explanation! Everytime I encounter the lambda I will think of the relation to the hat symbol. The latter we used a lot at school.
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 ago
  Different how?
  - mbivert 2y ago
    Well, 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

hun3 2y ago

Not much, but you immediately know it's a function (abstraction). Besides, the standard notation for the "arrow function" is maplet (x ↦ x + 1) in mathematics. I assume λ notation sees frequent uses in logic and computation theory.

DrDeadCrash 2y ago
The characters in OP's arrow function have the benefit of being available on a standard keyboard layout. This has proven to be an important syntactic feature, imo.
- quibono 2y ago
  Haskell uses a single backslash \ in place of λ

arethuza 2y ago

What about expressing something a bit more complex like: λf. (λx. f (x x))(λx. f (x x))

nuancebydefault 2y ago
It is so complex that i don't even know how to start reading it (not that i am a reference of mathematical complexity). Does the x each time mean something else?
- credit_guy 2y ago
  It’s a joke. That particular lambda expression is the Y combinator.
keithalewis 2y ago
f ↦ (x ↦ f (x x))(x ↦ f (x x))

charles-fox 2y ago

I heard it evolved from Bertrand Russell, who used a notation such as \hat{x}.(x+1) to mean similar in Principia Mathematica. Church and maybe others in the logic community originally used the same notation but Church's publisher, not having latex, was unable to print it. So they wanted to move the hat in front of the letter, such as ^x.(x+1). The hat symbol either wasn't available or the instruction got mangled during production into the similar-looking lambda.

nuancebydefault 2y ago
Lovely explanation! Everytime I encounter the lambda I will think of the relation to the hat symbol. The latter we used a lot at school.

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 ago
Different how?
- mbivert 2y ago
  Well, 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