Editing Lambda calculus (section)

=== Motivation ===
[[Computable function]]s are a fundamental concept within computer science and mathematics. The lambda calculus provides simple [[Semantics#Computer science|semantics]] for computation which are useful for formally studying properties of computation. The lambda calculus incorporates two simplifications that make its semantics simple.
{{anchor|anonymousForm}}The first simplification is that the lambda calculus treats functions "anonymously"; it does not give them explicit names. For example, the function
: <math>\operatorname{square\_sum}(x, y) = x^2 + y^2</math>
can be rewritten in ''anonymous form'' as
: <math>(x, y) \mapsto x^2 + y^2</math>
(which is read as "a [[tuple]] of {{mvar|x}} and {{mvar|y}} is [[maplet|mapped]] to <math display="inline">x^2 + y^2</math>").{{efn|name= mapsTo}} Similarly, the function
: <math>\operatorname{id}(x) = x</math>
can be rewritten in anonymous form as
: <math>x \mapsto x</math>
where the input is simply mapped to itself.{{efn|name= mapsTo|1= <math> \mapsto </math> is pronounced "[[maplet|maps to]]".}}

The second simplification is that the lambda calculus only uses functions of a single input. An ordinary function that requires two inputs, for instance the <math display="inline">\operatorname{square\_sum}</math> function, can be reworked into an equivalent function that accepts a single input, and as output returns ''another'' function, that in turn accepts a single input. For example,
: <math>(x, y) \mapsto x^2 + y^2</math>
can be reworked into
: <math>x \mapsto (y \mapsto x^2 + y^2)</math>
This method, known as [[currying]], transforms a function that takes multiple arguments into a chain of functions each with a single argument.

[[Function application]] of the <math display="inline">\operatorname{square\_sum}</math> function to the arguments (5, 2), yields at once
: <math display="inline">((x, y) \mapsto x^2 + y^2)(5, 2)</math>
: <math display="inline"> = 5^2 + 2^2 </math>
: <math display="inline"> = 29</math>,
whereas evaluation of the curried version requires one more step
: <math display="inline">\Bigl(\bigl(x \mapsto (y \mapsto x^2 + y^2)\bigr)(5)\Bigr)(2)</math>
: <math display="inline"> = (y \mapsto 5^2 + y^2)(2)</math> // the definition of <math>x</math> has been used with <math>5</math> in the inner expression. This is like β-reduction.
: <math display="inline"> = 5^2 + 2^2</math> // the definition of <math>y</math> has been used with <math>2</math>. Again, similar to β-reduction.
: <math display="inline"> = 29 </math>
to arrive at the same result.