Why use *λx.x* instead of *f(x)*?

Question

In my semantics class, we're learning about using (abusing?) lambda calculus. So far the professor hasn't imparted any reason for using λx.x instead of using f(x).

• Why use lambdas instead of basic functions?
• Why do mathematicians notate lambdas, λx.x; wouldn't λ(x) suffice?

Thank you

Answer 1

An alternative notation for $\lambda$ is $\mapsto$. For instance, the following mean the same:

• $x\mapsto x+3$
• $\lambda x. x+3$

You could have defined it like $f(x) = x+3$, but then $f$ becomes an entity in your discourse. And having to name every function you need becomes cumbersome after a while, especially if they are of no particular interest and you just want to convey their correspondence.

To understand why you need to write $\lambda x.x+3$ instead of simply $\lambda(x+3)$, consider the following function definition (given in the three different notations we are addressing):

• $x\mapsto (y\mapsto x-y)$
• $\lambda x.(\lambda y. x-y)$
• $f(x) = g(x,\cdot)$ where $g(x,y) = x-y$

In all of the above definitions it is clear that $x$ is the first argument and $y$ is the second. But, if you write $\lambda(\lambda (x-y))$, then that information is lost precisely because we have not associated with each $\lambda$ its corresponding argument, i.e. $\lambda(\lambda(x-y))$ could stand for both $\lambda x.(\lambda y.x-y)$ and $\lambda y.(\lambda x.x-y)$.