# Programming related calculus & math symbols and questions

I am reading a textbook for my next semester just for fun. I didn't study so hard during the high school so I have missed out many vital information.

Questions:

1) What is λ -calculus and λ (in calculus)?

2) The text book illustrates if there is a function i.e. f(1) and f(2) the equation is 2, 3 because f(x) = x + 1
But the books says "This declaration combines the description of the function for adding one, and its naming as f . The issues can be separated by considering nameless functions, using the Greek letter λ instead of the symbol f and rearranging the syntax to produce

λ x.x+1 .


Now the definition of f above can be rewritten as

f =λ x.x+1


"

I can't understand how λ x.x+1 came from f(x) = x+1. I understand that they are trying to give the function a name.

I hope you have understood my question. Thank you.

-

When you do algebra, you are mostly used to pushing variables around. Occasionally you pass a variable to a function and label the result as another value, as in the statement "let $y=f(x)$".
Lambda calculus is a formalism in which functions become "first-class citizens". In standard algebra, you can define a function with respect to an argument i.e. $f(x) = x+1$, but there is no way to write $f = \underline{\hspace{1cm}}$ to convey the same meaning.
A mathematician might use the notation $x \mapsto x+1$ to represent the function of $x+1$. Basically what a lambda expression does is assigns such a map to a symbol. Writing $f = \lambda(x.x+1)$, you establish that the function $f$ accomplishes the operation of "adding 1".
I've work with $\lambda$-calculus for some time and have never seen parentheses like $\lambda(x. M)$. In fact, in many places symbol $\lambda$ would be an indicator that what follows immediatelly after it (until the dot) is a variable. When you add patterns you might want to write things like $\lambda(4,x,\mathtt{true}). M$, but separating the dot and the lambda is very confusing. –  dtldarek Dec 21 '12 at 11:01