A basic question about recursive computation I was going through the following paper: http://www-formal.stanford.edu/jmc/recursive.pdf

In the last part of the second section (at the end of page 7) he speaks about the inadequacy of lambda notion for naming functions described recursively.

The right hand side of the following expression
$$\text{sqrt} = \lambda((a,x,\epsilon),(|x^2 -a|&lt;\epsilon \to x,T \to\text{sqrt}(a,\frac{1}{2}(x+\frac{a}{x}),\epsilon)))$$ cannot serve as an expression of function because there would be nothing to indicate that the reference to $\text{sqrt}$ within the expression stood for the expression as a whole.
I did not get what it meant. Why should there be a further indication that the expression stood for whole? Please explain this point.
In pure untyped lambda calculus functions are anonymous. Generally, when you want to define a function recursively, you can simply have it &quot;call&quot; itself, for example:

if $n=0$ then $f(n) = 1$
if $n&gt;0$ then $f(n) = nf(n-1)$.

In lambda calculus we don't have the same luxury, functions don't even have &quot;names&quot; in the first place, they are defined in terms of variable binding and variable substitution.
Still, there is a way to implement recursion, discovered by Haskell Curry, by using a particular lambda-term, called fixed-point combinator (if you are interested in details, any reasonable source on lambda calculus covers this aspect).
With the new notation, the authors are effectively introducing a mechanism to give names to functions, removing the problem entirely.