denotional semantic for while - fixed points In my book it is written: 
$$[[\text{while b do S}]] = \text{FIX F}$$, where $Fg = cond (\beta[[b]], g\circ [[S]], id). $
What is cond ?
$[[\text{if b then S1 else S2}]] = cond (\beta[[b]], [[S1]], [[S2]])$.  

I don't understand why we should use $FIX$. What it is FIX ?
For me it is sufficient:
$S[[\text{while b do S}]] = cond(\beta[[b]], [[\text{while b do S}]]\circ [[S]], id)$$

DEFINITIONS  

$$S_{ds}[[\text{while b do S}]] = FIX F$$
where $Fg= \text{cond}(\beta[[b]], g\circ S_{ds}[[S]], id)$

$$FIX: ((State\rightarrow State)\rightarrow (State \rightarrow State)) \rightarrow (State \rightarrow State)$$
In brackets functions are partial.
Your proposed alternate definition is no good.  It tries to define $[[\text{while b do S}]]$ in terms of $[[\text{while b do S}]]$.  That's a circular definition: you can't define something in terms of itself.

"FIX" typically refers to the least fixpoint, and is a way of avoiding circularity.  See your textbook for background on fixpoints and the precise definition of the "FIX" operator.