Can Add and Multiply On Cipher Text achieve all operations? A Fully homomorphic encryption scheme needs to support an evaluate function that can do add and multiply operations on cipher text.

Can we do all kinds of complex operations on cipher text like search, sort etc using just add and multiply ?
If addition and multiplication are preserved, then we can evaluate arithmetic circuits over the encrypted data.

It seems that your confusion about the class of problems that we can compute stems from the fact that you are envisioning boolean circuits in which we perform operations on the bitwise representations of the input. This is not an FHE question per se, but rather a question regarding the difference between the computational models of boolean circuits vs. arithmetic circuits. To understand the problems that can be computed with this model of computation, take a look at this paper. A relevant portion:


  The definition above shows an evident difference between arithmetic circuits and Boolean circuits.While Boolean circuits can perform operations on the "bit representation" of the input field elements,that are not necessarily the arithmetic operations, arithmetic circuits cannot. Nevertheless, most algorithms for algebraic problems fit naturally into the framework of arithmetic circuits.


In regards to why they defined FHE this way, it follows the definition of  Ring Homomorphism. By definition, a ring homomorphism is a function between two rings that respects addition and multiplication. 

In practice, recent research on FHE studies lattices since ideal lattices yield both additive and multiplicative homomophisms.