  First of all we give some reasons that "natural proofs" built not a barrier
to prove P $\not=$ NP using Boolean complexity. Then we investigate the
approximation method for its extension to prove super-polynomial lower bounds
for the non-monotone complexity of suitable Boolean functions in NP or to
understand why this is not possible. It is given some evidence that the
approximation method alone cannot be used to prove a super-linear lower bound
for any function $f \in {\cal B}_n$. Additionally, an overview on the methods
for proving lower bounds of the non-monotone and the monotone complexity of
Boolean functions is given. Finally, a personal opinion how to proceed the
research on the P versus NP problem and also on proving a super-linear lower
bound for the non-monotone complexity of a Boolean function in NP is given.
