  The class NP is very important practically and philosophically.  It is the class of problems, \(S\), such that any input \(w\) is in \(S\) iff there is a proof, \(p(w)\), that \(w\in S\) and \(p(w)\) is not much larger than \(w\).  Thus, very informally, we can think of NP has the set of intellectual endeavors that may be in reach: if we find the answer to whether \(w \in S\), we can convince others that we have done so.