There are indeed several reasons to suspect that the resolution of \(\textbf{P} \neq \textbf{NP}?\) will prove to have far reaching practical and theoretical consequences outside of computer science. Perhaps the most significant of these revolves around the possibility that despite the various heuristic arguments which can currently be offered in favor of the hypothesis that \(\textbf{P} \neq \textbf{NP}\), there remains the possibility that \(\textbf{P} = \textbf{NP}\) is true after all. Recall that \(\textbf{P}\) can be characterized as the class of problems membership in which can be decided efficiently, whereas \(\textbf{NP}\) can be characterized as the class of problems for which membership can be verified efficiently once an appropriate certificate is provided. If it turned out that \(\textbf{P} = \textbf{NP}\), then the difficulty of these two tasks would coincide (up to a polynomial factor) for all problems in \(\textbf{NP}\). Some consequences of this would be that the task of finding a satisfying valuation \(v\) for a propositional formula \(\phi\) is no harder than constructing its truth table, the task of factoring a natural number would be no more difficult than verifying that a given factorization is correct, etc.