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text: '[summary:\n\n - Proving $P \\neq NP$ is harder if true than proving $P=NP$ if true.\n - After 50 years we haven't found any efficient algorithm for $NP$-complete problems. \n - The polynomial hierarchy is compared to the arithmetical hierarchy, and the analogy implies $P \\neq NP$\n]\n\n\n\n#Natural proofs\nIf $P \\neq NP,$ then there are results showing that lower bounds in complexity are inherently harder to prove in a [natural_proof technical yet natural sense]. In other words, if $P \\neq NP$ then proving $P \\neq NP$ is hard.\n\nThe opposite proposition for $P=NP$ is also expected to be true. That is, it would make sense that if $P=NP,$ then it should be easier to prove, since we could build far more efficient [ theorem provers].\n\n#Empirical argument\nWe have been trying to get a fast algorithm for [ $NP$-complete] problems since the 70s, without success. And take into account that "we" does not only comprise a minor group of theoretical computer scientists, but a whole industry trying to get faster algorithms for commercial purposes.\n\nOne could also argue more weakly that if $P=NP$, then evolution could have made use of this advantage to sped up its search process, or create more efficient minds to solve $NP$-complete problems at thoughtspeed.\n\n#The arithmetical hierarchy argument\nSome authors have drawn an analogy between the [ polynomial hierarchy] and the [1mg arithmetical hierarchy].\n\nThere are results showing that the [ arithmetical hierarchy is strict], and if the analogy holds then we would have that the polynomial hierarchy is strict as well, [collapse_of_the_polynomial_hierarchy which automatically implies $P \\neq NP$].',
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