i'm not sure what you mean by the 'spirit' of Big-O; i find the question fascinating because it is relatively easy to understand the concepts involved, yet it's

i'm not sure what you mean by the 'spirit' of Big-O; i find the question fascinating because it is relatively easy to understand the concepts involved, yet it's been 40+ years since we understood enough to be able to ask the question, and we still don't know the answer. when you couple that with the fact that the prevailing wisdom that our inability to find an algorithm implies that it can't exist, the contrarian in me goes nuts: the academic attitude is always "we're so smart, if we can't find it, it must not exist," and i get a kick out of arguing for more humility on their part. even though a polynomial-time algorithm with a massive polynomial would still be computationally intractable, even a proof that such an algorithm exists, but is too complicated to express using all the atoms in the universe as bits would probably open the doors to all kinds of proofs and techniques we cant' even imagine.