You may want to take a look at https://cs.stackexchange.com/questions/38357/is-it-really-possible-to-prove-lower-bounds https://cs.stackexchange.com/questions/3

You may want to take a look at https://cs.stackexchange.com/questions/38357/is-it-really-possible-to-prove-lower-bounds https://cs.stackexchange.com/questions/38357/is-it-really-po... Something along the lines of this is what you would likely need to do. Take some problem that is NP-complete. Prove it has an exponential lower bound on time complexity. You've just proved P != NP. Proof of impossibility is sufficiently common in mathematics that one of the oldest and best-known proofs of anything, Euclid's proof of the irrationality of sqrt(2), is an example of such a proof (it is impossible to express sqrt(2) as the ratio of two integers). The philosophical issue relates more to physical non-existence. If I claim leprechauns don't exist, likely few people will argue with me, but I can't actually examine the universe outside of my own past and future light cones, so I can't really know for sure there is no part of larger existence that contains leprechauns. Even events that violate accepted physics may become possible if we live in a false vacuum that collapses to a state with a different physics at some time in the future. Proving mathematical non-existence is an entirely different animal. Trivially, any universally quantified proposition "for all X, predicate(X)" is logically equivalent to "for no X, not predicate(X)," yet proofs of such propositions abound.