Irreversible hashing algorithms

Hashing algorithms are based on those algorithms being impossible to be reversed (that is, given the hash and the algorithm, it is impossible to calculate the reverse function of that algorithm, and thus it is impossible to run the hash through the reversed algorithm in order to obtain the original string).

My question: how "impossible" is the "impossible" in my previous statement? Is this impossibility based on the computational infeasibility of calculating the reverse algorithm, meaning that today's math geniuses - with the help of their computers - cannot come up with a way to figure out how to reverse the algorithm, or is this an actual impossibility? Are there algorithms that have been mathematically proven to be irreversible? If so (and I hope not, because I can't wrap my brain around that concept), how? If not, can there ever be an algorithm that is absolutely irreversible?