Solvabilty of some combinatorial problems

Some of the most famous results in mathematics involve a proof of the intrinsic unsolvability of certain problems - such as the roots of a general quintic. In mathematical physics such results are largely unknown. I will describe a powerful numerical technique that provides compelling evidence for (but is not a proof of) the unsolvability of a wide variety of classical, unsolved problems in Statistical Mechanics and Combinatorics, in terms of the ``standard'' functions of mathematical physics (including D-finite functions).