Perhaps it would not be out of place to quote Miles Reid's Bourbaki seminar on the McKay correspondence here:
"The physicists want to do path integrals, that is, they want to integrate some "Action Man functional" over the space of all paths or loops $ \gamma : [0; 1] \rightarrow Y $. This impossibly large integral is one of the major schisms between math and fizz. The physicists learn a number of computations in finite terms that approximate their path integrals, and when sufficiently skilled and imaginative, can use these to derive marvellous consequences; whereas the mathematicians give up on making sense of the space of paths, and not infrequently derive satisfaction or a misplaced sense of superiority from pointing out that the physicists' calculations can equally well be used (or abused!) to prove 0 = 1. Maybe it's time some of us also evolved some skill and imagination. The motivic integration treated in the next section builds a miniature model of the physicists' path integral,..."