Gödel, truth & proof

The usual way of interpreting Gödel's (1931) incompleteness result is as showing that there is a gap between truth and provability, i.e. that we can never prove everything that is true. Moreover, this result is supposed to show that there are unprovable truths which we can know to be true. This, so the story goes, shows that we are more than machines that are restricted to acting as proof systems. Hence our minds are 'not mechanical'.