http://plus.maths.org/content/picking-holes-mathematics
Early on it starts off in this vein:
"The logician Kurt Gödel proved in the 1930s that if you set out proper rules for mathematics, excluding leaps of faith or intuition as admissible moves, you lose the ability to decide whether certain statements are true or false. And this isn't because you chose the wrong rules: for any set of rules, as long as they're strong enough to make sense of whole number arithmetic, there will be statements you can't prove or refute.
"This is rather shocking and you may wonder why Gödel's result hasn't wiped out mathematics once and for all. The answer is that, initially at least, the unprovable statements logicians came up with were quite artificial and didn't touch on ordinary everyday mathematics."
...and then it goes on to recount some of the work of Harvey Friedman of Ohio State University in the arena of Gödel incompleteness.
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