The limits of this truth become obvious when you try to store the value (2^N)+1 in an N-bit integer.

Most of the time I'd like my language to allocate a nice new N+1 bit bigint off the heap so I can worry about my integer overflows in the same place I worry about running out of memory :-)

Trying to prevent that is what we programmers call a 'hard' problem

No! Problems are never "hard" - they're just non-trivial!

(or - even better - they're left as a problem for the reader and/or an unfortunate postdoc assistant :-)

So what does this have to do with functional programming? Two words: dynamic scoping.

I remember many years ago, when such creatures were more common, explaining the difference between dynamic and lexical scoping to a non-programming mathematician.

Complete blankness for some time then she suddenly said "oh - it's like frames of reference". Not having a mathematical background I went "Huh?". She proceeded to scribble stuff on the whiteboard which I failed to understand. However since we both ended up with the right answer when we went through some example code I guess it made sense to her that way :-)