A circle has an observable, finite area and yet to define it, we must use that pesky little open-ended bugger, π. We truncate the result, since π's chaos inconveniences a tidy solution; not good math, that...not good scientific method at all.
And how to we justify ourselves? By falling back on our own mathematical system, rather like proving the Bible by quoting its own words in a classic example of circular logic.
What if an error lies in our whole concept of mathematics?