IMHO, exactness is what makes math fun, but its ability to be used
to create fuzzy statements is the only thing that makes it of value.
If I say that traveling at close to the speed of light makes your time
appear to pass more slowly to a stationary observer and makes you
flatten out like a raindrop or a pancake, that is much more useful than
knowing that t' = t/sqrt(1 - v^2/c^2) and x' = x sqrt(1 - v^2/c^2).
Those equations are only useful if they can be reduced to some kind of
statement in human terms of what will happen -- which is necessarily
Even statements like, "If Mary has two apples and gets three more, then
Mary has five apples," are fuzzy; "apple", "Mary", "have", and "get"
are not defined with anything approaching mathematical precision. It's
entirely possible to have two apples, get two more, and have ten
apples, if you have six apples at home that weren't initially included
in the two that you have in your lunchbox.
You might argue that this is fuzziness in the language; the underlying
concepts of having and apples are not at fault. I think that the
fuzziness in the langauge stems from the fuzziness of the underlying
-- <firstname.lastname@example.org> Kragen Sitaker <http://www.pobox.com/~kragen/> According to my medieval text in the seventh century a finalizer raised a dead object named Gorth who infected every computer in Cappidocia ending Roman rule in the region. -- Charles Fiterman on email@example.com