  There is a problem with the foundations of classical mathematics, and
potentially even with the foundations of computer science, that mathematicians
have by-and-large ignored. This essay is a call for practicing mathematicians
who have been sleep-walking in their infinitary mathematical paradise to take
heed. Much of mathematics relies upon either (i) the "existence'" of objects
that contain an infinite number of elements, (ii) our ability, "in theory", to
compute with an arbitrary level of precision, or (iii) our ability, "in
theory", to compute for an arbitrarily large number of time steps. All of
calculus relies on the notion of a limit. The monumental results of real and
complex analysis rely on a seamless notion of the "continuum" of real numbers,
which extends in the plane to the complex numbers and gives us, among other
things, "rigorous" definitions of continuity, the derivative, various different
integrals, as well as the fundamental theorems of calculus and of algebra --
the former of which says that the derivative and integral can be viewed as
inverse operations, and the latter of which says that every polynomial over
$\mathbb{C}$ has a complex root. This essay is an inquiry into whether there is
any way to assign meaning to the notions of "existence" and "in theory'" in (i)
to (iii) above.
