  While it is widely acknowledged that quantum theory is strange, and that the
world view it implies is elusive, there is considerably less consensus as to
why this is the case. This article will explore the possibility that the theory
may be opaque because the mathematical language it employs is inarticulate. To
address this possibility, three interconnected mathematical theories will be
developed, each specifically designed to handle aspects of quantum theory's
subject matter. The first will describe dynamic systems in terms that are
general enough to be applicable to any kind dynamics, including both classical
and quantum. That description will then be used to construct a theory of
experiments, with the aim of understanding what types of experiments scientific
methodology allows. Finally, the theory of experiments will found a theory of
experimental probabilities applicable to any collection of experiments, so long
as their probabilities are internally consistent. The resulting over-arching
mathematical language will then be applied to quantum theory. The initial
application will investigate whether quantum theory can be used to describe
experiments. It will be found that, in general, it can not; the difficulty
lying in its ability to describe recording devices. Next, they will be used to
investigate the types of worlds that satisfy quantum theory. To understand what
quantum theory means is to understand all the ways in which which the theory
can be satisfied. The nature and range of the possibilities will be delineated
by first using our mathematical language to describing a very simple type of
world that satisfies quantum theory. That construction will then be used to
derive a range of different types of quantum worlds; a range that will be seen
to be sufficiently inclusive to raise the possibility that all types of quantum
worlds can be constructed from the original, simple one.
