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While it is widely acknowledged that quantum theory is strange, and that theworld view it implies is elusive, there is considerably less consensus as towhy this is the case. This article will explore the possibility that the theorymay be opaque because the mathematical language it employs is inarticulate. Toaddress this possibility, three interconnected mathematical theories will bedeveloped, each specifically designed to handle aspects of quantum theory'ssubject matter. The first will describe dynamic systems in terms that aregeneral enough to be applicable to any kind dynamics, including both classicaland quantum. That description will then be used to construct a theory ofexperiments, with the aim of understanding what types of experiments scientificmethodology allows. Finally, the theory of experiments will found a theory ofexperimental probabilities applicable to any collection of experiments, so longas their probabilities are internally consistent. The resulting over-archingmathematical language will then be applied to quantum theory. The initialapplication will investigate whether quantum theory can be used to describeexperiments. It will be found that, in general, it can not; the difficultylying in its ability to describe recording devices. Next, they will be used toinvestigate the types of worlds that satisfy quantum theory. To understand whatquantum theory means is to understand all the ways in which which the theorycan be satisfied. The nature and range of the possibilities will be delineatedby first using our mathematical language to describing a very simple type ofworld that satisfies quantum theory. That construction will then be used toderive a range of different types of quantum worlds; a range that will be seento be sufficiently inclusive to raise the possibility that all types of quantumworlds can be constructed from the original, simple one.

A formal theory of experimentation and its applications to quantum theory