  As physics searches for invariants in observations, this paper looks for
invariants of probabilistic observation without assuming physical structure.
Structure emerges from the basic assumption of science that new information
shall lead to more accurate knowledge of the invariants. This leads to
statistically unique random variables for expressing observed information:
Complex probability amplitudes. Predictions are also just random variables
computed from observed data, and must become more accurate with more
observations as input. This singles out the quantum mechanical superposition
principle. The external conditions of a probabilistic experiment can themselves
be monitored at the most detailed level, resulting in observation of coicidence
probabilities. The invariants of any multi-coincidence experiment are the same
as those of a one-event experiment with the same number of possible outcomes.
An observable probability turns out to be controllable by two independent
experimental conditions, naturally parametrized as a direction in a
3-dimensional space. In summary, the probabilistic paradigm itself defines a
unique method of forming concepts and making predictions. The method appears
irrefutable within probability, because, whenever a prediction turns out wrong
the existence of an as yet unmonitored condition is postulated, and a formal
way to incoroporate it is shown. The Hilbert space formalism of quantum theory
seems to be isomorphic to this method.
