As far as I understand, the Hilbert space formalism can be derived using functional analysis and representation theory (not familiar with those) from the requirement that observables (their mathematical models) form a C*-algebra and the possible states of a system map the members of the algebra to the actual measurement results (that would be R). This needs clarification. Hence, questions: 1) At some point the result of a measurement must be translated (from a state of a probe) into a symbol that we can write down on paper - a number perhaps. Real numbers seem to be intuitively optimal: they have ordering, some familiar metric can be used, and they can also describe diagonals and circumferences. In pairs, they can even describe complex numbers. Very convenient. An ordered set which is a metric space allows some predictive power for models built (getting results that could not be ordered at all would be...restrictive). In reality, we always get a result in Q, right? This is not familiar ground at all. Am I missing something here? 2) For each observable (the set of possible numerical(?) results above) a mathematical model/entity is needed (e.g. a matrix). What experimental results or intuitive motivations do we have that require the mathematical objects modeling observables to form a C*-algebra? (=Abelian additive group, a ring with multiplication, the required existence of an adjoint element A* (and A*A - does this relate to positivity?), and so on) Any pointers to the right direction appreciated.