hand, I refer to [*observability*]{} questions (or [*measurement*]{} problems) as the operational explanation of how to obtain quantitative values for the observables of a system, preferably within the framework of a well-defined mathematical theory. This part is usually messy, and it is the part that I propose to avoid until the other part is satisfactory. (This also saves me from entering a discussion about the meaning of words like ‘operational’ or ‘obtain’.)

For example, in classical real analysis, infinitesimals are an ill-defined, meaningless concept, though they can be approximately realized, while random sequences are a well-defined, meaningful concept, though there is no constructive way of finding one, except approximately.

A historical case supporting the power of this view of separating the analysis of meaning from that of observability is the uncertainty principle. It was argued by [Heisenberg]{} [@Hei] in terms of observability, and later by [Robertson]{} [@Rob] in mathematical terms as a simple consequence of the canonical commutation relation $[p,q]=-i\hbar$. The mathematical argument takes only a few lines and is simple and compelling; the discussion of observability is complex but shows that the results of the mathematics cannot be in conflict with what can in principle be measured by experiment.

