Dirac writes in page number 4 of his "Principles of Quantum Mechanics":
It is usually assumed that, by being careful, we may cut down the disturbance accompanying our observation to any desired extent. The concepts of big and small are then purely relative and refer to the gentleness of our means of observation as well as to the object being described. In order to give an absolute meaning to size, such as is required for any theory of the ultimate structure of matter, we have to assume that there is a limit to the fineness of our powers of observation and the smallness of the accompanying disturbance-a limit which is inherent in the nature of things and can never be surpassed by improved technique or increased skill on the part of the observer. If the object under observation is such that the unavoidable limiting disturbance is negligible, then the object is big in the absolute sense and we may apply classical mechanics to it. If, on the other hand, the limiting disturbance is not negligible, then the object is small in the absolute sense and we require a new theory for dealing with it.
A consequence of the preceding discussion is that we must revise our ideas of causality. Causality applies only to a system which is left undisturbed. If a system is small, we cannot observe it without producing a serious disturbance and hence we cannot expect to find any causal connexion between the results of our observations. Causality will still be assumed to apply to undisturbed systems and the equations which will be set up to describe an undisturbed system will be differential equations expressing a causal connexion between conditions at one time and conditions at a later time. These equations will be in close correspondence with the equations of classical mechanics, but they will be connected only indirectly with the results of observations. There is an unavoidable indeterminacy in the calculation of observational results, the theory enabling us to calculate in general only the probability of our obtaining a particular result when we make an observation.
I want to make sure that I correctly understand this:
What we have are two observations and those observations do not seem to be causally related (meaning in an identically prepared system we are not guaranteed to have the same observations and therefore we cannot set up a deterministic mathematical equation for the system).
In such a context we assume that the system evolves determinsistically after an observation and it is only due to the indeterminacy introduced by the next observation that our result of the observation can no longer be predicted with accuracy.
1) How does this line of reasoning explain that consecutive measurements give the same result? And that the measured value will only be one of the eigenvalue of the operation? It seems to me that the indeterminacy is not in any way similar to the disturbance inherent in measuring "fragile" or "sensitive" properties of classical systems as those disturbances are totally random.
2) Is it possible to set up a mathematical framework for quantum mechanics that does not make this assumption. (For example, the new framework may postulate that a system non-deterministically takes many paths of evolution but it is only on subsequent observation that it settles on one particular value.)
3) Does this line of thought by Dirac ascribe to any of the interpretations of quantum mechanics (Copenhagen, Many-World, Bohmian, etc.)? Does it go against any of the interpretations?