``All metaphysicians (physicists) are therefore solemnly and legally suspended from their occupations till they shall have answered in a satisfactory manner the question, how are synthetic cognitions a priori possible (is it possible to learn from observations)?''Clearly this quotation must be taken in a playful way (at least as far as the invitation to suspended activities is concerned...). But, joking apart, the quotation is indeed more pertinent than one might initially think. In fact, Hume's criticism of the problem of induction, which interrupted the `dogmatic slumber' of the great German philosopher, has survived the subsequent centuries.2.1We shall come back to this matter in a while.
In order to build a theory of measurement uncertainty which does not suffer from the problems illustrated above, we need to ground it on some kind of first principles, and derive the rest by logic. Otherwise we replace a collection of formulae and procedures handed down by tradition with another collection of cooking recipes.
We can start from two considerations.
``the process of induction from experimental observations to statements about physics quantities (and, in general, physical hypothesis) is affected, unavoidably, by a certain degree of uncertainty''.
These two starting points seem perfectly reasonable, although the second appears to contradict the criticisms of the probabilistic interpretation of the result, raised above. However this is not really a problem, it is only a product of a distorted (i.e. different from the natural) view of the concept of probability. So, first we have to review the concept of probability. Once we have clarified this point, all the applications in measurement uncertainty will follow and there will be no need to inject ad hoc methods or use magic formulae, supported by authority but not by logic.