On the Scientific Method: The Role of Hypotheses and Involved Mathematics
The paper investigates the role of data, hypotheses and mathematical methods that can be used in the discovery of a law y=fo(u), relating variables u and y of a physical phenomenon, making use of experimental measurements of such variables. Since the exact knowledge of the function fo cannot be expected, the problem of deriving approximate functions giving a small approximation error, measured by some function norm, is discussed. The main contributions of the paper are summarized as follows. At first, it is proven that deriving a reliable approximation, i.e., having a finite error, is not possible using measured data only. Thus, for deriving a reliable approximation, hypotheses on the function fo and on the disturbances corrupting the measurements must be introduced. Second, necessary and sufficient conditions for deriving a reliable approximation are provided. If such conditions are satisfied, suitable accuracy properties of the approximation can be defined, called theoretical properties. Third, it is shown that it is not possible to verify the conditions necessary for deriving a reliable approximation, but it is possible to verify that hypotheses on fo and on the disturbances are falsified by experimental measurements, showing that no function and disturbances satisfying the given hypotheses exist, able to reproduce the measurements (this is called falsification property). The above properties are then discussed for hypotheses belonging to the following classes: Parametric Probabilistic, where fo is assumed to be a function depending on a vector p and the disturbances are assumed to be stochastic variables; Set Membership class, where fo is assumed to be a bounded smooth function and the disturbances are assumed to be bounded variables; Parametric Set Membership class, able to integrate Parametric Probabilistic hypotheses with Set Membership hypotheses.