Generation of Hypotheses by Ampliation of Data

Abstract

The issue of the ampliation of knowledge, in particular the generation of hypotheses by ampliation of data, is not effectively treatable from a logical point of view as it is affected by the multiplicity (nay infinity) of hypotheses that can be generated by the same data. This infinity is unavoidable as a consequence of the underdetermination of data by hypotheses. The paper argues that the issue of the generation of hypotheses is, instead, treatable from a heuristic viewpoint. More specifically the paper argues that the crucial step in the generation of hypotheses is the ampliation of data, that is the integration of the data of a problem with something not contained in them. The process of ampliation is crucial in the formation of hypotheses as it narrows the infinity of possible hypotheses that explain the data. It is essentially based on ampliative inferences, in particular analogies. The paper shows that there are three main ways to ampliate data and examines and compares two case studies of generation of hypotheses. The first one is the Black-Scholes-Merton equation, namely the hypothesis that the price of an option over time is given by the partial differential equation (PDE):

$$\begin{aligned} \frac{\partial O}{\partial t} + \frac{1}{2}\sigma ^{2}X^{2}\frac{\partial ^2 O}{\partial X^2}+rX\frac{\partial O}{\partial X}-rO=0. \end{aligned}$$

The second one is the generation of the Feynman Path Integral, a hypothesis about behaviour of quantum particles (about trajectories of quantum particles), namely the hypothesis that paths followed by electrons are all possible (infinite) paths, not just the ‘classical’ ones, which can be described by the functional integral:

$$\begin{aligned} K[a,b]= \int _a^b e^{^i/_\hbar S[a,b]} Dx(t) \end{aligned}$$