The selection of an appropriate mathematical representation is of central importance in the analysis of a physical system. Often the choice of the respective model depends on essentially two criteria: the particular characteristics to be abstracted and, perhaps more importantly, our ability to specify the representation quantitatively. A complication for determining the standard material functions of viscoelastic plants, such as the Maxwell and Kelvin models or the relaxation and retardation spectra, is that these problems are undetermined and ill-conditioned. Due to the noise or truncation of the experimental data, many models may fit the relaxation modulus experimental data adequately, but small errors in the data may lead to large changes in the models determined. Four motivating examples are given. Thus the practical difficulty in the identification of these models is rooted in a theoretical difficulty. The mathematical difficulties can be overcome by the reduction of the set of admissible solutions or by the respective regularization of the original problem. The remedy is a subject of the two next papers.