Use and misuse of statistics seems to be the signum temporis of past decades. But nowadays this practice seems slowly to be wearing away, and common sense and responsibility recapturing their position. It is our contention that little by little statistics should return to its starting point, i.e., to formalizing and analyzing empirical phenomena. This requires the reevaluation of many traditions and the rejection of many myths. We hope that our book would go some way towards this aim. We show the sharp conflict between what is needed and what is feasible. Moreover, we show how slender are the links between theory and practice in statistical inference, links which are sometimes no more than mutual inspiration. In Part One we present the consecutive stages of formalization of statistical problems, i.e., the description of the experiment, the presentation of the aim of the investigation, and of the constraints put upon the decision rules. We stress the fact that at each of these stagesthere is room for arbitrariness. We prove that the links between the real problem and its formal counterpart are often so weak that the solution of the formal problem may have no rational interpretation at the practical level. We give a considerable amount of thought to the reduction of statistical problems.