This theoretical paper explores the decision-making process involved in modelling and mathematizing situations during problem solving. Specifically, it focuses on the authority behind these choices (i.e., what or who determines the chosen mathematical models). We show that different types of situations involve different sources of authority, thereby creating different degrees of freedom for the problem solver engaged in the modelling process. It also means that mathematics plays different roles in these problems and situations. This epistemological analysis on the meaning of modelling implies that we should reconsider the mathematical status of realistic solutions and raises questions on the validity of some traditional choices of mathematical models and their use in diagnosing children's conceptions. It also suggests constructing modelling tasks by choosing a certain variety of situations that might lead to a better understanding of the roles of mathematics. (Contains 1 table and 2 figures.)