In this series of papers, a theory of functional organization is proposed for biological systems (formal biological system, FBS), which is based on the concept of 'functional interaction', and on a 'functional self-association hypothesis'. From the specific properties of functional interactions, i.e. non-symmetry, non-locality, and non-instantaneity, it is shown that a biological system can be considered as constituted by two hierarchical systems: (i) the (O-FBS) that describes the topology of the FBS, i.e. the functional organization, with a hierarchical directed graph; (ii) the (D-FBS) that describes the continuous non-linear dynamics of the FBS with a field. In the framework of this theory, the problem of the relation between structure and function is considered to be due to the distinction between structural organization and functional organization. Some advantages of this approach are: (i) the description of the time evolution, during development, of the organization of an FBS with an optimum principle, which leads to a clear comparison with a physical system (paper II); (ii) the description of the space-time dynamics as the variation in space and time of field variables in a hierarchical 'space of structural units'; and, consequently, the relation between topology and geometry, and the existence of non-locality in these hierarchical spaces (paper III). In this paper, the basic concepts of functional interaction, hierarchical functional organization, and physiological function are discussed from a mathematical viewpoint, and arguments for the validity of the self-association hypothesis are given. Specifically, it is shown that, for a particular class of biological systems that are taken as an example, the domain of stability of the (D-FBS) is increased after functional association. This property, which is specifically due to the nature of the biological system, corresponds to an increase in complexity. It will be shown in the second paper that such a self-organization corresponds also to an optimal principle for the (O-FBS). The case of real biological systems (RBSS) is considered in relation with the present theory, which leads to a new hierarchical representation in terms of fields. Such representation could be a base for integrative physiology. As an example, some physiological functions, respiratory and cardio-vascular, are considered and it is shown that the heart shock emerges from the formulation as a cyclic sub-graph.