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"Boltzmann's problem" of statistical thermodynamics, is that of eliminating the paradoxical incompatibility of structure, existing between the irreversibility of the classical phenomenological thermodynamics, and the reversibility of any purely kinetic model, one could ever think of for these phenomena. One finds that, in order to construct kinetic "analogs" to the laws of phenomenological thermodynamics, the dynamics of large assemblies of molecules (Liouville theorem, etc, ) must be completed by some hypotheses of randomness. Once established, this randomness can be followed up in its development, with no new conceptual paradox (although with great technical difficulty: there is a great amount of current work on this topic). But the introduction of randomness still raises entirely uncleared problems. Since, therefore, the kinetic foundations of thermodynamics are not sufficient in the absence of further hypotheses of randomness, are they still quite necessary in the presence of such hypotheses? Or else, could not one "short-circuit" the atoms, by centering upon any elements of randomness, for example those introduced by the process of observation? Our aim is to show (partly after Szilard) that a substantial part of the results, usually obtained through kinetic arguments, could be obtained by postulating from the outset a statistical distribution for the properties of a system, and following up with a purely phenomenological argument. The spirit of the theory is extremely close to that of the conventional (Copenhagen) approach to quantum theory, and the results are quite parallel, although the mathematics is quite different. Randomness is introduced by following the modern statistical theory of the estimation of non directly observable intensive variables of state, such as the temperature. The discussion of the methodological foundations of modern statistics can thus be translated into a full-fledged, and possibly significant , counterpart of the d- iscussion of the kinetic foundations of thermodynamics. Statistics is thus provided with a particularly concrete example for some of its more involved methods; thermodynamics appears clarified in its classical aspects, and is further completed with an apparently new uncertainty relationship. It may also be of interest to the communication engineer to have a unified treatment of the foundations of fluctuation phenomena, and of methods of fighting noise : a discussion of entropy and information performed in this spirit, will be given in Part II of the paper.