The entropic and symbolic components of information

At the turn of the 19th and 20th centuries, Boltzmann and Plank described entropy S as a logarithm function of the probability distribution of microstates w of a system (S = k ln w), where k is the Boltzmann constant equalling the gas constant per Avogadro's number (R N_A^-1). A few decades later, Shannon established that information, I, could be measured as the log of the number of stable microstates n of a system. Considering a system formed by binary information units, bit, I = log₂bit From this, Brillouin deduced that information is inversely proportional to the number of microstates of a system, and equivalent to entropy taken with a negative signal -S or 'negentropy' (I = k ln (1/w) = -S). In contrast with these quantitative treatments, more recently, Barbieri approached the 'nominal' feature of information. In computing, semantics or molecular biology, information is transported in specific sequences (of bits, letters or monomers). As these sequences are not determined by the intrinsic properties of the components, they cannot be described by a physical law: information derives necessarily from a copying/coding process. Therefore, a piece of information, although an objective physical entity, is irreducible and immeasurable: it can only be described by naming their components in the exact order. Here, I review the mathematical rationale of Brillouin's identitification between information and negentropy to demonstrate that although a gain in information implies a necessary gain in negentropy, a gain in negentropy does not necessarily imply a gain in information.