The chronologic age classically used in demography is often unable to give useful information about which exact stage in development or aging processes has reached an organism. Hence, we propose here to explain in some applications for what reason the chronologic age fails in explaining totally the observed state of an organism, which leads to propose a new notion, the biological age. This biological age is essentially determined by the number of divisions before the Hayflick’s limit the tissue or mitochondrion (...) in a critical organ (in the sense where its loss causes the death of the whole organism) has already used for its development and adult phases. We give a precise definition of the biological age of an organ based on the Hayflick’s limit of its cells and we introduce a desynchronization index (the cell entropy) for some critical tissues or membranes, which are mainly skin, intestinal endothelium, alveoli epithelium and mitochondrial inner membrane. In these actively metabolising interface tissues or membranes, there is a rapid turnover of cells, of their cytoplasmic constituents such as proteins, and of membrane lipids. The boundaries corresponding to these tissues, cells or membranes have vital functions of interface with the environment (protection, homeothermy, nutrition and respiration) and have a rapid turnover (the total cell renewal time is in mice equal to 3 weeks for the skin, 1.5 day for the intestine, 4 months for the alveolae and 11 days for mitochondrial inner membrane) conditioning their biological age. The biological age of a tissue is made of two major components: (1) first, its embryonic age based on the distance (in number of divisions) between the birth date of its first differentiated cell and the time until it reaches its final boundary at the end of its development and (2) second, its adult age whose complement until its death is just the lapse of time made of the sum of remaining cell cycle durations authorized by its Hayflick’s limit. From this definition, we calculate the global biological lifespan of an organism and revisit notions like demographic survival curves, duration and synchrony of cell cycles, living boundaries from proto-cells to organs, and embryonic and adult phases duration. (shrink)

We analyze the notions of indiscernibility and indeterminacy in the light of the Galois theory of field extensions and the generalization to \(K\) -algebras proposed by Grothendieck. Grothendieck’s reformulation of Galois theory permits to recast the Galois correspondence between symmetry groups and invariants as a Galois–Grothendieck duality between \(G\) -spaces and the minimal observable algebras that discern (or separate) their points. According to the natural epistemic interpretation of the original Galois theory, the possible \(K\) -indiscernibilities between the roots of a (...) polynomial \(p(x)\in K[x]\) result from the limitations of the field \(K\) . We discuss the relation between this epistemic interpretation of the Galois–Grothendieck duality and Leibniz’s principle of the identity of indiscernibles. We then use the conceptual framework provided by Klein’s Erlangen program to propose an alternative ontologic interpretation of this duality. The Galoisian symmetries are now interpreted in terms of the automorphisms of the symmetric geometric figures that can be placed in a background Klein geometry. According to this interpretation, the Galois–Grothendieck duality encodes the compatibility condition between geometric figures endowed with groups of automorphisms and the ‘observables’ that can be consistently evaluated at such figures. In this conceptual framework, the Galoisian symmetries do not encode the epistemic indiscernibility between individuals, but rather the intrinsic indeterminacy in the pointwise localization of the figures with respect to the background Klein geometry. (shrink)