Variations on the theme of invariants: conceptual and mathematical dualities in physics vs biology

In a work of Bailly and Longo (2010), a comparison between the invariants in physics and some particular invariants in biology is presented through conceptual dualities that we highlight and further specify here in their main theoretical differences. Many controversies in relating physics and biology, for instance the question of a physicalist approach in biology, derive from strong and well-established epistemological and cultural guardrails. Our purpose is to identify at least some of the patterns or principles that may guide theory building in biology by physico-mathematical methodologies which simultaneously takes into account its autonomy. In other terms, we discuss biological extensions of physical theories that are based just on "conceptual practices" from physics and not necessarily on its laws and objects. To this purpose we stress the divergence between mathematical invariants and variables in physics with respect to biological variability (and evolvability), which is the leading idea of this work. We shall also stress the unstable instability of living entities, in reference to the notion of extended criticality, a notion which goes beyond the physical theory of point-wise critical transitions. Protensive activities will then be mentioned as proper to the "living state of matter" and extraneous to physical theoretizing.