Principles of biological organization: Local–global negotiation based on “material cause”

Organization of living things is characterized by dynamical hierarchical structures inheriting discrepancy among levels. It can be expressed as a system consisting of two layers; the microscopic perspective (Extent) defined by a collection of elements and the macroscopic perspective (Intent) defined by the property as a whole, and the interplay between them. First we show that if the microscopic and macroscopic perspectives are consistent with each other (an ideal case), then the operation between the two layers can be expressed as a sheaf between a lattice and a quotient lattice, where a sheaf is a mathematical operation representing the gluing process. Second, we introduce an observer who cannot look out over the whole world, and this reveals a discrepancy between the two layers. Third, we introduce a new mathematical construction, called skeleton, that is derived by the sheaf operation. The skeleton reduces the discrepancy between the micro- and macroscopic perspectives, and that reveals the perpetual transition between the perspectives. This process yields a basic framework of biological organizations. Finally, we argue that the skeleton mediating between the two levels is a particular expression for the material cause.