Appears in collection : Misha Gromov : Mathematical Description of Biological Structures

We shall try to assign  mathematical meaning to the language used by biologists for describing   basic structures and processes in  living organisms, from the   (sub)cellular level up to  evolutionary dynamics of populations.  In particular, we shall elucidate  the mathematical as well as biological meaning of the following concepts. ●Biological (non-Shannon) information, ●descriptional (non-Kolmogorov) complexity, ●biological structure, ●biological function (performed by a particular structure),  ●biological purpose (of a function), ●information/program encoded and stored by a material structure (DNA, RNA), ●information/signal transmitted by a matter/energy process/flow,  ●information/program, which controls such a "flow", ● biological structures build by (networks of) matter/energy flows, e.g transcription --> translation -->  protein folding. Also we indicate a potential use of formalisation of  biological language  in genetic engineering, e.g. in the analysis/applications of CRISPR and of  phage assisted continuous evolution.