Lieber 



states and fields. Microstress fields mean here fields of stress that extend 

 over finite domains, but which have small magnitudes everywhere within these 

 domains. A macrostress field instantaneously prevailing within a finite domain 

 of a material body, is conceived here as a superposition of a collection of mi- 

 crostress fields, each of which instantaneously agree with (a) force equilibrium 

 conditions, (b) the instantaneously prevailing geometrical constraints in all their 

 categories, (c) with the condition of the universe as it is posited by forces im- 

 parted to the material domain, and (d) with the constitutive properties of the do- 

 main, which can be translated into certain of the categories of geometrical con- 

 straint; in particular, the categories of active and passive conditionally stringent 

 constraints. At each instant, the macrostress field is sustained over duration, 

 and thus constantly evolves into new states which are derived by the selection 

 and development of one of the fields of microstress which belongs to the imme- 

 diately preceding field of macrostress. The selection and development of a 

 particular microstress field that belongs at a particular instant to a macro- 

 stress field is determined by the condition invoked by the principle of maximum 

 uniformity. Each and every actual microstress field that belongs to the collec- 

 tion that instantaneously corresponds to an actual macrostress field, is what we 

 may call here an actual-possible state, in the sense that each is an actual mi- 

 crostate and is endowed with the possibility of subsequently (in time) evolving, 

 according to the principle of maximum uniformity, new states of macrostress. 

 We see that the above statement of the principle of maximum uniformity ac- 

 commodates the evolution in space and time of various and distinct micro- and 

 macrostates of stress which reveal in experience the various hierarchies of 

 uniformity. This principle, which is understood here to be universal and conse- 

 quently not contingent, refers to a whole hierarchy of aspects of uniformity, 

 rather than as in the case of force equilibrium, to a particular member belong- 

 ing to the hierarchies of uniformity. From the present statement of the princi- 

 ple of maximum uniformity, we see that it accommodates the emergence in 

 space and time of various members of the hierarchies of uniformity. 



An Observation Concerning Turbulence 



Turbulence, i.e., its evolution, appears from the hydrodynamical study 

 cited above, to derive from an aspect of the principle of maximum uniformity 

 which is not embraced in the propositions of classical mechanics, and conse- 

 quently not by the Navier -Stokes equations. 



In this sense, it follows that the information which can resolve the enigma 

 of turbulence is not contained in the Navier -Stokes equations, and correspond- 

 ingly, neither do the equations contain the information necessary to construct 

 analytical solutions to them, which derive from realistic boundary conditions. 

 One has to prescribe more information than the Navier -Stokes equations contain 

 in the category of implicit information, in order to construct these solutions. 

 This seems to be related to Godel's theorem, which is here interpreted to be a 

 particular aspect of a general law that concerns the accessibility and inaccessi- 

 bility of information in its various categories. 



Some fundamental aspects of the principle of maximum uniformity derive 

 from an interpretation of force, which ascribes to all of the forces in nature 



476 



