Thus a living cell consists of molecules, organisms are formed from cells; the latter in turn unite to form populations; populations form communities which in turn can be considered as elements of the biosphere. That is a fragment of the biological hierarchy.

A version of the geological hierarchy: molecules, minerals, rocks, sediments, terrains.

The geographic hierarchy: molecules, sediments, faces, landscapes, physical-geographic districts, provinces, zones, countries, continents, the dry land, the geographic shell (according to I.A.Solntsev, from the book by I.I.Mikhailov (1985)).

A sketch of the astronomical hierarchy: molecules, bodies, planet systems, star associations, galaxies... (All these examples are not rigorous constructions, rather, these are just illustrations to enlighten the general considerations.)

I would like to mention the general processes for objects belonging to the canonical example of biological hierarchy.

The general process in living cells is metabolism, the process of replacing molecules forming the cell. For multicellular organisms the general process is growth, during which new cells appear and the existing ones are replaced or disappear. The number dynamics, summing up the births and deaths of individuals, is the general process for a population. Species replacement, called succession, is a manifestation of the general process in ecological communities. The change of associations of species in the Earth's biosphere is called the evolution process.

"In the descriptions of the measurement process, so essentially simple, one can notice a significant reticence in many courses of mechanics and physics which have become classic. It was my task to establish more determinacy in the problem and, along with that, to show what a great arbitrariness is present in establishing a measurement" (Friedmann 1965, p.16).

Thus "any class of properties can be arithmetized; if these properties are made intensities (by our definition), then we can... estimate them by numbers; finally, if the intensities are made (again by our definition) measurable intensities, then we can... measure them; a measurement will contain certain arbitrariness, removed by establishing the zero point and the measurement unit" (Friedmann 1965, p.15).

The necessity of such an agreement is realized by natural scientists (Milne 1948): A priori we can take any dynamic phenomenon and use its development to define the time scale. However, a uniform natural scale does not exist, since we cannot say what is meant by the word "uniform" with respect to time; we cannot catch the present minute and put it side by side with the next one. It is sometimes said that a uniform time scale is defined by periodic phenomena. However, allow me to ask a question: can anybody tell us that the two periods, following one another, are equal?

In physics the role of a uniformity agreement is played by Newton's first law: the time intervals during which a body, moving without interaction with other bodies, covers equal distances, are called equal (Thompson and Tait 1890).

The proper time (Vasilyev et al. 1974) or the proper age (Zotin and Alexeyeva 1984) of an organism can be defined from the counts of consumed oxygen molecules.

The proper age of an organism can be measured by the number of newly formed cells; by wound healing area (Noüy 1936); by growth of specified organs of the body, e.g., the size of eye crystalline lens is considered to be one of the best biological age markers for mammals (Shaher 1982); the number of separated cells of yeast, being their only stable age characteristic, unlike any chronological dating (Voitenko 1985).

The scale of age stages of a plant (Cheburayeva 1977): germination, juvenile, immature, young vegetative, young, mature, old, subsenile, senile stages) is treated as a specific form of ontogenetic time accounting, such that the intervals between the neighbouring age stages are taken to be equal (Uranov 1975). The same is the case for the scale using the instants of pea alternate leafs appearance (Thornwaite 1953).

The dynamics of microalgae populations is well described in terms of consumed biogenic elements which are limiting factors of community development (Levich 1980; Levich et al. 1986). The maturity parameter of an ecological community (a concept close to ecological age) is introduced by M.E.Vinogradov and E.A.Shushkina (1983): the maturity index is connected with the ratio of community destruction (commonly measured by the amount of biogenic chemical elements leaving the community) with respect to primary production (proportional to the amount of biogenic elements entering the community).

In paleontology the analysis of large groups of organisms on the basis of measuring the number of taxons is quite common. One usually takes into account those characteristics which refer to a single stratigraphic division, namely, the overall number of taxons and the numbers of those appeared and died out (Dmitriev 1978).

The amount of oxygen consumed by an animal from its "first sigh" and "last breath" can serve as its physiological age. If the Rubner rule is valid (Zotin and Alexeyeva 1984), then this quantity is the same for all representatives of a single biological species. Moreover, if one assumes that the oxygen consumption rate for oxidation of a unit mass of food is constant, then the physiological age corresponds to the amount of consumed food, i.e., one speaks of a certain substrate-energetic proper time scale of an organism. The Rubner rule is confirmed by experimental results obtained mostly on rodents, insects and unicellulars: food limitation leads to life prolongation (McCay 1935; Bauer 1935).

In quantum-mechanical field description there exists a deep correspondence between the representations of phenomena in the language of frequencies and that of particle replacements ("creation" and "annihilation"). It is involved in the second quantization method.

The ideas that time is more than one-dimensional repeatedly emerged in natural science. "For life viewed from the geochemical standpoint, time is expressed in three different processes: firstly, the time of individual existence, secondly, the generation changing time, with the form of life unchanged, and, thirdly, the evolutionary time, i.e., that of changing forms along with generations" (Vernadsky 1988, p.231). "The existence of many time scales is without doubt the most significant aspect of life.... For instance, there exist the physical time (in equations of motion), the catalytic time (necessary for describing fermentative reactions), the time of cellular fission, the ecological succession time and, finally, the evolutionary time..." (from G.Patti's letter addressed to C.H.Waddington. On the way..., 1970, pp.177-178). G.E.Mikhailovsky (1982) introduces the complex time of living organisms (see also G.E.Mikhailovsky's chapter in this book). Its real part is the ontogenetic time of an organism while the imaginary one is determined by the stage of the self-reproduction processes. N.I.Moiseyeva (1980) insists that a three-dimensional biological time should be introduced.

Note that the physical processes aspiring to be used as time standards (the rotating Earth, the ephemeris time, the "second world time" taking into account the seasonal corrections to the rotation of the Earth, the tropic year, radiation of caesium atoms) represent the clocks of significantly different uniformities from the standpoint of the modern level of accuracy (Martynov 1961).

The scale connected with wound healing rate (Noüy 1936) turns out to be non-uniform with respect to the chronological age: a five-year-old child's wound is healed ten times faster than that of a person of fifty.

A nonlinear transformation of planetary time (Backman 1943) applied in a description of growth curves for a broad class of living organisms made it possible to discover the elementary durations, "life quanta"; their density is uniform in Backman's "organic time" and is much greater at the first stages of development from the viewpoint of the ordinary time scale (that is why it is so hard to catch them).

E.Milne (1948) removed the postulate of congruence between time intervals shown by clocks of one type, namely, mechanical and atomic ones, and introduced the logarithmic scale to measure the cosmological time of the Universe. The time transformation eliminated the gravitational interaction from the fundamental equations of motion and greatly simplified the description of the non-stationary universe.

Speaking of specific time, for instance, of physiological, ontogenetic or evolutionary times, we mean that the first one is measured by the number of absorbed oxygen molecules, the second one by the number of newly formed cells of a growing organism while the third one by the number of taxons in the reconstructed biosheric annals.

The geological, biological, geographic and other specific times are proper time pyramids constructed in the corresponding natural hierarchies.

Thus, if an ecosystem is studied as a unity of molecules, of which all the biotic components and all the inanimate substance pools consist, we can say much about the substantial functioning basis of such systems. However, the molecular language is hardly applicable for formulating the ideas of trophic connections between species, of age, sexual and other structure of the population, of seasonal and other successions, on the behaviour of individuals and other non-molecular processes.

In physics the role of a factor which "wipes up the structures", is frequently played by energy.

Thus, for instance, application of the metabolic time scale, connected with the number of substrate molecules absorbed by cell populations in microalgae culture development, made it possible to discover the growth and consumption stages which had not shown themselves in any way when ordinary growth curves had been analysed (Levich et al.1986).

When the durations of development stages of animals was measured in detlafs (Dettlaff and Dettlaff 1982; see also a chapter in this book), it provided a way to compare development times both for different species and for the same species in different development conditions (e.g., at different temperatures).

The problem of the existence of individual life duration bounds is very hard to analyse if the age is measured in ordinary chronological units. Unlike that, application of a proper time, for instance, in terms of the number of molecules of matter consumed by the organism, gives a hope to discover the natural bounds connected with a limiting total amount of oxygen, absorbed by an organism in its whole life, different for different species (provided the Rubner rule is valid - see Zotin and Alexeyeva 1984).

Already in quantum mechanics there exist timeless events: emission and absorption of electromagnetic quanta by atoms, i.e., transitions to other energy states of atoms; wave packet reduction; changes in quantum numbers of one part of a quantum system resulting from a measurement process executed over the other, arbitrarily remote, part (the Einstein-Podolsky-Rosen paradox).

A good illuminating image of such a motion is that of a "running advertisement", or a moving picture in the screen of a kinescope.

For instance, molecule replacements in a cell can be spoken of in the language of the time flow (an image: a stationary cage, with entering and departing molecules counting the steps of time - a "cell-centric" viewpoint). Or the same phenomenon can be spoken of as a cell's motion in the molecular space, where the word "motion" means a transition from inclusion of one environmental molecule to the cell to inclusion of another one (an image: motionless point molecules and a cell moving from one molecule to another - a "molecular-centric" viewpoint).

Table 1. A physical interpretation of system-theoretical concepts.

The life of a cell is connected with replacements of molecules including many chemical elements and their replacement rates are drastically different. A metabolic clock of a cell can be defined via the so-called "limiting" element, or a sum of all sorts of molecules, which is convenient in certain cases and unsatisfactory in others.

The cells of an organism are differentiated, so that cell replacement rates in different tissues, subsystems and organs are not equal. Thus, which are the cells whose replacements count the biological age of an animal, epithelial, neurons or erythrocytes?

An ecological community of individuals belonging to different species can be well described by the structure of a partitioned set with classes corresponding to populations which form the community.

The proximity and remoteness concepts in the empirical space are described by the topological structure.

The aggregate of atomic states can be described in terms of vectors in the infinite-dimensional Hilbert space or, equivalently, by the field of infinite matrices.

A community of living organisms which inhabit a certain location, for instance, all the individuals living in a pond; a whole set of indications divides them into populations belonging to certain biological species.

In the set of whole numbers any two elements determine the third one, their sum.

For an ecological community morphisms conserving partition into species, are transformations consisting of individual births and deaths. These transformations transfer a species into itself.

For the metric structure one of the morphisms is a transformation of rotation in the three-dimensional space, since rotations conserve the spatial metric.

The rotation properties of the geometric space can be effectively studied using products of certain matrices.

The whole measure theory, including measurements of lengths, areas and volumes in geometry, is an example of quantitative parametrization built on the basis of representations of geometric structures into the numerical structures of the analysis.

Equivalent mathematical descriptions are frequent in physics: atomic states in quantum mechanics can be described both by vectors in the infinite-dimensional Hilbert space (the Schrödinger approach) and by infinite matrices (the Heisenberg approach).

It can be noticed that a category-theoretic description of systems does not necessarily require an explication of a natural system by a mathematical structure. One can have a “qualitative” category description of systems, i.e., the system states can be enumerated and described, along with all transformations between states (morphisms), not in the mathematical language but in some internal language of the relevant field of knowledge.

To obtain a formulation of a full dynamical theory it only remains to determine its T and L components, i.e, the variability parametrization method and the variability law.

Entropy of systems. Let us introduce the entropy of a state A with respect to a state X in terms of the invariants of structured sets in the following way:

. (2.1)

Here Q is the category of structured sets and is the category of sets with a "wiped off" structure. X marks the states of a system belonging to its certain macrostate (the concept of a macrostate is introduced in the subsequent paragraphs). The entropy of a whole macrostate is determined by a proper summation over all X.

Expression (2.1) for the entropy can be interpreted as a measure of structuredness of state A, i.e., a measure of a deflection of the structure of state A from its structureless counterpart.

This expression generalizes the traditional ways of introducing entropy to statistical physics. Commonly the entropy of a system is defined as the logarithm of a specific number of its different microstates corresponding to its given macrostate. I would like to interpret a macrostate as a class of states among which transformations are admissible from some consistent considerations, for instance, some system macroparameters are conserved (energy, particle number, etc.), while a microstate is a result of an arbitrary transformation of the system.

Thus to calculate the entropy of a system it is necessary to calculate the number of its transformations-morphisms, . This number depends on both the system structure and the number of elements contained in it. The multiplier has been introduced in order to normalize the entropy to a single element of the system, i.e., to introduce a specific invariant of structure.

The "entropy-like" functions of system state, having appeared in thermodynamics, have penetrated via statistical physics, information theory and cybernetics to the quantitative methods of a wide range of sciences. The success of its applications is entirely determined by the possibilities of calculating the entropy in the cases of interest for the researcher. The purely thermodynamic approaches to calculating the entropy are very restricted: "... a formulation of the second law from a modern physicist's viewpoint is a programme rather than a statement admitting an unambiguous interpretation, since neither Thompson nor Clausius indicated a precise procedure of expressing an entropy variation in terms of observed quantities" (Prigogine 1986, p.93). Only Boltzmann's interpretation of entropy through the number of ways to reach a given macrostate of the system yields a constructive way of calculation. Its interpretation in terms of the number of morphisms gives further possibilities for calculations.

As it has already been mentioned, the invariants of many (if not all) mathematical structures are expressed in terms of the invariants of partitions associated with the structures. The invariants of partitioned sets are multiplicative with respect to invariants of each partition class, therefore the logarithms of invariants involved in the entropy, are additive and have the characteristic "entropy-like" form of sums over partition classes.

The entropy expressions are obtained apart from any statistical prerequisites. For instance, if the admitted morphisms are mappings, then the expression

,

where and , is valid for all, even very small, n. Introducing and representing the set of values p_i as a distribution function for a state A, one can interpret the entropy formula as a generalization of Boltzmann's H-function

,

which (Prigogine 1985, p.303) serves as a probability measure of the deflection of a system's state at an instant t from its equilibrium state.

It should be noted that the entropy expressions given in the present section as examples (as well as the generalizing invariant expressions of Table 2) correspond to the simplest (two-level) systems. In the problems where greater numbers of hierarchic levels of systems are significant, it becomes necessary to calculate invariants for hierarchic structures (e.g., for the armade structure - Levich 1982).

In natural science an equation of motion is usually a postulate, generalizing the experience of mathematical descriptions of a certain piece of reality, and invented by a genius whose name becomes that of the equation. There is another way of obtaining the equations: to replace a postulate in the form of an equation by a postulate in the form of a functional. We are speaking of the extremum principles of natural science, claiming that only those states of systems take place in the reality for which a given numerical function (functional), whose argument contains the trajectories of motion, takes extreme values.

If an equation of motion is known, it can be a standpoint for determining the form of the functional whose extremal would be the above equation. And conversely, if a functional is given, the variational method of finding its extremum leads to the equations of motion. Thus building of dynamics from postulated equations and postulated functionals leads to the same results. However, the extremum principle has a greater heuristic and generalizing power.

The extremum principle: For natural systems described by mathematical structures,the extremum principle is suggested in the following form: a system passes from a given state X to another state A for which the entropy is maximum within the limits admitted by the conditions of system functioning (e.g., the available energy or other resources).

Thus, in the language of category-theoretic descriptions of systems, the L component of the theory appears.

In the present context entropy is not an initial, undefined concept but an exact construction. This suggests additional interpretations to the extremum principle:

1. By definition, the entropy is considered as a measure of remoteness of the structured state A belonging to category Q from its counterpart belonging to category with a "wiped off" structure. According to the extremum principle, those states are realized which are most remote from their entirely structureless counterparts, i.e., more structured than the others.

2. In the entropy formula the denominator is an invariant of the structured set A. The inequality means that the structure of set A is weaker than that of set B (with respect to object X), in the sense of the order relation "structure strength". Assume that for two system states A and B the numerators in the entropy formula are equal, i.e., the states A and B are different realizations of a certain structure at equicardinal basic sets, e.g., different partitions of the same set, or groups which are equicardinal as sets. Then the maximum entropy and the real state correspond to a minimum (by structure strength) state of the system. Thus entropy plays the same role as the invariants of mathematical structures: it orders the structures themselves along with natural system states described by them.

3. On the example of partitioned sets (so that morphisms are bijections) one can see that the entropy is maximum if the following two conditions are simultaneously valid: the number of elements of the system (the numerator) is maximum while the number of transformations preserving the structure (the denominator) is minimum. A small set of transformations admitted by the system structure can be treated as a high stability of the state. As for the extremum principle, it realizes both a maximum expansion of the elements of the system and its most stable state.

4. Note that without additional restrictions the maximum expansion requirement, created by the extremum principle, leads to an infinite growth of the number of elements in the system. The life of natural systems is always connected with limited resources (energy, substrates, space, information, etc.). For instance, for substrate and energy factors these restrictions can be written formally as the balance inequalities

where are functions of a state and are resources consumed by the system in its expansion. Therefore the real states of systems are solutions of conditional extremum problems.

It can be shown that the entropy extremum principle with resource-factor restrictions (2.2) is equivalent to the minimum consumption principle with respect to any of the limiting resource factors, with restrictions taking into account the minimum admitted value of the entropy characteristic of the system.

5. Both mentioned variational problems are equivalent to an unconditional extremum problem (with the same Lagrange function for both problems). For instance, the problem "the entropy is maximum, , where E is the system energy" is equivalent to the unconditional minimum problem for the functional where d is a Lagrange multiplier. Recall that in statistical physics for perfect gases the Lagrange multiplier turns out to be inverse proportional to the temperature, d = - 1/T , and the relation F = 1/T (E - TH) is valid. In such a condition the extremum principle is treated as that of minimum free energy of a system.

6. The entropy of a system expressed in terms of the invariants of the corresponding mathematical structures, can be interpreted as the amount of information connected with the system structure (Levich 1978). Therefore the above extremum principle can also be interpreted in terms of information.

In the subsequent sections other interpretations of the extremum principle are also discussed: the maximum ecological variety, a comparison with the second law of thermodynamics, the aspects of growing system complexity and self-organization.

Has one to interpret the extremum entropy principle just as that of inevitable entropy growth to its maximum? G.E. Mikhailovsky singles out the world of the existing (Laplace's), with a stationary entropy value, the world of the emerging (Prigogine's), with entropy growing along the real trajectories due to dynamic instabilities, and the world of the developing (the world where the equifinality principle governs), with negative entropy increments; he ascribed the essentially biological and social systems to the world of equifinality.

The extreme structure realization principle put forward in the present paper is certainly open for further generalizations and new interpretations.

The appearance of entropy in the present context has some peculiarities compared with situations in which entropy has repeatedly occurred in different fields of natural science:

• Entropy is here endowed with a universal role of general scientific significance, that of a functional creating the variability laws for natural systems.
• Entropy acquires a meaning similar to that of the invariants of mathematical structures, namely, a quantitative representation which orders the states of natural systems, and is monotonic with state ordering with system homomorphisms. Since the invariants of structures are calculated through the numbers of admitted transformations, the above treatment of entropy turns out to be a generalization of Boltzmann's entropy definition.
• Entropy is unambiguously treated apart from any probabilistic considerations. Its value does not depend on a mathematical structure explicating the system nor on transformations admitted by the system.

2.4. Example: a formula of species structure in ecology of communities

Let us describe an ecological community by the structure of partitioned sets: the elements of sets are individuals, the partition classes are populations of biological species. A state of the community is the vector whose components are the numbers of individuals ("sizes") of each of the w constituent species.

Fig.6 shows the transformations occurring with the biological organisms and their mathematical interpretation.

Let us specify the model for a community of organisms of the same trophic level, for instance, for a phytoplankton community without cell introduction from outside. Phytoplankton cells do not absorb their counterparts and do not merge; besides, the species do not interact with each other genetically or trophically. Therefore the admitted transformation are not necessarily everywhere defined (mortality), not always functional (cell fission), but surjective (no introduction) and injective (no cell merging) correspondences of each partition class into itself (absence of interaction between species).

Note that the model of a community is determined not only by the explicating mathematical structure (partitioned sets), but also by admitted morphisms. A choice of other morphisms for the same structure would create another model. Thus, bijections chosen as admitted morphisms would lead to the existence of states with only constant sizes of species. As illustrated below, changed properties of allowed morphisms drastically change the character of species size distribution. Therefore the present model is essentially category-theoretic rather than just using the language of that theory.

The functional, whose maximization is to determine the dynamics of the community according to the extremum principle, for the chosen admitted morphisms, has the following form:

where is the total number of cells in the community.

Let us also note that here we have used the global formulation of the extremum principle rather than the local one: we are looking for a final stationary state of the system for which only the entropy of a transition to itself is extreme, i.e., .

The community lives at the expense of the environmental resources which restrict the infinite expansion of species. Thus a real state of the community is an extremal of the variation problem

Here and is the requirement of the species i of the resource k, and is the quantity of the resource k present in the medium.

The problem has the solution

where are the Lagrange multipliers found together with n from the algebraic equations

It should be pointed out that such a solution is unique and always exists (Levich et al. 1994).

The species structure formula (2.5) adequately describes the empirical sizes of species in ecological communities (Lurie 1983; Gavrilov et al. 1986; Levich et al. 1986b; Levich et al. 1993a).

An analysis of the formula (2.5) has made it possible to draw a theoretical conclusion (and to confirm it by experimental studies) that the relative size of a given species takes its maximum value when the resource factor ratios in the medium is equal to the requirement ratios for the above species.

In one of the most recent works (Levich et al. 1994) the "limiting link rule" has been proved for the problem (2.4). The rule generalizes Liebig's minimum principle and Mitscherlich's combined action law for the environmental factors: the space of the consumed factors is decomposed ("stratified") into 2^m - 1 non-intersecting subsets, with a unique collection of consumed factors for each of them, so that in that subset they and only they

a) are consumed entirely,

b) are the arguments of the extremal (2.5).

Let us illustrate the potentialities of the method by choosing other admitted transformations. Assume that arbitrary correspondences are admitted, i.e., that, apart from deaths and reproduction, introduction of individuals and absorption of organisms by other organisms are admitted. A solution of the problem

is the extremal

.

It should be noted that the functionals (2.4) and (2.7) are applied in ecology irrespective of the described approach and are called species structure variety indices of ecological communities (Levich 1980). Thus the extremality of the functional H which creates the community dynamics, can be interpreted as the requirement that the variety of a system should be maximum possible in the condition of the restricted available resources.

Let us point out more precisely, which sort of dynamics is spoken of: the variability of a species size (number of individuals) parametrized by the amount of the consumed substrates. Indeed, the sizes depend, by (2.5), on the multipliers , which, according to (2.6), are, as well as the total number n, functions of only the resource flows through the community . Thus the dynamics in terms of the substitutional time of the system is implied.

The substitutional time has been defined as the number of elements replaced in the system. The entropy time is connected with the number of transformations of the system. Since (according to the variability unification principle) the transformations consist in just element replacements, at a qualitative level a relation between the metabolic and entropy times is evident. However, there is an unambiguous quantitative relation between them. Let us illustrate it on the example of the variational problem (2.4) of the previous section. The entropy where , taking into account that and , is converted into where the Lagrange multipliers are also -dependent. The connection between the entropy and the metabolic time components is strictly monotonic since it can be shown that (Levich et al. 1994).

Note that if one of the factors which is significant for the functioning of the system is the heat , then the corresponding Lagrange multiplier is 1/T where T is the absolute temperature. This implies that the thermodynamic entropy , as well as Boltzmann's statistical entropy, is a particular case of the category definition of entropy.

The relation demonstrates an important result: entropy plays the role of an "averager" of metabolic times of a system with several sorts of replaced elements. The Lagrange multipliers play the role of metabolic "eigenfrequencies" of the system. Thus the entropy parametrization solves the problem of different sorts of elements in universums, emerging in the metabolic approach.

The entropy parametrization has appeared due to a mathematical trick: system structure ordering was replaced by number ordering. Therefore one can say that the entropy parametrization is secondary with respect to the way of system ordering by homomorphisms. In the above formalism we have compared the systems using monomorphisms (similar to injections for comparisons of unstructured sets). In general, structures can be compared with other morphism choices than monomorphisms in the category. In addition to the main single-place functor which has created the entropy, other functors can also be monotonic with respect to structure ordering, or it may turn out that there are no such functors. Thus time parametrization by structure numbers (let us call it category time) is a more general construction than the entropy method. However, as it has been mentioned, the structure numbers are only partially ordered, therefore the time introduced by them has rather exotic properties.

The entropy (and, accordingly, metabolic) times are brought to an agreement with structure ordering with the help of injective morphisms (i.e., embeddings). Therefore, according to the extremum principle, the evolution of a system proceeds from subobjects to objects and further to super-objects. Such an evolution can be called conservative or causal: the achieved states do not disappear (a subobject is "conserved" in an object), while new states emerge not on a blank space but from their "weaker" predecessors (in the sense of ordering by "structure strengths").

If one refuses to use monomorphisms as ordering morphisms in describing a certain system, it may lead to alteration of the just formulated principle of conservative evolution of natural systems. Apparently the choice of ordering morphisms is a matter of explication of the operational ways of establishing the belongings of elements to objects. The concept of a signal can apparently be a natural-scientific interpretation of the ordering morphism construction.

The theorem of category theory establishing the monotony of structure ordering and their entropy invariants (Levich 1982, p.113) actually plays the role of Boltzmann's H theorem, since the former also proved the monotony of entropy but in category time instead of the physical one. The above monotonic connection between the entropy and metabolic times plays a similar role.

Since in the course of system variability its metabolic time interval strictly increases, the system entropy also increases along its trajectory in the space of states. That means that entropy can be used to parametrize the variability, or to determine a quantitative measure of changes in its states. Thus the T component of the category-theoretic description of systems appears, which can be called the entropy time of systems.

As it has already been pointed out, in the development of a dynamical theory for a certain fragment of reality one has to solve a number of concrete methodological problems: to construct an elementary object of the theory, to enumerate the allowed states of objects and the ways of their variability, to construct the ways to parametrize the variability and, finally, to formulate the variability law which should select the real generalized motion of objects in the space of states from all the possible ones. The results of the present chapter suggest the following:

As elementary objects - a hierarchy of structured sets which are open with respect to element substitution at some levels of the hierarchy.

As the space of states - a category whose objects (the elementary objects of the theory) are interpreted as states of the studied system while the morphisms as the ways of system variability.

As a clock - variability parametrization given by the entropy of system states, calculated via the values of a one-place functor of the categories. The entropy parametrization is co-monotonic with the metabolic one, determined by the number of elements replaced in the system.

As the variability law - the maximum entropy law, under the restrictions connected with the finiteness of the resources available to the system.

The suggested way of describing natural systems is certainly insufficient for building a theory and presents just certain foundations of a potential programme of studies. Such a programme should, in particular, provide the consistent possibility of obtaining the traditional dynamic equations in specific fields where these equations are already known, in other words, it should provide the fulfilment of the correspondence principle. A derivation of equations can be thought of as a conseqence of the maximum entropy principle, unlike I.Prigogine's (1985) programme of studies where the dynamic equations were considered to be initially given and the point was to supplement them with the ways of introducing the entropy (as a Liapunov function of the dynamic equations) and with allowed state selection principles on the basis of the second law of thermodynamics.

Having learnt to parametrize time by substitutional or entropy clocks, are we able to say something about the very origin of variability, or, in other words, about the nature of the time flow, or about the origin of the general process?

In the framework of the weak hypothesis (see Section 1.3), i.e., dealing with the over-molecular levels of, say, the biological hierarchy (molecules, cells, organisms, populations, communities, the biosphere), we can answer the question of origin: the energy and substrate flows in the biosphere of the Earth are created by the initial flow of solar photons which causes and creates all the other flows in the biological hierarchy.

Such a viewpoint is generalized by the strong hypothesis. Not only every motion has a metabolic nature (a motion of an object in a medium is absorption and emission of the elements of that medium; the space is a medium made of pre-elements), but, in addition, the very origin of element replacements is due to an external flow piercing through the objects of the natural hierarchy. In other words, if an analogy between the biosphere and the Universe as systems is appropriate (the Universe is here to be understood as the domain available to instrumental study), then the time of the Universe is created by a certain flow of pre-elements of rather deep levels of the physical hierarchy, i.e., the Universe is unisolated, open, non-equilibrium and, moreover, it is the absence of equilibrium that creates the time flow in it.

Such an assertion is nearly evident for biological hierarchies, interpretable for geological and geophysical universums but seems quite unacceptable for physical objects. It is quite easy to imagine equilibrium systems for which the physical time flow is evident: free motion of a body, oscillations of a pendulum, motion of noninteracting bodies in a closed and isolated system (e.g., planets around the Sun). However, here one can also find a flow and to discover it, it is sufficient to just change the language describing the mechanical motion. Let us imagine displacements of bodies as absorption of some points of the space and emission of others. Formally such a description can be formulated quite similarly to that using geometric trajectories.

The associations connected with the idea of a creating flow, are not new in philosophy and natural science. One could mention the world picture in daosism, the modern physical vacuum, and J.Whitrow's quasi-substantial view concerning "a hypothesis ... by which there is a unique basic rhythm of the Universe" (1964, p.63). One could also mention Newton's absolute time flow (though, Newton's flow is entirely matter-independent (Newton 1687), while a generating flow is to be directly coupled to material objects). There is also N.A.Kozyrev's (1982) time flow which forms causality in the world, provides one of stellar energy sources and causes a new "meta-interaction" of the world objects (Yeganova 1984). I.A.Yeganova's paper mentioned a number of experiments whose results apparently require inclusion of N.A.Kozyrev's ideas (see also a succession of recent experimental works carried out by M.M.Lavrentyev with co-authors, 1990, 1991, 1992). The second volume of this book is dedicated to reviews and developments of N.A.Kozyrev's ideas.

Identification of the time flow with the existence of a substantial flow often meets the objection that the very notion of a flow already includes a tentative notion of time: one usually calls a flow a change of matter or field concentrations in the context of physical time. However, a more general understanding of a flow is possible. This understanding is fixed within the substitutional construction of time: entry of elements into a certain standard system is considered as element flow in the system, the entering acts are called the instants of time, the number of entries determines a measure of the "flow time". In other words, element number changes in a standard system are parametrized not by some external process (not by "physical time") but by the process of element substitution in the same system. For the generating flow a standard system is the Universe, open to entries of pre-elements of a certain deep levels of matter structure. Changes in the number of those pre-elements, i.e., the pre-element flows, determine the time of the Universe.

Having appeared as a logical extrapolation of the properties of metabolic time, the generating flow gives an opportunity to seek constructive approaches to the problems of natural science. A connection between the time flow and absence of equilibrium, between stream dissipation and irreversibility is trivial: the absence of equilibrium in a system, i.e., the existence of a substantial stream across it, is just the flow of time. What is non-trivial is the fact that a generating flow should exist for that triviality. The question of the "nature" of time and the "reasons" of its course, of the mechanisms of its formation, is transferred by the substantial flow hypothesis to the question of the origin and status of the substrate and energy "feeding" of our Universe.

In the construction of time the generating flow is a fundamental, primary standard object for the ordering principle (see Section 1.4), in other words, it creates the sequence of time instants. Time turns out to be linearly ordered because the generating flow has these specific properties. The irreversibility of time is not its inherent property but is inserted into the Universe along with the directional character of the pre-particle stream: the irreversibility of time exists as long as the generating flow has not been reversed.

The generating flow hypothesis is very radical and can influence many fields of natural science. A reader of restrained views can be offered to consider the flow hypothesis as just a convenient means for describing such phenomena as formation, development and the course of time, i.e., to transfer the flow concept from the field of ontology to the methods of gnoseology. However, a constructive set up of the problem is apparently not the question of whether the generating flow exists or not but the question of how to reveal it operationally, i.e., how to measure in a reproducible way some properties of the generating flow, distinct from its basic manifestation - the flow of time.

A reformulation of the properties and problems of time in the language of the generating flow (whose properties are to be postulated in the same way as the primary properties of time) can be useful at any rate at least in two respects. First, new heuristics can be discovered. Second, the set of primary entities is reduced, therefore the picture of the world is simplified. The generating flow hypothesis is sufficient for constructing time, space, particles and interaction.... I would like to demonstrate some heuristics, reminding the reader that these are not rigorous results but rather just a programme of studies in the path of the metabolic approach.

Let us identify the sources of generating flow entry to the universe (or flow discharge from it) with the charged particles. Existence of several generating flows can be admitted, so that the Universe is open to pre-element flows of several deep levels of its structure. Particle charges are dynamic characteristics of the flows whose sources are the particles themselves. Various sorts of charges are created by generating flows of different hierarchy levels.

A superposition of the deep-level elements of the natural hierarchies forms the space (synonyms: a medium for particles, a substantial ether, a physical vacuum). Metabolic motion of objects in the space (ether) is accomplished not by "moving aside" of ether elements but as these elements penetrate the objects and replace the elements formerly situated in the objects. Thus there is no "ether friction" or "ether wind". A pictorial image of object structure would consist of "fountains" or "springs" spouting in a pond rather than eddies or tores. Sequences of pre-elements "emitted" by the particles form systems of neighbourhoods of the particles and create the notion of "proximity" in space, i.e., the construction of its topology.

The assumption that the particles of matter are "entrances" and "exits" of the generating flows in our Universe, suggests the Lesage mechanism (Rosenberg 1935) for particle interaction: "The modern proof of the Newton theorem rests upon the hydrodynamic considerations dating back to Laplace: the point is that the only spherically symmetric motion of an incompressible fluid is radial motion with velocities inversely proportional to squared distances from the center... Thus the attracting force field of a point mass mathematically coincides with the field of velocities of an incompressible fluid" (Arnold 1987). Pre-element emission and absorption can be treated as pre-element exchange between particles, in correspondence with the construction of interaction traditional for theoretical physics. Different generating flows thus correspond to both different sorts of charges and different types of interactions.

Depending on which of the generating flows is chosen as a standard for measuring time (different generating flows can turn out to be non-uniform with respect to each other), the corresponding interaction can be eliminated from the description of motion. For instance, for the perfect physical time of the laws of mechanics the role of the imperativity postulate is played by the axiom of inertia claiming that those time intervals are equal, during which a body covers equal distances in the absence of interaction with other bodies. Thus the uniformity of the time scale is connected with the absence or presence of object interactions. Any nonlinear transformation of the temporal variable in the equations of mechanics leads to the appearance of terms responsible for additional interactions (Misner et al. 1977). Thus interaction can be described in terms of nonlinear time scales.

Non-uniformity of the time scale can be treated as inhomogeneity of time, implying, by Noether's theorem, energy nonconservation. For the metabolic construction of time such a nonconservation is not "criminal" since the metabolic time flow emerges only in systems open to an external pre-particle flow, so that the question of uniformity or homogeneity of that flow is reduced to a reference level where time has been defined. If the standard is chosen so that the flow is uniform, then in the corresponding equations of motion there will be no interaction and the energy will be conserved. If the choice implies a non-uniform flow, then the equations will contain forces and energy nonconservation will be "legalized".

The substitutional approach enables one to try to introduce a construction of causality. Let us consider a situation where a pre-element of the generating flow is emitted by a source particle and absorbed by some other object. Let us call such a process a causal link, the event of emission of a fixed pre-element - the cause in the causal link and the event of absorption of the same pre-element - the consequence. If the "generating flow propagation velocity" is finite (according to the interpretations of Section 1.3), it corresponds to a certain average number of pre-elements in elements. The cause and the consequence turn out to be separated by a space-time interval. A more detailed discussion of the relations between the "time flow" and the ideas of causality is contained in the second volume of this book, dedicated to N.A.Kozyrev's ideas.

The generating flow propagation with respect to particles-sources can have different signs depending on whether emission or absorption occurred. A preferential flow direction, apart from the course of time, inequivalence of the right and left-handed coordinate frames and also different signs of charges. A gedanken operation of flow "reversal" should lead to the three connected effects: to time reversal ("temporal parity" nonconservation), to interchanging of left and right-handed coordinate frames ("spatial parity" nonconservation) and to changed signs of particle charges ("charge parity" changing; an antiparticle, as it is adopted in quantum electrodynamics, turns out to be a particle moving "against time"). A generating flow reversal (and the corresponding combined changes of the charge, spatial and temporal parities) apparently does not change the state of the world.

While the physical processes take place in a medium of pre-elements, which are not particles and whose emission creates the particles possessing mass and other charges, the overmolecular hierarchies, in particular, those connected with life, are determined by media constructed of particles themselves.

The general process, underlying the metabolic construction of time, has a number of properties which stress its similarity to the process called life and considered as a complex of connected changes: matter exchange, growth, development, reproduction, evolutionary changes.... The general process is a concept of less unidirectional nature (concerning the processes of life) than that of, e.g. E.Bauer's (1935) "stable non-equilibrium", at least because it also includes such processes as degradation and destruction. Nevertheless, attempts to understand the properties of biological time can turn out to be the same as a description of the essence of life. V.P.Voitenko (1985, p.74) cames to a similar conclusion: "the connection of the time measurement "technology" with such fundamental processes as cell growth and fission or matter exchange point out that the biological clock is not an annex of the biosystem building but the building itself". P.B.Gofman-Kadoshnikov (1984) also paid attention to the similarity between life and the general process; he stressed that self-renovation by element replacement is the basic system property of life which distinguishes biological systems from physical and chemical ones. In G.E.Mikhailovsky's opinion (1989, see also a chapter in this book), it is just the temporal organization of biological systems that determines the qualitative peculiarity of life. V.I.Vernadsky (1967, p.65) also wished to stress the identity of the two processes which are of interest to us; he mentioned the following words belonging to the classic: "Life is a more or less rapid, more or less complicated vortex whose direction is always the same and which always captures the molecules possessing certain properties; however, individual molecules continuously penetrate it and leave it, so that the shape of a living body is more significant for it than matter. As long as this motion exists, the body where it takes place, is alive.... When the motion finally stops, the body dies" (Cuvier 1817).

The generating flow hypothesis admits more radical assumptions concerning the specific nature of life than the analogies with metabolic motion. As well as the particles-charges are sources of generating flows which create physical interactions, the living organisms are sources of specific pre-element flows of some matter structure hierarchy levels (Levich 1993). Different flows may be responsible for the properties of life, consciousness, passionarity (on passionarity see L.N.Gumiliov 1989). The properties of life by their origin become similar to those of, e.g., electric and baryonic charges. The difference between life and unanimate matter turns out to be connected with the specific nature and quantities of the generating flows existing in the phenomenon of time.

An experienced reader can suspect in those hypothetic substantial flows of the metabolic approach a revival of the "vital forces" of vitalism, for instance, Aristoteles' entelechia, Wolf's vis essentialis or Blumenbach's nesus (Drisch 1915). However, the assertions of the metabolic construction are much more prosaic: we are dealing with material, although undetected by modern scientific instruments, structure levels of natural systems. The hypothetic pre-element flows of those pre-elements are required not in order to introduce some kind of vital forces but are in accord with the derivation logic of the metabolic approach leading to a whole circle of constructions related to the time phenomenon in natural sciences.

If one recognizes the generating flows, he thus removes the opposition between the second law of thermodynamics and the existence of development since the second law is valid only for isolated systems. The applicability of the second law to the open part of the Universe, where the generating flows create the course of time, is also avoided; however, this conclusion does not make a discovery for both physicists and astronomers: "...the everyday experience convinces us that the properties of nature have nothing in common with those of an equilibrium system, while the astronomical data show that the same is valid for the whole enormous part of the Universe available to our observations" (Landau and Lifshitz 1964, pp. 45-46). Moreover, "separate celestial bodies and their systems are so well isolated from each other that thermal death must noticeably approach them before any external system could interfere. Therefore degraded states of systems ought to be predominating, whereas in practice they are nearly never observed. Thus the problem is not only to explain why the Universe as a whole is far from equilibrium, it has a much more concrete meaning: to gain an understanding why separate systems and even separate bodies continue to live despite their short relaxation times" (Kozyrev 1963, p.96).

The theory which based on the maximum entropy principle (see Section 2) does not return to the second law of thermodynamics since it is applied to a system open to substrate and energy. Moreover, the entropy formula has been obtained apart from any probabilistic prerequisites, therefore the interpretation of the extremum principle as that requiring to pass from a less probable state to a more probable one, disappears. The restriction of system development by available substrate and energy flows in solving the variational problem (2.4) do not lead to a homogeneous distribution. On the contrary, extremely non-uniform distributions of system characteristics appear, like that of Gibbs (formula (2.5) in Section 2.4); in addition, opportunities of structure formation, i.e., selforganization, are implied.

The interpretations of the extremum principle from Section 2.3 in the language of generating flows can be supplemented by such formulations as "generating substance" economy and increased complexity in the process of natural evolution of open, unisolated systems, or increased self-organization.

The entropy extremum principle applied to the Universe open to generating flows, drastically changes the view of world evolution. Here is the future of the Universe drawn by I.D.Novikov: "... If there are no appreciable amounts of matter, unobserved by some reasons, between the galaxies, then the Universe will expand forever.... In about hundred thousand billion years the very last stars will die out.... Despite the present-day absence of direct experimental data, the totality of our physical knowledge indicates that the matter of the Universe is unstable and, although very slowly, decays.... Black holes, the remaining after the death of some massive stars and also existing in the galactic cores, also evaporate due to quantum processes.

Thus both the cooled stars and the dilute gas, and later the black holes, will disappear from the Universe in its remote future... and only rare electrons and positrons, spread over huge distances from each other, will remain".

The future of the world is described still more dramatically according to "the Helvetius principle" (The time whose tooth chews up iron and pyramids, sees only the death which it brings" - Helvetius 1974, p.114) by Yu.B.Molchanov (1990, p.133): "...everything disappears in time, and disappears tracelessly, and that's the true essence of time".

The generating flow hypothesis gives an opportunity to put forward, in contrast to the Helvetius principle, another one, worth calling "the Kozyrev principle" ("Evidently some possibilities of struggle against the thermal death must be contained in the most basic properties of matter, space and time, there must be processes opposite in direction, those which can be called the processes of life. Those are the processes which keep the eternal life of the Universe." - Kozyrev 1963, p.96): as long as the Universe is open to generating flows, its evolution is accompanied by self-organization, increasing inhomogeneity and complexity.

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