The Nature of Time: Geometry, Physics and Perception. Edited by Rosolino Buccheri, Metod Saniga, and William Mark Stuckey. Kluwer Academic Publishers: Dordrecht / Boston / London (published in cooperation with NATO Scientific Affairs Division), pp. 427-435.
PARADIGMS OF NATURAL SCIENCE AND SUBSTANTIAL
A. P. LEVICH
Department of General Ecology, Biology faculty,
M.V. Lomonosov Moscow State University
Vorobyovy gory, 119992 Moscow, Russia
1. The birth of the paradigm of an open World generated by "time"
One can formulate some implicit premises which dominate in the present natural science:
During the 20th century there appeared some tendencies in natural science which lead to an alteration of the existing scientific paradigm . It often happened in the history of science that the most difficult problems of natural science required a revision of the time concept for their solution. This is demonstrated, in particular, by the works of L. Boltzmann, A. Einstein, N. Kozyrev. Ludwig Boltzmann's  studies of the contradiction between the time reversibility of the equations of mechanical motion and the irreversibility of the equations of statistical physics have led to the statement of a question: "Is the time reversible"? Albert Einstein  was able to reach an agreement between the velocity addition law in classical mechanics and the light propagation phenomena by refining the operational procedure of establishing simultaneity between remote events; in doing so, he introduced a new type of clocks, "light clocks", or Langevin clocks. Astronomer Nikolai Kozyrev [21, 9], investigating the problem of origin of stellar energy, arrived at the hypothesis of the existence of a new physical essence, which does not coincide with matter, or field, or space in their usual understanding. He named this essence "the time flow".
It became clear by the second half of the last century that scientists deal with times rather than time. In my view, one of the significant reasons of the increased attention of the specialists to the temporal aspects of their specific areas is the comprehension of the fact that there can be different clocks that measure the variability of objects. Clocks are not only physical processes on the basis of gravitational forces or the electromagnetic radiation of atoms. There are biological clocks: the processes of breath, cell fission, growth of organisms, exchange of generations or species... There are geological annals, the processes occurring in psychology, society, history... The main property in which the possible types of clocks can differ is the uniformity of their course . More rigorously, the time intervals which turn out to be equal when measured by one type of clocks, become different when using other clocks. The conventional nature of the choice of clocks has been understood long ago by the methodologists of science [22, 19], but only in the recent decades the natural scientists have comprehended the importance of such a convention. A natural motivation for using nonphysical methods of measuring time when studying objects of nonphysical nature was a hope to discover the laws of their variability, or their "equations of motion". The construction of dynamical equations describing natural systems remains one of the basic tasks of scientific research. Generalized motion of systems, looking complicated and involved when described with the aid of physical clocks, can become simple and natural when described using specific time units, adequate to the nature of the system itself.
One more tendency of the present natural science, washing out the existing paradigm, is the rebirth of the substantial outlook. A wide circle of substantial ideas employs the active properties of physical vacuum: there multiplies the set of scalar, vector and tensor fields, suggested in order to explain the phenomena of cosmology, particle theory, biology, psychology, communications. They claimed to exist ontologically. The above-mentioned Kozyrev's conception is in essence substantial. Prigogine, solving the time irreversibility problem, introduces additional terms into the equations of general relativity which describe "creation of matter from space-time" in the form of particles with the Planck value of mass . I would like to note that the substantial conceptions mentioned here and many others tend to the notion of an open rather than isolated World with respect to matter or energy. The revival of the substantial outlook is a kind of response to a long-term predominance of the relational paradigms. However, as a rule, one does not return, for instance, to the elastic light-bearing ether of the 19th century. The question is a search for understanding the physical structure of space and fields. There exist substantial approaches to the nature of time along with relational ones (see, e.g., [2, 3]). These approaches are not confronted with each other but are rather complementary.
There is an obstacle on the way of including the problems of time into the problems of natural science as such, and this obstacle is more and more clearly realized. Time in modern science is an initial and undefinable concept. Therefore a basic task of both time researchers and disciplinary specialists is to create an explicit construction of time, or its model. In other words, it is necessary to replace time in the initial conceptual basis by other basic postulates. After such a replacement it will be possible to formulate the properties of time itself in the form of theorems of a deductive theory instead of axioms. It becomes possible to discuss any properties of time only in the framework of its certain model (see, e.g., [2, 6, 18, 24, 25]).
Creation of a construction of time actually leads to reworking of the whole substructure of the conceptual means. It is quite clear that such reworking inevitably affects a wide circle of basic concepts of natural science. Among such concepts are space, motion, matter, energy, interaction, charges, entropy, life... One thus cannot speak of a specific study, but rather of a vast and deep research programme , and its implementation may require the effort of several generations of researchers. At present, however, the main point in this problem is to understand that it does exist. A number of centuries have been necessary for that.
Now, at the beginning of the new century, certain features of a new scientific paradigm are becoming visible. Its comprehension may be able to help both temporology and natural science as a whole in their development:
2. The components of scientific theories
The purpose of my further presentation is to suggest an example of a substantial construction of time. My main reason for studying time is a hope to find the ways of derivation rather than guessing the laws that govern the variability of the World objects, or in other words, derivation of the fundamental equations of generalized motion.
The dynamical theories in science necessarily include a number of components, whose development, deliberately or more often implicitly, forms stages in the creation of theories :
O-component: choice and description of an idealized structure of the elementary objects of the theory.
S-component: description of the set of admissible states of the objects, called the space of states of the theory.
C-component: description of the ways of object variability.
T-component: mapping of the object variation process into variation of the chosen reference (clock).
L-component: formulation of the law of object variability, which elicit their real motion in the space of states. The variability law has, as a rule, the form of an equation of generalized motion of the theory.
I-component: the use of a set of interpretation procedures which put into correspondence the formal concepts of the theory to specific objects of the actual research area, and the latter to quantities measured in the experiment.
I would like to pay special attention to the stages of the choice of elementary objects, the space of states and the ways of variability in the theory. Newell and Simon  have named these stages the qualitative structure principles of the theory. Here are some examples of such principles: the atomistic doctrine; material points in the phase space of positions and velocities in classical mechanics; probability amplitudes in the infinite-dimensional Hilbert space of quantum mechanics; the structure of an atom and an atomic nucleus; the geo- or heliocentric system of the near space; Everitt's parallel Universes; the cellular theory of organisms; the bacterial nature of infectious diseases; the population, trophic and other structures of the Earth's ecosystem and biosphere; plate tectonics in geology; the class theory of the society.
The structure principles specify the frameworks for functioning of the whole sciences. Creation of structure principles is an area of intersection of empirical generalizations, science, intuition, images of art, philosophy and natural philosophy. Since the structure principles are postulates for a future theory, the way of their foundation is not so important as the adequacy of the resulting theory.Not a single scientific theory exists without the above components. One may lack a comprehension of the existence of qualitative structure principles when applying a ready theory, but one cannot avoid their explicit formulation when creating new approaches.
In the present work I propose to discuss the structure principles of substantial temporology and the approaches to variability parametrization, i.e., to the choice of clock.
3. Postulates and consequences in the metabolic time construction
The term "time" includes at least three different nuances of meaning: time as a phenomenon, a synonym of the World variability; time as a concept, a construct of human thought and time as a clock, a method of measuring the variability. Choosing the first interpretation, we say that time is a reality and a phenomenon, with the second one it is a convention and a noumenon, with the third - an operational procedure. One can deny the use of one or other meaning mentioned as an interpretation of the word "time", but this convention will only concern the word usage rather than the underlying reality. It will be convenient for me to use the term "time" in all the three senses, indicating the context in question. If time is described as a phenomenon, then it is necessary to indicate its natural reference, i.e., a process or "carrier" in the material world, whose properties might be identified or put into correspondence with the properties ascribed to the time phenomenon. The term "metabolic time" traces back to Aristoteles, who described alteration as motion in the widest sense and called it "metabolД›".
Now I would like to present the postulates of the metabolic approach to the description of the substantial nature of time [12, 13]:
Let us briefly formulate some consequences of the metabolic approach.
4. Metabolic clock
I call the measurement of variability, i.e., mapping of an alteration process into a numerical set, "parametrization of time". In the framework of the metabolic approach, the variability is formalized by element substitutions in a system on different levels of its hierarchic structure. It is suggested to measure the variability of the master process by counting the number of replaced elements of the system. We thus introduce the following definitions :
Clock: a reference object belonging to a certain level of the system structure.
Imperativity principle: changes in the set of elements of the reference object by one element are regarded to be equal and may be appointed a unit of time.
Metabolic clock: a natural object whose element replacement is chosen as a uniform variability reference.
Event (time instant): the event of element replacement in the system.
Metabolic time interval: the number of elements replaced in the system between two events.
One can discuss the properties of metabolic time in the framework of the suggested model:
5. Mathematical problems of studying the variability, or the entropy parametrization of time
The quantitative aspect of the metabolic approach consists in counting the number of elements in sets that form the system. The comparison procedure between sets (both finite and infinite) by the number of elements is correctly elaborated for structureless sets. However, the whole theoretical natural science is built on the basis of modelling objects by some mathematical structures (sets with relations, algebraic constructions, topological spaces, etc.). In particular, systems created by several generating flows cannot be presented as hierarchies of structureless sets. Therefore a nontrivial application of the metabolic approach requires the ability to compare sets possessing structure. This ability must replace the method of comparing structureless sets by numbers of elements.
The necessary ability can be reached using the method of functor comparison of structures [10, 12, 17]. The method consists in a consecutive generalization of the concept of cardinality, applicable to structureless sets, to sets with structure. The analogues of cardinality for structured sets, "structure numbers", turn out to be only partially ordered. The next step on the way to generalization consists in building a representation of the category of structured sets into the category of structureless sets with the aid of a one-place standard functor. The images of structured sets according to this functor representation are sets of morphisms admitted by the specified structure. The images of "structure numbers" turn out to be the cardinalities of the sets of admissible morphisms, and the extension of the ordering of the above sets onto "structure numbers" turns out to be linear.The method consists in a consecutive generalization of the concept of cardinality, applicable to structureless sets, to sets with structure. The analogues of cardinality for structured sets, "structure numbers", turn out to be only partially ordered. The next step on the way to generalization consists in building a representation of the category of structured sets into the category of structureless sets with the aid of a one-place standard functor. The images of structured sets according to this functor representation are sets of morphisms admitted by the specified structure. The images of structure numbers turn out to be the cardinalities of the sets of admissible morphisms, and the extension of the ordering of the above sets onto structure numbers turns out to be linear. Thus the number of admissible morphisms replaces the notion of the "number of elements".
Since every admissible morphism transforms the system to a new state without changing its general structure, the number of structure-preserving morphisms can be interpreted as the number of "micro-states" of the system preserving its "macro-state", i.e., its general structure. Such an interpretation makes it possible to call the logarithm of the specific number of transformations of the system state its generalized entropy.
The above entropy parametrization of metabolic time allows one to approach a derivation of the variability law in systems theory. This derivation rests on postulating an extremality principle which determines a "trajectory" of the system in its space of states: from a given state, the system transforms to a state with the "strongest" structure admitted by the available resources. The above functor method of structure comparison makes it possible to give an entropy formulation of the extremal principle: from a given state, the system transforms to a state with the greatest generalized entropy admitted by the available resources.
The entropy extremal principle, along with the ability to derive rather than guess the form of the functional to be extremized, generalizes the formalism of Janes  and makes it possible to formulate variational problems. Variational modelling, being applied to the description of open systems, leads to useful theorems in systems theory and enables one to find new interpretations of the extremal principle [16, 1, 15]. The solutions of variational problems either directly describe the dynamics of systems in metabolic time  or create the Euler-Lagrange equations of motion.
This work is supported by Russian Humanitarian Foundation (grant No 00-03-00360a).