Space and Time

Emancipated from philosophy since the beginning of the XIX century, physics still suffers from the common malady of all the new-born nations, exaggerated egocentrism. In the rage of self-determination, it claims its rights to all the gains of culture and puts forth its method as the only path of any true science, while all that is not science (in this degenerate sense) must be at most tolerated with disdain. Science is deemed to be the supreme arbiter in any dispute, the all-potent source of definitive answers, and the only road to the future. As soon as something becomes a physical problem, the others automatically are out of business. In particular, the ideas of space and time have come all the way downward from universal categories to narrow physical concepts.

A closer look will reveal that the emancipation of physics from philosophy was spurious and ephemeral, and all the conceptual breakthroughs in the history of modern science came from philosophical considerations rather than rigorous study and formal derivation. Scientists have generally been too philosophically ignorant and too self-conceited to admit that what they do is not exactly science, but often a kind of amateur philosophizing eclectically mixing popular ideas and political propaganda. Luckily, the world is made so that, moving in any direction, you will necessarily find something interesting and worth a deeper investigation. That is why philosophical misconceptions were as productive in science as most profound wisdom, and no official decree can stop the progress and forbid further discoveries.

The ideas of space and time originally expressed the general aspects of human life and activity, from the topography and growth of the human body to the geographic range and a sequence of generations. These germinal categories reflected the cyclic nature and the scope of activity, production and reproduction. No wonder that space and time also served as a universal expression of measurement, producing the two primary mathematical notions, a range (collection, set, segment) and a number. In the course of activity, space and time were mutually reflected and penetrated each other; thus sets became measurable (countable), and numbers were grouped according to various qualitative criteria. Practical needs gave birth to the three principal branches of ancient science: geography (dealing with space), chronometry (measuring time), and astronomy (the synthesis of the other two, the abstraction of motion).

The qualitative notions of space and time preserved their importance up to the Age of Enlightenment, and they have not yet entirely disappeared in our days. However, the development of natural sciences preparing the industrial revolution of XVIII-XIX centuries gradually spread a purely quantitative approach, overemphasizing the idea of measurement. The number became the king and god of the new epoch, while mathematics and mechanics were put in the basis of any knowledge at all. Starting from Descartes, space and time were generally identified with the length of a segment (the dimensions of a body) and a measurable duration. The Newtonian idea of infinitesimal elements naturally extended this numerical treatment. In the lines of Cartesian mechanism, the ideas of space and time were detached from any activity and made the designations of an absolute frame for any mechanical motion, which was declared to be the only conceivable form of motion at all, and the degree of understanding any other kind of existence was related to the possibility of its expression in mechanical terms. Though modern physics has gone a whole way away from this purely mechanical picture, its basic elements remain intact in most physical theories, including the most advanced chapters of the quantum theory of everything. The formal unification of space with time and further with the dynamics of masses does not change the principal assumption of a common measure applicable to the whole Universe and embracing all the special manifestations of spatiality or temporality. Popular books on cosmology are rich in picturesque descriptions of the world's development on the same time scale, from the supposed beginning to the possible end. Despite the obvious incompatibility of such a chronological order with the principle of relativity, the very attempts to extrapolate the regularities of the observable Universe to the whole world are nothing but the relics of the old anthropomorphic views, if not a kind of religious mysticism.

A slightest philosophical observation reveals a number of weak spots in radical physicalism. First of all, the ideology of measurement does not tell much about the moments of time and space points, despite the general notion of their being the elements of space-time. A physical system cannot be infinitely small (or large); otherwise, it would not be measurable. Representing physical characteristics with mathematical abstractions is only an approximate model applicable to the systems with characteristic dimensions within definite limits. For instance, solid bodies interact as point masses at distances much greater than their sizes; to describe the dynamics of close bodies, one has to account for their shape. In general, there is a hierarchy of characteristic lengths (durations) determining the applicability areas of different formal representations. This does not deny the importance of measurement as such, but prevents us from too wide generalizations absolutizing one of the natural scales in the detriment of the others.

On the other hand, the numerical value of anything makes little sense on itself. A thousand miles, is it far away or close at hand? It depends. A couple of seconds, does that mean "just a moment", or "for ages"? Both interpretations are equally possible. That is, any measurement is only meaningful within a qualitative description of the system, assuming an overall conception of the typical spatial and temporal relations. Of course, all the special manifestations of space and time assume some commonality that we address speaking about space and time as universal categories. In particular, one could imagine a most general measure associated with such universals. But why these universal ideas should coincide with the notions of physics, reducing all the diversity of space-time to mere numerical values? Even in physics, we encounter the elements of hierarchical understanding of space and time irreducible to mere coordinate systems; for instance, in quantum mechanics, the coordinates and the moments of time as the parameters of the wave function (or a state vector) have nothing to do with observable positions and moments of time (dynamic variables), which are rather represented by translation operators transforming wave functions, or state vectors, which, in general, do not need to be their eigenstates. Why not admit that reality is much wider than our limited experience, and that it may comprise formations irreducible to mere physics? The available answers all repeat the same: because space and time are physical notions. This is not ultimately true. The notions of science never come from within science; they reflect the common modes of conscious activity, the way we see and transform the world.

In general, the hierarchy of the possible forms of motion will produce the corresponding hierarchy of spatial and temporal relations, and physical space-time reflects one of the lowest levels of this hierarchy. Thus, an organism is primarily a physical body, and it will obey the laws of physics. As a chemical system, it will certainly obey the laws of chemistry. However, life cannot be reduced to mere physics or chemistry, this is a special arrangement of otherwise inanimate things that makes them behave as living creatures. Physics and chemistry make such higher-order organization possible, but they do not imply it. On the contrary, life restricts the range of appropriate conditions and thus modifies the effect of physical and chemical processes. This means that, while the physical notions of space and time are still applicable to the body of a living organism, there are also manifestations of spatiality and temporality characteristic of life and absent in dead bodies. Similarly, conscious activity cannot be reduced to any physical or biological motion, and, along with any physical and biological space-time relations, there must be something that belongs to this very level of hierarchy, and which is absent outside the totality of cultural phenomena. Of course, the levels of any hierarchy are mutually reflected and the same hierarchy can be unfolded in many ways to produce quite different hierarchical structures. That is, the distinction of physical, biological and subjective time is also hierarchical: there can be intermediate levels, or some intricate differentiation within a single level. But this does not change the situation in principle: there are many kinds of space and time that cannot be reduced to pure physics.

Well, we could just shut our eyes and dismiss the unpleasant complexity as philosophical nonsense. Sooner or later, the cultural necessity will oblige us to somehow cope with these delicate issues, and then we'd better have a sound philosophical basis for serious research than start inventing anything from scratch, following our commonsense stuffed with philosophical prejudice of the past. Who knows? If we are lucky enough, philosophy could suggest nontrivial solutions to some of the difficult problems of today.

For instance, the problem of distinction and unity of space and time. Mere declaration that there are no separate space and time, but rather a combination of spatial and temporal dimensions, is not enough to deny the fact that space and time are differently treated in real life, despite all the possible resemblance and interdependence. However radical, physical theories are essentially asymmetric, they may contain lots of spatial dimension, but only one temporal. The sporadic attempts to develop a "many-times" formalism have remained an exotic technical trick far beyond any fundamental significance. While a scientist can always sweep the garbage under the carpet calling such questions non-physical (well, nature is designed that way, and the job of a scientist is to describe it as it is, without ever asking why), the evident discrepancy between the declarative adherence to the strategy of unification and the lame ground principles does not add to the beauty and clarity of the result.

In philosophy, we turn to practical activity in order to find the answers. How do we get aware of space? Through time, collecting our experience of moving from one point to another. How do we get aware of time? Through space, observing the others' motion. That is, space and time are always interrelated, but the experience of time comes from a different level of reflection. As long as we keep within the same cultural environment, we can project our own experience to the presumable experience of the others, thus coming to the notions of space and time apparently independent of the observer (the principle of relativity). The rest of related physics hence follows. All we need is to compare the kinds of reflection involved in the formation spatial and temporal experience. However, one important consequence is ready before any closer investigation. The organization of the physical space-time reflects the physical aspects of our activity; in an entirely different culture built on a different physical background one is bound to find some other notions of space and time, thought as distinct and as interrelated. In a way, the transition from the Galilean culture of mechanical motion to the new culture based on electromagnetism and quanta has caused a revolution in the physical picture of the world in the beginning of the XX century. Maybe the computer revolution we face today will significantly influence the notions of space and time as well.

Unfortunately, modern philosophy borrowing the ideas of space-time from physics (possibly in the form of their negation) cannot suggest much in revealing their universal content. In the following, I will outline an approach that could restore space and time in their rank of philosophical categories rather than special concepts. Otherwise, any productive cooperation of science and philosophy would be impossible. It is only after comprehending the place of space and time in an integral picture of the world that the peculiarities of physical space-time can be considered as a special manifestation of the universal features. However, discussing general issues, I will certainly have to use illustrations from physics, where no other source is available.

Any definition is primarily comparing different things in a common respect. This is how space and time are defined in real life, in any practical situations. For instance, one can measure time by the quantity of food eaten, by the subjective feeling of tiredness, or by the sum accumulated on a bank account. Similarly, there are different measures of space, like muscle effort, the weight of a standard piece of the same material, or the quantity of certain lexical elements of the local dialects. All such "measurements" (or, rather, evaluations) reveal some important features of space and time and extend our understanding of the matter. However, in philosophy, we must seek for a universal mode of comparison incorporating all the possible distinctions and dependencies. That is, we need to indicate a philosophical category (or a categorical scheme) producing the ideas of space and time in a comprehensive and necessary manner. Since the possibility of such an attribution is related to the current level of cultural development (assimilating a certain range of activities), there is no final answer for all epochs to come. So far, space and time seem to be appropriately referred to as the universal aspects of any motion.

In a very general sense, motion is a level of existence along being and development, which is expressed by the triad

being → motion → development,

assuming that any being comes as a result of development, so that the whole picture gets reproduced in he next cycle, or on the next level of hierarchy. For completeness, let us recall that existence is a level of reflection in the hierarchy

existence → life → activity,

while reflection is one of the universal aspects of the world as the only unity of all the possible distinctions:

matter → reflection → substance.

Within this categorical framework, we can introduce the categories of space and time to express the relations of motion to being and development respectively.

This definition has a number of far-reaching implications. Space and time have been characterized by V. Lenin as the universal attributes of motion as early in 1908 in the context of criticizing the attempts of idealistic interpretation of the relativistic revolution in physics. The phrase has later been replicated by millions of Soviet authors and has become an obligatory element of the official high school philosophy. Still, its interpretation remained tied to physics, as if there were no other kinds of motion. Paradoxically, Lenin's words were often mentioned along with the references to F. Engels' description of the hierarchy of forms of motion including the biological and social levels. Thus official Marxism has yet another time demonstrated its inability to assimilate the basics of dialectical materialism, nothing to say about any further development. Lenin was not acquainted with Engels' Dialectics of Nature which has first been published in 1925, after Lenin's death. However, he stressed the idea of the unity of all kinds of motion (including the human society) in his philosophical notes as the only consistent kind of materialism. This means that he understood of space and time was much wider than mere physical notions; the idea of the objective necessity of the specific forms of spatial and temporal aspects of motion on different organization levels has been entirely overlooked by the public.

In this section, the hierarchy of the word is associated with the forms of reflection rather than the forms of motion. There is no principal difference, since the categories are all mutually reflected, and the hierarchy can be converted in other contexts, to satisfy some practical needs. In particular, the levels of reflection can be derived from the hierarchy of motion, in accordance with the classical Marxist approach. The principal conclusion remains the same: the world is hierarchical, and hence any spatial and temporal aspects must reflect the same hierarchy. However, the indirect relation of motion to reflection could explain, and probably justify, the monopoly of physics in the study of space-time. Indeed, if motion is an aspect of existence, space and time as its attributes refer to existence as well, keeping a low profile on the levels of life and conscious activity. In other words, any motion is primarily physical motion in the wide sense, as contrasted with biological and social phenomena. This means that the peculiarities of space and time on the higher levels of hierarchy should mainly be observable as modifications to the physical space-time, some unusual behavior that cannot be attributed to mere physical reasons. Such higher-level effects may be hard to detect, as they have to be separated from the forms of physical motion, which shape the inner hierarchy of existence. Thus, some philosophically inclined physicists tried to explain quantum effects by the direct interference of consciousness; this is a typical example of methodological confusion. The influence of the human reason on microscopic interactions is indeed important, since our practical activity comes to involving certain kinds of quantum systems and therefore prepares them so that they can exhibit essentially quantum behavior. Still, similar types of behavior can also occur without any human interference, like in the objects of the distant space; their observation, however, is not entirely free from subjectivity, since it will necessarily put forth some specific aspects while neglecting anything else, in accordance with the current trends of cultural development and human practical needs.

For a physicist, the above abstrusities suggest at least one important corollary: any physical system is regarded as such in respect to the current organization of human activity, the level of cultural development in general, and technological development in particular. Consequently, there is no physical theory, however fundamental, that would not, sooner or later, be modified or replaced by a more general approach, reflecting (and reflexively enhancing) the new realities of everyday life. Scientific revolutions of any scale are always a result of industrial and social development, and never an arbitrary invention of a supreme genius. Formal constructions may lead to incredible guesses, but, first, the very formal method reflects the options of development inherent to the present culture, and second, any novelty can only be accepted by a society prepared enough for discoveries of that kind. Thus, the objective necessity in the heliocentric system appeared well before Copernicus; similarly, the laws of genetics only provided a concise expression of what people practiced for centuries. Mathematicians keep saying that complex numbers were historically introduced for purely technical reasons, as the solutions of algebraic equations. But higher-order equations did not come from nothing, they were needed to solve quite practical problems. Mathematical abstractions are as culture-dependent as any other scientific models.

While waiting for the new physics to come, let us look closer at the categorical background of the scientific treatment of space-time, which might probably extend our present notions, at least adding a new interpretation to the already known.

First of all, defining space to characterize motion from the aspect of being makes this category essentially static, associating it with the ideas of stability, constancy, permanence etc. On the contrary, time as the link between motion and development is an expression of change, both quantitative and qualitative. In this context, time is the opposite of space, and one can never be reduced to the other, though, as any opposites, they are mutually reflected and impossible without each other.

In a way, this returns us to the classical picture treating space and time as essentially different. On the other hand, a distinction like that could explain the traditional asymmetry of physical theories in respect to the dimensionality of space and time. The commonly felt arrow of time is readily associated with the overall direction of development, from the primitive forms to more complex formations.

In physics, the unity of space and time is usually imposed in a static manner, reducing time to space through a standard motion, the propagation of light in vacuum (understood as empty space). Such an approach is a concession to the common sense; for a plain reasoning, spatial relations seem to be much more tractable due to their immediate presence "here and now"—the prejudice relativistic physics claims to ruin. The attempts to turn it the other way round and derive space from time did not receive much academic (and public) interest, mainly due to the necessity to explain three spatial dimensions from the single-dimensioned time. However, in the modern algebraic formulations of quantum field theory, one could discern a prototype for the possible reconstruction of space through a series of symmetry violations (that is, unfolding space in time).

The traditional relativistic reductionism involves a kind of logical circularity, since the constancy of the speed of light in vacuum as the logical basis for the geometric treatment of time is based on the assumption of an empty space spanned by propagating light; no wonder that relativistic physics is perfectly explaining facts from which it was basically derived. To discover other kinds of physics, one needs an entirely new kind of interaction that would serve as a measure of space-time independent of the propagation of light.

The unity of the opposites can only be established through yet another category embracing the both poles in a synthetic manner and hence mediating their mutual transformations. In particular, we need something different from space and time while containing them both. Like space, this something must present a static arrangement of elements; like time, it must admit an inner orientation, the direction of development. In unism, we make a bold assumption that this unity is expressed in the category of organization, taken as both the process and result. Now, in the triad

space → organization → time,

time is understood as organization of space, and space is understood as organized time. In any case, space and time appear to be the modes of organization, in addition to other possible organization forms. While space refers to something between being and motion, and time refers to the link between motion and development, organization links motion to itself, providing a kind of inner reflection.

However sudden, this unification of space and time with organization well agrees with common intuitive ideas as well as the ways space and time figure in physics. From the ancient times, the very word "space" referred to the nearest environment, the portion of the world already assimilated by the culture, organized according to the range of common activities. Similarly, time was associated with the expansion of the organized activities to involve more of the objective world (nature). The simplest form of temporal organization, mere repetition, determines a spatial point; the simplest form of spatial relation, mere distinction, determines a moment of time. The combination of the two, repeated distinction, or distinction in repetition, forms a scale as the simplest kind of the inner organization of any motion. Different kinds of motion run on different space-time scales, and this the first knowledge we get about any new physical phenomena; quite often, establishing the scale for a physical system allows to attribute its behavior to quite definite physical processes and explain it at least on the qualitative level. Characteristic times and lengths are of fundamental importance in physics and are certain to play as fundamental role on the other levels of the hierarchy of reflection.

Scales combine both qualitative and quantitative features and provide measure for a class of phenomena. A scale does not need to be an ordered set, it may admit no quantitative estimates at all. For example, a music scale is a collection of notes (the points of a pitch space) that are distinguished by their quality rather than quantity, and their usage does not necessarily depend on any numerical estimates. The relation of musical scales to pitch perception is nontrivial, and the commonly known "well-tempered" 12-tone scale is a result of long evolution of musical hearing; there are other scales used in certain cultural areas, and a number of new scales can be theoretically predicted. In any case a point in the musical pitch space is rather a zone allowing small variations without changing the quality of the tone; moreover, musical motion develops in several pitch dimensions implied by the same scale (scale embeddings and super-scales). Though this example refers to esthetics and psychology, similar types of organization are quite admissible on the physical level as well.

Scales can be differently ordered, depending on the cultural context. Thus, in European tradition, musical tones are normally ordered by pitch; however, many cultures (including the music of Ancient Greece and medieval modal systems in Europe), admit only a partial order, while for people with absolute pitch sense there is no order at all, as any tone has its own qualitative definiteness that does not require comparison with other tones. On the other hand, the elements of the standard 12-tone scale can be ordered either by pitch (which gives the ordinary piano keyboard), or by the interval of the fifth (which is the standard in the traditional treatment of harmony and the principle of most combinatory scale theories). In physics, the "natural" order of spectral lines says nothing about the sequence of transitions producing that very observable pattern. However, an ordered scale corresponds to a higher level of hierarchy, introducing the ideas of distance and delay. Primarily, these are merely qualitative estimates reflecting the fact that some point (moments of time) immediately follow or precede each other, while others requires several hops to reach. In the terms of activity, one might consider the overall effort needed to achieve the result. It is only when all the spatial points (and the moments of time) are treated as equivalent (in some respect) that we come to the idea of translation (transition from one point or moment to another) as a level of motion.

Note that this logic is the inverse of the usual physical approach, which first assumes space and time, and then imposes the requirement of spatial and temporal invariance as the basis of momentum/energy conservation, but primarily of the very possibility to define a frame of reference for the notions of momentum and energy to make sense.

Now, as we got to the idea of translation, there is all we need to define measurement as comparison of translations. An attentive reader will immediately notice that this implies yet another level of hierarchy, treating translations as points of some space. Anyway, we finally obtain the familiar notions of length and duration as quantitative estimates of distance and delay in respect to some standard measure. And here is where regular physics begins (but in no way the end of the row).

I dwell so much on this stage only to show that the basic physical ideas are not as elementary as they might seem. On each step, we could decide to unfold hierarchy in a different direction, thus getting to space and time of a very uncommon sort. So far, physics did not need them. But who knows? Our future experience may require a revision of the most elementary conceptions; philosophy is to indicate how we could cope with that.

The integration of space and time with organization is a kind of research program aimed to establishing the interdependence of the corresponding special notions on all the levels of motion, including physics. That is, any discussion of space and time implicitly assumes a specific organization of the system of interest, and the numerical (or other formal) expressions are necessarily restricted to this problem area. In particular, this might mean that the future generalizations of space-time in physics will include a third element related to the system's organization (for instance, statistical characteristics, or inner symmetries). In a way, the general relativity theory is a step in that direction, since it tries to bind the structure of space-time to the arrangement of material fields and conversely, the distribution of matter to the spatial and temporal characteristics. However, reducing time to mere spatial coordinate, it remains too geometrical, while organization is wider than mere geometry. A new physical theory of space-time will certainly need an enhanced language overcoming the geometrical or temporal bias in terminology.

In the rest of this section, a number of traditional spatial and temporal notions will be projected into the basic categorical framework, without too much concern about formal consistency. Any categorical scheme must be unfolded in the context of a specific practical problem; within a general ontological discourse any detailed construction would bear a touch of arbitrariness, merely illustrating the general ideas rather than really developing them.

Like in any triad, the mediating position of organization between space and time assumes two complementary forms of organization that could be called spatial and temporal organization to refer to the outer manifestations of motion; as usual, there is also a third element, a kind of inner organization that is both determines the features of the observable motion and reflects any external determination. Thus we get an organization triad:

spatial organization → inner organization → temporal organization,

or, in the inverted form,

temporal organization → outer organization → spatial organization,

with inner and outer organization being the aspects of some dynamic organization in general. This distinction is quite intuitive, saturating all the modern physics. Thus, the description of spatial organization includes overall dimensionality, topological assumptions, and a hierarchy of natural measures. Temporal organization specifies any kinds of sequencing, like the order and contiguity of time moments, parallel processes and cascades, typical durations etc. Dynamic organization is represented in physics by a formal description of matter, including its distribution in space, evolution in time, as well as the law of dynamics, asymptotic conditions and higher-level constraints. Basically, this corresponds to a model of the experimental setup, or the accepted level of observation. It is through inner organization that space gets related to time, and conversely, temporal organization in certain conditions leads to a definite spatial organization.

A frame of reference is an important example of organization in physics. It combines some spatial organization (represented by a coordinate system) with a definite temporal organization (represented by the value of time). The choice of the frame of reference is determined primarily by the outer conditions (the observer), and the transitions between different reference frames are described in terms of motion. However, the parts of a physical system can play the role of observer for each other, and hence there are preferable reference frames associated with the inner organization of the object. The reduction of time to space in relativistic physics assumes a kind of outer organization, the propagation of light, and is meaningful only within this light-bound framework. However, a relativistic frame of reference is more than mere coordinates; even in an entirely geometrical approach, it assumes a clear distinction of space and time, albeit on the local level and in a relative way. Physicists tend to ignore the ubiquity of this distinction and the obvious asymmetry saying that this is how nature is organized, and that is why we need a particular group to describe it. But a mathematical construct is merely a form of expression, it does not imply any physics and cannot explain anything.

The hierarchy of organization in the triad

space → organization → time

can be unfolded in yet another direction, stressing the static nature of space and developmental origin of time. Thus we come to the fundamental triad of organization levels:

structure → system → hierarchy.

An extensive discussion of the hierarchical approach is available elsewhere. Here, we only indicate that the structural aspect of any object takes it in a kind of simultaneity, all the structural features are present at once and hence mutually comparable; a system transforms structures according to its inner structure; the levels of an object's hierarchy reflect the stages of its development and manifest themselves as hierarchical structures and hierarchical systems. This triad is fully applicable to all kinds of organization, though some situations would accentuating one of these aspects. In particular, spatial or temporal organization can be considered on the structural, systemic or developmental level. The triad of organization will also apply to the category of motion, which lead to the three fundamental levels of motion, or the general kinds of change: transition → transformation → reproduction. Roughly, the category of transition is used to express the spatial aspect of motion, while the temporal aspect is associated with reproduction; the category of transformation is the synthesis of the poles, as well as the necessity of their mutual reflection.

Space, as suggested, refers to the link between being and motion in the triad

being → motion → development.

This means that both being has an aspect related to motion, and motion has an aspect related to being; the category of space serves to express the unity of these aspects. The element of a space as a kind of individual being is readily identified with a point, a minimal portion of space. The spatial aspect of motion must obviously be an abstraction of displacement, transition from one point to another. Taken apart from development, this transition does not make any difference between the distinct points, and hence the idea of symmetry. Any sequencing requires the idea of time, which is basically expressed by the terms like "after" and "before". A pair (set) of points produces a space. An ordered pair (sequence) produces time.

Time assumes spatial differentiation, at least regarding the same point in two complementary aspects, as the starting point and the destination point. That is, time compares two instances of the same space (structure), thus producing a layered, hierarchical structure. The layers in this hierarchy are characterized as "previous" and "next", which is already a kind of development. But the mechanism of structural transformations (and structural comparison in particular) is known as a system. Any motion is impossible without this systemic core. Hierarchy appears in reflexive systems, mapping a structure into itself. Such systems can be unfolded into a pair of transformations, one from the initial space into some inner space, and the other restoring the initial space from its inner representation. In the meanwhile, a complex inner motion may take place, which will result in a non-trivial character of an elementary displacement in the initial (outer) space. This possibility may, for instance, lead to quantum dynamics. However, in general, there is no restriction on the number of levels, and a quantum state can be decomposed into inner hierarchical structures, as well as the "macroscopic" space-time can serve as an inner space for some higher-level motion.

The general concept of space point as an aspect of being is much wider than the traditional geometrical notion; it could be compared to the notion of a configuration space, which is well known from physics but could be applied to any levels of motion, including biological and social systems. However, the hierarchy of being admits many conversions exhibiting very unusual types of spatiality. Thus, one could imagine a space with no points at all. For instance, a topological space is defined as a family of open spheres, with a number of formal properties; such a sphere can be pictured as a collection of points, but this not necessary, provided we know the structure of the family. A spatial point, in this model, can be defined as the limit of a sequence of intersecting spheres; but, for some spaces, such sequences may have no limit at all, being a kind of filter without the least element, which is equivalent to an infinitesimal region of space that cannot be reduced to a point. In a way, this mathematical abstraction well agrees with the physical understanding of space-time, since there are no abstract point objects, and any "physical point" is only a region of space of a negligible size. Still, in respect to any elementary motion, we still need the idea of the initial and final states, and hence generalized points possibly far from the traditional graphical visualization. Conversely, the abstract idea of an entirely continuous motion that cannot be decomposed into elementary transitions (however infinitesimal) and hence does not require the notion of a spatial point comes from a different categorical context, where the category of motion is not directly related to being and development. For instance, for some practical reasons, one could study interrelation between matter, motion and form and discover different hierarchical structures in either category.

As indicated, a space point can be understood as folded time, when some activity would repeatedly reproduce forms of being that we (for some reasons) consider to be the same. Such a reflexive activity is what we call observation. Psychologically, this corresponds to an elementary act of categorization (classification, grading), when a number of the possible forms of being is treated as a single form, thus producing a new level of hierarchy. In other words, any space point is a hierarchy of the modes of reproduction, implying multiple changes that eventually restore the initial state. Thus any spatial point will involve time and develop an inner organization. A common way to represent this inner organization in physics is to treat measurements in a statistical sense, so that a spatial point would be characterized by a distribution of results rather than a single coordinate. This leads to the notion of a point as a representation of the whole space, a specific position of hierarchy (in the sense of hierarchical conversion), in this case manifesting itself as a spatial position. Any particular type of dynamics will produce a different inner hierarchy of the spatial points thus understood as folded motion. The organization of the whole space is reflected in the inner hierarchy of each point, but the two hierarchies does not need to coincide. One could associate the global organization with some higher-level dynamics taking the whole space as a point of another space. The notions of inner and outer organization thus become relative.

Any distinction within a space is associated with a moment of time. Since a spatial hierarchy can be unfolded in many ways, there is a corresponding hierarchy of time. The same motion may look differently on different time scales. What seems chaotic dynamics on one level may be fully deterministic on another, manifesting an entirely new type of behavior on yet another level.

The unity of space and time exhibits a hierarchy of organization forms. For instance, spatial organization may take the form of a trajectory, while temporal organization takes the form of history. One could say that a trajectory is spatially represented time, while history is a temporal aspect of space. On a different time scale, a trajectory may become chaotic; with a different spatial scale, history may become fragmented. In general, such organization forms assume projecting a dynamic hierarchy onto a different hierarchy, a structural approach to systemic or developmental phenomena.

Since time is closely related to development, it can never "turn back", and the next moment of time is never like any previous moment. However, due to hierarchical conversion, several levels of a hierarchy can be folded in a single level, thus becoming a kind of inner development hidden on the higher-level scale. As an inner distinction, time will manifest itself as space, and become formally revertible. While physics can ignore development, physical time will remain equivalent to space. As soon as it comes to self-organization or decay, the notion of time has to be reconsidered. That is why thermodynamic time is essentially different from space dimensions, and complex values of time (or energy) are employed in quantum physics to describe decay.

With time related to the direction of development (and hence to hierarchical organization), one could conclude that time is essentially one-dimensional, thus providing the universal basis for distinguishing spatial directions and determining the dimensionality of space. However, such a view is only locally valid, it refers to a particular level of hierarchy, while the general picture may be much more intricate. Since any hierarchy can be unfolded in many ways, the "flow of time" gets split into numerous alternative channels, each being a cluster of parallel threads. This also leads to a complex spatial organization assuming hierarchical dimensionality. For an illustration, consider two one-dimensional oscillations with very different periods. The moments of coinciding spatial positions will form a level of hierarchy manifesting a kind of chaotic dynamics. While they virtually cover all the incident (one-dimensional) space, there is no adequate notion of direction, and a fractal (dynamic) dimension may differ from unity. Similarly, electronic and ionic components in plasmas move on very different time scales, and hence the complex picture of plasma waves.

One could argue that such "statistical", or "collective" effects are not significant since they can be deduced from a number of "fundamental" interactions that are all compatible with the traditional idea of dimension as an oriented scale. But nature does not need to adhere to our notions of fundamentality; the directions in space-time could as well be explained by the type of dynamics producing ordered scales on average. This is yet another example of hierarchical conversion.

Modern physics has long since gone beyond the common picture of three-dimensional space. Today, physicists deal with all kinds of configuration spaces, and even the conventional classical dynamics has been shown to produce nontrivial manifolds far from reproducing the plain Euclidean structure. Quantum theories operate with infinite-dimensioned Hilbert spaces and their extensions, with the classical space-time used to represent the "degrees of freedom" in such dimensional infinities; in principle this geometrical representation is not necessary, since one can choose any set of dynamic variables (not necessarily independent) to parameterize the configuration space, so that the difference between configuration spaces and phase spaces becomes relative.

Nevertheless, the fact that the 3+1 geometry is still saturating physical theories as a kind of symmetry required in any case, possibly in combination with other (inner) symmetries, may indicate that the roots of space-time dimensionality problem lie outside physics, probably being related to the most universal features of the world as a whole, requiring a philosophical treatment.

One possible explanation binds the observed properties of space-time to the organization of human activity, which is a partial reflection of the overall organization of the physical world admitting the existence of other forms in certain objective conditions. In other words, we impose the organization of our culture (as a local circumstance) onto the world in general, extrapolating our previous experience to what we expect to encounter in the Universe. This differs from Kantian apriorism in that we always are aware of the natural origin of our ideas, however ancient and ubiquitous.

On the other hand, the development of the world will always manifest some features in common for all the individual worlds coming as the different positions of the same hierarchy. Though, at any moment, we can only observe one of the possibilities, their universal core will shape our activity in a quite definite way, eventually brought to awareness in the form of a philosophical category, or a categorical scheme, as a basis of any special notions and concepts.

Thus, the resemblance of the traditional 3+1 space-time with the overall topology of a tetrad may be purely incidental; but it might also express a fundamental feature of any reflection at all, including the world's self-interaction and self-development. In the latter case we must expect the same structure to be found in all the possible instantiations of space-time.

The universal phases of reasoning in diathetical logic, syncretism → analysis → synthesis, must be followed by lifting the synthetic whole in yet another syncretism (a logical operation of lift-in, or anairesis, or Aufhebung), thus allowing the reproduction of the same chain in a new context, and hence introducing a kind of time, so that the previous stages would play the role of space in respect to this particular direction of development. This logical tetrad might be considered as a subjective counterpart of the objective organization of the world's development in triadic spatial forms lifted in time.

The primary hierarchical position of a tetrad taken in the linearized form as A → B → C → A' is the most general description of reproduction through a hierarchy of inner distinctions and outer motion. In this respect, time is understood as a form of cyclic reproduction. In the simplest form, motion returns to the original point (A = A'), spanning a spatial area in the meanwhile. This leads to the usual "geometrical" time. In extended reproduction, the organization of the whole will change, and the new cycle will span a different space. In any case, the act of reproduction provides a natural measure of time, and in particular, a time scale.

Since the hierarchy of reflection corresponds to the same hierarchical organization of matter, the distinction of things and their motion is relative. This means that any spatial and temporal relations will always be expressible in "material" terms similar to the well-known duality of space-time and momentum-energy in physics. In this picture, the inner organization of a physical system corresponds to the notion of mass. Since inner hierarchies are nothing but folded development, the origin of mass can be found in cyclic reproduction on a lower-level time scale, similar to the appearance of self-energy (or effective mass) in virtual interactions of a particle with the medium (e.g. vacuum). However, simple reproduction is only the lowest level of reproduction in general, which is impossible without systemic reorganization and true development. The world will necessarily change, and the space-time aspects of motion will change as well. First, merely quantitative changes will lead to the world's expansion, in the most general sense, concerning all the aspects and manifestations. The famous cosmological expansion does not need to be explained by some huge explosion; the world expands in every point and at any moment just because "a point" and "a moment" are folded hierarchies that must get unfolded due to inner development and the overall development of the world. The way of such unfolding depends on the current environment and virtually on the state of the whole world. For the same reasons, each cycle of reproduction may have its own measure of time. That is why the rate of expansion may vary both in space and in time, though the overall trend will remain the same.

But expansion does not exhaust the range of possibilities. Extended reproduction starts where quantitative changes surpass the current measure and situation will qualitatively change, thus opening new directions of expansion. Within each expansion mode this will look like Big Bang; but all that happens within the same world, and the cosmological catastrophes in one respect are accompanied by simple reproduction in another. This indicates that the universal categories of space and time are to be redefined in respect to a particular direction of development, with its own chain of quantitative changes and qualitative leaps. Thus we come to a hierarchy of organization that would differently unfold itself in every instance of motion, and the world's self-reflection in general.

The universality of the subject as a level of mediated reflection implies that there is nothing in the world that could not be involved in the subject's activity. If the humanity happens to belong to one of the branches of the world's hierarchy, this does not mean that we cannot get aware of the other branches. Our philosophical discussion of space and time gives us at least the assurance that such hierarchically converted forms must certainly exist. To practically assimilate them, we need to develop our activity to the point where mere expansion is no longer possible and a qualitative leap becomes inevitable. Of course, if we do that on the global scale, the resulting reorganization of the world may put an end to our existence in favor of some other implementations of consciousness. This would only mean that we were not reasonable enough and could not find an optimal operation scale never exceeding our power of coping with the products of our own activity. Of course, this is a question of all the aspects of motion on any level of reflection, physical space-time being a very special case.