Scale Dimensions in Nature

Ivan M. Havel

"It was much pleasanter at home," thought poor Alice,
"when one wasn't always growing larger and smaller."



1. Introduction

Both in physics and in science the concepts of space and time are intrinsically related to their phenomenal counterparts. Phenomenal space and time are strongly dependent on certain cognitive faculties of the mind, the most relevant of which are vision, kinesthesis, and memory. The interactions among these faculties in an individual subject through its experiences personalize, in various adverbs of place and time (e.g. "here", "there", "now", "already", "not yet"), a spatio- temporal order of and establish speaker relative reference for the surrounding world.

     The scientific quest for objective knowledge, on the other hand, systematically suppresses everything that is subjective or observer-dependent. Phenomenal space and time are superseded by "objective" space and time. There has, over time, evolved a scientific map of physical reality that enlarges our conceptualization of space and time far beyond our common-sense perceptions. Instead of an individual perspective, centered always around some specific personal "here" and "now", there is a geometrically homogeneous three-dimensional (3-D) space continuum and one-dimensional (1-D) time continuum with impersonal and arbitrarily placed reference points.

     This distinction between the phenomenal and the objective seems to provide for interesting applications of ordinary space and time perspectives to other "dimensions of order" in Nature. The aim of this preliminary study is to propose two such applications: the first is related to the order of spatial scales and the second is related to the order of temporal scales. We can formally introduce two additional coordinate axes, one for the spatial scales and one for the temporal scales, to represent these orders.

We shall first focus our attention on ordinary space, from which space we should be able to develop useful visual metaphors. The fact that among various intuitive tools of theoretical reflection the visual metaphor is, at least on the most basic level, the most-favored one, suggests that our analysis of scale dimensions may have also some deeper philosophical significance.

2. Phenomenal Order of Space

     Textbook geometry and physics make most of us feel like inhabitants of a 3-D continuum (predominantly Euclidean) in which our actual location within that continuum is of secondary or of no significance at all. For convenience, let us keep this Euclidean intuition in the back of our minds as we ponder what additional structures of this space we should consider if we were not to suppress human, individual presence.

     Imagine, for example, a wanderer roaming freely over some arbitrary landscape. This landscape extends around him in all directions, in front of him as well as to each of his sides. His spatial world is essentially two-dimensional; he can move forward or backward; and he can move to his left or to his right. (For the purpose of this discussion, we shall disregard the third, vertical spatial axis since it offers no extra degree of freedom for him.)

     I propose that what is near to our wanderer in this two- dimensional space is, in general, more relevant to him (either positively or negatively) than what is far. In a certain sense, the relative significance of a typical object depends on its distance from him.

     Thus "nearness" imposes a priori an implicit geometrical form on the personal space. We can visualize this form as a function, the value of which characteristically decreases with distance, diminishing entirely at the horizon. To allow even more general cases, when other aspects of the environment may be significant, we shall use the term field of relevance (technically, let us say, a real-valued function over space). The "nearness" function is a particular (and the simplest) case (see Fig.1).

     When the wanderer strolls from place to place, his "here", "there", his horizon, as well as part of his field of relevance (the purely geometrical "nearness" function) shifts with him; its implicit form is, however, unchanged (see Fig. 2). Thus we have similarities in different (because shifted) relevance fields. We may, as a result, talk about a certain natural order; let us call it the phenomenal order of space.

3. Cognitive maps

     Let us now investigate how we can understand the ability of our wanderer to remember his past in terms of fields of relevance and the phenomenal order of space. Imagine that various past fields of relevance (associated with various "here"s) are superimposed over one another, each with its weight depending perhaps on frequency of "visits" of the corresponding local "here", or on the cumulative personal "impact factor" of all such visits.

     A new field of relevance emerges, with no single central "here". Let us call this new field, borrowing the term from urban psychology, the cognitive map.

     Thus the individual cognitive map is shaped more by memories of the past than by immediate visual and/or kinesthetic experiences. It is a roughed "landscape" with a variety of salient peaks of Joycean recollections interspersed with large lowland areas corresponding to never encountered or entirely forgotten places (Fig. 3). Such a cognitive map is an essential part of one's personal identity.

     It is not difficult to visualize a collection of cognitive maps of many individuals, aggregated to form a collective cognitive map of some community. The salient features of individual persons are more or less smoothed out and what remains is a "public" map which can be plotted and printed for everyone's use.

     Of course, in order to match up individual maps we need to make sure we are matching corresponding relational levels or scales. This point brings us to the concept of a scale.

4. The Spatial Scale Dimension

     Perhaps we can best start with an example: consider ordinary, geographical, 2-D maps of different scales (from small- scale street maps of towns to country maps to large-scale maps of continents). Assume various maps of the same region, superimposed one on top of another, with larger-scale maps on top of smaller-scale maps (Fig. 4).

     We can straightforwardly apply the notion of a scale (expressed, say, by a real number) without limits both "downward", to the small, microscopic and submicroscopic scales, as well as "upward", to the large, astronomical scales and beyond. Moreover, we can consider an arbitrarily dense sequence of different scales: between any two we can imagine an intermediate one. The result is a natural continuum of scales. Together with the 2-D maps we thus obtain a 3-D "space" with a vertical coordinate axis corresponding to scales. Let us call this axis the (spatial-)scale axis. Thus we have a new abstract space with two (or, if somebody likes, three) spatial dimensions and one additional (spatial-)scale dimension.

     A movement in the upward direction along the scale axis resembles zooming out: objects shrink and patterns become dense. A downward movement, on the other hand, corresponds to zooming in: objects are magnified and patterns become dispersed or lost (Fig. 5).

5. The Spatial Metaphor for the Scale Dimension

     Somewhere in the "middle" of the scale axis, near or at the 1:1 or "natural" scale, exists our habitat, the world of our (human) magnitudes, our "scale-here". We can, of course, look out in both directions, at least in our imagination. Thus we might be able to talk about the "scale-here", as well as about the "scale-horizon". Consequently we can apply, in a natural way, the notion of a field (or function) of relevance also to the scale dimension, obtaining the (spatial-) scale field (or spectrum) of relevance.

     However, there are several interesting differences. First, unlike the spatial "here", the "scale-here" normally has the same position (the 1:1 scale) on the scale axis for all people (and perhaps for animals of about our size). Of course, we can "move" up and down along this axis, using, for example, telescopes and microscopes (or fantasy); but these instruments are not tools of our everyday perception.

     Thus we do not have a scale analogue of the roughed spatial cognitive map, peculiar to each individual. There is only a collective picture of the world with natural peaks at the 1:1 scale. Only gradually have physical sciences, on the basis of laborious experimental and theoretical investigation, extended the picture to the neighbouring scales.

6. The Geometrical Viewpoint

     Euclidean geometry is scale-invariant: its laws are independent of the absolute size of objects. That does not mean, however, that the idea of scale dimension is not relevant to geometry.

     Consider for instance the concept of shape. Most common geometrical objects (triangles, squares, circles, etc.) have an easily distinguishable and a unique shape. Intuitively we can, therefore, recognize instances of such objects at a glance, certainly without inspecting them in different scales. Let us call such objects scale-thin (see also Sec. 10).

     We also know that there are abstract geometrical objects with much more complex shapes, such as the recently popular self-similar objects or fractals (Mandelbrot 1978, Edgar 1990). They represent a recursive scale order. Their descriptions, therefore, can utilize the concept of scale dimension quite naturally.

     In Fig. 6(a), I diagram a possible representation of a self-similar fractal as a construction sequence, rather than as a single figure. The desired fractal is the limit object in this sequence. Each step in the construction introduces a smaller scale. Each smaller scale can thus be considered as having a significant relevance to our fractal object. The scale fields of relevance corresponding to successive steps in the construction (schematically diagrammed in Fig. 6(b)) show how the partial fractal structure gradually penetrates through the scales. In the limit, the "complete" fractal penetrates through arbitrarily small scales, i.e., its scale relevance function does not diminish as one moves downward on the scale axis. Self-similar forms are common in nature, albeit over limited ranges of scale only (cf. e.g. McGuire 1991).

     There are, from the scale point of view, other geometrical objects of interest. For instance, there are objects whose apparent dimensionality varies over scales: ideal foam (2-D/3-D), thin sheet (3-D/2-D), thread (3-D/1-D) or ball of thread (3-D/1-D/3-D), etc.

     It may be interesting to examine the differences between the "ideal" geometric conception of the scale dimension and the conception that physicists have about real physical space. For instance, modern physics has given up the assumption of scale invariance. It is well-known that, for example, the quantum domain behaves quite differently than the macroscopic domain, even if the nature of the difference is not yet well-understood and even if no one knows where the boundary between the two domains is situated. There are speculations (e.g. Kaluza-Klein theory) of different dimensionality (number of dimensions) of physical space at extremely tiny scales (cf Pagels, 1986, p.324, Rucker, 1983, pp. 24-34). Of course, one may pose the question of physical (or metaphysical) meaning of an imagined infinity of the scale axis in one or both directions.

     Thus, the geometrical version of the four-dimensional space-scale continuum introduced above may diverge from its physical version in essential ways at the scales far from our scale-like "here".

7. Phenomenal Order of Time

     There are interesting similarities between the phenomenal order of space and the experienced subjective order of time. In addition, space and time are intrinsically connected in two of the previously mentioned relevant cognitive faculties, namely, in kinesthetic sense and memory.

     The spatial "here" has an obvious analogue in the temporal "now". We can talk about something being "near" or "far" in time in a manner similar to how we use these terms in spatial reference. A measure of distance from "now", the temporal "nearness" function, can be taken as the first approximation of a temporal field of relevance (Fig.7) The main difference is, of course, the essential asymmetry of our attitude towards the past as contrasted with the future. Yet, in both cases, an event of a given type seems to be more relevant if it is "near" than if it is "far" in time.

     Another difference concerns freedom of movement. One can wander in space from place to place as one likes, both physically, in the real world, and imaginatively, in fantasy, over the cognitive map. With respect to time, on the other hand, only our imagination (using memory or fantasy) is relatively free.

     "While I roam through space, time flows through me."

     This point brings us to an interesting difference between the spatial and temporal experiences. I shall describe it, for the sake of brevity, in terms of absolute space and time.

     Whenever I am projecting my subjective "here" into the supposed (or invented) absolute space, I am inclined to treat my "here" entirely as my private affair, independent of anyone else's presumed projection of his or her "here". On the other hand, when I am projecting my "now" into absolute time, I have a strong conviction that with me also everybody else projects his "now" exactly to the same temporal coordinate.

     This intersubjective synchronism of time lends a collective status to the temporal "now", and thus also to such temporal adverbs as "already" and "not yet". This synchronism holds, of course, only for actual "now", not for the imagined one.

8. The Temporal Scale Dimension

     In the same way as we introduced the scale coordinate axis for spatial dimensions, we can introduce a scale coordinate axis for time. Imagine, for example, a vertically ordered sequence of time arrows, each with its own specific time scale. One would correspond to microseconds, another to minutes, hours, years, centuries, etc. As in the spatial case, let us imagine the continuum of all conceivable arrows below, above, and between those already used (see Fig. 8).

     Having already some familiarity with the spatial scale dimension we should have no difficulty pondering the temporal case.

     Let us observe first the expanded segment of the temporal scale axis, from the millisecond scale to the hour scale (Fig. 8 right). These temporal scales correspond (presumably) to our cognitive and brain processes.

     D.C.Dennett (1991, pp. 151-2, 168) argues against the point-like character of consciousness (as an objective, physiological process) by referring to "smearing" of time on a scale of hundreds of milliseconds due to the size of the human brain, to the highly distributed nature of its functioning, and to the relatively slow transmission speed of neuronal signals. Thus the subjective "here" and "now" have no objective geometrical interpretation below a certain scale (see Fig. 5 and 8).

     As in the case of spatial scales, we can associate the temporal-scale axis with a function representing the temporal- scale field of relevance (which should not, of course, be confused with the temporal field of relevance). Its span depends on what we intend to concentrate on: we could talk about a field of relevance for cognition (as above), for individual life (from hours to tens of years), for human cultural history (from years to ages).

     Again, there is a temporal-scale analogue of "here": scales of the temporal intervals most common to our everyday experience. Consider a sophisticated video that can be arbitrarily slowed down or accelerated (let us put aside technical difficulties). Normal speed represents the scale 1:1 - the temporal-scale "here". We can easily notice even a relatively minor change of speed; though within certain limits we would feel that we are observing a natural scene, with only slight distortion in terms of time. Now assume very large shifts in the temporal-scale dimension - causing the speed of the video to be either greatly retarded or accelerated. We would, then, suddenly visit an entirely different, strange and unknown reality. In one case we would witness soap bubbles slowly breaking up, or bullets cautiously making their way through glass. In the other case we would see trees rising from the ground and growing in front of us, or planets frantically whirling around the Sun. Of importance, however, is that we cannot, in the same moment, perceive these different scales.

9. Three Perspectives: Absolute, Objective, and Subjective

     We have constructed an extraordinary "space": three spatial dimensions, one temporal dimension, one spatial-scale dimension, and one temporal-scale dimension. All of these dimensions put together yield a six-dimensional abstract space. Such abstraction conforms to the traditional ideal of a scientific approach: the scientist forms concepts and formulates laws which are, to the greatest possible degree, independent of any conscious observer. The space is uniform: no a priori preference is given to any particular place, time, or scale. Perhaps we can say that God, or "a universal transpersonal observer", sees the world in this way.

     On the other hand, from the very concept of a scale we learn that we, human beings, see the world differently. We have noticed that in the spatial scale dimension, as well as in the temporal scale dimension, there is a collective human "here" - a relatively short interval of scales accessible to human consciousness. Most of our concepts are inherently associated with these intervals - let us call such concepts "scale-local". The chair in which I sit relates to my size, i.e., is defined by human proportions. It is a chair for us as humans, and only for us; for wood-worms a chair is a kingdom, not a chair. It turns out that even some typical concepts of science, like 'particle' or 'wave', are scale-local.

Scale-locality does not effect the objectivity of a concept. 'Objective' means 'shared by people' or 'collective'. In science it is quite common for concepts and laws (and even whole disciplines) to be restricted to a limited range of spatial and/or temporal scales. Nobody objects so long as the concepts and laws can be shared by other people and thus the requirement of objectivity is satisfied.

     I propose to make a clear distinction between the objective perspective and the absolute perspective. The absolute perspective is what is sometimes called "God's-Eye View": complete independence of an observer, exemplified by the perspective of mathematics. The objective perspective, on the other hand, assumes independence merely from the personal, subjective view of some particular observer. If there were a collective human mind, the objective perspective would be "subjective" with respect to this mind. It is the first-person plural.

     The subjective perspective is assumed by a participant, an individual who is situated in space, time, as well as in spatial and temporal scales. He is always confined to his spatial, temporal, and scale horizons, and limited by the windows of his own consciousness.

     We agree that it would be rather embarrassing for a scientist if his concepts and laws were meaningful only in his private spatio-temporal neighborhood; this situation would make him unable even to communicate with his colleagues. Why, then, are we not embarrassed that most of our concepts and laws are meaningful only in our "neighborhood" in the scale dimensions? Is our reaction a manifestation of the Baconian "idola tribus" - idols of the tribe?

     Of course, we can "move" to other scales, either in imagination or with the help of sophisticated devices. In such visits we often have an illusion of expanding our scale-horizon, but, in fact, we are only shifting our local scale horizon. There is something more: when roaming in science or fantasy over scales, we are faithful to our scale-local concepts and resort to them as much as possible, sometimes cautiously calling them 'metaphors'. In fact, instead of saying that we move over scales to visit another reality, it might be more appropriate to say that we import objects from that other reality closer to us, into our natural conceptual environment.

     We encounter particular difficulties, for instance, when we try to apply our macroscopic scale-local concepts to the already mentioned quantum scale. Even the two most relevant concepts, that of a particle and a wave, do not apply there very well. The so much celebrated quantum mechanics is in essence an art by which we translate phenomena from the inaccessible world of extremely small scales to the accessible world of large things, measurement devices, and ourselves.

The role of mathematics in this translation is particularly interesting. While most people would agree that abstract mathematical tools are, for the most part, safe from the idols of the tribe and thus that they function in domains distant from our scale-local domain in the scale dimension, it is not clear, after all, to what extent mathematical concepts and assumptions are really independent of human scale location and scale locality.

10. Things, Events, and Processes: Their Distribution over Scales

     In this section we shall, more or less, assume the absolute perspective with respect to scales. We shall, however, take examples from "our" world.

     The scale-local objects (entities of size and/or duration accessible to humans) are a special case of objects that I call scale-thin (or, in particular, spatial-scale-thin or temporal-scale-thin), i.e., objects occuring in a limited range of spatio-temporal scales.

     In Sec. 6, we introduced the term scale-thin for geometrical objects. It is, however, a common feature of abstract geometrical properties to have counterparts in physical space. Thus we may perceive also extended (physical) entities in our common-sense world as more or less scale-thin. This is probably related to the fact that our senses have scale-limited ranges.

     One consequence is that we fragment real-world entities into several basic categories: things, events, processes, etc. By things, we typically mean those entities which are separable, with identifiable shape and size, and which persist over time. We do not exclude dynamical processes, those processes which occur on lower scales. Events, on the other hand, typically have a relatively short duration and are composed of the interaction of several, perhaps many things (of various sizes). Processes are, in this last property, similar to events, but have, like things, a relatively long duration.

     However simplifying and vague these descriptions are, we can see that two important aspects of them are their relatively specific position on the spatial and/or temporal scale axis and the mutual relation of their positions. In the world of all scales there is no essential difference: things are just long-lasting events and events are just short-lived things (where "long" and "short" are relative with respect to our temporal scale perspective). Many other entities (vortices, clouds, flames, rivers, networks, sounds, bubbles, winds, ceremonies, meetings, wars) have an intermediate character.

There is an obvious difference between generic categories and particular entities (individuals). A category may be scale-thin in two distinct ways: generically scale-thin, when all members of the category have (more or less) the same scale location (e.g. atoms, birds, chairs), or individually scale- thin, when each member is scale-thin, but with arbitrary scale location (e.g. geometrical concepts: squares, cubes). Homogeneity (of spatial structures) is an example of a scale-relative property: a structure that is homogeneous on a certain scale usually loses this property on neighbouring scales (exceptions are fractals, which can be homogeneous in space as well in spatial scale).

     There is an interesting asymmetry with respect to either of the two scale axes: we have a different attitude towards what exists or happens inside things or events (i.e. on lower scales, temporal or spatial) as contrasted with our attitude towards what exists or happens outside of things or events (on higher scales). This asymmetry seems to exist in all areas. Whether it is absolute or (only) objective is an interesting question., but one that is beyond the scope of this paper.

     The insides and outsides of spatially extended objects with shape are readily distinguishable. We can often assign a specific size to such objects. In terms of the language of this study: one would say that they are scale-thin with a salient peak in their spatial-scale field of relevance.

     In Figure 9, I plot the most relevant spatial scales for various significant structures in the universe (horizontally they are ordered according to their masses, which, however, is not important for our purposes). In a first approximation (without regard to atomic and subatomic structure), we can identify three classes of objects according to the distribution of relevance of different scales - in particular, according to the distribution of sizes of their components. Asteroids, planets, and stars (box A) are single bodies; only their global size is relevant. Secondly, star clusters, galaxies, and galaxy clusters (box B) are already structured, that is, they are composed of lower level clusters or single bodies. Accordingly, their scale field of relevance has two or more isolated peaks. Thirdly, there are complex systems or organisms (box C), characterized by a great number of intermediate relevant scales. I shall return to the category of organisms in the next section.

     Figure 10 shows examples of different spatial-scale fields of relevance of natural objects. I add case (D), which represents scale-homogeneous structures, i.e., objects with continuous spectra of relevant scales. An example of such an object is water in the state near its critical point: there are fluctuations of density at all possible scales (drops of liquid are interspersed with bubbles of gas, both of all sizes from single molecules up to the entire volume of the container (cf., Wilson 1979). By their scale spectrum such objects resemble fractals (cf., Fig. 6).

11. Artifacts, Messages, and Pictures

     Into which class of objects according to their scale relevance distribution can we place human artifacts (tools, engines, computers, houses, cities)? In the first place, I do not think it is proper to talk about "their" scale relevance because what matters is much less the relevance from the "point of view" of those objects than the relevance from the point of view of those who conceive, construct and use them. There are, often only a few, relevant (in this sense) scales for such objects, occasionally separated by large gaps. Consider the steam engine for which the most important scale is that of macroscopic machinery (it is the scale on which the engine is designed and meaningful); the second relevant scale is much lower: the scale of molecules whose behavior supports the thermodynamic cycle.

Whatever the scale spectrum in the designers' perspective is, there is always one and only one relevant scale (peak in the spectrum) for most artificial objects, including computers. It is the local scale "here" of us, the users, where the meaning of the object is located.

Basically the same holds for language and symbolic communication. Symbols and symbolic patterns are suitable for conveying meaning only within a narrow range of scales, beyond which they are incomprehensible (Fig. 11).

Pictorial representation (technical drawings, photography, figurative art) presents a similar discontinuity in scales. For each picture there exists a dominant scale (within the scale span of our visual perception) in which the picture is properly "readable". It is the only relevant scale: pictorial language is based on form, not on substance, and thus an over-enlarged detail does not carry any relevant meaning (Fig. 12).

The fact that we are used to pictures with uniform scale has an interesting consequence: local variations in scale in a picture may offer a special expressive device (Fig 13). (Another case is when nonuniformity in scale is part of a pattern-generating procedure, cf., Fig. 14).

Particularly interesting is a scale "loop" - change of scale reveals similar or relevant content (Fig. 15).

12. Levels and Their Hierarchies

In the case of complex objects, there is a close relationship between their distribution over scales and a hierarchy of their structural, functional, or descriptional levels. In many situations in which we find it convenient to talk about various levels, we can also distinguish corresponding scales or ranges of scales, spatial as well as temporal. Accordingly, Salthe (1991) uses the generic term scalar hierarchy whenever the levels are characterized by different scale (as opposed to the specification hierarchy of levels based on degree of specification or generality). While a scalar hierarchy may be based on the part-whole distinction, a specification hierarchy is typically based on the token-type distinction.

Cases B and C in Fig.10 illustrate the appearance of levels with respect to the spatial scale dimension. To illustrate this point, let us consider a cluster (for example, in terms of its overall shape and size and in terms of its global properties, like opacity). Such a cluster may be studied on the global level (overall shape and size of the cluster) or on the level of its components (their distribution, density, interaction). Natural levels correspond to salient peaks (or elevations) over the field of relevance.

One may, of course, ask about the actual reality of levels as such. Do they exist independently of our analysis and description of objects and events? I believe that sometimes they do, at least partly, but that sometimes they are, again partly, our mental constructs. The scale-thin world of our perception and thought makes it difficult for us to grasp more than a certain limited range of scales at once. For this reason, and other reasons, we tend to decompose objects of our concern into structural levels and events and processes into functional levels. Obvious differences of individual levels yield different descriptions, different languages and, eventually, different disciplines. If all is done properly, the decomposition may match something which approximates the real differentiation of nature.

Two types of difficulty immediately emerge: one caused by our limited understanding of whether and how distinct (possibly distant) levels of a system can directly interact and the other related to the language barriers developed over decades of specialization of scientific disciplines. The language barriers are caused by the traditional research style and may be gradually overcome with by the help of an appropriate transdisciplinary methodology. In the following section, we shall make a few comments on the interaction accross levels.

13. Interaction across Levels

In systems science, the scalar hierarchy is commonly treated under the tacit assumption that "constitutive dynamics at different scalar levels are largely screened off from each other (non-transitivity of effects across levels)" (Salthe 1991, p.252), and this assumption results in the belief that "three contiguous levels should be sufficient to understand most of the behavior of any real system" (ibid, p.253). This assumption and belief are, in fact, included already in the term 'hierarchy' (in contrast to 'heterarchy'). According to systems scientists, occasional influences from distant levels are generally considered as "perturbing fluctuations" that need not be included in a dynamical description of the system in question.. I think this view is inherently connected with the explanatory role of causality in science.

Scientists base their understanding of the processes of nature mostly on causal interaction; it is the principal explanatory apparatus. Yet the scale-thin conceptual field restricts our experience and intuition more or less to an infra-level, "left-right" causation (that is, we refer to an earlier event to explain a later event). There are, in fact, cases in which it is quite legitimate to employ inter-level causal explanations, usually from a lower level to an adjacent higher level. For instance: the properties of molecules cause the growth of a crystal to a specific global shape, the disorganized movement of molecules causes Brownian motion of larger particles, etc. But one has to be careful about generalizing this way of thinking. For example, it would seem to be a mistake to say that "mental phenomena are caused by neurophysiological processes in the brain and are themselves features of the brain" (Searle 1992, p.1). Such an assertion assumes too liberal an interpretation of the term "are caused by", even if we agree, for the sake of understanding, on a strong assumption, namely, that there is a natural hierarchy of levels above the neurophysiological one with some higher level attributable to the mental phenomena.

One of the first mathematical theories in science that dealt with inter-level interactions was Gibbs-Boltzmann's statistical physics (thermodynamics and the study of collective phenomena). It succeeded by eliminating the lower (microscopic) level from the macroscopic laws by decomposing the phase space to what is considered macroscopically relevant subsets and by introducing new concepts, such as entropy (which is, of course, a wonderful trick). As an indirect result, we obtain, for example, an "explanation" of macroscopic asymmetry of (physical) time. Can we, however, really say that the time asymmetry is caused by the behavior of particles?

There is a well-known technique of renormalization (cf., e.g., Wilson 1979) that deals with problems having multiple scales of length. This technique is particularly suitable for phenomena near critical points (see Section 10) and has applications in various branches of physics. But it is not a descriptive theory of nature and, therefore, has little ontological relevance.

Another relevant area is the study of deterministic chaos. Here people become more and more used to situations in which extremely tiny fluctuations are almost instantaneously amplified to a macroscopic scale. What seems to be a purely random event on one level appears to be deterministically lawful behavior on some lower level. This is (mathematically) a recurrent situation. We can therefore take the deterministic description as something permanently hidden behind the scale horizon, albeit always formally available for explanatory purposes.

I have already mentioned quantum mechanics (see Sec. 9), where the question of measurement is actually the question of interpreting macroscopic images of quantum-scale events.

These are all samples of accepted scientific approaches for dealing with interlevel interactions. (Various further approaches can be found in systems science; cf., e.g., Goguen and Varela 1979/1991, p. 298, for some references to literature on multilevel interaction). We could, of course, continue with various ideas on or behind the frontiers of traditional science, most of which are, unfortunately, still resistant to deeper rational analysis and understanding. On the one hand, there are Jungian synchronicity, the Bohm's (1990) concept of active information, and various holistic theories; on the other hand, scientifically inclined philosophers of the mind have recently paid a good deal of attention to the concepts of emergence or supervenience.

Particularly interesting, and surprisingly much neglected, is the compelling question of the asymmetry of interactions with respect to the scale axis. Why is the arrow of causality or of other influences usually assumed to have a direction from lower levels to higher levels? Is the direction of causality part of the heritage of a clock-work mechanistic conception of the nature? Or is it one of its inherent asymmetries?

In this respect, we can perhaps find some inspiration in the notion of information (as an ontological category) and, even more important, in the notion of mind (cf also Sec.15).

14. Organisms: Multilevel Interactional Structures

We have characterized organisms in terms of the multiplicity of intermediate relevant scales. To be more precise, we should talk about a large hierarchy of structural and functional levels which are more or less dense (i.e., which have a high value of relevance over a continuum of scales) and which, in addition, mutually interact. Both density and interaction are crucial features. In the steam engine, for example, there is (some) interaction; but there is no density. In the fractal, there is, in contrast, density; but there is no (physical) interaction.

Let us state then our key question: is it not the very existence of such a hierarchy of interacting levels that makes living organisms different from steam engines, clouds, and fractals? Is it not perhaps just this property that makes it so difficult to reduce their behavior to a mechanistic explanation?

     The function, meaning, and being of living organisms cannot be associated with some chosen level or scale. Each of these aspects of living organisms takes place at the level of molecules as well as at the level of cells, organs, individuals, social groups or ecosystems, i.e., on different scales of space as well as time (cf., Fig. 16).

At present, the relative density of structural and functional levels in organisms seems to be more an observational fact than a logical necessity. We can imagine hierarchical systems with large gaps between levels, which would nevertheless have other features of organisms. There is one important property, however, which is typical of all organisms: their organic growth. Only in the first approximation we can compare organic growth with non-organic inflation (that is a shift in the overall size of objects to larger scales, while the size of their components remain unchanged, cf., Fig.17a). The actual growth of most natural organisms is entirely different (Fig.17b): as soon as the distance between two neighboring relevant levels gets sufficiently large, a new intermediate level emerges. Thus the density of interacting levels seems to be one of the parameters subject to homeostasis in organisms.

In general, our observations seem to be inconsistent with reductionism in biology. Indeed, to understand life does not mean to reduce it to some basic components, but rather to appreciate various influences, bounds, and interactions between all structural and functional levels, close as well as remote with respect to the the scale dimension.

15. Who are we?

Perhaps a similar expansion of the scope of our vision might help us to understand better the nature of the mind and consciousness, or at least to avoid certain related fallacies. For instance, there is a belief, held by some cognitive scientists, that the mind is nothing but a collection of processes occuring on a certain specific level above the level of neurophysiological processes in the brain. This belief is probably facilitated by at least three misconceptions.

The first misconception follows from an inability to think in "scale-extended" way. Consider the relatively simple and obvious fact that mental properties can be ascribed only to the system as a whole and not to its microscopic parts. But here we are used to identifying the "whole" with just a certain, chosen upper level, namely the level of the overall behavior or composition of the system. The more appropriate scale-holistic view, dealing with many or all levels at once, is usually not taken into account.

The second misconception may be the overjudgement of the computer metaphor. If someone maintains that the brain is a (sort of) computer equipped with programs and that the mind is a collection of such programs (or processes controlled by them), then he is immediately drawn to the language metaphor (that the programs may be "written" in some language). And, since languages happen to be scale-thin, it is natural to take mind to be scale-thin too, i.e., restricted to a specific level, namely the same as the level of (human) communication.

The third misconception may be the confusion of the intentional content of mental states with those states themselves. This content, i.e., topics of our thoughts, objects of our perceptions, images of our fantasy, and outcomes of our intended actions, are primarily things of ordinary size - "things which a baby can handle and (preferably) put into his mouth". Mental states are believed to be neurophysiological states. Because it is absurd to think that neurons can handle the same things as babies, it is inferred that the mental level is sufficiently above the neural level. A similar attitude is held even by some of those philosophers of mind who are bravely opposing eliminative and reductionistic materialism. For instance, J.R.Searle (1992) claims that "conscious states are simply higher-level features of the brain".

But is it proper, in the context of mental phenomena, to talk about "levels" at all? Even if we did not restrict ourselves to the scalar hierarchy (cf., Sec. 13), it would be a mistake, I believe, to treat mentalistic terms as something the use of which should be carefully confined to a certain "level", "domain" or "subject area".

D.R.Hofstadter (1979) was one of the first authors to discuss the connection between mental phenomena and a hierarchy of levels. He rightly points to the importance of inter-level interaction (even cyclic or what he calls the "strange loop"), but he seems to err by embedding mental levels into the scalar hierarchy of functional levels and to confuse the scalar hierarchy of the brain with the specification hierarchy of indirect reference. Moreover, his functional "holism" is not quite consistent with his previous warning against use of nondifferentiated language for different levels of description, to which use he ascribes the many confusions in psychology.

It may be the other way round. Perhaps these confusions are consequences precisely of such a strict fragmentation of concepts into levels. If, for instance, consciousness is to be understood as a property of a body, it certainly should not concern a "level-x-body" but the whole living organism that "penetrates" through many mutually interacting and cooperating levels, including, perhaps, even the level of quantum physics.

Cognitive theorists should not be surprised that their theories are crucially limited in their applicability to certain scales and that they may require, for a different scale domain, a revision or extension. A recent example is Dennett's (1992) replacement of what he calls "Cartesian Theater" theory by new "Multiple Drafts" theory, aspiring to cover microscopic temporal scales (cf., Pylkkänen (1993)).



The ideas considered in this paper were greatly influenced by conversations with several friends and colleagues, particularly with Zdeněk Neubauer and Jiří Fiala. I am especially grateful to Jonathan Zimmerman, who read the entire manuscript in draft form and made many helpful comments and corrections.




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