The
Once and Future Wallace
A Theory of Spatial Systems:
Discussion of the Spinozian Attribute "Spatial Extension"
In papers published some years ago I introduced a model combining aspects of the philosophy of Benedict de Spinoza (1632-1677) with modern systems theory concepts [Smith 1986a, 1986b, 1987]. The 1986a paper has just been reproduced here in highly abbreviated/edited form: that is, both to express what I now feel to be its relevant aspects (to study of the Spinozian attribute "thought"), and to serve as a logical introduction to this discussion, which deals with the Spinozian attribute "spatial extension." The model leads to empirically examinable predictions, and following this writing we will take a look at a number of these. Nothing more pervades everyday awareness than the notion
that the three-dimensionality of the world we perceive is not distinct from
the physically relative locations of the objects "contained within it."
On first consideration this seems a safe assumption; on the other hand,
it is by no means a trivial question as to how three-dimensionality PHILOSOPHICAL FOUNDATIONS Spinoza is not frequently consulted these days for advice on the nature of space; though philosophically elegant, his ideas on natural organization have steadfastedly resisted application within an applied, physical context. Moreover, and even more significantly, it was only shortly after his time that Locke and others provided such a context based on other presuppositions. Most efforts have gone into exploring the implications of the latter route--Empiricism--ever since. Spinoza's concept of spatial extension in effect admits
an additional structural level into the fundamental framework of natural
organization. While we habitually view things that "fill space" as having
properties such as color and weight that directly characterize their physical
essence, Spinoza saw the situation rather differently. In his In Spinoza's system, substance is understood as "operationalized" (my term) through some number of "attributes" (his term) that constitute, as it were, governing rules of expression; i.e., it is these rules that actualize substance in a fashion creating a worldly milieu characterized, ultimately, by individual objects with directly tangible properties. Spinoza believed that there might be any number of such attributes operating, but that human powers are limited to the appreciation/recognition of but two: "thought" and "spatial extension." It must be emphasized that the two primary Spinozian attributes are not physical properties in the sense that "color" or "weight" are in the Empiricist tradition; instead they are better viewed as entirely cryptic organizing principles--thus the notion of the "rules of expression" mentioned above. Further, present-day thinkers are advised to allow the concepts "thought" and "spatial extension" as appropriated by Spinoza a good deal of room: clearly, his designation of these terms only very imperfectly hints at current usage, either in general, or in my writings here. Within Spinoza's natural system there is also terminology to describe the tangible reality to which we actually attach meaning (as constructed on the basis of the evidence of our senses): he referred to the elements of this as the "modes" and "modifications" of the particular attributes. Spinoza's model of natural order can be diagrammed
schematically as given in Figure 1, where S represents substance, A Figure 1. No one has ever (to my knowledge) been able to suggest and empirically defend a working model of reality based on this conceptual structure. Complicating the picture is that Spinoza apparently argued that it is impossible to isolate specifics from the continuum of reality--that is, each modification of an attribute is in some sense both unbounded and unrelatable to other modifications. This interpretation of nature seemingly casts it as having order but no distinct parts, or at the very least parts that extend infinitely. Gottfried Wilhelm Leibniz (1646-1716) attempted to amend Spinoza's position with the "monad" concept, which provided for a kind of elementary particle in each of which was reflected the characteristics of all other such particles. Both men's positions support a hierarchical representation of nature and are to a considerable degree complementary--Leibniz's view does not contradict the idea of there being universal "rules of order," while Spinoza's tacitly accepts the possibility that one might yet be able to "see the whole of the universe in a flower." Importantly, Spinoza argued that each attribute, My extension of these ideas is actually rather straightforward.
I have conceived each "modification" of a given attribute as reflecting
an internal organization into subsystems according to a set plan. This
"set plan"--not the overtly manifest properties of the individual system
entities themselves--constitutes the underlying attribute. In this way
the subsystemization of all extended space entities takes place as a condition/function
of "rules of expression" that are common to all such subsystemizations.
This commonality, I thus suggest, Spinoza seemingly does not rule out the possibility
that the two fundamental attributes are in In the paper preceding this one my use of the term "natural system" was meant as a near synonym to Spinoza's "modification," and I used Spinoza's own term "mode" in parallel fashion as my name for the subsystems whose relations with one another sustain the encompassing system by observing the "rules of order" constituting the attribute. In that work I also explained a rationale for believing that such modes may be hierarchically organized. I then produced a most-probable-state (and set theory) based model for such an inclusion series, arguing that this represents a conservative starting point for trying to get at the nature of the hierarchical relations involved. Following all this out produces the inclusion series shown and described in Figure 2, originally appearing in the first of the earlier works. While I still feel this approach is relevant to advancing the cause of an "updated" Spinozian interpretation, it leads to no immediate progress on how a three-dimensional "physical space" milieu might be produced, or how to test any such notion (and, as I stated in the last paper, I now believe it pertains mainly to the attribute "thought"). I have therefore more recently been following another track which seems more productive in this regard. To appreciate this approach we need first study a manner of complementarity of the two fundamental attributes that I feel is consistent with the double "complete representation" restriction of Spinoza's model. SPACE AS LOGICAL REPRESENTATION The hierarchical inclusion series sketched above rests
on the assumption that each mode within it be Human population within the typical Western regional area distributes itself hierarchically, according to socioeconomic forces governing the population size of the constituent towns and cities; the sustaining inverse function relates size class of center to number of centers present in each such class. Consider the contrived regional setting diagrammed in Figure 3. This consists, as represented, of one large "central" city, five smaller cities, ten large towns, and twenty small towns, coded as such in Figure 3.A. Such classifications are often based on socio-economic function within the overall system, a function that is usually closely related to population size of center. Figures 3.A, 3.B & 3.C: Fig. 3.A. Fig. 3.B. Fig. 3.C. Figure 3.B shows how the same basic classification can be arrived at by looking at interaction data. Here, we might be dealing with, for example, commuting data (where dotted lines between centers represent thresholds of fifty trips, dashed lines five hundred, and solid lines five thousand). In this instance, the total number of commutes emanating from the center would identify it with one of the four size levels. In Figure 3.C, however, the same trips data are used to specify a subregional classification, since, independent of size considerations, certain towns and suburbs fall within one or another of the several general "commutersheds" in the region. In this instance, absolute size of center is not such a relevant consideration as compared to nearness--to subregional evolution. The classification shown in Figure 3.C is of a historical type, whereas that shown in 3.A describes organization of an "ecological" type. Either form of classification of centers into groups
can be followed by the calculation of group (class)-level averages. The
latter values may then be arranged in matrix form. In the case of the
subregional delineation, an idea of the subregional interaction structure
is obtained by looking at totals of trips/calls between and within regions.
This form of representation (Figure 3.C) can produce an additive inclusion
series of (subregional) relations interpretable in hierarchical terms
of the type discussed in the last paper (if one wishes to split the regions
into finer units, or keep lumping them together into broader ones). Economic
function according to size/rank, however, is best examined in relational
terms by linking the This ecological classification form has its analogs
in various other kinds of systems, whether we measure the internal associations
supporting them on the basis of gravitational relationships (solar/planetary
systems, topography in drainage basins, etc.), economic functions in social/political
systems, or rates of chemical, biochemical, or organismal interaction
in physical and biological systems. Indeed, I propose that Note that Spinozian thinking will cause us to focus
on the relations among the subsystems of Structural relations between particular subsystem pairs in a given subsystemization can be expressed in terms of "partial identities" (the varying degrees of similarity or dissimilarities between particular subsystem pairs), or perhaps most ideally as actual net flows of materials or services (as in the earlier commuter trips example)--or information--between/among them. The summary representation of flows or similarities is easily rendered in matrix form; the result is a depiction of subsystem structure that reflects its natural differentiation in terms of such actual or implied flows of information. But again, such initial depictions do not provide
an entropy-maximized view of structure, which dwells on the logical/functional
Although the preceding concepts are admittedly not (yet)
organized about a rigid mathematical or logical framework, they are reasonably
tidy conceptually and have the advantage of readily lending themselves to
simulation. Simply, one can investigate beforehand what range of measurable
(magnitudes of) relations must obtain among the interacting elements within
The easiest way to introduce the rationale for the
simulations I have performed is to compare them to the general manner
of use of the well-known multivariate statistical operation multidimensional
scaling (MDS). This starting point helps us in our efforts to "work backwards"
from the Let us now imagine that we have performed an analysis
of the spatial pattern affinities among the classes of towns/cities as represented
in Figure 3.A, concentrating on the rural-urban hierarchy (i.e., It will, I submit, if three general conditions are met. First, and most fundamentally, if the fact of system subdifferentiation is related to spatial extension in the sense I am suggesting. Second, if we have adequately isolated the extent of the system as it exists and chosen a "complete and unbiassed" means of representing it through appropriate surrogates. Third, if our technical means of measuring the system is itself adequate and our secondary calculations (involving, for example, pattern measures) are reasonably unbiassed. To summarize: I suggest that "natural" subsystemization
within any physically-extended system might correspond to--and actually
produce--its projection as a three-dimensional entity. The fundamental limitation
on system organization is therefore that the numerical representation of
its subdifferentiation of elements For the present, I must skip over a number of very interesting related and extenuating subjects (e.g., how the "elemental bits" of the sum relations might be organized, why we should expect reality to be expressed as a three-dimensional Euclidean space rather than through some other dimensionality, the representation of different system equilibrium states, etc.) and proceed immediately onward to the aspect of the model focussed on here: its inherent testability. TESTING THE MODEL: PRELIMINARIES I believe that the model presented here can be tested
through the combination of simulation and natural experiment. A properly
conceived set of simulations should identify the range of all possible
conditions of system subdifferentiation corresponding to numerical matrix
representations that meet the relevant constraint (i.e., producing an
unambiguous dimensional outcome). It was hoped initially that many of
these matrix representations will I shall describe the exact simulations I have conducted in the next several write-ups; a few conceptual matters that look ahead to these and empirical applications must be dealt with first. Consider the following 4 by 4 similarities matrix:
This matrix of values can be mapped directly through
MDS into a three-dimensional output configuration, with no "fuzziness" of
representation (i.e., no "stress," or lack of correlation between the original
data and the representation) resulting. Although the set of coordinates
produced is unambiguously The relational data in the input matrix are usually indicative of measurements of interaction characteristics between classes of things. Just because there may be a greater than average magnitude of "overt" interaction between a given class and all the others doesn't necessarily mean, however, that the class involved is "more important" to defining the system: there may simply be a larger built-in element of redundancy in its information processing function. Transforming the matrix into an entropy-maximized version of itself in which all row and column vectors are standardized assures representation of the classes as logically equivalent entities with respect to the integrity of the sum system (a procedure routinely applied in many fields, e.g. econometric analyses of intra-regional commodity flows or trips data to and from vastly differing-sized places). The entropy maximization procedure I have been using is double-standardization, in which all rows, and then all columns, of an x by x matrix are alternately standardized to z scores as many times as it takes to force the matrix elements to converge to stable values (see Smith [1983a, 1983b] and Slater [1976] for examples of use; many others also exist). Any double-standardized 4 by 4 matrix consisting of symmetric z scores (i.e., where all ij values are the same as their corresponding ji values, and all i=j values are the same) can specify a set of unambiguously-placed three-dimensional coordinates. Such coordinates will either project a (two-dimensional) rectangle centered at 0,0,0, or two pairs of equally spaced and opposed points centered three-dimensionally at 0,0,0, in either case describing an underlying space of the type appropriate to present considerations, as such configurations are three dimensionally symmetric (i.e., their four coordinate points are geometrically indistinguishable from one another once one removes axial affinities). The similarities matrix displayed above, when double-standardized, converges to the following symmetric set of z scores, which do in fact meet all the criteria just stated:
So far I have been speaking as though subsystems were no more than abstractions akin to MDS configurations--points "in space" located at simple linear distances from one another. In the real world, however, the elements of the subsystems (for example, the differing-sized towns in the example given earlier, or the varying elevations that may be areally sampled from within a stream basin) will be distributed in some non-singular fashion. To generate a matrix of relations in such instances that reflects conditions of relative location, it will be necessary to equitably sample, and then apply some measure of average (group level) spatial autocorrelation (or, alternately, use some measure of actual energy flow from location to location). Thus, once the sampled elements are classed into groups on the basis of some independently measurable property (such as population size or elevation), the resulting groups' degrees of spatial autocorrelation with one another can be calculated, and the matricized spatial autocorrelation scores double-standardized. Under the present model, if the resulting z scores are symmetric (all ij = ji, and all i=j scores identical, and highest), the results conform to what is necessary to define a three-dimensionally extended "natural system." Clearly-defined surrogates that mirror internal system
differentiation properties are not that easy to come by. I have found,
for example, that population of centers In sum, empirical analyses bearing on the validity of the present model will only be possible if certain precautions are taken. To proceed at all we should begin with a system with recognizable boundaries; we must then have some characteristic of internal differentiation of the system that can be measured (or at the least, logically distinguished) such that a sampling over the areal extent of the system will lead to a classification into subsystem units. If the empirical focus is on spatial patterns (as opposed to straightforward flows data), the spatial autocorrelation properties of the subsystems can then be measured, matricized, double-standardized, and the results examined for conformance to the criteria mentioned. SIMULATIONS OF THREE-DIMENSIONAL SYSTEM STRUCTURES One should remember that the "historical" (subregional) and "ecological" (urban hierarchy) conceptualizations of the regional population system sketched in Figure 3 can be based on exactly the same data. The ecological structure in particular is derived from a global hierarchical arrangement; usually, and in the example, socioeconomic function. Let us now ask the question of whether the overt pattern of population distribution associated with such a function--or any analogous one--may not only be "set" in two/three dimensional space, but actually define (be projectable as) that space. Let us suppose for the moment that we have a perfectly symmetric pyramidal figure consisting of four apexes and four faces. Once we have set such a figure geometrically within a three dimensional coordinate system and retrieved all the distances among point pairings in the projection, we can represent this system of distances as a symmetric matrix whose primary diagonal consists of zeros, with all other elements being some other single value. Now imagine that we are instead given only the matrix of distances, and from it asked to construct the figure. One way of doing this, as alluded to earlier, is to input the matrix elements into a metric multidimensional scaling program and specify a three-dimensional solution. The resulting projection of the distances will yield an unambiguously defined set of coordinate locations identifying the relative positions, in geometric space, of the apexes. Now imagine any other set of numbers arranged as a symmetric i=4, j=4 matrix, and consider the question: How many such sets will inequivocably project a three-dimensional space, as in the pyramid example above? To examine this matter, crucial to present considerations, I devised a series of simulations. These are described in the next several write-ups.
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dimensions." _________________________
Copyright 2006-2014 by Charles H. Smith.
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