Space as Neither Vacuum nor Plenum

by Robert S. Brumbaugh

Robert S. Brumbaugh is professor of philosophy at Yale University, New Haven, Connecticut.

The following article appeared in Process Studies, pp. 161-172, Vol. 7, Number 3, Fall, 1977. Process Studies is published quarterly by the Center for Process Studies, 1325 N. College Ave., Claremont, CA 91711. Used by permission. This material was prepared for Religion Online by Ted and Winnie Brock.


Space is symmetrical in its mathematical, abstract form: — isotropic, static, one-modal. But concrete process finds space entangled with acting entities and with time, and in this concrete domain, the symmetries of abstract fields do not exactly match the facts of location.

Historically, process philosophers have been fascinated by time, and rather bored by space. There are reasons for this: spatially oriented models and plans lead naturally to philosophies which "spatialize" time and change. It is a mistake, however, to overlook the notions of space and location: it is not necessary to leave these concepts to the abstractness of a static formal model or to unanalyzed technological "common sense." In fact, it is necessary not to do so, if we want to philosophize well, for inattention to spatial concepts may interfere with both the theoretical consistency and the practical efficacy of process philosophy.

Aristotle, an excellent observer, thought that every change of location in terrestrial space took place through a medium that had some resistance. If the space which related places were wholly empty, he thought, it would be pure nonentity, and so unable to have contents or contain relations; while a wholly filled space would be a plenum, an extended substance in its own right, and thus unable to contain or transmit anything else. In regard to places and the space that contains and relates them, this Aristotelian empirical notion of a field which is neither vacuum nor plenum seems to me to be right. Interaction must take account not only of intensity and distance, but of the space through which the distance is measured, and its coefficients of conduction and insulation.


Between the time when Aristotle included the "where" in his categories and the time when Whitehead criticized "simple location," the question of places "where things are" attracted rather slight attention from philosophy. The tacit assumption was that the physical or metaphysical places in question could be identified with sets of mathematical points, or with knife-edged states, or with quasimathematical monads. And Whitehead’s reopening of the theme used this concept only in passing, as an arbitrarily chosen case-study of the difference between scientific abstractions and adequate common-sense. Nevertheless, it was an important case-study. It was important because the abstract scientific concept of location that Whitehead found dominant in current common-sense led to further notions of "inside" and "outside," "here" and "there," that were inadequate philosophically and inefficient practically.

Whitehead himself, particularly concerned to protect the primary role of time in his physical theories, made space abstract and derivative. By contrast, my discussion will center on space as primary, as a kind of venture in metaphysical topology; the project is Whiteheadian in inception, but my own in execution. I propose to show the inadequacy of two extreme notions of "space" and "place" by testing them against our experiences of social space, in situations in which we are the entities contained by the social field. This assumes as a premise that such social fields are in fact spaces, and that the behavior of contained entities in such spaces is relevant to any philosophical theses about location in general.


A space is a continuous field made up of "distance" relations, which can be defined geometrically. Whether the space is something substantial that contains related entities, or whether the field is simply generated by the relations of the things that create it, will not matter for the present discussion. Space can contain entities, and entities can interact in and through space. That interaction can be thought of apart from dynamic time, as forces at a moment, state, or small section. Space is alike in all directions; the entities and relations it contains are all actual; and for "distances" such as S, S (x,y) always equals S (y,x). Further, an object moved through space, then returned to its starting point, is not affected by its shifts of position.

Unlike space, time is a set of relations of entities which form a sequence. A sequence is not the same in every direction, but has an irreversible "temporal" direction of its own. The terms are transitively connected, but they are not reversible. Some terms of a sequence are actual (its past), some only potential (its future). The actual terms are entities which are causally connected.

An actual entity is a substance or set of properties located at a point of intersection of a spatial distance scheme and a temporal causal sequence. (Thus the definitions of space and time also specify what sorts of entity the spatiotemporal system can include.) A process is a sequence in which a special kind of causality, "creative activity," gives direction to the constituents.

In physical spaces we can disregard temporal effects. There remains, however, interaction along "distance" lines, or lack of it, which pure mathematical space does not have. And an adequate general characterization of space, able to include natural as well as mathematical types, must be able to accommodate spaces that are neutral, that are insulators, and that are conductors.

In fact, at the very limits of the natural world, there seem to be fields in nature which would correspond to perfect conducting and insulating spaces. At the one extreme lies the superconduction of the field at absolute zero temperature; at the other, the lack of radiation in a field of "black hole" entities with infinite density (so that they no longer exert even gravitational influence mutually). By and large, however, the spaces we are interested in are "normal" in the sense that their counterparts in nature fall somewhere between the infinitely full and the infinitely empty.


A first experiment with a concept of space follows a suggestion of Whitehead’s. He pointed out that the idea that physical things are each "simply located" in a Cartesian pure space is a case of a past technical concept diffused into present common sense (SMW ch. 3). What happened seems to have been that the past technical notion of "location" from Cartesian physics was made into a metaphysical notion of location generally, and this generalization was what common sense uncritically accepted. Whitehead described this differently, saying that current common sense was the heir of past metaphysics. But in the present case, what we have seems rather to be a current common sense (and metaphysics) resulting from generalization of past technical science.

In this current version, all spaces are presupposed to be perfect insulators. Thus, two things in distinct places are totally irrelevant to each other, unless those places are in immediate or mediated contact. For some purposes, this conception of things, each in its proper place, will work quite well: it seems quite congenial to a context of mechanics, invention, and technology. But in other contexts, it runs into difficulties, both theoretic and practical. A case of the practical problems that convinced me that something was wrong was the implication of this "sensible" view for educational practice. In a classroom there is a concrete space, as in the library there is a kind of logical space that contains items of information. The classroom space contains as its local entities students -- active, sensitive, and restless organisms. In the simple location model, the function of us as teachers is to give the students "ideas" which they are to "keep in mind." The ideas are thought of as though they were workbench parts and gears, the minds on the analogy of separate cabinets in which the gears and parts are stored after they have been sorted.

The aim is, for the teacher, "covering the material"; for the student, speed and accuracy in "information retrieval," The rigid application of this paradigm is well illustrated by the teaching of classical languages in the nineteenth century. There was initial rote memorization of the words that, in their various inflections, were the elements of the language -- usually more of them than any ancient speaker ever used, or ever knew. Then followed analysis of the syntax of single sentences. Finally paragraphs and verses were read, and classified in terms of style and meter. In this logically impeccable form, the scheme killed off interest in the classical languages so effectively that it had to be modified. In fact, the description of Greek and Latin as "dead" languages was sometimes felt as more than metaphor. But the modifications were thought of concessively, as sugar coating to "enrich" what was essential.

Though we think we know better now, I suspect that every teacher has worried about not covering the material, has speeded up coverage, and come away wondering what went wrong. But it has seldom occurred to anyone that what went wrong was the presupposed notion that education meant covering the material.

Another damaging consequence of this technology of education, in addition to what we have seen, is its deduced practical rule that maximum learning efficiency should involve minimum student-with-student interaction. For if, realistically considered, each "mind" to be stocked with information is as separate from every other one as one Kansas silo waiting to be stocked with grain is from one other in a rail-side row, contact and conversation between students can only be regarded as inefficient -- as "noise" in both the ordinary and the technical sense. One inspired noise-reducing device is to nail each desk to the floor and confine each student to an insulated "proper place" in the classroom. This is considered part of "discipline."

The whole model and technology rest on a mistaken view of where things are, which has led by strict deduction to an equally mistaken view of what is "really" going on. But this sort of deduction about entities in space, based on this presupposition about its character, still is widespread, still fancying itself "sensible."


At this point, it may be helpful to go back to ancient Greece. There we find, in the classical atomic theory, the first appearance of the idea that space is a neutral insulator; and at about the same time, the antithetic view that it is a perfect superconductor. This historic topic, at any rate, is not overstudied; in 1963, Max Jammer noted that his history of concepts of space was the first he knew of, and the field has seen few others since then. One virtue of this going back to classical concepts is that they are often "classical" in their logical consistency and their purity.

Classical atomism was generated by an intersection of formal logic, ordinary sense experience, and technology. In defense of his thesis that "only Being is," Parmenides argued that if there were plurality or change, there would have to be a nonbeing to divide the single whole of Being into simultaneous or successive separate "parts." Since he held that the statement "non-Being is" was unintelligible and without any referent, that, he thought, ended the matter. But this doctrine had the very counterintuitive consequence that plurality and change, which seem to pervade our world, not only are unreal, but are nothing at all, hence do not even seem to be! In partial defense of this view, Parmenides’ student, Zeno, showed that both science and common sense ran into paradoxes when they assumed (as they did at that time) that space and time are made up of elementary units, but still are continua. His particular line of attack was to show that taken together these views made it impossible to define and describe motion with finite velocity.

The atomists began their reasoning here. Since change does appear, they reasoned, nonbeing must exist as well as pure being. And that pure being must come in a plurality of packets that are indivisible These indivisible atoms escape Zeno; the nonbeing postulated to accommodate Parmenides’ argument becomes the "space" in which the being particles move. That space, to satisfy Parmenides’ demands, must be a perfectly neutral nothing: if it had positive properties, it would be a kind of "something," and the old difficulty of explaining plurality would arise again.

The world view of atoms of pure being in a spatial sea of nothing still needed to offer some explanation of causality and change. Here mechanics supplied the final notion needed to complete the theory. Quite clearly, the only change a classical atomist can admit as real is mechanical change -- particles can transfer momentum, rebound, cling together, but that is all. And in principle, the theory held, all events can be given this type of mechanistic causal explanation.

The seventeenth century revived this notion of an insulating space, via Gassendi and Descartes, and that revived notion became generalized and built into succeeding Western common sense. (Newton’s ideas were not so easily generalized and simplified.)

As Whitehead suggested, if this view of location is taken as a complete and concrete account, it works badly: for, taken in this way, it equates a highly abstract selective construct with a full concrete reality. As we saw when we considered the application of this view to education, it runs into practical failures which show that something must be wrong. The theory runs into theoretic failures as well. These difficulties facing the classical atomic theory are well known: secondary qualities remain inexplicable; no meaning can be given to the notion of an external world outside of the sense organs of the observer; organic time must be reversible -- which it is not; we can never choose among hypotheses, since all of our mental states follow "from necessity," so we don’t have theories, but can only report autobiographies, and so on.

With the atomic theory, Greek philosophy proposed a world of atoms and the void as a way to respect logic, yet save appearance. An alternative way to escape Zeno was, of course, to have space and time be continuous, with no atomic particles whatever. Such a continuous space would be a sort of "impure" being: a theater of overlapping, mixing qualities, with no postulated "substances" underlying them (because such underlying substances would be like the impassive pure being of Parmenides). This line of thought was pursued by Anaxagoras at about the time Democritus was working out his atomism. As the theory develops, it becomes clear that Anaxagorean space is a perfect conductor.

It is very important for us, at this point, to recognize that the rejection of a Democritean view of space does not constitute a proof that space must be Anaxagorean. If there were no intermediate options, it would do that, but there are. A comparable case in modern thought would be the idea that Whitehead’s rejection of simple location establishes the metaphysics of Hegel.

In some ways, Anaxagoras anticipates process philosophy. His spatial field (setting aside "mind" as a special case) is a continuum of overlapping qualities. There are no lines of nonbeing to cut this up, since an "empty space" is a contradiction in terms on this view; and thus the field is a pure conductor. like its dialectical opposite, this view breaks down pragmatically, and theoretically as well. Pragmatically, the idea that size and distance make no difference in the efficiency of central control fails to work. A single central accounting office for the postal service, the army quartermaster corps, or the Soviet Russian economy, however economical it seems in Anaxagorean theory, is cumbersome and self-destructive in actual operation. On the theoretical side, we find that in a world of this kind, there is no more chance of our discovering objective theories than there was in the world of atomism. The trouble this time is not that the theorist is walled off from the subject-matter, but rather is melted into a fusion with it. Since when any two things are brought close together in this space (whatever "close together" can mean) they blend and a single new third thing emerges, there is no standpoint that can qualify as "objective." (Anaxagoras kept nous "unmixed" to avoid this objection: perhaps, indeed, a divine Mind, in his scheme, would have a Pythagorean, nonperspectival view of space and time.) There is also an aesthetic, intuitive objection: that I see the desk as next to me, but I do not -- as on this theory I should -- identify the desk with myself. Most important, there is a moral problem, parts lose their separate identities when they enter larger wholes -- the exactly the converse of the atomic theory’s savage individualism: since State, the Church, the CIA -- we cannot defend an ethical conviction that individuals have freedom or that persons each have responsibility.


Well, then, if both of these antithetic classical views have failed, where, indeed, are things? Neither in isolated boxes, nor in one great porridge melted together. Things extend out from centers of identity, overlapping and influencing each other variably, depending on (1) the intensity of each property a thing has, (2) the relevant "distance" between centers in a field, (3) the resistance, or conductivity of the field in question. It seems that each entity in space has a center of identity which holds together ("prehends" in Whitehead’s vocabulary) its aspects or perspectives that spread out into the places of other persons and other things. This accounts for my feeling when I encounter a desk that the desk is really where I am, but that it has a center of identity different from mine.

It is not easy to construct a static spatial design that is an adequate diagram of this sort of "modal location," and even Whitehead, expert mathematician that he was, seems never to have found a general one. Perhaps he thought that his early graphs for physics did supply the structural design for elementary cases, and that his new humanistic vocabulary in Process and Reality indicated a transferability from the simpler cases to more complex ones. If that is so, it seems to me he was mistaken. Of the two projects, the transfer from complex to simple entities is more plausible and clear.

While location that is neither simple nor diffuse may be hard to imagine in any detail when we populate it with atomic particles or physical aggregates of molecules, it becomes transparently clear when we substitute guests at a dinner for the abstract "entities" in a general location formula. ". . . Joe and George had better be separated by as many places as possible; they get noisy when they are close together . . . Joe and Jane go well next to each other, but not Jane and Anne; they just gossip together." Authors and hostesses have always been sensitive to the complex overlaps and interactions of characters who meet "in" convivial or literary situations. But they have usually not cared about the validity of metaphysical generalizations based on their expertise. (An exception is Empedocles, whose six cosmic "elements" included, in addition to Earth, Air, Fire, and Water, Love and Hate.) I propose to regard these "social" interactions as typical cases of the way entities in space relate to each other, except that in these instances the "entities" have more than average sensitivity.

While I am sure that Empedocles was right in thinking that attraction and repulsion are experienced directly, I am not clear whether there is an additional literal "telepathic" overlapping of thoughts and feelings between selves. But even if there is not, it is clear that the extreme location concepts will fail to match either sound theories, or sound practices, or sensitive intuitions.


Up to this point, I have been focusing on the relation of "and": item by item adjunctions of "entities." The ideas and implications are very similar for the relations between containers and contents, the "in" relations. When we wonder how sharply a boundary separates space into "inside" and "outside" regions, simple location again offers a tempting model. By uncritically accepting the model it offers, we can make mistakes in our attitudes toward such relations as those of organisms to their environment, of a human self to other selves, of one sovereign state to other nations.

I want to glance at this pair of ideas -- inside and outside -- from the standpoint of pure mathematical topology, where they are not simple even though the spaces and entities involved are purely mathematical. Then a look at pragmatic operations with inside and outside locations will be followed by a conclusion about the container-content relation in general.

A look at pure mathematics shows how far from obvious the theorem that "every actuality has an inside and an outside" is. For, once we have defined inside and outside by specifying that every closed curve in a plane or curved surface in a space divides the plane or space into two regions in such a way that pairs of points in different regions cannot be joined without crossing the boundary, very odd things happen. A long tube, for example, turns out to be one-sided, since by going around the open ends, the points on the interior and the exterior can be connected without passing through the wall. The familiar one-sided Moebius strip is an even more exotic example. Common sense suggests that we can generalize our description by saying that "pairs of points differing in inside-outside location can be joined by a straight line that cuts the boundary once"; but that is not true at all. For complex, zig-zag polygons, such as illustrated in FIGURE 1,1 the general condition is rather that such a connecting straight line cuts the boundary an odd number of times. Thus already, in the pure nonconducting and noninsulating space of mathematics, the concepts of inside and outside prove far from "simple."




That these relations might nevertheless be simple for the "pragmatic" social spaces we inhabit is shown to be false by the evolution of the British castle. Here, from the Norman Conquest on, a need for insulation was reflected in spatial designs intended to keep persons and property safely "inside" a fortification. In the eleventh century, the castle design projected a simple notion of inside and outside. The inside of the castle was insulated from the rest of the world by a single high wall, with square towers at the corners. Unfortunately, this is a design where a single break at any point completely destroys the inside-outside distinction. (And the higher the towers, the more easily the break can be made by mines and fire.) The prize of the British castles -- one that Cromwell could never take -- is not Chepstow, high above the Severn, with its single wall; but rather a flat, polygonal fortress, Beaumarais (see FIGURE 2),2 last-built of the castles in Wales, which to the commonsense tourist looks like the most vulnerable of the lot.



The castle builders progressively mastered the rules of pragmatic space. Thus, in a relatively peaceful time in the district of Kent, they devised plans in which each part of a castle is "inside" every other; while in the hostile world of Beaumarais, they contrived plans in which every part of a castle is "outside" of every other part. (To do this, Beaumarais takes advantage of such devices as D-shaped half-towers, open on the inner side: so far as arrows and missiles are concerned, an enemy who has captured an outermost D-shaped half-tower, but not the inner ones, is just as effectively outside of the central castle as before. The same relation holds between different levels of archers’ platforms and battlements, the inner higher and overlooking the outer. The "sociable" design, by contrast, arranges its space about a central circular stairwell so that the rooms are all connected, while openings in the boundary -- windows, gateways, gardens -- break down the barrier between outer world and inner castle -- see FIGURE 3)3

Although pure mathematics and impure practice thus combine to suggest that living things, human selves and societies, should not be pictured on the model of Chepstow Castle -- as though they were ping-pong balls, single shells that either insulate or shatter -- our generalized common-sense notions of inside and outside by and large remain early Norman in their simplicity. (Two exceptions are our recognition that "separately located" cells are the best design for espionage and revolutionary



organizations and that "round-table" patterns work best for educational and convivial purposes.)

An individual self is, like a social group, neither wholly private nor wholly public, but with a location vague about the edges. Clearly, a good deal of our development as selves takes place socially: through learning a language, imitating roles, identifying our feelings with those of other persons, and so on. (This is important to remember in connection with the political metaphor of "national self-determination" a self, there too, must to some extent be created and grow; it is not innate and ready-made.) The boundaries certainly are not sharp: looking from the inside, I am rather constant in seeing my body as part of myself, but I am not so sure about my property. Is that property a "substantial" part of my identity? And what about my family, or my country?

If I suppose a self, in this case mine, to be a simple, defensive castle, with "my" experience inside it, insulated abruptly from a "not mine and "not me" domain outside, then what will count as important for me will be my own "inner" interests, pains, and pleasures. A social order, while it may be useful to protect me from invasions of privacy by other persons or from other damage by the environment, will always be external, operating by coercion. It will seem clearly realistic to judge that my actions, as well as everyone else’s, are and must be guided by calculations of self-interest, looking toward some maximum excess of pleasure over pain for some segment of an expected lifetime. While it would give one a warm emotional feeling to believe that the individual pursuit of private good and the general welfare will coincide, if the insulating location model we are using is right, we can prove that this coincidence is infinitely unlikely. Suppose that the consequences of an action that will give me pleasure are nevertheless undesirable in other ways. For example, suppose that if everyone chose to act in this way, the end we aim at would be destroyed; or that if my choice were repeated often enough, the human race would disappear. On the present view, these suppositions are ethically irrelevant, except for some inner perturbing effect they just might have. For I am not proposing that everyone else do as I do, nor even that anyone else be permitted to; their experience, like a remote future in which the human race might disappear, lies outside of my own self-interest, and the inner "I" is uniquely important to me.


Practically every area of human knowledge and behavior may find itself faced with problems created by mistaken notions of location. In the present discussion, I have concentrated primarily on the consequences that follow from the "simple location" version. That is because I think these consequences are more widespread and less easily recognized than the errors that follow from the "diffuse location" view. In any case, the problems generated by diffuse location concepts are directly derivable, simply by asserting the contrary, from the problems consequent on the adoption of the notion of simple location.

Setting up an archive by mentioning in passing some of the relevant confusions, we can begin by noting that in ethics we have just seen simple location leading to radical ethical egoism. In politics it leads to ideas of nationalism-of sovereignty and self-determination -- that are unrealistic and brittle. In religion, the misapplication of ideas of location leads to problems of God’s relation to the world, to history, and to time that become insoluble. In technology, consistent application leads to the self-destructive techniques of the efficiency expert’s point of view. In aesthetics, alone, the notion has been relatively ignored and harmless: creativity has managed to remain outside the range of misguided sensible or metaphysical strait-jacketing.


One function of the philosopher is to be "a critic of abstractions." Our notions of location are a case where, in the absence of that criticism, common sense may lead us into mistaken and disastrous ideas and plans involving space and time. And since the misleading notions in question not only seem so sensible, general, and familiar, but carry a penumbra of scientific respectability, we are often either unaware of them or wholly indisposed to question them.

Another function of the philosopher is to offer speculative alternatives to traditional positions and hypotheses. If we are not satisfied with what our current practice, science, and philosophy have to say about such a topic as space and place, can we propose an alternative that is more general, more coherent, more potentially effective in practice? I think that we can.

Space is symmetrical in its mathematical, abstract form: isotropic, static, one-modal. But concrete process finds space entangled with acting entities and with time, and in this concrete domain, the symmetries of abstract fields do not exactly match the facts of location. (It is not a mismatch, but a nonidentity between the generic abstract scheme and its correct specification.) Yet the fact that there is a distance and difference between concrete existence and philosophic abstraction does not guarantee the applicability of notions that derive their appeal from a rejection of all intellectual abstractions in favor of some anti-rational alleged "intuition."

In the twentieth century, process philosophy seems to offer the most promising context for a discussion of new notions of space, place, and location -- physical, social, or formal. The process orientation is not yet committed to school-wide orthodoxy regarding either the spaces of science or those of common sense, nor to any dogmatic identification of physical place with mathematical.

Since there are no such commitments, we are free to entertain the idea that the attempt to understand space by thinking away all contents may yield an eccentric abstraction. Though it seems that the resulting concept does match a space in nature, it is the space at a temperature of absolute zero, and the match with our social, pragmatic spaces is not good at all. An alternative route of abstraction, by thinking away all energy that sets up physical relations, and thinking this away by having substances too dense to release radiation, also leads to a space concept that may be approximated somewhere in nature. But if it is, this matching space is a field between super-dense "black holes," too dense to permit any light or gravitational influences to move between them. And our normal pragmatic spaces, with usual temperatures and densities, do not match this concept either. What space is conceived to be must depend to some extent on what it contains and what it does; oversimplified abstractions defining location are a danger when science erects them into "theoretical" or when common-sense makes them into "practical" generalizations that are put forward as exhausting the concrete situation.



1 Richard Courant and Herbert Robbins, What Is Mathematics? (London: Oxford University Press, 1941), p. 245.

2 B. H. St. J. O’Neil, An Introduction to the Castles of England and Wales (London: H. M. Stationery Office, 1954), pp. 49, 56.

3 Ibid.