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Place as Container
Aristotle’s Physics
Everything remains naturally in its proper place.
—Aristotle, Physics 212b34–35
No one thinks or speaks—even when the thought or word is erroneous—without recognizing, from this very fact, the existence of place.
—Henri Bergson, “L’Idée de Lieu chez Aristote”
I
That place was a continuing cynosure of ancient Greek thought is abundantly evident in Aristotle’s treatment of the topic: for Aristotle, where something is constitutes a basic metaphysical category.1 Except for the extraordinary cases of the Unmoved Mover and the heavens (ouranos) taken as a single whole, every perishable sublunar substance (including the earth as a whole) is place-bound, having its own “proper place” as well as existing in the “common place” provided by the heavens.2 Thanks to this stress on the importance of place for each particular “changeable body”—that is, changeable with respect to motion or size—the Stagirite situates his most scrupulous examination of place in the context of physics rather than of cosmology. Cosmology is of decidedly less interest to Aristotle than to Plato; and of cosmogony only the barest traces survive in Aristotle’s text, typically in the form of bemused and skeptical citations from pre-Socratic figures. The at least quasi-mythical aura of the Timaeus—its ambiguous status as a mixed “third genre” (triton genos) of discourse (Timaeus 48e, 52a)—gives way to the sturdy, no-nonsense attitude of the Physics, wherein place is conceived in the cautious, finite terms of container and limit, boundary and point. Chōra yields to Topos, the bountiful to the bounded.
It is precisely because of its indispensable role within the physical world that, for Aristotle, place “takes precedence of all other things” (Physics 208b35). In particular, it assumes priority over the infinite, void, and time.3 Place is requisite even for grasping change itself (kinēsis), with which the study of physics is always concerned; for “the most general and basic kind [of] change is change in respect of place, which we call locomotion.”4 Locomotion, after all, is movement from place to place.5 On Aristotle’s view, one simply cannot study the physical world without taking place into account: “A student of nature must have knowledge about place” (208a27). For wherever we turn in the known universe—outside of which there is “neither place, nor void, nor time” (De Caelo 279a18)—we find place awaiting us and shaping any move we might wish to make. Remember that even a void, were it to exist, would be a “place bereft of body” (208b26).
Given this perception of the pervasiveness of place, it is not surprising to find Aristotle offering his own version of Archytas’s archetypal argument for the primacy of place—an argument whose other advocates include Zeno, Parmenides, Gorgias, Plato, and, much more recently, Whitehead. Aristotle puts it this way:
For everyone supposes that things that are are somewhere, because what is not is nowhere—where for instance is a goat-stag or a sphinx? 6
It is at this very point that Aristotle makes a rare gesture toward muthos by citing the Theogony as an early testimonial to the inevitability of implacement. Having just argued for this inevitability from the various phenomena of anti-peristasis (i.e., the replacement of one body by another: despite the exchange of bodies, the place remains the same), natural movement (whereby different kinds of bodies move to “distinct and separate” regions [208b18]), and the void (in its empty placelikeness), Aristotle observes,
These are the reasons, then, for which one might suppose that place is something over and above bodies, and that every body perceptible by sense is in place. Hesiod, too, might seem to be speaking correctly in making Chaos first; he says
Foremost of all things
Chaos came to be
And then broad-breasted Earth
suggesting that it was necessary that there should first be a space (chōra) available to the things that are, because he thinks as most people do that everything is somewhere (pou) and in place (en topō). (208b27–33)
Here Aristotle rejoins the analysis of chaos at stake in the last two chapters. Rather than a species of no-place, of sheer void, chaos is for Aristotle a kind of place, however inchoate and formless it may be. Indeed, it is just because chaos is some sort of place and not a void that Aristotle can exclaim that “the potency of place must be a marvelous thing, and take precedence of all other things.” For, adds Aristotle, “that without which nothing else can exist, while it can exist without the others, must needs be first.”7 In these last words, the Archytian axiom is literally reinscribed in Aristotle’s text as he prepares to make his own case for the primacy of place in the physical world.
Before he can make this case, however, he must come to terms with Plato on the subject of place. He does so by an ambivalent admixture of praise and critique. The praise is straightforward: “While everyone says that place is something, [Plato] alone tried to say what it is” (209b16–17). The critique, however, is less than straightforward. For one thing, it rests on the supposition that for Plato “matter and space are the same thing” (209b12) and thus that place is also reducible to matter: inasmuch as “place is thought to be the extension of the magnitude [of a physical thing occupying that place], it is the matter” (209b6–7). For another, in the Physics “space” as chōra is no longer an independent term designating a vast extent such as that found in the Receptacle. Considered as “magnitude” (megethos), space is brought down to the scale of “place” qua discrete topos—given that place is coextensive with the magnitude of a particular thing-in-place.8 As W. D. Ross puts it bluntly, “The doctrine of place in the Physics is not a doctrine of space. Neither here nor elsewhere does Aristotle say much about space, chōra, and he cannot be said to have a theory about it.”9 Not to have a theory of chōra, to replace it with considerations of megethos and topos, is tantamount to a rejection of what had been most important, or in any case most challenging, in Plato’s cosmology.
Beyond this, Aristotle levels at Plato the general charge that “we should ask Plato why the Forms and numbers are not in place, if place is the ‘participative’ (to metalēptikon), whether ‘the participative’ is the great and the small or whether it is matter, as he writes in the Timaeus" (Physics 209b34–36). The charge is unanswerable; not only does the term “the participative” not occur in the Timaeus (which limits itself to claiming that the Receptacle “partakes in some very puzzling way of the intelligible” [Timaeus 51a–b]), but, more important, the Forms and Space, along with the items of Becoming, are posited by Plato as ultimate metaphysical givens, necessary postulates of any adequate cosmology. Elsewhere, notably in On Generation and Corruption, Aristotle takes Plato to task for failing to “say clearly whether the omnirecipient [i.e., the Receptacle as all-receiving (pandeches)] is separated from the elements” (Physics 329a14–15; see also 329a23–25) and for “making no use of it” in that Plato does not show precisely how, apart from Demiurgic intervention, the matrix of Becoming is transubstantiated into the geometrically configurated primary bodies (Physics 329a15–23).10
II
Place is thought to be some surface and like a vessel and surrounder.
—Physics 212a28–29
Having laid Plato to rest—albeit in an unquiet grave—Aristotle proceeds to make his own case for the priority of place. Although he makes this case in the text entitled Physikē akroasis (Hearkening to Nature), a text considered by Heidegger to be “the basic book of occidental philosophy,”11 Aristotle operates as much like a phenomenologist as a physicist, carefully investigating “in what way [place] is.”12 In so doing, he inaugurates an alliance between physics and phenomenology that extends into the recent past: the very word “phenomenology” was coined by Lambert in 1764 to designate the study of physical phenomena as they appear to the senses; Mach and Einstein continued to draw on this sense of the term.13 What is unique in Aristotle’s enterprise is its concern for general principles of change and motion—a concern combined with a scrupulous description of concrete phenomena. As Aristotle says in opening the Physics, “Start from the things which are more knowable and obvious to us and proceed towards those which are clearer and more knowable by nature” (184a17–18). To be “more knowable and obvious to us" is to be the potential object of a descriptive, phenomenological investigation, since such an investigation considers how things present themselves to the human observer in his or her immediate life-world.
A first instance of Aristotle’s protophenomenological description is found early in book 4 of the Physics.
These are the parts and kinds of place: above, below, and the rest of the six dimensions. These are not just relative to us. Relatively to us, they—above, below, right, left—are not always the same, but come to be in relation to our position, according as we turn ourselves about, which is why, often, right and left are the same, and above and below, and ahead and behind. But in nature each is distinct and separate. ‘Above’ is not anything you like, but where fire, and what is light, move. Likewise, ‘below’ is not anything you like, but where heavy and earth-like things move. So they differ not by position alone but in power too.14
Notice the fine balance here struck between matters of physics proper—which considers place as something “distinct and separate” and as having its own “power” (dynamis) when considered “in nature” (en de tē phusei)—and matters of phenomenological description: for example, the relativity of right versus left to our own particular position at a given moment. A complete consideration of place will have to take both matters into account: how place is “in itself and how it is relative to other things.
Much the same dual focus is evident in Aristotle’s treatment of two basic aspects of place: (a) just as in Husserlian phenomenology the method of “free variation” helps to discern how many basic kinds of a given phenomenon there are, so Aristotle does not hesitate to project two variant kinds of place: the “common place” (topos koinos), “in which all bodies are” (209a33), and the “special place” (topos idios) that is “the first in which a body is” (209a34); (b) since each kind of place involves an “in” as an integral component, Aristotle proceeds to specify eight senses of being in something.15 Two of these can be considered logical or classificatory, two are metaphysical, one is political, two delineate part-whole relations, and a final one is expressly descriptive: “as [a thing is] in a vessel and, generally, in a place” (210a23–24). It is striking that this last sense of “in,” the most manifestly phenomenological sense, is also declared to be “the most basic of all” (ibid.).16 To be in a place is very much like being in a vessel, and the question becomes just how this is so—thereby calling for further descriptive refinement.
It is the analogy of the vessel that allows Aristotle to refute the persisting temptation to regard either form or matter as providing the key to the nature of place: “Since the vessel is nothing pertaining to that which is in it (the primary ‘what’ and ‘in which’ are different), place will not be either the matter or the form, but something else” (210b27–30). Matter and form inhere in the body that is located in a given place—the matter furnishing the substratum, the form providing shape. The form belongs primarily to the surface of the located body, not to the place locating it, even if the two are contiguous and coextensive.17 As Aristotle states with phenomenological precision,
It is because it surrounds that form is thought to be place, for the extremes of what surrounds and of what is surrounded are not in the same [spot]. They are both limits, but not of the same thing: the form is a limit of the object, and the place of the surrounding body. (211b10–14)
But this leaves unanswered just how place is “thought to be some such thing as a vessel” (209a27–28). The answer is clearly to be sought in the containing and, more specifically, the surrounding, capacity of vessels: their power to hold (things) in. By carefully describing this capacity of holding-in, Aristotle is able to determine the exact definition, the “what-is-it” (211a8: ti estin) of place. The definition itself is set forth in two stages. In the first, Aristotle concentrates on the factor of containment as such by observing that we are located in the celestial system by virtue of being surrounded by air, which is in turn surrounded by the heavens. We are placed in this system by being located “in the air—not the whole air, but it is because of the limit of it that surrounds us that we say that we are in the air.”18 Place in its “primary” sense is thus “the first thing surrounding each body.”19 It is this immediately environing thing taken as a limit. But the limit here belongs to the surrounder, not to the body surrounded (the limit of the latter is determined by its form, i.e., its outer shape: see 209b3–6). As a vessel, such as a glass or a jug, surrounds its content—say, air or water—so place surrounds the body or group of bodies located within it. “Surround” translates periechein, which means “to circumscribe without including as a component part” literally, it signifies to “hold” (echein) “around” (peri-, as in perimeter). As a vessel holds water or air within it, so a place holds a body or bodies within it in a snug fit.
But Aristotle does not rest content with this first definition of place. For one thing, the analogy with a vessel is imperfect. While a vessel can be transported, a place cannot: “Just as the vessel is a place which can be carried around, so place is a vessel which cannot be moved around” (212a14–15). Still more serious, there is the problematic fact that a river is a place for a boat and yet the content of the water immediately surrounding a boat continually changes. Hence the inner surface of the surrounding water, that which delimits the boat’s place, is not selfsame from moment to moment. Since a minimal requirement of place is to be selfsame—to be the same place for different things located in it—Aristotle must add to the first definition the rider that a place cannot itself be changing or moving: it must be “unchangeable” (akinēton). This allows him to move to his most definitive formulation: “That is what place is: the first unchangeable limit (peras) of that which surrounds” (212a20–21). In the case of the river, it is thus “the whole river” that is the place: a phrase that Simplicius and others interpret to mean the banks and bed of the river, its fixed inner surrounding surface.20
Place thus construed is “the inner surface of the innermost unmoved container of a body.”21 As such, it contains-and-surrounds the body by furnishing to it an environment that, if not always stable (the immediate “spot” of a boat in the river is only a momentary locale, not a lasting locus), is nevertheless a defining locatory presence. Thanks to this presence, place is actively circumambient rather than merely receptive.22 It is just here that Aristotle’s departure from Plato becomes most manifest. In the Timaeus, space qua chōra—including both regions and particular places—is held to be receptive: indeed, it is “omnirecipient.” Precisely as such, it can be qualified by sensible qualities and can serve as the medium in which physical bodies will appear. But these bodies receive their definition, that is, their limit or shape, from geometric figures. Hence the limiting factor comes from the active infusion of forms by the Demiurge.
On Aristotle’s account, the limiting power is already in place; it is of the essence of place itself to provide this delimitation by its capacity to contain and to surround: to contain by surrounding. Where Plato’s interest lay in the shaping of the outer surface of physical bodies, Aristotle’s concern is with the fixed contour of the inner surface of environing places. For Aristotle, the limit is found within place, indeed as part of place itself. Limit is ingredient in place from the beginning—indeed, as the beginning of an ordered natural world—and is not imposed by an external ordering agent. Hence there is no need to invoke a deific regulator, a divine inseminator possessing a logos spermatikos. Places have their own independent potency. As Aristotle puts it in a characteristic understatement, place “has some power” (208b11). But the result of this modest proposal is quite sweeping: the world is always already fully implaced; it is never without those determinate topoi whose limits circumlocate particular things within their immediate environments.
III
Given the choice between Whitehead’s two models of creation—“Immanence” versus “Imposition”—Aristotle, in revealing contrast to Plato, opts unambiguously for a model of immanence. This is to be expected, for the Aristotelian scheme of things does not contain anything even remotely resembling chaos (the word itself appears in the Physics only as a vestigial term). Only by a process of conceptual prescinding does Aristotle reach the level of “prime matter” (prōtē hulē), which is as close as he allows himself to come to chaos. But prime matter is too indefinite in status to exist by itself. Instead, in the physical world—and that means effectively everywhere, since “everything is in [this] world” (212b18)—we encounter only matter that is already informed. In this world, material bodies have their own integrity thanks to their indissociably hylomorphic character. There is thus no need to explain the infusion of form into matter, much less the generation of an entire well-formed cosmos. The invocation of the Demiurge may have been essential in a situation in which sheer sensible qualities had to be transformed into full-fledged material bodies with stereometric shapes, but any such invocation is now pointless. Since the physical world takes care of itself by appearing from the start as fully formed, the only pertinent deity is an utterly stationary Mover who is (despite the appellation) eternally at rest outside the world and thus in effect nowhere at all. All places belong to the world, but the world-all itself has no place of its own.23 We have come a long way from the temptation to posit a primordial no-place: now the only philosophically legitimate null place is located neither before creation (as in ex nihilo accounts) nor between bits of created matter (as in the infinite void of the Atomists) but in the very being of the Unmoved Mover. If it is indeed true that there is “no place or void or time outside the heaven” (De Caelo 279a12–13), then the Mover itself is placeless.
A crucial paradox emerges from this situation.24 In a text such as the Ti-maeus, a quasi-diachronic account of creation leads both to the positing of a preexisting Space (along with its various regions and places) and to the need for demiurgic intercession in order to give regular shape to formless sensible qualities. Space is thematized in an account whose narrative nature entails Time. In the Physics, a nonnarrative account plays down place at the origin: placelessness obtains “outside the universe” (212b18). The paradox is thus double-sided: where a time-bound tale such as that told in the Timaeus requires deity to interpose itself literally in place—to give shape to qualities in particular places so that “the ordered whole consisting of them [can come] to be” (Timaeus 53a)—the timeless tale told in the Physics gives to its deity no place to intervene, given that this deity exists outside the world-whole of perceptible bodies in a metaphysical Erewhon of its own. In the one case, time and place conspire to draw deity into the world—at least during the critical event of creation. In the other, deity remains out of the world in a timeless and placeless state. The conception of a richly regionalized and still unordered world, spatially inchoate even if not strictly chaotic, gives way to the idea of a world at once coherently placed and formally shaped—a world having an immanent order that is the rigorous counterpart of the independence of the Unmoved Mover.
One important corollary of this shift in outlook concerns the role of mathematics and of geometry in particular. If the created world of the Timaeus involves what might be called an “ingrafted geometrism”—that is, the introduction of plane triangular figures as the primary structures of the surfaces of solids—there is no trace of any such externally infused geometrization of material things in the Physics. What had been essential to Platonic cosmology (creation necessarily includes geometrization in the Timaean account) is viewed with deep skepticism by Aristotle, who might well have applied to this cosmology Eugène Minkowski’s sardonic pathognomonic label “morbid geometrism.”25 If the world already possesses an inherent ordering that includes form or shape as well as place, to call for a separate act of geometrizing is an otiose gesture.26
I dwell thus on the disparity between Plato and Aristotle—especially in the contrasting terms of imposition versus immanence, geometrism versus physicalism—in an effort to indicate that two deeply different ways of regarding place are already present in ancient Greek thought. Moreover, in contrast with the two other most important early Greek paradigms—Hesiodic Chaos and the Atomistic void—the Platonic and Aristotelian conceptions of place have a significant posteriority in contemporary nonscientific thinking on the subject. Geometry provides a model for several early modern notions of space that are even today, in the twentieth century, pervasively operative at the level of common sense, if not of scientific thinking. And the Aristotelian alternative is the active ancestor of those phenomenological approaches that, in the writings of Husserl and Merleau-Ponty, question the superimposition of geometry and call for a recognition instead of the world’s immanent shapeful order.
The critical question for Aristotle as a protophenomenologist is how (not why) the world possesses such deeply inherent placeful order. The answer is: “Place is together with [every] object,” for “the limits are together with what is limited” (212a30–31). It is the “together” (hama) that is the clue to the “how” of place, to the manner in which place is “the most basic way” in which one thing can be in another: “Things are ‘together’ in place when their immediate or primary place is one.”27 A material thing fits snugly in its proper place, a place that clings to that thing, since thing and place act together in determining a given situation. I say “act together” in view of the power of place to actively surround and to situate what is in it—that is, a physical thing or body, which is not there as a mere passive occupant: as actually or potentially changing or moving, and as changing or moving precisely in/to its proper place, it, too, has power.
The double immanence, the reciprocal belongingness, of thing and place is summed up in an axiomatic formula that quite appropriately incorporates two uses of “in”: “Just as every body is in a place, so in every place there is a body” (209a25–26). This is not a merely empty or redundant statement. The Atomists were not the only ones to posit a place without a body (i.e., qua void); Plato did so as well: none of the primal regions at play in the Receptacle contains a full-fledged physical body. (Nor is it to be taken for granted that there are no bodies without place: what of the circumstance of being between places?) It remains that, according to Aristotle, to be in motion or at rest is to be in place, however momentary or transitional that place might be. And this continual implacement is itself the result of the closely cooperative action of places and things. Just as things are always (getting) placed, places are themselves always (being) filled—and filled precisely with things.
Such cooperation is the main way in which the limit acts together, hama, with what is limited: the outer limit of the contained body rejoining the inner limit of the containing place. Not only can one limit not exist without the other, but each actively influences the other, helping to shape a genuinely conjoint space, a space of mutual coexistence between container and contained. This co-constituted, coincidental, compresent double limit is what defines place in its primariness.28
IV
A point is that which has no part.
—Euclid, Elements, Book 1, Definition 1
The point is projected in imagination and comes to be, as it were, in a place and embodied in intelligible matter.
—Proclus, A Commentary on the First Book of Euclid’s Elements
It is not necessary . . . that there should be a place of a point.
—Aristotle, Physics 212b24
Despite its double delimitation, place is something unchanging vis-à-vis the changing things that are its proper occupants. “For,” as Aristotle warns us, “not everything that is, is in a place, but [only] changeable body” (212b27–28). In fact, four things lack place within the Aristotelian system: not only the heavens and the Unmoved Mover but also numbers and points. The most exalted physical and metaphysical entities join forces with the minimal units of arithmetic and geometry in a common circumstance of placelessness. The specter of no-place that haunts cosmogonic accounts of creation now characterizes not just a God who is impassively (and impassably) beyond changing and moving things—and even beyond the heavens that encompass these things—but the very numbers and points by which these same things come to be grasped arithmetically and geometrically. Contributing to the strangeness of the situation is the double paradox that (a) God as the Unmoved Mover might seem to be the ultimate place since, existing outside the heavens or at its outer edge, He might be thought to contain or surround (and thus to provide place for) the physical universe itself; (b) numbers and especially points, as formal constituents of a material world that is knowable scientifically, might seem to require a certain intrinsic placelikeness in order to play their proper roles in any mathematical understanding of this world: roles that rely on order and position. But if metaphysical and mathematical “places” are thus strongly suggested within the system of Aristotelian physics, they just as surely are denied within that same system.
Without trying to resolve this doubly perplexing circumstance—leaving God and numbers for the delectation of the Neoplatonists and the heavens for the construal of Copernicus, Kepler, and Galileo—I want to focus in this section on Aristotle’s treatment of the point in relation to place. The question of whether points have places (or, alternatively, are places) is more complex and intriguing than it first appears. To begin with, there is the basic question of how to distinguish point from place.
Since a body has a place and a space, it is clear that a surface does too, and the other limits, for the same argument will apply: where previously the surfaces of the water were, there will be in turn those of the air. Yet we have no distinction between a point and the place of a point; so that if not even a point’s place is different [from the point itself], then neither will the place of any of the others be, nor will place be something other than each of these.29
The premise in this line of reasoning is that the series of “limits” (perata) represented by lines, surfaces, and solids is ultimately dependent on the point as their non plus ultra constituent or progenitor. Where Plato prefers the indivisible line as a basic unit in cosmology, Aristotle states that “it is common ground that a point is indivisible.”30 But if points lack places, how will places accrue to everything constructed out of points: lines, surfaces, and three-dimensional bodies? No one, least of all Aristotle, wishes to deny that solid bodies lack place.
Inasmuch as “a point is that which has no part,”31 we might think that it cannot occupy space at all, much less be surrounded by a container, since to contain or surround normally requires that what is encompassed possesses at least one part. A passage from Plato’s Parmenides is illuminating in this connection.
If it [the one] were in another thing, it would presumably be surrounded all around by that in which it was, and that would be in contact with it, with many parts, at many places; but it is impossible to be in contact all around in many ways with something that is one and without parts and that does not partake of a circle.32
But, isn’t a point something that is always surrounded—indeed, totally surrounded in the space in which it is placed and thus as fully ensconced in its own surrounder as any sensible body? Is not the point a paradigm of being in place, precisely on Aristotle’s own view of place as a matter of strict containment? What could be more completely contained or surrounded than a point, whether it occurs in isolation or as part of a line or a surface or a solid?33
In attempting to resolve the issue, it will not help to claim that points are simply nonphysical, as is suggested by the idea of their indivisibility and by their status as a “limit.” Such may well be true of Euclid’s notion of point: “‘Point’, then . . . is the extreme limit of that which we can still think of (not observe) as a spatial phenomenon, and if we go further than that, not only does extension cease but even relative place, and in this sense the ‘part’ [of a point] is nothing.”34 This may hold for points as they figure into plane geometry proper—Euclid’s primary concern—but it is hardly adequate to their role in the physical world, where they certainly can be observed: for example, as the center or at the extremity of a given perceptual phenomenon (to cite instances given by Aristotle himself).35 If it is the case (as Proclus asserts in the exergue to this section) that a place for points can be projected by our imagination into “intelligible matter,” places for points surely can be discerned in physical matter as well.36 Indeed, does not Aristotle’s own ingrained immanentism and physicalism—his conviction that “spatial magnitudes cannot exist apart from things” (Metaphysics 1085b35) and thus his antipathy to any imposed geometrism—require us to find a valid role for points precisely within the physical world?
Indeed it does, and Aristotle’s preferred solution to the present predicament—whereby points are at once indispensable (as the minimal units of any plane or solid figure), observable (in physical nature itself), and yet place-less—is found in his distinction between place and position. If points do not possess place stricto sensu, they do exhibit location or “position” (thesis). In this respect, they are to be contrasted with the “one” (monas) to which Plato alluded in the passage cited above from the Parmenides; the one, as the basic arithmetical unit, is definable as “substance without position,” whereas the point is “substance with position.”37 This view, whose ultimate roots are to be found in the Pythagoreans,38 allows Aristotle to accord to points a spatial determinacy that exists despite their placelessness. Beyond sheer locatedness, this determinacy consists in an inherent bipolarity of direction, as when points aid us in distinguishing right from left, above from below, front from back. The determinacy is also evident in the way that points demarcate the limits of given spatial intervals as well as the shapes of figures of many kinds (including nongeometric figures).
That the determinacy yielded by position is limited in scope, however, is indicated by (i) the linguistic fact that the word thesis can mean merely “convention” or “orientation” as well as “position”39 (ii) the geometric fact that intervals between points call for lines to connect them, as do also the bipolar directions mentioned above (if not explicitly drawn, then at least imputed); (iii) the phenomenological fact that directions, and even intervals, are usually relative to the percipient’s own position: “Relatively to us, they—above, below, right, left—are not always the same, but come to be in relation to our position, according as we turn ourselves about” (Physics 208b14–16; my italics), where “our position,” being the position of a physical body, is a position with its own proper place.
There are three telling arguments against the implacedness of points that Aristotle does not set forth but that are worth considering here.
1 The first of these bears on position: if position is a necessary condition of place, it is not a sufficient condition; thus points, having position alone, are still not full-fledged places. This is not to deny that points can characterize places: for example, boundary markers at the edges of fields (ranging from Mesopotamian kudduru to concrete posts of more recent times), the points where the walls of a room come together, or the corners of a basketball court or a football field. In each of these cases, points establish determinate positions—they “pinpoint” them—and are invaluable, indeed indispensable, in this very role. (In fact, it is thought that Pythagorean points or dots were at first representations of boundary stones.)40 But it would be straining the point to say that they establish the place itself. For this to happen, something else must occur or be present within the interior of the field, the building, or the court, whether this be a specific activity of raising crops or playing a sport, a generalized action such as dwelling, or a sheer potentiality (e.g., a forthcoming event scheduled to occur in that very place). Points, then, as physically determinate—that is, as fixed in world-space—can serve as crucial demarcators of place even if they do not, solus ipse, bring about place as such. Thus we can agree with Proclus’s encomium that the point “unifies all things that are divided, it contains and bounds their processions, it brings them all on the stage and encompasses them about”41—so long as we do not go on to claim that the action of points is sufficient to bring about places themselves.
2 Points cannot constitute depth, an uneliminable dimension of all places.42 Points, taken by themselves alone, do not give rise to depth as an actual dimension of surfaces, much less solids composed of surfaces, or fields populated by solids; and by the same token they only rarely give rise to the perception of depth on such surfaces or solids or fields. Thus even in perceiving a highly complex composition of city lights seen from an airplane, I still may not grasp the recession in depth of the city below me: it remains a sheerly pointillistic scene. The perception of depth requires the co-perception of several shapes qua surfaces, for example, the profiles of city buildings in the distance.43 In making this observation, I am only rejoining a familiar passage from the Timaeus: “All body has depth. Depth, moreover, must be bounded by surface” (Timaeus 53c). We need not claim (as Aristotle imputes to Plato) that all physical masses are generated from a dialectic of the “deep and shallow”44 to concede the basic point: that a minimal requirement of depth is surface and that a precondition of surface in turn is line. And even if we concede that “a moving line generates a surface and a moving point a line” (De Anima 409a4–5), the point remains only indirectly constitutive of a surface and hence even more indirectly constitutive of the depth that a surface brings with it.45
3 If we grant that points are capable of being wholly contained—strictly surrounded by their immediate environment and thus themselves fully in place on Aristotle’s own criterion of implacement—we cannot aver the converse: namely, that points contain in turn. In fact, points, regarded as discrete entities, do not contain anything other than themselves; they are, quite literally, self-contained. As such, they cannot be analogized to “a vessel which cannot be moved around” (Physics 212a15). To fail the test of this analogy is to fail the Aristotelian test of place, for it is to fail to embody the criterion of containership. A point can be extended, that is, at once manipulable and visible, and yet, in its very compactness and density, still be incapable of surrounding in the manner of a vase or jug or river.46 For surrounding to arise, two conditions must be met: there must be both a plurality of units, and it must be possible to draw lines between them. Either way, we must move beyond any single point if a circumstance of containing is to obtain. Though sine qua non for containership (i.e., as constituents of surfaces), points are not themselves containers.47
This discussion leads us to distinguish between boundary and limit. We can grant that a point is a “limit of localization”48—precisely the lower limit, beneath which we cannot (and need not) go. For limit, like shape,49 belongs primarily to what is limited and only secondarily to what does the limiting (e.g., a container). At least this is so in Aristotelian physics, given its resistance to any externally imposed mathematization. In such a physics, as Proclus suggests, “the limits surrender themselves to the things they limit; they establish themselves in them, becoming, as it were, parts of them and being filled with their inferior characters.”50 Indeed, in a properly Aristotelian physics, the point can even be regarded as a paradigm of the limit because of its compressed and self-contained state. As Proclus says, “All limits . . . subsist covertly and indivisibly in a single form under the idea of the point.”51
To be a boundary, by contrast, is to be exterior to something or, more exactly, to be around it, enclosing it, acting as its surrounder. As such, a boundary belongs to the container rather than to the contained—and thus properly to place conceived as the inner surface of the containing vehicle, that is, as (in Aquinas’s formulation) “the terminus of the container.”52 Like place itself, a boundary “shuts in and closes off something from what lies around it”53—which is precisely what a point cannot do. Even if it is composed of points, a boundary must be at the very least linear in character if it is to function in this simultaneously en-closing and closing-off manner: hence its affinity with the idea of a “borderline.” But, as linear, a boundary is the boundary of a surface or a solid, not of a point. A point is surrounded by space as immersed in it, not as bordered by it; to be itself part of a boundary, a point must be conjoined with other points so as to constitute a line.
Two possible outcomes are suggested by the distinction I have just made between boundary and limit. On the one hand, the case for Aristotle’s denial that a point is itself a place is strengthened: if a point is indeed a limit, it does not constitute a boundary; and since it is the latter that is essential to place on Aristotle’s own model, a point cannot be a place or perhaps even an integral part of place. Self-limited in its splendid isolation and other-limiting only as part of a continuous line, a point lacks the crucial criterion of containership. On the other hand, place itself is more like a boundary than like a limit. Not only is a place two-sided in the manner of a boundary—insofar as it is inclusive and exclusive at once—but it is also like a boundary in the special signification that Heidegger detects in the ancient Greek conception of horismos, “horizon,” itself derived from horos (boundary): “that from which something begins its presencing.”54 For a place is indeed an active source of presencing: within its close embrace, things get located and begin to happen.
In view of place’s considerable boundarylikeness,55 one move seems clearly indicated: if Aristotle’s definition of place is to avoid leaking like a sieve, that is, like a vessel that has been moved one time too many, we ought to substitute “boundary” (horos) for “limit” (peras) in its formulation. Then the definition might hold water once again, and in so doing it would also put point itself finally in its proper place. But what is this place?
V
Now in imagined and perceived objects the very points that are in the line limit it, but in the region of immaterial forms the partless idea of the point has prior existence. . . . Thus it is at once unlimited and limited—in its own forthgoing unlimited, but limited by virtue of its participation in its limitlike cause.
—Proclus, A Commentary on the First Book of Euclid’s Elements
A point is a nexus of actual entities with a certain “form.”
—Alfred North Whitehead, Process and Reality
Suppose no feeling but that of a single point ever to be awakened. Could that possibly be the feeling of any special whereness or thereness? Certainly not. . . . Each point, so far as it is placed, [exists] . . . only by virtue of what it is not, namely, by virtue of another point.
—William James, Principles of Psychology
The comparison of point and place has more of a point than the skeptical reader might imagine. For one thing, point is at stake in any cosmogenesis of place that is of recognizably geometric inspiration, whether by way of conspicuous presence (as in Pythagorean accounts and in Euclid as read by certain Neoplatonists) or because of an equally conspicuous omission (as in Plato’s case). For another thing, points are invoked in concrete descriptions of place that lack any cosmological or geometrical overtones: as in such descriptive phrases as “meeting point,” “the point of the peninsula,” “the point of overlap [between two adjacent areas],” or “the point of no return.” Indeed, Aristotle himself, ignoring his own precautions, sometimes adverts to point-language in describing movement between places.
As it is with the point, then, so it is with the moving thing, by which we become acquainted with change and the before and the after in it. The moving thing is, in respect of what makes it what it is, the same (as the point is, so is a stone or something else of that sort); but in definition it is different . . . [i.e.,] different by being in different places.56
That the point is a unit by which place, and still other regions of space, can be conceived and even experienced has been of perennial interest. If Plato regarded the point as a “geometrical fiction”57 contra the Pythagoreans, Aristotle reinstated the abiding importance of the point, considering it to be as indispensable in geometry as it is problematic in physics. By the time of Proclus (A.D. 410–485), the point had assumed an almost irresistible allure that has continued to capture the attention of thinkers as diverse as Descartes and Hegel, Leibniz and Bergson, Whitehead and Derrida—each of whom devotes himself to the fate of the point in space and time.
In this tradition of continuing attention to the topic, Proclus represents something of a watershed. For him, the point is both cosmically and geometrically generative. It is this not as something aggressively imposed on an underlying matrix by some theurgic power but as itself a procreative principle. As Proclus says, “Although its being is determined by the Limit, [the point] secretly contains the potentiality of the Unlimited, by virtue of which it generates all intervals; and the procession of all the intervals ‘still’ does not exhaust its infinite capacity.”58 “Intervals” include lines and distances of all kinds (i.e., the very basis of many modern conceptions of place as metrically determinate), and their dependence on the point represents a reversal of the Platonic view that a point is nothing but the beginning of a line.59 No wonder that Proclus is able to proclaim, “We have expanded somewhat largely on these matters in order to show that points, and limits in general, have power in the cosmos and that they have the premier rank in the All.”60
On this expansive view, points come to replace place itself as “the first of all things.” Just as Aristotle reacts against Plato by espousing an immanent physicalism in which place and not space is paramount, so Proclus proposes a view of the created universe in which the point and not place is the most effective immanent generative principle. Indeed, we witness in Proclus the first appearance of a distinctive pointillism of place wherein points, regarded as cosmically primary, give rise to places as if by natural extension. For Proclus, the question is not whether there are such things as points (as Plato wondered), or whether points themselves are places or placelike (as Aristotle ponders), or whether points are superimposed on indifferent space (as Descartes will speculate), but instead how points generate lines, surfaces, solids, and ultimately places themselves by virtue of producing “all intervals.”
Where Aristotle is concerned to put point in (its) place—to confine it to a status as a limit-concept in a geometry that reflects, rather than informs, the physical world—Proclus insists on the place-making power of the point, a power that exceeds what Aristotle calls “the power of place [itself]” (Physics 208b34). That which has (much less is) strictly no place at all in Aristotelian physics becomes a cosmogenetic force that “unifies all things that are divided,”61 including all places and regions in the known universe. The point becomes a first principle, an archē, in the process of cosmic procreation.
Echoes of such a principle still resonate in Hegel’s philosophy of nature, where the movement of space (conceived as Being-outside-itself), from an initial situation of sheer undifferentiation into a first moment of determinacy, is effected precisely by the point.
The difference of space is, however, essentially a determinate, qualitative difference. As such [the point] is first the negation of space itself [insofar as] this is immediate, differenceless self-externality.62
Derrida comments tellingly on this passage.
The point is the space that does not take up space, the place that does not take place; it suppresses and replaces the place, it takes the place of the space that it negates and conserves. It spatially negates space. It is the first determination of space.63
For Hegel, the point is determinative from within the spatial world itself and is not the result of any supervening action on the part of a separate deity. It is determinative of place in particular by its internal negation of sheer space; thus it precedes place, which comes after space and time in the Hegelian dialectic.64 Point “replaces” place by its very position before place in the final scheme of things; it is thus pre-positional, not by being put over place but by being posited as the abstract moment that gives rise to place—to begin with.
We might contrast this Proclean-Hegelian vision of immanent point-power with the very different vision of Marduk, whose lethal pointed arrows “split the belly, pierced the gut, and cut the womb” of Tiamat. I have argued that Tiamat, whose writhing body is “too deep for us to fathom,” is the mythic progenitor of the Receptacle. As such, she is deeply threatening to the world-ordering interests of Marduk, who must subdue her from without by martial maneuvers and by the pointed power of arrows. Only by the application of such power can the Tiamatian ur-place become a well-ordered place-world with determinate locales.65 In this protogeometric act of creation—which we have seen to be remarkably analogous to the actions of the Demiurge in the Timaeus—we witness the point as an alien power, as something that ravages space, indeed annihilates it from a position of aggressive exteriority. Instead of respecting and preserving space—instead of taking “the place of the space that it negates and conserves [i.e., by an act of Aufhebung]”—it is as if Tiamatian space is too dangerous to live with, much less to conserve: thus it must be eliminated. This is accomplished by a sharp-tipped point that draws away the vital force of space qua primal Place. The dot destroys the matrix—in poignant contrast with the composite dot-matrix solutions proposed by Aristotle (who promotes place over point) and by Proclus (who makes point primary within place itself).
It is instructive to learn that Aristotle is the last of the early Greek thinkers who consistently used the word stigmē for “point.” Stigmē connotes a puncturing point,66 a point that includes the arrows of Marduk, punctuation points, and the insistent isolation of separated geometric points. Becalming the ambition and hostility of the stigmatic point—embedding stigmatism within the ambience of place—Aristotle inaugurates an astigmatic era in which a more irenic relation between dot and matrix will become possible.67
VI
Yet how can there be a motion of void or a place for void? That into which void moves comes to be void of void.
—Physics 217a3–5
Aristotle repeatedly assimilates theories of void to theories of place.
—Edward Hussey, Aristotle’s Physics, Books III and IV
It is a striking structural fact that Aristotle, having disposed of infinity in the opening chapters of book 4 of the Physics, treats the void in between place and time in the same book. Void, then, exists between place and time: as if to say that to get out of place is to get into the void and to get into time is to get out of the void. Time is therefore one way of avoiding, indeed of devoiding the void—emptying its emptiness by introducing measured cadences and reliable rhythms into its abyss. These cadences and rhythms are dependent on motions and magnitudes that belong in turn to place.68 Thus to go from place to void to time is in the end to return to place; it is to travel in a topoteleological trajectory that keeps coming back to place even as it departs from it.
In view of this circular topology it is hardly surprising that Aristotle argues for the indissociability of place and void.69 He does so at two levels. First, at the level of endoxa, or common belief, “those who say there is a void suppose it to be a kind of place” (213a16). They do so because of a seemingly commonsensical (but in fact paralogical) line of reasoning: “People think that what is, is body, and that every body is in a place, and that void is place in which there is no body; so that, if anywhere there is no body, then there is nothing there” (213b32–34). Second, at the level of conceptual analysis, Aristotle takes over this paralogic of ordinary belief for his own purposes. He assumes the possible truth of this belief in order to discern its implications for place: void, were it to exist, would be placelike. As placelike, however, it cannot exist as “separated,” that is, in its own right: for a place is always inseparable from its occupant. And yet an unseparated void—a void dependent on its contents—is no void at all. In short, to the extent that void is placelike, it cannot be a true void; conversely, insofar as a place is vacuous, it cannot be a true place. Referring to his own discussion of place in the immediately preceding chapters of the Physics, Aristotle concludes that “since an analysis of place has been made, and void, if it is, must be place deprived of body, and [as] it has been stated in what sense there is and is not place, it is manifest that in this sense there is no void” (214a16–18). Even when we regard void merely as “extension between bodies”—that is, as the interval (diastēma) posited by the Atomists—we find that it remains placelike, for such an extension is a place of possible occupancy by bodies.70
Consider the leading argument for the void as set forth by the Atomists: the void is “responsible for” change in that it provides the setting for all change (including motion), being “that in which change occurs.”71 But, given that the void is nondifferentially structured, it cannot explain the inherent directedness or the differential speed of natural motion—indeed, it cannot explain why anything moves to begin with—and its invocation in physics is otiose: “For what then will the void be responsible? It is thought to be responsible for change in respect of place, but for this it is not.”72 Place, on the other hand, explains any change—including velocity and direction—that involves locomotion. Thanks to its stationariness, it also explains rest. While the void renders motion as well as rest incoherent, for Aristotle place qua container accounts for both of these phenomena economically and effectively.73 Similarly, if we consider condensation or rarefaction, or the displacement of substances, the void will explain nothing: worse, if it were in fact to exist, it would render such changes senseless.74
For all of these reasons, the void as a concept (and not merely as a belief) is regarded as dispensable by Aristotle. Fascinating as its idea may be and compelling to the Atomists as it doubtless was, it is finally a gratuitous fiction—a ghostly double of that which is not gratuitous at all, namely, place. Place suffices to account for all that the vaunted void purports to illuminate. As Edward Hussey comments, “The implication of the argument is that a void which is not an explanatory factor of anything is pointless and therefore cannot exist.”75
Pointless as well is any effort to associate the point with the void—an effort stemming from the Pythagorean association between the point and the Unlimited.76 As Aristotle says brusquely, “It is absurd, if a point is to be void; for [void] must be [place] in which there is an extension [within] tangible body” (214a4–6). Just as we can neither imagine nor think a void that is unplacelike, so we cannot imagine or think point as void—or, for that matter, void as point. Therefore, not only does Aristotle deconstruct the point as a candidate for place, but he ends by eliminating both point and void as competitors with place in the determination of location. In such determination, place takes first place; and in this privileged position it takes care of itself, needing neither the point nor the void as explanation or support. If everything is fully placed—if nothing, at least nothing sensible, is without a place of its own—then no void need exist, actually or potentially, and things do not require points to specify their status.77 Otherwise put: to be a physical body is to occupy a determinate topos, a place-pocket as it were, that is filled by this very body and that (at another time) can be reoccupied by another body of the same dimensions. To Freud’s dictum that “the finding of an object is in fact a refinding of it,” we can add Aristotle’s rule that every implacement is in effect a reimplacement.78 And if everything in the physical world is not only placed but also displaceable and replaceable, then we have to do with a world in plenary session—a lococentric world-whole. This is a world in which points and the void are not so much absent (particular points and discrete vacua may still occur) as superfluous. As Bergson says, “All is full in Aristotle’s world.”79
Aristotle conceives this place-world not by expanding but by restricting his field of inquiry. In contrast with the logical and rhetorical excesses of Zeno, Parmenides, and Gorgias—each of whom extols the ubiquity of place without ever telling us anything specific about place itself—Aristotle’s nuanced descriptions attempt to say just what place is and how it differs from other constituents of the physical world. And in contrast with Plato, Aristotle confines his efforts to describing the exact characteristics of just one of the three sorts of spatial entities distinguished in the Timaeus. The Physics concerns itself only with the most particular such entity, that is, topos, while general regions and chōra are made marginal. The amplitude of the Receptacle gives way to the stringency of the container; and within place-as-container, concrete issues bearing on boundary and limit, line and surface, point and void, are addressed in scrupulous detail.
VII
It is obvious that one has to grant priority to place.
—Archytas
This is not to claim that Aristotle’s idea of place is without complications and difficulties. To begin with, there is the fact that he changed his model of place in a major way in the period between the early composition of the Categories—where place qua chōra is construed as equivalent to empty “interval” (diastēma)—and the text of the Physics, where this very model is decisively rejected.80 More important, there are at least four serious problems in Aristotle’s mature view of place as the immobile inner surface of a container, (1) By its emphasis on surface (epiphaneia), this view is confined to a two-dimensional model of place, despite the fact that place itself is manifestly three-dimensional inasmuch as it surrounds solid objects. (In comparison, Aristotle’s fascination with the point can be taken as an incursion into one-dimensional or even zero-dimensional space and, for all its interest, is foredoomed as a fitting model for volumetric containment.) (2) There is an unresolved tension between the localism of the container model—which points to physical things as “place tight” in their immediate environs—and the globalism implicit in certain of the Stagirite’s descriptions of the physical universe.81 Even if it is true that “everything is in the world” (212b17) and that there is nothing outside the world—no external void—the world-whole encompasses any particular place of any given changeable body and must be a global Place for that place-cum-body. (That the total world is a Place follows from the fact that it contains and surrounds all more particular places within it.) A place is not only a place for a body but a place in the larger world-Place.82 In addition, only such a cosmic Place can make sense of Aristotle’s insistence on the irreducibility of the up/down dimension. Construed as cosmic, this dimension signifies that the earth is at the center of the universe and the heavens at its outer limit.83 But to make this latter claim—to say that the earth is always and only at the center of the universe—is to call for a sense of space as absolute or global that is not allowed, strictly speaking, by the container model in its constrictive, localizing character. (3) The full determination of the “first unchangeable limit of that which surrounds” remains moot. In the case of the floating vessel, is this limit the immediately surrounding water regarded as an ideal perimeter (yet as flowing water, it is constantly changing, with the result that the place of a stationary boat will be continually changing), or is it the river’s bed-and-banks or even the river itself as a whole (in both of these last cases, two boats equidistant from two banks but heading in opposite directions will occupy the same place)?84 This seemingly trivial but in fact momentous question was to engage over two thousand years of debate in Western philosophy: it is still a live issue for Descartes in the seventeenth century A.D. (4) Finally, we must inquire as to what it means to contain something. Is it merely a matter of “holding,” as is implied by the verb periechein—in which case, the emphasis is on the act of delimitation, that is, of surrounding? Or is it a question of establishing a boundary—which stresses the surrounder? Where the former interpretation directs us to what is surrounded, the latter points to what is other than, and beyond, the surrounded object (and perhaps even beyond the surrounder itself). How are we to choose between these two interpretations—one of which stresses the container as limit, the other the container as boundary? And if we cannot choose effectively, are we not confronted with an essentially undecidable phenomenon?
Despite these perplexities and still others,85 we need to retain what is most original—and most lasting—in Aristotle’s mature vision of place. This is the acknowledgment of place as a unique and nonreducible feature of the physical world, something with its own inherent powers, a pre–metric phenomenon (thus both historically and conceptually pre-Euclidean in its specification), and above all something that reflects the situation of being in, and moving between, places. It is just this accommodating and yet polyvalent model of place that became lost in Euclidean and post-Euclidean theories of strictly measurable space.86 Aristotle was able to resist this mensurational view even as he was drawn to it early in his career: he came to realize that, regarded as extension or interval, place becomes merely an item of exact quantitative determination. For what matters most is not the measurement of objects in empty space but the presence of sensible things in their appropriate and fitting places.
In effecting this tour de force—whereby a focused, forceful description yields what may well be the most astute assessment of place to be found in Western philosophy—Aristotle proceeds with a phenomenologist’s deft sensibilities.87 This is most evident in his resolute refusal to restrict the phenomenon of place to atomistic or formal properties. Just as he rejects Plato’s attempt to regularize sensible bodies by the imposition of elementary geometric figures (he takes such bodies to be straightforwardly “what is extended in three dimensions”),88 so he approaches place on its own terms. His preoccupation with the propriety of place is evident in his telling remark that “each thing moves to its own place” (Physics 212b29), that is to say, to its proper natural place. That each such place is encompassed by the common place of the firmament—and that this latter is conceived as having constant circular curvature—does not mean that Aristotle has “spatialized” place in the manner of the spatialization of time decried by Bergson and Heidegger alike.89 Problematic as we have just seen it to be, the very nesting of special topoi within an overarching Topos has the virtue of conceiving the cosmos not as an empty and endless Space but as an embracing Place, filled to the brim with snugly fitting proper places. The firmament that encircles the world-whole is at once a paradigm for all lesser places and filled with these very same places. Everything, or almost everything, is in place. To be an existing sensible thing is never not to be in some place. Place prevails. Archytas stands vindicated.
Aristotle surpasses Archytas, however, in his eagerness to show just how “it is obvious that one has to grant priority to place” and just why “it is the first of all things.”90 He does so by demonstrating that place, beyond providing mere position, gives bountiful aegis—active protective support—to what it locates. Defined as a bounding container, place in Aristotle’s sure hands takes on a quite dynamic role in the determination of the physical universe. Place indeed “has some power.” It has the power to make things be somewhere and to hold and guard them once they are there. Without place, things would not only fail to be located; they would not even be things: they would have no place to be the things they are. The loss would be ontological and not only cosmological: it would be a loss in a kind of being and not merely in the number of beings that exists.
