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
