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
