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
