Points, boundaries and the horizon around the presence of place, in Edward Casey's The Fate of Place
We can grant that a point is a "limit of localization" - precisely the lower limit, beneath which we cannot (and need not) go. For limit, like shape, 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." 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."
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." Like place itself, a boundary "shuts in and closes off something from what lies around it" - 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. [p. 154]" For a place is indeed an active source of presencing: within its close embrace, things get located and begin to happen.