This Point in Time

I

What is the present? The most obvious and natural answer to that question may be provided by a simple graphical representation: a point on a line. Time-lines like that used by the Hammer Museum for its second Biennial are common enough to suggest that they involve an inevitable metaphor, time as a linear progression from point to point. In principle, a point denotes position without regard to size, and therefore the increments into which a line is divided may stand for days, months, years, or even centuries. Though it is not uncommon to use the term present even for the very largest of these units, it is probably most natural to think of points as designating the smallest parts of time. The very concept of punctuality suggests that we think of temporal precision in terms of such points, measured as finely as possible. An aural equivalent is provided by the ticking of a clock, which can sound, under certain circumstances, like time itself steadily dripping away, bit by bit.

Expressing time in terms of such bits is a convenience and maybe even a necessity whenever it comes to synchronizing things. It is a habit that has also acquired a certain authority over the years. Kant, for one, thought that our inner sense of time itself has no shape, and thus we must resort to analogies, the most basic of which is the line, along which points would represent the fundamental relation of successiveness. Aristotle, whose analysis of time in the Physics is the basis of most discussions to follow, frequently uses the phrase "point of time" when he means to speak of the smallest division of it. The representation of time in terms of two-dimensional geometry is a feature of some of the most basic tools of modern science, including the Cartesian coordinate system, whenever time serves as one of the axes. In Newton's physics, time is divided into "temporal places," indivisible moments that correspond to geometric points, except that they are spatially invariable, as spatial points are temporally invariable.

While geometric points might be useful in the representation of time in the abstract, they pose certain problems when it comes to time as it is experienced by human beings. In its purest form, the temporal point is a time period without any time in it, and this does not seem to afford much scope for experience. Even in Newton's day, philosophers differed as to whether time could be sensibly divided into moments without any extension. Locke divided time into discrete, uniform instants arrayed, at least in principle, along a single line. These instants were thus points in every respect except that they could be understood to have some dimension. Hume, however, argued that in space and time the least possible increments have no magnitude, so that the briefest moments of time are point-like in being unextended and nondimensional. These are recognized in experience by their successive relation to one another, not by virtue of their own temporal breadth. This argument about points and time-lines has never really died down. In the 20 century, Henri Bergson argued on one side that the geometric metaphor is fatal to any real understanding of time, while Gaston Bachelard argued on the other that "duration is made up of durationless instants, just as a straight line is made up of dimensionless points."

The notion of "durationless instants" does have a few logical arguments in its favor. Hume's contention that the smallest imaginable increments of anything cannot possibly have any magnitude is one of these. Aristotle used another such argument when he pointed out that a now with some dimension to it will be divisible into parts, some of which will be past in relation to others, so that the present will turn out to have another time-line within it and not be the present at all. One of the oldest and still most influential definitions of the present, that of Saint Augustine, begins with this inevitable and yet counterintuitive idea of a moment in time that takes up no time: "In fact, the only time that can be called present is an instant, if we can conceive of such, that cannot be divided into even the most minute fractions, and a point of time as small as this passes so rapidly from the future to the past that its duration is without length." By Augustine's reasoning, the present moment must be indivisible and dimensionless, since if it had dimension it would then contain within it past and future and no longer remain simply the present.

Though Augustine's reasoning seems sound, it has left behind it a number of logical problems. Kant was to insist rather stoutly that "any part of time is a time," and thus it would seem that any moment must itself occupy some time. There is an elementary inconsistency, in other words, in dividing time up into instants that themselves take up no time. How could any series of such temporal zeroes add up to a positive amount? In any case, how could anything happen in a dimensionless instant? What sort of experience could exist in a temporal space so small it can include no change? As Aristotle had insisted long before, "if there is in the time-continuum a time so small as to be absolutely imperceptible, then it is clear that a person would, during such a time, be unaware of his own existence, as well as of his seeing and perceiving." Aristotle put a great deal of effort into redefining the now so as to avoid these logical absurdities. He notes that a line is not really made up of points but rather is infinitely divisible by them. Points, that is to say, are not uniform stations situated along a line, like beads on a string. Between any two points on a line there are as many more points as anyone might care to mark. The line, in other words, is a continuum and, as such, is infinitely divisible. The point on the time-line we call the present is, therefore, not a part of time but a limit to it, more like a cut or division than a discrete bit. As such, it is itself indivisible and dimensionless, and so it does not contain any time within it. An important benefit of this redefinition for Aristotle is that it helps to refute the reasoning behind the famous paradoxes of Zeno, some of which derive their power from a conception of time as composed of indivisible parts. Zeno contends that a flying arrow must always be in one of these parts and, while in it, must be at rest. Therefore, the arrow can never hit its target, since it must always be in one of these parts. Aristotle responds that "time is not composed of indivisible nows any more than any other magnitude is composed of indivisibles." Time is not composed of nows but is a continuous magnitude divided in two by a particular now.

Though this solution may have dispelled some of the intellectual tension that had built up around the paradoxes of Zeno, it also created some problems of its own. If we agree with Aristotle that time is not composed of nows, we may then wonder exactly what it is composed of. And if we agree that the...