[Neri]

Time presumes the existence of separate objects (the reality of multiplicity), for time is precisely that condition that allows for differing distances between any two objects.


This does not mean that nothing lies between separate objects; for, if that were so, they would be so densely contiguous that the whole of reality would consist of a single impenetrable object.


The “something” that lies between objects is called space. We notice that when space lies between objects and ourselves, the objects may be closer to us or further from us. This can only mean that space may be quantified. This is what we have called called distance.


The idea that space may be quantified necessarily presumes that space consists of intervals that may be multiplied or divided. There does not seem to be anything like a natural interval. Intervals are quite arbitrary. Yet, the fact that one interval may easily be converted another interval makes us confident that this method determines real distance.

Intervals necessarily involves the idea of points. The latter are said to be those unique locations in space that determine intervals. Since arbitrary intervals are so determined, we conclude that the distance between any two points can be perfectly determined by arbitrary intervals.


It is apparent that when two objects are traveling towards us from the same place, one may reach us before the other. Therefore, we say that one is moving faster than the other. We find that to determine exactly how much faster it travels, we need intervals of time in addition to intervals of space. This involves the use of points in time. Just as points in space have no extent, points in time are conceived have no duration.

The temporal locus about which we seem most concerned is the one called Now or “the present.” However, there is something perplexing about Now when taken as a temporal point. If the duration of Now is naught, how can we possibly experience it? We are forced to admit that Now must have some minimal duration in order to be sensible. Certainly, to be experienced, any sensation must have some temporal extent. But, exactly what is the duration of Now? Is it a constant sewn into the fabric of reality or is it a psychological variable? If the latter, the perfect measurement of time would seem to be disallowed, since all points in time hinge on the present as a perfect locus or at least as an invariable interval. Nonetheless, we usually dismiss this consideration as pedagogic, for the idea of the point has served us so well that we are reluctant to scrutinize it too closely.


Yet, there is an aspect of time that is not so easily dismissed. Any spatial interval is determined by two points. Either may be called “the beginning” or “the end.” It makes absolutely no difference. Temporal intervals are also determined by two points, but there is a difference. For any temporal interval, only one point may be called “the beginning” and one “the end.” This condition has been referred to as the irreversibility of time or the arrow of time. It is the unique feature of time that makes it quite different from space.


Everyone agrees that events in the past are certain (fully determined). The view of classical mechanics is that the future is just as certain. This approach seems to attach no particular significance to Now; for Now, according to the determinist paradigm, is just another point in the inevitable progress of happening. Quantum mechanics, on the other hand, views the future as uncertain. This attaches great significance to Now, for Now then becomes a kind of crossroads of happening where various potentials become a particular actuality. This idea is essentially equivalent to the common-sense notion that the future is “not yet written.”
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The fact that spatial and temporal intervals require perfect loci (points) to be fully determined has necessitated the conclusion that both time and distance are infinitely divisible. Unfortunately--for the reasons set forth by Zeno, Hume and others—such a conclusion renders all action impossible as a matter of logic. This regrettable state of affairs allows only one of two possibilities:


(1) All action is illusory; or


(2) There are irreducible (and therefore fully determined) natural intervals (atoms) of time and space which together account for action.

Possibility (1) is unacceptable, because it would mean that time is unreal. Without real time and space, all human experience would be utterly delusional, for nothing could possibly happen to excite the senses. Sensations would be useless to warn us of danger, simply because danger could not possibly exist in the first place. Indeed, happening of any sort would be proscribed. Without time, we could not have been born, nor could we proceed from being young to being old or from being alive to being dead. Further, since thought itself is a time-dependent process, the absence of time would forbid all thought. The very fact that we are considering this problem, therefore, is conclusive proof of the existence of time and action; for our thoughts are themselves a kind of action.

Possibility (2) solves Zeno’s paradox. Let us take the simplest possible case and say that the course raced by Achilles and the Tortoise is 2d in length, where d is an irreducible atom of space. Let t be an irreducible atom of time. Say that Achilles is travelling at the speed of 2d per t and the tortoise at d per t. Achilles can never attain the distance d, for that would require a time less than t--and t by definition is irreducible. No matter how much longer we make the course, it will always be some multiple of d. If Achilles and the Tortoise continue to race at the same speeds past the finish line, Achilles will progressively widen the gap yet will always be running at twice the speed of the tortoise. Further, if the given speeds of Achilles and the tortoise are multiplied by any number, Achilles will always travel twice the speed of the tortoise, extending his actual lead according to the multiples involved.

If Achilles and the tortoise had reached their respective speeds at the start line (which would ordinarily not be the case), it would be impossible for them to be at the start at the same time, since, they would be separated by at least the distance d. If they were motionless with respect to the d of the start, they would have to accelerate to their respective speeds in one t. The tortoise would “make a jump” to d in one t and Achilles would “make a jump” to 2d in the same time.

If the tortoise is given a lead, it can only be a lead of d, since 2d would place it at the finish. That is, at the expiration of t, both the tortoise and Achilles will be at 2d, and neither will have won the race. It will be noted again that Achilles never reaches d, for he would have to do so in less than t; and this is impossible. If the contestants continue racing at their respective speeds past the finish, at 2t Achilles will be at 4d and the tortoise at 3d. At 3t, Achilles will be at 6d and the tortoise at 4t. This process will continue with Achilles extending his lead, as both he and the tortoise maintain their respective speeds.

The paradox of the flight of the arrow may be resolved in a similar manner. Let us say the arrow is one d in length and is motionless with respect to all other d’s but thereafter is propelled by a bow and attains the speed of 2d per t. This means that each d of its length traverses 2d in a single irreducible t. So that the arrow is never motionless at any time during its flight even at the least possible “instant of time” (t). Other such paradoxes may be resolved in the same way.

It will be noticed that the above describes a paradigm of absolute motion, since the atoms of space and time would be a general frame of reference for action of any sort and would allow absolute simultaneity as well as the addition and subtraction of velocities. However, it does not explain the constancy of the speed of light as judged from inertial frames of reference, nor does it explain why the effects of accelerated, curvilinear and rotational motion are the same as those of a gravitational field. Relativity theory explains these things by denying absolute motion and simultaneity and by maintaining that space and spatial relations are only observer-dependent manifestations of the structure of space-time. Yet relativity theory does not resolve the fundamental paradoxes of motion. Unfortunately, time does not so easily give up her secrets.