Zeno's Arrow and the Possibility of Motion

In the previous post I dealt with the famous paradox called dichotomy. This post shall deal with another paradox of Zeno of Elea. This one is called the arrow, and it can be described as follows:

The Arrow: During any interval of time a moving arrow occupies a fixed location in space at any instant. Consequently, it is fixed at all instants. Then, the arrow is fixed throughout the interval. That is, it does not move at all.

The most obvious flaw in the above argument is that while the arrow may occupy any one location at any instant and thus remain fixed at that instant, one should not infer that the arrow therefore occupies the same location in space at all instants during the interval. The arrow, while remaining fixed at any instant, can be at different locations at different instants. Thus the arrow would have changed its location, after all.

Well, the above does not satisfy all. One may continue to ask: when does an arrow, that is fixed at every instant, move? The effort to resolve the paradox leads one to defining motion, which can be done as follows:

If an object is at two distinct locations at two distinct instants then the object is said to have moved between those instants. Thus the motion of a body is defined over an interval of time. (Please note that occupying same location at two distinct instants does not negate motion between the two instants). To talk about motion at an instant is meaningless. Of necessity (which arises from the manner in which instants are assumed to be defined in Zeno's paradox) the arrow will be fixed at each instant. And yet, it may move over an interval.

The paradox is thus resolved by noting (a) that a fixed location at an instant does not imply the same fixed location at all instants, and (b) that the definition of motion involves comparison of location at distinct instants.

In dealing with this paradox we have assumed that an interval of time is made up of instants and also that an object has a fixed location of the same shape and size as the object itself at any given instant. Both the assumptions can be doubted upon. Thus what I have tried to do with this posting is to show just one very simple way of showing why this paradox is not really a paradox.

If instants are defined as Russell does (the class of all mutually overlapping events such that no event outside the class overlaps every member of the class), what consequences does it have on Zeno's Arrow.

The movement of the arrow from one point to another can also be regarded as an event. This we can do because it satisfies the axioms employed in defining instant. Name this event "E". Also, let "X" be an instant at which E exists, i.e. of which E is a member.

Consider the following options:
1. At X, the arrow has a fixed location in space.
2. At X, the arrow has an indeterminate exact position but its location in space can be fixed within a boundary.
3. At X, the arrow has a location such that occupation of a certain part of the location by the arrow is earlier than the occupation of a certain different part.

None of the above leads to any inconsistency in so far as that it rules out motion.

Can any Continuous Process be physically realized ?

For an apple to grow full, it must first grow half. To grow half, it must first grow 1/4. To grow 1/4, it must first grow 1/8. And the series can be carried on ad infinitum. This series shows that an apple must go through infinite steps to grow full. Can we then conclude, on account of the infinite in our argument, that an apple can never grow into a full ruddy fruit?

The above is clearly an alternate statement of Zeno's paradox. The paradox as stated by Zeno or me or any other version is equivalent to stating the following:
        "No process is possible if it can be broken into an infinite series of physically realizable states."

Basically, we are grappling with infinity. The problem arises from the intuitive idea that an infinite number of steps is not physically realizable, at least not in a finite duration.

This raises some important questions. First, whether the above is a problem at all. That is, whether the splitting of a finite magnitude of something into an infinite number of magnitudes (most of which are infinitesimal) should be regarded as an obstruction to the realization of the whole magnitude if the process of such realization entails realization of each of the component magnitudes.

Secondly, if the realization of infinite states indeed poses trouble then whether each of the infinite states are physically realizable at all. That is, whether the attribute whose magnitude is being split/divided is continuous or discrete or something?

In other words, to solve the problem stated above one must prove either of the following two:

    1.      that an infinite number of states in a process of finite duration poses no problem at all; or

    2.      that, physically, no process of finite duration can go through an infinite number of states. (i.e. processes occur in a discrete fashion)

Before dealing with this problem properly, it would be useful to develop a better understanding of the more elementary concepts. This we shall pursue hereafter.

Instants - points in time

The best definition of instant that I have encountered so far is given by Bertrand Russell. A good analysis of this subject can be found in his works - Human Knowledge and The Analysis of Matter.

Since Russell has already dealt with this topic in great detail and with a mathematical precision, I shall not describe the whole idea. However, I mention it because this is the definition of instant that I shall use while pursuing this blog.

Roughly speaking, an instant is defined as a class of events such that any two events from this class overlap and also, no event outside this class overlaps with every event in this class. The great advantage that this definition has is that it assumes that we already know what events are and then describes instants in a manner that removes all the troubles of questions like whether an instant is akin to a geometrical point or otherwise. Despite avoiding such questions the definition satisfies what one may intuitively demand from an instant. Plus, events are something that are allowed to follow, preceed or overlap each other. These relations, as one can note, arise out of what we see in this universe and, such notion of events is closer to commonsense.

The books referred above can be read for more precise, mathematical and, of course, beautiful discussion on this topic.

The point of repeating these things here is that this will be the definition that I will use in this blog.

The Topology of Time

The Big Bang Theory is one of the leading theories in contemporary cosmology. According to this theory the universe had a beginning. Many philosophers and scientists regard the Big Bang as also the beginning of time. Time, independent of the universe, is denied and hence the above conclusion. However, there are others who think that an infinite stretch of time existed before "now" even if the universe had a beginning. There are yet other philosophers who think they can prove that time must have a beginning.

There are other questions too regarding topology of time that continue to vex the philosophers and the scientists alike. These concern the structure of time. Some of these are here:

1. if time is one-dimensional, does it branch?
2. are all points of time strictly ordered?
3. whether the series of time-points is discrete, compact (like rational numbers) or continuous (like real numbers)?
4. can there be loops, i.e. closed curves in time?

All these questions have been framed in terms of time alone, and not in terms of spacetime. That does not mean that general relativity and other advances in physics have not been considered. If we restrict our attention to just one observer, the questions are both meaningful and significant.

Clock Must Be Periodic

The clock, or the standard process to be used for measurement of time must be either a periodic process or a predictable process. Why? Because only a process of one of these kinds will make it possible to assign a formula or an algorithm to be applied to the states of the process.

What if the clock were to be a non-periodic but a predictive process? How will the clock be used to assign the numbers to various states of this process in a manner that comparison of intervals makes some sense? Suppose it can be predicted that the states s1, s2, s3, s4, etc. will occur in that order as the process proceeds. While these states can clearly be used to assign order to any other set of events, assigning numbers arbitrarily to these states does not answer the problem of comparing time-intervals that do not completely overlap.

On the other hand, if a periodic process is adopted as the clock (while restricting the choice to such processes only that appeal to the observer as having limited/none human, random and arbitrary interferences or influences) the assignment of numbers can be done by mapping an arithmetic progression to those successive states of the process any two consecutive of which are separated by one time-period. With this process and mapping, the comparison of non-overlapping or partially-overlapping time-intervals can be done. One must, however, note that this system is as arbitrary as the system if the process adopted as the clock were predictable but not periodic.

Does this mean that any periodic process is a sound candidate for a clock? The answer is nearly yes. Nearly because the observer restricts his choice to only such processes that are seemingly independent of human/random/arbitrary influences. However, the knowledge of these influences changes as the understanding of the processes, their working, and effects of various factors on them changes.

What is amazing in all this is that so many periodic processes in this universe are so consistent with each other, thus lending the possibility of using the time co-ordinate with so much success in framing useful laws and equations of physics, etc. It is this consistency which finally justifies a particular choice of clock, even though the choice seems to be almost arbitrary theoretically (as seen from the discussion above).