Zeno's Arrow and Modern Physics[1]

Iyyun – 1997

In this article I wish to distinguish between two modes of relating to dynamic processes, and to discuss several implications of this distinction that pertain to areas entirely different from one another. At the outset I shall discuss Zeno of Elea's paradox of the arrow in flight, which nicely illustrates the basic distinction. Following that, I shall try to derive certain foundations of Heisenberg's uncertainty principle in quantum physics; we shall then continue to discuss the problem of the interpretation of time in Einstein's theory of relativity, and consider several implications of this distinction in other areas of thought.

A. The Paradox of the Arrow in Flight

The natural point of departure for our inquiry lies in Elea in ancient Greece, where Zeno attempts to undermine the concept of motion by means of several paradoxes.[2] One of those paradoxes, called the paradox of the arrow in flight, may be formulated as follows:

(1) "The flying arrow, in one indivisible moment of its flight, both flies and does not fly at once. And both claims are true: it flies, for were it not flying at every moment, its motion would not take place. And it does not fly, for within one indivisible and unextended moment of time it cannot perform any motion. Say, then: the arrow both flies and does not fly at once".[3] Samuel Hugo Bergman, who cites this formulation as one of the challenges to the law of non-contradiction, goes on to say that the same applies to every continuous process, such as the freezing of water and the like.

The paradox can be formulated somewhat differently: (2) At every indivisible moment the arrow is located at a definite place in space, which may differ from its previous locations. If so, at what moment of time does the arrow change its location? In other words, when does it move?

These two formulations appear, at first glance, to be equivalent. Both assume the proposition that if a body is stationary at a certain moment then it cannot also move at that same moment, a proposition based on the law of non-contradiction (though the second formulation does so implicitly). Later we shall see that these are not entirely equivalent formulations.

There have been attempts to solve the problem by using concepts drawn from infinitesimal calculus and the problematic nature of the concept of continuity. In simple terms, this may be put as follows: a continuous line is not composed of points but of infinitesimals as small as one wishes. The time axis is a continuous line. Therefore it is meaningless to discuss the state of a body at a point of time that is indivisible and unextended. Bergman's formulation there is very similar: "It is impossible to divide the continuum into moments that are indivisible." This claim, like the discussion that will follow in the continuation of this article, is formulated in the context of an intuitive approach to the concept of continuity and infinitesimal calculus.[4] In modern mathematical theory there are also descriptions of the continuum as a set of points satisfying certain properties (the set-theoretic approach), and in such an approach the solution to Zeno's paradox appears entirely implausible. This distinction between the continuous and the discrete has proved very fruitful in the mathematical description of processes by means of infinitesimal calculus. Still, it seems that the use of this distinction to solve the paradox does not accord with simple intuition. That intuition feels that a line can be formed from discrete points, and likewise that one can certainly speak of a discrete point in time and of the state of a body at such a point. The assertion underlying infinitesimal calculus merely helps us avoid technical difficulties in the mathematical description of continuity. That is, the above assertion, which denies the possibility of dividing the continuum into points, does not seem to be an ontological claim but rather a mathematical assumption that enables us formally to avoid the paradoxes of continuity.[5] It follows, then, that the philosophical difficulty in describing the flight of the arrow remains in place.[6]

I shall now try to present a different solution to the paradox of the arrow in flight. In what follows we shall see that this solution contains new and surprising insights in various fields.

B. The Arrow and the Uncertainty Principle

The implicit assumption in the two formulations above of the paradox of the arrow in flight is the application of the law of non-contradiction to the concept of motion. To be sure, the assertion that a body cannot both move and not move at one and the same time is a logical tautology, but I would like to argue that this is not precisely the assumption underlying the paradox. The various formulations of the paradox assume an identity between the claim that "body A has velocity" and the claim "body A changes its location." My contention is that the term "moves" is used in the formulations above in both of these senses, and that they differ from one another. In an entirely parallel way, one can distinguish between two senses of the term "stands" in a certain place: the first is "is located" in that place, and the second is "is at rest" in that place.

Quantum physics faced, in the first half of this century, a dilemma that resembles in several respects the one presented here. Physicists discovered that every elementary particle sometimes has the properties of a particle and sometimes the properties of a wave. Niels Bohr proposed, as a solution to this dilemma, the principle of complementarity, according to which every such entity (which we ordinarily call a particle) has both wave-like and particle-like properties simultaneously. The character revealed to the observer depends on the mode of measurement and on the quantities being measured. A formal and quantitative formulation of this principle is called Heisenberg's uncertainty principle. This principle is stronger than the principle of complementarity,[7] but in this article I shall address only the principled content of these claims and not their formal quantitative form. For our purposes, this qualitative formulation of the uncertainty principle will suffice: the velocity and position of a particle cannot be determined at one and the same time. In other words, when a particle has a definite place, its velocity is a completely undefined quantity about which one cannot speak, and certainly cannot measure, and vice versa.

This principle can be connected to the paradox of the arrow in flight on several levels:

(a) On the semantic level, according to this principle one cannot formulate propositions that include the concept of velocity at the moment when a particle has a definite position, since such propositions contain a concept that is not defined: "the velocity of the particle located at x." Hence argument (1), which leads to the conclusion that the arrow moves at the moment when it stands, is a meaningless argument. The claim that this argument is meaningless does not rely on the law of non-contradiction, which states that the proposition that a body both moves and does not move at the same time is invalid, but rather on the fact that nothing can be said about the motion of a body when its position is defined. Yet formulation (2) still stands, for one may still ask when the body moves if not at the very moment when it stands in some place. To answer formulation (2) by using the principles of quantum theory, we turn to the next level:

(b) On the physical level, a particle can be in different places at different times, even though when its position is known one cannot attribute velocity to it. This follows from the fact that the concept of the moving point-particle is not well defined in quantum theory. Within this theory, the particle is represented by a wave function that describes the probability of finding the particle in a certain place. The laws describing the motion of a point object are not the laws of kinematics, which relate an object's position to its velocity and acceleration through infinitesimal calculus (the Newtonian description). Those laws are indeed valid for large bodies at the macroscopic level, but the dynamics of small particles are described by the laws of quantum theory (the Schrödinger equation, etc.). Those laws describe the dynamics of the wave function that represents the particle. The particle's position, which is only an average quantity, can be calculated from this function. At any given time there is also another way of looking at the wave function, one that describes the probability that the particle has a certain velocity. A particle's motion between positions is not well defined in terms of a moving point-particle, but rather through the dynamics of the wave function that describes it.[8]

In what follows, however, I would like to focus specifically on a third, perhaps more surprising, way in which one may relate the paradox of the arrow in flight to quantum theory:

(c) Arguments (a) and (b) above cannot provide a philosophical solution to the problem of the flying arrow. They can only point to the connection between this difficulty and the one accompanying the uncertainty principle. For that reason, we shall now try to reverse the direction of explanation and say that the uncertainty principle does not explain the paradox of the arrow in flight, but is rather explained by it.

As is well known, the uncertainty principle has troubled many physicists and philosophers of science in this century, and it may be that a good understanding of the paradox of the arrow will help us develop some insight into this principle. Of course, to do so we need to develop a good intuitive understanding of the matter of the flying arrow, in a way that does not depend on quantum theory, and then to try to project from it onto the uncertainty principle.

The attempts to solve the paradox by pointing to the problematic nature of the concept of continuity, as presented above, are philosophically unsatisfactory. I shall therefore now try to develop another understanding of the flying arrow. Afterward I shall try to apply it back to the uncertainty principle, and then to other subjects.

 

C. Does a Dynamic Process Mean a Change of Static States

 

For the sake of the discussion I shall use a simple example from elementary mechanics, although the point holds for any two concepts related to one another through the derivative in infinitesimal calculus. The concept of instantaneous velocity is defined in mechanics as the ratio between the change in position and the span of time in which that change occurred, for time intervals as small as one wishes. Mathematically:, where x(t) and V(t) are the position and velocity of a body at time t, respectively. This is in fact the definition of the derivative of the position function with respect to time. We see that although, in order to calculate the velocity, we had to observe the body over a time interval and it was not enough for us to know its position at a single moment, the result of the calculation is the velocity of the body at time t, which is an indivisible moment. If so, contrary to our intuitive understanding, which says that velocity is a quantity defined only over a stretch of time, a body has a well-defined velocity at every indivisible moment of time. The way to calculate this velocity from the body's position is what forces us to use a stretch of time around that moment. Clearly, I do not mean to claim that the body changes its location at that indivisible moment,[9] but only that it has a definite velocity at that moment.

The definition of velocity through the derivative of position is called in physics the "definition" of velocity. This notion of definition denotes an operational definition (how one calculates velocity) and not an essential definition (what velocity is).

The reason for the mistaken intuitive understanding that velocity is defined only over a stretch of time is a confusion between the concept of "velocity" and the concept of "change of position." The "velocity" of which we are speaking here is a potential for a change of position and not the change itself. The change is only a consequence of the fact that the body has such a potential. From here one may continue and argue that even if the body is in a certain place at a certain moment and at the same time also has a definite velocity, this does not mean that the body both moves and does not move simultaneously, as Zeno claimed. The body certainly cannot change position while having a definite position, for that would be a proposition contrary to the law of non-contradiction, but that does not mean that the body has no velocity at that moment.[10]

In order to clarify somewhat the concept of velocity as potential and distinguish it from the change of position that follows from it, I shall try to give two examples from mechanics in which no change of position occurs even when a body has velocity. When a body collides with a wall, then even if its velocity has a certain value different from zero, the wall will not allow it to bring this potential of motion from potentiality to actuality. That is, this is a case in which the potential (the velocity) exists, but its consequence (the change of position) does not. In the same way, when a body collides with another body, some of the velocity it has may be transferred to the second body, so that not all of the potential is actualized by the body that carried it before the collision. In section E, further examples of this distinction, from other fields, will be given.

Having distinguished between "velocity," which is the potential, and "change of position," which is the consequence, one may add that macroscopic change of position does not necessarily entail that at every moment a microscopic change of position is taking place, as is assumed at the root of Zeno's paradox. The change of position is a consequence of the fact that the body has velocity. If we now relate to formulation (1) of the paradox, then we do not accept the assumption that were the arrow not flying (in the sense of changing place) at every moment, its motion would not take place. It would be more correct to say that were the arrow not to have velocity at every moment, its motion would not take place, and yet the fact that it has velocity at a certain moment does not contradict its being in a definite place. With respect to formulation (2), we shall say that the question of when the arrow flies is not well defined. In both formulations the problem lies in the interpretation of the term "flies": if "flies" means "has velocity," the answer is that at every moment it has velocity; but if the meaning is "changes place," we shall say that the question is not well defined. The body indeed changes place over every interval in which we observe it, but the concept of "change of position in an indivisible moment" is not a well-defined concept. This is similar to the question of when time itself changes. The concept of change does not admit the question of when (in the sense of "at which indivisible moment"). Change always occurs over an interval. The question that one may certainly ask is whether the body has some instantaneous characteristic when its position changes. The answer is certainly yes: it has velocity at every moment.

This is the place to add the reverse side of this argument: the fact that "the arrow is in a certain place" does not mean that it "is not moving at that moment," for "is located" does not mean "stands still." One can be in a certain place even while in a state of motion. "Stands still" means having velocity 0, while "to be located" means being in a certain place. When one says that something stands still, it is not necessarily required to specify its location; this is not so when one says that something is located, for then one must specify where it is located. This argument too leads, perhaps even more simply, to the rejection of the conclusion that the arrow is not moving while it is moving. The arrow merely is located in a certain place at every moment of its motion.[11]

Unlike the previous explanations, this explanation of the paradox of the arrow in flight is not connected to the problematic nature of the concept of continuity. In the argument presented here, the paradox rests on the conceptual confusion between "change of position" and "velocity." In this respect one may certainly continue to hold the view that time is composed of a collection of indivisible points laid densely one next to another. My intention here is not to argue for the correctness of this approach, which, as is known, raises other problems, but only to detach it from the difficulty posed by the paradox of the arrow in flight.

Let us now try to probe more deeply the causes of this conceptual confusion, which, as stated, is rooted in conflating the potential of motion with its consequence, namely motion itself. When we observe[12] a moving body, all we are really observing is that it is in different places at different times; we have no way of grasping the concept of velocity directly except through interpolation between the positions.[13] This is also the reason that the definition of velocity in mechanics is given by the derivative of the position function, which is usually treated as though it were the more basic concept. Human perception is more comfortable with static concepts, and therefore it also defines dynamic concepts by using those concepts. In other words, Zeno is entirely right in claiming that human perception cannot discern a body's velocity at an indivisible moment. Yet this difficulty arises from the static way in which we think. Our consciousness compels us to observe only the changes of position caused by velocity, and these, of course, occur only over intervals. We are not able to observe the potential (velocity) directly, but only its consequences. My claim here is that one should not infer from this that such velocity does not exist.

 

D. Back to the Uncertainty Principle

 

As noted in section B above (in way (c), by which the arrow was connected to the uncertainty principle), one may proceed in the opposite direction and attempt to project from the understanding we developed regarding the flying arrow onto the uncertainty principle.

For this purpose, let us formulate the paradox of the arrow in flight somewhat differently.[14] Suppose we look at a flying arrow and photograph it at different moments (indivisible ones: the theoretical camera has an exposure time of 0, an ideal camera). In every photograph, the arrow will appear stationary, except that its location changes each time. We may now ask: when does it move between these different places? Our answer is that the arrow moves and at the same time has a position (and not that it both moves and at the same time stands still, as in Zeno's contradictory definition). The fact that we do not see the arrow's motion in the photograph should be attributed to the fact that a camera is not the appropriate instrument for observing motion (or for measuring velocity). A camera is an instrument that measures (or observes) positions. One may now define, by analogy, another theoretical instrument: an ideal movie camera. This instrument measures or observes the velocity of a body at a single indivisible moment. As explained above, our consciousness operates in a static way, and therefore it is hard for us to imagine such an operation. An ordinary movie camera, as we know it, actually functions as a camera that takes a rapid succession of static images. Our consciousness creates in us the sensation of motion by interpolating between those images. By contrast, the ideal movie camera is not an instrument that measures velocity through differences of position by using the definition of velocity as the derivative of position (as do the movie cameras in our possession, which are subject to the static limitations of our consciousness), but rather in a pointwise manner. Let us continue the analogy and say that if we observe (film) through such an instrument a moving body, we shall be able to see it moving at every moment, but we shall not be able to discern its position. An example of this is that a camera with a long exposure time clearly shows that the object is moving by means of a trail formed in the image, but one cannot discern a definite position.[15]

It follows, then, that the information we receive about the moving body depends on the instrument through which we observe it. Our consciousness, which as stated is static at its foundation, serves us as a camera, and therefore only information about position is directly given to us, while velocity is obtained indirectly through interpolation between different positions. If we continue this line of argument, we may say that even if it were possible to know a body's instantaneous velocity by using a movie camera, one still could not speak simultaneously of its position. Let us now assume, as a plausible hypothesis, that human consciousness cannot operate simultaneously in the mode of an ideal camera and of an ideal movie camera. This statement is in fact a simple formulation of the uncertainty principle.[16]

This is the place to recall once again that the uncertainty principle also includes a quantitative determination of the uncertainty (Planck's constant). This, of course, cannot emerge from a qualitative argument such as the one presented here, and therefore there is no pretension here to ground the uncertainty principle fully on the basis of classical physics, but only to point to the principled basis of uncertainty between pairs of dual quantities.

A further remark concerning the reversal of direction from the arrow to uncertainty. Explaining Zeno's paradox by means of the uncertainty principle (the semantic and physical connections above) requires no additional assumption, whereas explaining the uncertainty principle by means of the understanding we derived from the paradox of the flying arrow (connection c) certainly does require an additional assumption. The discussion of the moving arrow led us to the conclusion that if a body has a definite position at a given moment, the question of when it changes its position is meaningless. But the other side of the uncertainty principle, which states that if a body has a definite velocity then a discussion of what its position is has no meaning, does not follow directly from the argument above. One may present a parallel argument based on the concept of the integral (the definition of position by means of velocity) and thereby show the other side. The route I have chosen here (the definition of the ideal instruments) seems more fruitful in the quantum context, as the reader will immediately see.

As is known to anyone familiar with quantum theory, the uncertainty principle implies that there are two ways to describe dynamic quantities: either in the momentum picture (usually proportional to velocity), or in the position picture. These are two pictures in which the properties of the physical system are characterized by the probability of being at a certain velocity, or the probability of being in a certain position. In the first picture, all physical quantities are described by using velocity (momentum) coordinates, whereas in the second everything is described in position coordinates. These two forms of description exclude one another (in scientific jargon: they are non-commutative). That is, in a manner parallel to the argument presented here, the inability to know position and velocity simultaneously stems from the fact that these quantities belong to different conceptual systems (pictures) that do not "speak" to one another. We may use our terminology and say: looking at the world with a camera yields the position picture, whereas looking with a movie camera yields the momentum picture. In other words, the uncertainty that exists between two quantities may be grounded in the fact that the two quantities belong to different conceptual worlds, analogous to the description of the instruments above. It follows that the argument presented here also gives meaning to the pairs of dual pictures that accompany, in quantum theory, pairs of physical quantities that stand in relations of uncertainty to one another.

At this point an interesting objection may arise to the meaning given here to the uncertainty principle: a view accepted today in the scientific community is that the uncertainty principle describes not us but matter itself. It is not our limitation, but a principled impossibility. Velocity and position are not only not measurable simultaneously; they are also not present simultaneously (that is, they do not both characterize the body at the same time). By contrast, the argument put forward in this article attributes the uncertainty to limitations of the human observer's consciousness. That is, according to the argument presented here, an observer who does not suffer from the limitation according to which consciousness cannot operate in the mode of an ideal camera and an ideal movie camera at the same time would be able to measure simultaneously the position and velocity of a moving body.

A possible solution to this problem, if we use Kantian terminology, lies in the distinction between phenomena and noumena. Clearly, uncertainty cannot be a property of the thing-in-itself (the noumenon), for we have no access at all to its properties. It can only be a property of the world of appearances, that is, of the object as it appears to our eyes (phenomenon). To be sure, even within the world of phenomena one must distinguish between the object measured as it appears to us and the measuring/observing instrument and the observer himself as they appear to us. The physical claim that uncertainty is a property of the object and not of the measurer concerns entirely a distinction within consciousness (or within the world of appearances). That is, even if we agree that this is not a technological problem but a principled one, we may still argue that even this principled problem originates in the properties of human perception, which defines the measured quantities and the instruments that measure them. A more detailed discussion of this issue belongs to a separate inquiry.

To summarize our discussion so far: velocity (or process) is a quantity that exists even in a single indivisible moment, except that its meaning is a potential for change of position (or state) and not the change itself, which is only a consequence of the process. The change itself, of course, cannot occur in an indivisible moment, but only over an interval. Human perception describes the process itself by means of the change of state caused by it, and this is due to the static character of consciousness.[17]

E. Implications of the Distinction Between "Process" and "Change of State"

As Henri Bergson already noted in his book Creative Evolution, it is entirely clear that the paradox of the flying arrow does not concern only the concept of velocity, but every process of change. In this section we shall try to see the implications of the analysis presented thus far for issues unrelated to mechanics.

In section C we defined the two concepts "process" and "change of state" in terms of potential for motion and actual motion. These two concepts, even if they are not identical, appear at first glance to be twins joined together. At first sight it seems that although, as we claim, these are indeed two different concepts, one cannot discern one without the other also occurring or existing in parallel. In section C we saw examples from mechanics (collisions) in which one can discern the existence of a potential without its actual consequence.

In this section I shall attempt to discuss additional cases in which one can separate what is joined together, and speak of one apart from the other. The discussion of the examples we shall bring will be only sketchy, and its purpose is merely to stimulate thought rather than to provide a complete analysis of each case.

A first example is the claim one occasionally encounters in books dealing with management, according to which the very process of change is good for an organization. That is, even if there is no problem at all in the present structure of the organization, some benefit will arise from changing it, and this because of the process itself, not because the next state will be better than the current one. Of course, a change that leads to a worse static state (for example, the wholesale destruction of the machines in a factory) does not improve the state of the organization. But if we assume that there is a group of organizational states none of which is preferable to the others, it is better for the organization to be in transition among them than to be frozen in any one of them. At first glance, this seems to be a clear case of benefit arising from the existence of a process even if it is not accompanied by a change of state. It seems that in order to describe such a benefit one cannot use the terms of "change of states" but only those of "processes," for we have no need at all of a different state (structure), but only of the very process of change.

Someone may argue that this is not an unequivocal example, for this claim can also be formulated as follows: the organization derives benefit from the very fact that its structure (its state) changes, that is, that it moves from structure A to structure B. In this formulation we again used only the term "change of state" and had no need for the second term. It is true that it makes no difference at all what the second state to which the change leads will be, but even so we are not compelled to avoid the term "change of state" and use only the term "process." This argument seems to be only a semantic evasion, for to say that any other state is preferable, no matter which, is in practice to say that it is preferable to be in a process of change.

Another interesting example in which one can use only one term and not the other arises in the discussion of the problem of divine perfection and self-perfecting.[18] Rabbi Kook formulates this problem as follows: "There is a perfection of added perfection, and this cannot exist in Divinity, for absolute infinite perfection leaves no room for addition. And for this purpose—that the addition of perfection itself should not be lacking in being—the existence of the world must come into being."[19] The basic assumption is that self-perfecting (that is, spiritual progress) is one of the perfections, and therefore it too must exist in Divinity. On the other hand, it is clear that one cannot speak of God's self-perfecting for two reasons: (a) no change can occur in God; (b) God is perfect, and therefore one cannot speak of Him in terms of spiritual progress. The solution I propose here to this dilemma parallels the one proposed for the organizations discussed above: the process itself (or at least its root) exists in God, even though the change, which usually accompanies it, cannot exist in Him for the reasons just stated. That is, the spiritual perfection expressed in self-perfecting lies in the process and not in the change of state, and this can characterize God as well.[20] It should be noted that in this example one cannot present an alternative formulation such as the one suggested in the management example, since with respect to God one cannot speak at all in terms of changes of state. Therefore this is a clear case of a process that is not accompanied by changes of state at all.[21]

There are also several examples from the Talmud in which one can apply this distinction between change of state and process. Only one example will be given here, which is discussed in greater detail elsewhere,[22] namely the giving of a get (bill of divorce) in the divorce of a woman. From an examination of the passages in the Talmud that characterize the giving of a get, it emerges quite clearly that attempts to formulate the criterion for valid giving (that is, a giving that succeeds in effecting the divorce) by means of "states," that is, by defining what the state is before the giving and what the state is after it, fail utterly. There are attempts to describe the giving as a transfer of ownership (a monetary conveyance) of the get, but this is contradicted by several explicit laws in the Talmud. The attempt to describe the giving as a physical act of giving (the transfer of the get from the husband's hand to the woman's hand) also fails. The solution I proposed in my article was that the Talmud is trying to characterize a process of giving and not a transition between states. The examples presented in the Talmud of valid and invalid giving of a get are very numerous. The reason is the inability to provide an explicit definition of the process of giving, owing to the static nature of human perception. The multiplicity of examples tries to convey to the learner an intuitive sense of what constitutes a valid process of giving a get.

F. Time in Relativity Theory: Bergson and Einstein

After presenting some of the implications of the distinction between the concept of "change of state" and that of "process" in areas unrelated to physics, we now return to another important point—again in the world of physics—in which this distinction finds expression. In this section we shall discuss the description of time within Einstein's theory of relativity, and the question whether it is exhaustive.

The most prominent characteristic of time, as opposed to space or extension, is its being a flow. We are discussing here one-dimensional extension, for there is another clear difference between space, which is three-dimensional, and time, which is one-dimensional. That difference is irrelevant to the discussion to be conducted here. Richard Taylor, in his book Metaphysics,[23] shows that space and time have identical characteristics by means of the following argument: in every sentence describing motion or the relation between space and time, one may replace the temporal relations with parallel spatial relations and vice versa, and the result is always a sentence with a clear and well-defined meaning (see there for various examples).

From this the positivist may conclude—and he generally does conclude—that time and space have identical properties. Everything that can be done in space can also be done in time, and vice versa. This is in fact also the picture yielded by Einstein's theory of relativity. In this theory, time is one of the four coordinates describing an event in Minkowski space-time. The basic object in this theory is the body's "world-line," that is, the line describing the body's positions at all times. In this picture all of space-time is static. The flow of time is a phenomenon that physical theory simply ignores.

One of the common philosophical arguments for rejecting the feeling of the passage of time and classifying it as a mere illusion is the following argument: "A changes" means "A is in one state at one time, and in another state at another time." In particular, to say "A moves" means "A is in one place at one time, and in another place at another time."[24] To place time as the subject (A) of one of these sentences is meaningless, since time itself cannot be in some state/place at one time or another.[25]

One may rescue the intuition that feels the passage of time from this strait by claiming that the passage of time is described along a second time axis, which serves as an index describing the state of ordinary time and its motion. But here the claim of infinite regress arises, since the new time axis will be subject to the same attack if we also wish to attribute to it the same property of constant flow. One can stop this regress by asserting that there is one time that serves only as an index and does not flow, and another time that flows across the first.[26] One of the most prominent defenders of the conception that time flows (creative time or "duration") was Henri Bergson in his book Creative Evolution,[27] who even engaged in a debate with Einstein, who held the opposite view. Each side in that debate claimed exclusivity for its own conception, whereas I propose here that each represented a different aspect of time—or, in other words, one of the two kinds of time defined above. A somewhat different description of the two aspects (or the two kinds) of time was proposed by McTaggart already in 1908,[28] and there is even an interesting attempt to give it a certain mathematical-physical dress.[29]

What concerns us here in this description of time is the relevance of these two kinds of time to the description of processes as opposed to changes of state, as defined in the previous section. My claim is that flowing time "carries on its back" the process, whereas the changing static states are characterized by an index that is also called time, which indicates their "location" on the time axis. That index is time of the second kind.

Here too, apparently, one may ask—parallel to the two formulations of the paradox of the arrow proposed at the beginning of our discussion—how time flows. That is, at every given indivisible moment of index time (Einsteinian time), flowing time (Bergsonian time) is at another time. If so, when does it change the index? This question seems meaningless, since Bergsonian time is not a quantity that flows in time; rather, it is the flow itself. Every dynamic quantity is borne along Bergsonian time, and creates change in the static states characterized by Einsteinian time. Our feeling that time passes means that we pass across Einsteinian time, and Bergsonian time carries us. The consequence of our flowing in time is that each time we are in a time-state with a different Einsteinian index.

At the beginning of my remarks I presented a definition of velocity in mechanics as the derivative of position. Later I argued that this definition is an indirect way of deriving the process from the change of states. The use of such a form stems from the limitations of human perception, which usually operates by way of static apprehension, like a camera that takes in changing states and not processes. If it were possible for us to apprehend in the manner of a movie camera, that is, the process itself, we could describe velocity directly as borne on a Bergsonian time axis. This is also the reason that Einstein's theory of relativity describes only index time, since that is the time that can be directly apprehended in our consciousness. The ideal movie camera defined in section D operates along Bergsonian time, and can therefore observe velocity (or change) at a single time-point. This is a point of Bergsonian time, which in static perception appears as an infinitesimal.[30]

A question may arise here: with respect to time, how can our consciousness operate simultaneously as a camera and as a movie camera, and apprehend both kinds of time? Two lines of thought may be presented in order to answer this question. First, it seems more fitting to intuition to say that with respect to time itself, our direct experience is precisely the dynamic one and not the static one. We feel time flowing, and artificially mark upon it static index-points. Perhaps this itself is the essence of the difference between space and time: one is apprehended only in a static way, and the other only in a dynamic way. Another line of thought that may be suggested here is that the apprehension of Einsteinian and Bergsonian time takes place on two different levels of consciousness. Einsteinian time seems to have a more objective (or intersubjective) meaning, whereas Bergsonian time is apprehended as more subjective. By its very nature, science describes our "objective" experiences, and therefore in physics the conception of time as an index predominates. In any case, the claim that our consciousness, operating as a camera, cannot apprehend processes directly but only through changes of state refers to objective consciousness that can be quantified mathematically. I do not wish to argue against the existence of an immediate experience of our consciousness that experiences the passage of time.[31]

[1] I wish to thank Amnon Levav, Dalia Drai, and Gadi Prudovsky for a careful reading of this article, and for their helpful comments, which contributed not a little to its writing. My thanks also to Avshalom Elitzur for a discussion I held with him at the very early stages of formulating the ideas presented here, and for reading the almost final version of the article.

[2]   cf. F. Cajori, “History of Zeno’s Arguments Against Motion”, American Mathematical Monthly, 22 (1915);   W. C. Salmon, ‘Zeno’s Paradoxes’, Indianapolis, Bobbs-Merrill, 1970.

[3] This formulation is cited by Samuel Hugo Bergman, Introduction to the Theory of Logic, Jerusalem, Bialik Institute, 1975, in chapter 3, section 18.

[4] This approach is more characteristic of the Leibnizian formulation of differential calculus, as opposed to the Newtonian formulation, which is less intuitive. For an intuitive formulation of differential calculus, see H. J. Keisler, ‘Foundations of infinitesimal calculus', Boston, Prindle, c1976.

[5] There are attempts to present formal solutions to the paradox of the arrow by means of an assumption of a discrete structure of space and time. For a more detailed discussion, see W. C. Salmon, Space, Time and Motion, Dickenson, 1975

See also the book mentioned in note 2 by the same author.

[6] My thanks to an anonymous reviewer of this article, who drew my attention to the approach of synthetic differential geometry, in which the continuum is not composed of points. See, for example, J.  L. Bell, “Infinitesimals and the Continuum”, Mathematical Intelligencer 17 (1995): 55-57; idem, “Infinitesimals”, Synthese 75 (1988): 285-315

Admittedly, the logic underlying this approach is intuitionistic, but, as argued in the body of the article, common sense still sees the concept of a point of time as evident.

[7] This principle also establishes a relation between the measures of uncertainty of the two complementary properties, beyond the principled claim that these two values cannot be known simultaneously.

[8] One can of course expand Zeno's paradox to every situation in which there is change (see below at the beginning of section E), and not specifically to the concept of velocity. In this way one can raise, regarding the dynamic laws that describe the change of the wave function, the same difficulty that Zeno raised concerning the flying arrow: the function changes its "location" (in the space of functions) over time. If so, at every indivisible moment of time the function changes and is at the same time also static. Discussion of this question involves a discussion of the ontological status of the wave function, an issue that, as far as I know, is far from being understood.

[9] When it moves at a finite velocity, this is self-evident. But even when its velocity is infinite, it seems impossible to change position in a single indivisible moment of time. In that case one may change position within a stretch of time as short as one wishes, but not in an indivisible moment. A point-body cannot be in two places at the same indivisible moment of time: this is a logical contradiction, and not a physical inability to reach infinite velocity.

[10] In field theory in physics one sometimes deals with an ultra-static and ultra-local model. This is a theory that describes a field defined at one indivisible moment of time, and one indivisible spatial point. One can define a velocity field (or momentum) in such a model. Clearly, if velocity is change of position over time, such a field has no meaning within such a model.

[11] See a similar discussion in Salmon's books mentioned in note 5. Unlike the discussion presented here, he does seem to connect this argument to problems involving the concept of continuity.

[12] "Observe" in this paragraph is used to describe cognition and not perception. I am aware that the distinction between them is not always sharp.

[13] This claim is one of the main axes of argument in Henri Bergson's book Creative Evolution, Jerusalem, Magnes, 1978. He even uses it there in the context of the discussion of Zeno's flying arrow. See also section F below.

[14] See also Bergson's discussion in Creative Evolution, chapter 4.

[15] An example of this kind was brought in Aharon Pinker's book The Atom Book, Jerusalem, Reuven Mass. My thanks to Itamar Pitowsky of the Hebrew University, who brought this source to my attention.

[16] For anyone who does not sense the plausibility of this hypothesis, one may again reverse the direction of the argument here, and use the uncertainty principle as support for the hypothesis that consciousness cannot operate simultaneously in the mode of an ideal camera and an ideal movie camera.

[17] One can say that velocity and change of position (or, more generally, the process and the change it causes) are not two things that exist on the same plane. The process is the thing-in-itself, while the change that follows from it is the phenomenon. If this claim is correct, then it is impossible to speak of these two quantities as duals in such a way that one is apprehended by means of a camera and the other by means of a movie camera. The thing-in-itself cannot be apprehended by any instrument. A detailed analysis of this claim raises several problems, which there is no place to discuss here, and therefore no such relation is established in the body of the article.

[18] For a discussion of this problem, see Yosef Ben-Shlomo, "Perfection and Self-Perfecting in Rabbi Kook's Theology," Iyyun 33 (1984): 289.

[19] Rabbi Kook, Orot HaKodesh, Jerusalem, Mossad Harav Kook, 1985, vol. 2, p. 531.

[20] The solution proposed there by Rabbi Kook himself appears, on its face, to be different. In my opinion, his solution is very close to the one proposed here, but that is a matter for separate discussion.

[21] Let us note here that if we adopt the conception presented in note 17, according to which the "process" is the thing-in-itself of the "change," then in God the thing-in-itself is present in such a way that its appearance in the eyes of consciousness (the change) is expressed in created beings. This is a panentheistic conception more similar to the one proposed by Rabbi Kook there. See Ben-Shlomo's article mentioned in note 18 above.

[22] See Michael Abraham, Daf Shavu'i, the Department of Basic Studies at Bar-Ilan University, Parashat Ki Tetze, 1995, Torah and Science section.

[23] Translated into Hebrew by Adam, Free University, 1983. See there chapter 7.

[24] See Avshalom Elitzur, Time and Consciousness, Tel Aviv, Broadcast University Press, 1994, chapter 4.

[25] In the second formulation it is also impossible to place space as the subject of the sentence. This follows only from the fact that we chose to discuss an example of change, namely motion, and not the concept of change itself as in the first formulation. The general statement is the first formulation; the second is only a concrete example intended for illustration.

[26] One can speak of two aspects of time rather than of two different times. McTaggart's formulations (see note 28) seem closer to such a formulation.

[27] See note 13.

 [28] Philosophical Studies, ed. S. V. Keeing, London, E. Arnold, 1934, chap. 5; ,McTaggart J. E.

And see reference no. 30 in the article by Horwitz, Arshansky, and Elitzur cited in the following note.

[29] L. P. Horwitz, R. I. Arshansky and A. C. Elitzur,  “On the Two Aspects of  Time: The Distinction and Its Implications", Foundations of Physics, 18, 12 (1988): pp. 1159-1193

[30] Berkeley's expression regarding the concept of the infinitesimal is well known: ”ghosts of recently departed quantities”                   (“The Analyst”, reprinted in James R. Newman’ ed. The World of Mathematics, New York, Simon and Schuster, 1956) ; cf. note 6

[31] This aspect recalls Schopenhauer's conception, which claims that a person who looks inward into himself can apprehend the thing-in-itself (the soul/spirit), whereas in the direct apprehension of objects outside him he is, of course, limited to the phenomenon alone (the body). Physics describes the intersubjective aspects of perception, and is therefore limited to describing index time alone. Philosophy (in some of its forms) and mysticism are, of course, freed from these constraints, and for that reason these disciplines often rebel against the narrowness of the scientific conception.