A Systematic Look at the Static and the Dynamic (Column 705)

With God’s help

Disclaimer: This post was translated from Hebrew using AI (ChatGPT 5 Thinking), so there may be inaccuracies or nuances lost. If something seems unclear, please refer to the Hebrew original or contact us for clarification.

In the previous column I discussed Sam Bak, his memoir, and his paintings, and I argued that they express the complex relation between the static and the dynamic. I dealt with this relation in the past, mainly in my article “Zeno’s Dichotomy and Modern Physics,” and later also in column 32 about repentance, and in column 170 about “Avodah Tzorekh Gavoah” (“service for the sake of Heaven”), among others (some of which will be cited below). Here I wish to present a more systematic account of this delicate issue, which has philosophical, mathematical, and scientific aspects. More specifically, I will discuss representing the dynamic by static frames (as in cinema), and the mathematical, scientific, and philosophical problems this creates.

Analytic philosophy teaches that philosophical difficulties always stem from an imprecise, imperfect language (on this view, there are no genuine philosophical problems). I think that is greatly exaggerated, but clearly there is something to it. Language represents reality, and as a representation it is not always successful or precise. Sometimes the problem is not the language but our mode of thought, which also finds expression in language (see the series of columns 379–381, column 662, and more). When we represent a dynamic process by static frames, it is important to remember that this is a representation and not the thing itself, since sometimes the model misleads us. We must not be prisoners of the model; we must remember that it is a model and that what interests us is the reality the model is meant to represent. This is true in science, which is built entirely on models representing reality (and which we tend to mistake for reality itself), but it is also true in philosophy and, of course, in mathematical representations of reality in general. Representing dynamics by static frames is no exception, and as we shall see it raises severe problems in several fields.

Let me preface by saying that this column is a bit difficult, since it relies on some basics of infinitesimal calculus, quantum theory, and even a bit of special relativity. I tried to explain in a way that laypeople can also follow, and I hope I succeeded. I think the effort is worthwhile, because there are some very important links and insights here, which in my opinion even professionals in these fields often miss.

The Arrow-in-Flight Paradox

Zeno of Elea tries to undermine the concept of motion by several paradoxes. One of them is the arrow-in-flight paradox, which can be formulated roughly as follows. Consider an arrow in flight. At every moment it is located at a different point. If so, when exactly does it pass from one place to another? When does it move? The assumption is that at any discrete moment the arrow cannot move, and therefore it is clearly at rest at the point where it is located.

The Arrow and Infinitesimal Calculus

There are attempts to resolve the problem by using notions from infinitesimal calculus and the problematic nature of the continuum. In simple terms: a continuous line is not composed of points but of infinitesimals as small as we please. The time axis is a continuous line. Therefore, it is meaningless to discuss the state of a body at a discrete instant. Over an interval of time, however small, the arrow can move.

This definition of the continuum is necessary in mathematics. Viewing a continuous axis as a dense set of discrete points is problematic, and one cannot define the continuum that way. And yet it is hard to accept that one cannot speak of a discrete point on a continuous axis. The axis is composed of discrete points, but with the added property of density. Even if the assumption that the continuum consists of infinitesimal segments rather than points is useful mathematically, it does not look like an ontological claim (about reality). For our purposes, it is implausible that we cannot speak about the state of the arrow at an instant, and therefore Zeno’s problem remains.

The Arrow and Quantum Theory

Quantum physics teaches that every elementary particle sometimes has particle-like properties and sometimes wave-like properties. Niels Bohr proposed the principle of complementarity as a solution to this dilemma: for any such entity (which we colloquially call a “particle”—see my remark on this in the previous column), there are times when it exhibits wave properties and times when it exhibits particle properties; and how it appears to the observer depends on the measurement and on which quantities are measured. A more formal, quantitative statement is Heisenberg’s uncertainty principle. That principle is stronger than complementarity, but here I will refer only to the conceptual content rather than the formal quantitative form. A simple formulation of the uncertainty principle: a particle’s velocity and position (like other such dual pairs) cannot be determined simultaneously. In other words, when a particle has a defined position, its velocity is entirely undefined—one cannot speak of it, let alone measure it—and vice versa.

You can immediately see how to relate this principle to the arrow-in-flight paradox on several levels:

1. On the semantic level, according to this principle, one cannot formulate sentences that include the concept of velocity at a moment when a particle’s position is defined, since such sentences include an undefined concept: “the velocity of the particle located at x.” Hence the claim that the arrow moves at the very moment it is standing still is meaningless.
2. On the physical level, a particle can be in different places at different times, even though when its position is known one cannot ascribe to it a velocity. This stems from the fact that the notion of a point particle in motion is not well defined in quantum theory. In this framework, the particle is represented by a wave function describing the probability of finding it at a given place. The laws describing the motion of a point-like object are not the kinematic laws linking position to velocity and acceleration via infinitesimal calculus (as in Newtonian mechanics). Those laws are indeed correct for large, macroscopic bodies, but the dynamics of small particles are described by the laws of quantum theory (the Schrödinger equation, etc.). These laws describe the dynamics of the wave function that represents the particle. The particle’s position, which is only an average quantity, can be computed from this function. A “real” position exists only after measurement.

Seemingly, this gives us a physical solution to the arrow-in-flight paradox. The formulation of the paradox uses claims or concepts that are not well defined, and those are presumably what generate the difficulty. One may argue, of course, that if we are speaking of an arrow rather than an elementary particle, quantum theory is irrelevant. It deals only with microscopic entities, whereas macroscopic bodies obey Newtonian mechanics. But that should not trouble us, since we can formulate the paradox for an elementary particle, in which case quantum theory resolves it. If we then move to the formulation for an arrow (a macroscopic body), it does not really change the picture. Our ability to use Newtonian mechanics for arrows and ignore quantum theory derives only from our lack of interest in position and time down to arbitrarily exact precision. If we demand perfectly exact values of position and velocity, they cannot coexist simultaneously even for an arrow. When we speak of an exact position at an exact instant, that does not exist for an arrow either, not only for elementary particles.

However, this is a dubious solution to our paradox. The reason is that quantum theory itself is philosophically unclear to us. Using it to explain the arrow’s paradox can help only to clarify that the arrow indeed flies—but we already know that. To provide an interpretation that resolves the philosophical problem, we cannot appeal to quantum theory, which itself calls for such an interpretation. My favorite analogy for this difficulty is an English–English dictionary that explains one word we do not understand by ten others that we understand even less.

I now wish to relate the arrow to quantum theory in a third, perhaps more surprising way. We saw that arguments (a) and (b) above cannot supply a philosophical solution to the arrow’s flight, but only indicate a relation between this difficulty and the one accompanying the uncertainty principle. For this reason, let us now reverse the direction and say that the uncertainty principle does not explain the arrow-in-flight paradox; rather, it is explained by it:

1. As is well known, the uncertainty principle has embarrassed many physicists and philosophers of science in this century, and we saw that this is precisely why we cannot use it to explain and dissolve a philosophical paradox (again, it is that English–English dictionary). If so, assuming I can find a satisfactory solution to the arrow paradox, I can reverse the direction and, by means of the arrow, explain quantum theory and perhaps dispel some of the fog around it. But this, of course, requires solving the arrow paradox without presupposing quantum theory (a solution within classical concepts and thinking).

A Body with Non-Constant Velocity

For the discussion I will use a simple example from mechanics, although the points hold for any two quantities related via the derivative (in infinitesimal calculus).

Consider a body moving at some constant velocity V. Its position (X) changes constantly, and there is a linear relation between its position and its velocity:

X(t) = X0 + V*t

where X0 is its position at t=0. The graph describing the relation between position and time (when velocity is constant) is, of course, a straight line. What is the body’s velocity? Rearranging gives:

V = (X − X0)/t

Velocity is the difference in positions traversed divided by the time it took. It is the slope of the above graph.

What happens if the body’s velocity is not constant (changes with time)? Then the position–time relation is not a straight line, and one cannot define a single slope. At each point it has a different slope. In that case we define the instantaneous velocity at time t, denoted V(t), computed as the ratio between the change in position and the time during which that change occurred, over a very small interval around time t. In mathematical notation:

V(t) = lim (delta(x)/delta(t))

delta(t) –> 0

This is the slope of the tangent at that point. We can see that here too velocity is a ratio between distance and the time to traverse it, but distance and time are very small segments around the time at which the velocity is computed. The resulting V(t) is the velocity at that instant. This means that at each moment the body can have a different velocity, since velocity is the slope of the position–time graph at that instant, and because that graph is not a straight line, each point has a different slope.

This is, in fact, the definition of the derivative of position with respect to time (the derivative is the local slope of the function’s graph). Likewise, one can speak of derivatives of other quantities as functions of other variables. For example, one can speak of height as a function of location, and represent it by a graph. That graph has a slope given by the derivative of height with respect to place. In everyday language we also call this a “slope.” For instance, a steep mountain slope means that height changes very rapidly with small horizontal displacement. A gentle slope means the change is not so sharp. When the slope is not a straight line but varies—sometimes steeper, sometimes milder—we define the local slope by the derivative at that place.

What This Description Means: Velocity vs. Change of Position

We saw that to compute a body’s velocity we had to observe it over a time interval (however short, infinitesimal), and it was not enough to know its position at a single instant. This would seem to reflect the fact that one cannot speak of velocity at an instant, just as Zeno claimed. On the other hand, the result of the computation is a velocity function V(t) that gives, at every time t, a possibly different velocity V. Note that this function gives us a value of velocity at each discrete instant, which means that we can indeed speak of velocity at an instant. Therefore Zeno’s assumption is false: observing a discrete instant does not imply that the body is at rest. Contrary to our intuition that velocity is defined only over a time interval, a body has a well-defined velocity at every discrete instant of time. The way to compute this velocity from the body’s position forces us to use an interval around that instant. In other words, one can define a slope at a discrete point of a graph, but computing it requires looking at a neighborhood of the point, not only the point itself.

It goes without saying that I am not claiming the body changes its place at that discrete instant, only that it has a well-defined velocity at that moment. Its transition to another place necessarily takes time—however short—i.e., it must occur over an interval.

The conclusion is that although a body does have a velocity at a discrete instant, the change of position caused by that velocity will appear only if we observe the body over an interval (and not at a single instant). Why is this so confusing? Because the definition of velocity in mechanics is given via the derivative of position, i.e., it requires an interval and is not made at a point. But this is an operational definition, not an essential one. It is the way to compute velocity, not its essence. The intuitive mistake—that velocity is defined only over an interval—stems from conflating the concept “velocity” with “change of position,” and from confusing the operational definition with the essential definition.

Thus, the velocity we are speaking about here is not a change of place but a potential for change of place. The change of place is only a consequence of the fact that the body has a velocity—i.e., such a potential. The transition from potentiality to actuality occurs when the body changes its place.

A Solution to the Arrow Paradox

The conclusion is that even if a body is at some place at some moment, there is no obstacle to saying that at the very same instant it has a well-defined velocity. It does not change its place at a discrete instant, but it has a well-defined velocity. Hence it is incorrect to infer, as Zeno assumed, that at every instant the body is at rest. The body certainly cannot change position while having a defined position—since that would violate the law of non-contradiction (it is both here and not here)—but it certainly can have a velocity at that instant. We have located the error in Zeno’s argument. At each discrete instant the body indeed occupies a well-defined place and does not change its position, but it has a velocity. To the question “when does it move?” the answer is: at that very instant that it is at place X it also has a velocity; that is, it is also moving. The fact that it has a velocity will be manifested in a change of place at subsequent instants. Motion does not occur at a different moment than being at some place; it is simultaneous.

Importantly, the solution I propose to Zeno’s arrow paradox does not require infinitesimal calculus or quantum theory. It is a correction of a conceptual mistake (just as analytic philosophy would claim). Zeno asserted that at each discrete instant the body “stands” at some specific place, but that is a mistake: at each instant it “is located” at some specific place, but it is not necessarily “standing” there. The difference between “being at a place” and “standing at a place” is that being at a place can also be true of a moving body (i.e., one that has a velocity). Saying that the body is standing at some place means it is there and its velocity is 0. Zeno conflated being at a place with standing at a place, and thus fell into paradox.

Unlike the previous explanations (mathematical and physical), this explanation of the arrow paradox is unrelated to the problematic notion of the continuum. On the present view, the paradox rests on the conceptual confusion between “change of position” and “velocity,” or between “being at a place” and “standing there.” In this sense, one can indeed continue to hold that time consists of an assembly of indivisible instants densely adjacent to one another. I am not arguing for that view (which, as is known, raises other problems), but only disconnecting it from the difficulty raised by the arrow-in-flight paradox.

Between “Standing in Place” and “Being There”: Examples

Our conclusion thus far is that there is a difference between being at some place and standing there; likewise, between having a velocity and changing place. Velocity causes change of place (or is a potential for change), but velocity itself is not change of place. To clarify these distinctions further, I will now give two examples from mechanics in which there is no change of place even when a body has a velocity. Such cases illustrate the claim that one must not identify velocity with change of place.

When a body collides with a wall, even if its velocity has a value different from zero, the wall stops it—that is, it does not allow that potential for motion to be actualized as forward motion (change of place). This is a case where the potential (velocity) exists, but its consequence (change of place) does not follow. Similarly, when a body collides with another body, part of its velocity can be transferred to the second body, so not all of the moving body’s potential is actualized by that body itself. Part is actualized in the second body (which starts to move—i.e., to change place—following the collision). In inelastic collisions (where the body’s momentum—i.e., its velocity—is not conserved), the velocity is converted into heat dissipated into the environment or other forms of energy, at the expense of change of place. In such cases it is not actualized as a change of place at all.

Back to the Relation Between the Dynamic and the Static

Let me add another point that brings us back to the question of representing dynamic processes statically. When we observe a moving body, all we truly observe is that it is in different places at different times. Our eyes perceive static pictures: at every discrete instant the body is at a different place. The dynamism of the picture, the film—the sense that there is motion here—is created by our brain, which interpolates between the static images (the different positions at different moments). This is why the definition of velocity in mechanics is by the derivative of the position function. Human perception prefers static concepts (like position at a moment), and therefore it defines dynamic concepts (like velocity) by using the static ones. This is exactly what confused Zeno, who identified velocity with change of place. Changes of place indicate that a body has a velocity, but they are not velocity itself. We have no direct way to perceive that a body has a velocity. That is why a film is always created by a sequence of static frames, as I explained in the previous column.

In other words, Zeno is quite right that human perception cannot detect a body’s velocity at a discrete instant. But that does not mean the body lacks velocity at such an instant. It certainly has velocity, and our difficulty in perceiving it stems from the static mode of our thought. Our consciousness forces us to attend only to changes of place caused by velocity, and those occur only over intervals.

Note another point. In a film, the frames are projected at high frequency, but there is a tiny time gap between each frame and the next. It is still discrete and not continuous. As explained, the dynamic picture is created only in our brain. Therefore Zeno’s paradox does not arise with respect to film, because the transition between frames truly occurs during the brief moments in which nothing happens (the empty instants between frames). Consider a film of a flying arrow. There the arrow truly has no velocity, since at every instant it is standing (not merely located) at a different place. The philosophical problems arise when we consider events in reality itself, not in a film. An arrow in the real world keeps flying at every moment; there are no gaps between frames. The frames are dense with no gaps between them (this is the density property noted above). In such a case, motion is continuous rather than discrete; this is a genuine dynamism existing in the world itself (not only in our mind), while only our representation of it is discrete (via frames). Note the difference: in a film what is real is the static (the collection of frames), and the dynamism is created in our mind. In the real motion of a body, reality is dynamic, and the frames exist only in our inner cognition. In film the static is not a model of the dynamic; the dynamism is an illusion produced in us. For the velocity of a real body, the static representation is a model that represents an occurrence that is in fact dynamic.

This brings us back to quantum theory.

Implications: Back to Quantum Theory

As I explained above, if we have succeeded in resolving the arrow paradox, we can now proceed in the opposite direction and try to infer from our understanding of the flying arrow to the principle of complementarity in quantum theory. To do so, let us reformulate slightly the arrow-in-flight paradox.

Suppose we look at an arrow in flight and photograph it at different discrete moments (the hypothetical camera has exposure time 0—one discrete instant. This is an ideal camera). In each photograph the arrow will appear to be standing still, though its position changes each time. Zeno’s question then arises: when does it move between these different positions? Our answer is that the arrow moves (has a velocity) and at the same time has a position. The fact that we do not see its motion in a photograph should be ascribed to the fact that a camera is not the right device for observing motion (or measuring velocity). A camera measures (or observes) positions—just like our eye. By analogy we can now define a hypothetical device that I will call an ideal movie camera. This device measures or observes a body’s velocity at a single indivisible moment. As explained above, our consciousness operates in a static mode, so it is hard for us to imagine such an operation. Hence an ordinary movie camera, as we know it, actually functions as a camera that takes static pictures (frames) rapidly in sequence. By contrast, the ideal movie camera does not measure velocity via differences of position using the definition of velocity as the derivative of position (as our cameras do, constrained by the static limitations of our consciousness), but rather in a point-like way. The movie camera “captures” a body’s instantaneous velocity. Continuing the analogy, if we observed a moving body with such a device, we would see it moving at every instant but would not be able to discern its position. (An analogy: a camera with a long exposure time shows that an object moves by a streak in the image, but we still see only an assembly of positions and do not truly know the velocity.) Likewise, with the ideal movie camera we could know the body’s velocity exactly but would have no idea of its position. By analogy, my sense is that even with an ideal movie camera, if we increase exposure time (or open a wider aperture) we might glimpse some position as well.

Thus, the information we get about the moving body depends on the device through which we observe it. Our consciousness, which is static at its base, serves us as a camera and therefore gives us directly only information about positions, while velocity is obtained indirectly by interpolation between positions (what the derivative does). If we continue this line we may say that even if it were possible to know a body’s instantaneous velocity using a movie camera, one could not simultaneously speak of its position. Now let us assume, as a reasonable hypothesis, that just as our consciousness is forced to function as a camera, there could be a consciousness of another creature that functions as an ideal movie camera (for it, filming would not be done via frames but directly). I now add that there is a contradiction between these two modes: one cannot operate as an ideal camera and an ideal movie camera simultaneously. You can now see an intuitive explanation for complementarity, which denies the possibility of knowing a body’s position and velocity simultaneously. The uncertainty principle further adds that the more precisely we know one, the less precisely we can know the other (a quantification of complementarity). I think the level of certainty corresponds to the exposure time of the camera or movie camera, as explained above.

Our discussion of the moving arrow led us to the conclusion that if a body has a defined position at a given instant, it is meaningless to ask when it changes its position. But the “other side” of the uncertainty principle—which states that if a body has a defined velocity it is meaningless to ask what its position is—does not follow directly from the preceding argument. We reached it by analogy between camera and movie camera, and between position and velocity. In fact, I think one can present a parallel argument based on the notion of the integral (defining position via velocity) that would show the other side.

As everyone familiar with quantum theory knows, the uncertainty principle implies that there are two ways to describe dynamic quantities: in the momentum picture (usually proportional to velocity) or in the position picture. These are two “pictures” or languages in which the system’s properties are characterized by the probability of finding a given velocity, or the probability of finding a given position. In the momentum picture, all physical quantities are described using velocity coordinates. Thus, for example, energy E would be described as a function E(V). In the position picture everything is described using position coordinates; e.g., energy would be a function E(X). It is known in quantum theory that these two languages are dual, i.e., they exclude one another (in scientific jargon: they are non-commutative). We now have a very simple interpretation of this surprising result: the inability to know a body’s position and velocity simultaneously stems from the fact that these quantities belong to different conceptual systems (pictures, or languages) that do not “speak” to each other. Position is seen with a camera (which gives us the world in the position picture) and velocity with a movie camera (which gives us the world in the momentum picture). We saw above that one cannot use both languages simultaneously—i.e., one cannot observe the world with a camera and a movie camera at the same time. This argument also sheds light on the dual pairs of pictures accompanying, in quantum theory, pairs of physical quantities that stand in uncertainty relations with each other.

As an aside, one could suggest that velocity and change of place (or, more generally, the process and the change it causes) do not exist on the same plane. The process is the thing-in-itself, whereas the ensuing change is the phenomenon. If this is correct, then we cannot really speak of these two quantities as dual in the sense that one is grasped via a camera and the other via a movie camera. The thing-in-itself cannot be grasped by any device. A detailed analysis raises several problems that are beyond our scope here.

Implications of the Distinction Between Process and State Change

As Henri Bergson already noted in Creative Evolution, the arrow-in-flight paradox does not concern only velocity but every process of change. In this section we will see the implications of the foregoing analysis for issues unrelated to velocity or mechanics at all.

I begin with the fact that in the picture presented here there is a passage between states, which illustrates that there is a dynamic process in the background. Velocity is a potential for change of place, and change of place is the result of actualizing that potential. We can now examine, in any dynamic situation, whether we are speaking about velocity or about changes of state. They usually come together, but as we shall see, there is room to discuss which is primary and which is only an accompaniment.

A first example is a claim encountered in management literature: that the very process of change is good for an organization. That is, even if there is no problem with the organization’s current structure, benefit accrues from changing it because of the process itself, not because the resulting state will be better than the current one. Dynamism is required not in order to reach an improved static state; the goal is not the changed state but the very dynamism. Of course, a change leading to a worse static state (e.g., the general destruction of the plant’s machines) does not improve the organization, but assuming there is a set of organizational states none of which is preferable to the others, it is better for the organization to be in transition among them than frozen in one. I am not taking a position on whether this claim is true; I am analyzing its meaning. This is a clear case of a benefit arising from the existence of a process even without an accompanying change of state (velocity without change of place). It seems such a benefit cannot be described in the language of “state change” at all, but only in terms of “process,” since we do not need a different state (structure), only the very process of change.

Another interesting example where only one term can be used and not the other is the discussion of repentance (teshuvah) and the problem of divine perfection versus becoming (see both in column 170). Regarding penitents, the Sages tell us that a penitent is greater than a perfectly righteous person. If the goal of repentance were only to improve our spiritual state, then the penitent could at best be equal to the perfectly righteous. If he fully succeeds, he reaches a perfect state and becomes perfectly righteous. It is quite clear that the Sages see repentance as a dynamic process with value in itself, not just as a means to reach an improved spiritual state.

In that column I compared this to Rav Kook’s problem of perfection versus perfecting. He formulates it thus (Orot HaKodesh, vol. II, p. 531):

There is a perfection of added perfection, which is impossible in divinity; for absolute infinite perfection allows no addition. For this purpose—that added perfection not be lacking in existence—the worldly existence must come to be.

The premise is that perfecting (spiritual progress) is itself one of the perfections and thus must exist also in divinity. On the other hand, we clearly cannot speak of God’s becoming more perfect for two reasons: (a) no change can occur in God; (b) He is perfect, so we cannot speak of spiritual advancement. We saw there that Rav Kook’s solution parallels the organizational case above: the process itself (or at least its root) exists in God, though the change that usually accompanies it cannot exist in Him for the reasons just noted. That is, the spiritual perfection manifested in perfecting lies in the process, not in the changed state, and this can characterize God as well. Change of state is impossible for Him (He cannot become more perfect), but the process exists in Him—just as we saw with bodies that have a velocity that cannot be actualized (e.g., when they hit a wall).

In column 519 I discussed the relation between dynamism and staticity in the context of the dispute between Heraclitus and Parmenides (within a critique of Assaf Inbari’s words on secular revelation). There too I pointed out the link between this topic, the arrow, and the notion of the derivative in mathematics. There are also several examples from the Talmud where this distinction between change of state and process can be applied. In the next column I will discuss such an example in detail and its implications (see a brief note here).

On Two Time Axes: Bergson and Einstein

Having presented some implications of the distinction between “change of state” and “process” in non-physical domains, I return to another important point—in physics again—where this distinction is manifest: the description of time in Einstein’s theory of relativity, and whether it is exhaustive.

The most salient feature of time, as opposed to space, is that it flows. Space is perceived as static and frozen, whereas time flows constantly. Time is seen as the very essence of dynamism, and some even identify them (i.e., take time not as an existing entity but as a way of describing dynamic occurrences). One could even go further and claim that what truly exists in the world is only a sequence of static states, and that time and dynamism are concepts we invented for our needs to handle such sequences (see a bit on this in column 33). Such a claim relies on the modes of perception described above. I assumed these are the results of the limitations of our cognition, whereas proponents of these views claim that this is reality itself.

Richard Taylor, in his book Metaphysics (ch. 7), shows that space and time have identical features, by this argument: in any sentence describing motion or a relation between space and time, one can replace the temporal relations by parallel spatial ones, and vice versa. If this is done consistently and appropriately, the result is always a clear, well-defined sentence (see different examples there and also in column 33).

From this the positivist can—and usually 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 also the picture arising from Einstein’s relativity. In that theory, time is one of four coordinates describing an event in Minkowski spacetime. The basic object is the “worldline” of a body—the line describing its positions at all times. In this picture, the worldline is perceived as static and ignores the phenomenon of the flow of time. Time appears there as just another dimension of space.

A common philosophical argument for rejecting the feeling of the passage of time and classifying it as mere illusion is this (see also that column): “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 at one place at one time and at another place at another time.” If we wish to discuss the flow of time, we must place time itself as the subject (A) of such a sentence. But what results is meaningless, since time itself cannot be in some state/place at one time or another.

In column 33 I proposed that the flow of time is described along a second time axis serving as an index describing the state of ordinary time and its movement. A regress threatens, because the new time axis would be subject to the same challenge if we wanted to ascribe to it a constant flow. This regress can be stopped by positing that one time serves only as an index and does not flow, while the other flows along it. A prominent defender of the flowing-time view (“duration”) was Henri Bergson in Creative Evolution, who even debated Einstein, who argued the opposite (I mentioned his discussion of Zeno’s arrow above). Each side claimed exclusivity for its conception, whereas I suggest here that each represented a different aspect of time—in other words, one of the two kinds of time defined above. A slightly different description of these two aspects (or kinds) of time was proposed by McTaggart already in 1908, and there is even an interesting attempt to give it a certain mathematical–physical garb. See also columns 465–466.

Our concern here, in this description of time, is the relevance of these two kinds of time to the description of processes as opposed to state changes, as defined above. My claim is that flowing time “carries” the process, whereas the static states that change are characterized by an index—also called “time”—that marks their “location” on the basic time axis. This index is time of the second kind.

Here too one might ask, as Zeno wonders in his arrow paradox: how does time flow? That is, at each discrete instant of index time (Einstein time), the flowing time (Bergson time) is at a different time. So when does it change the index? This question seems meaningless, because Bergson time is not a quantity that flows in time; it is the flow itself. Every dynamic quantity is carried by Bergson time and produces changes in the static states characterized by Einstein time. Our sense that time passes means that we pass along Einstein time, and Bergson time carries us. The consequence of our flowing in time is that we find ourselves each time at a state with a different Einstein-time index.

This means that the feeling of the passage of time expresses the dynamism at the basis of the reality around us, but we have difficulty noticing it because of the static nature of our cognition. Therefore we mark ticks along the time axis and thus create for ourselves a static representation (frames) of its dynamism. If we could perceive in the mode of an ideal movie camera—that is, observe the very process—we could describe velocity directly as carried on Bergson time. This is also why Einstein’s relativity describes only index time: that is the time directly accessible to our consciousness. The ideal movie camera described above operates along Bergson time and thus can observe velocity (or change) at a single instant.

I think the feeling that Einstein “defeated” Bergson in their debate about time is mistaken. Einstein had the advantage as a mathematician and physicist and presented a scientific theory with empirical consequences for his concept of time, so people feel that Bergson clung only to philosophical horns and opposed empirical findings. Here I argue that indeed science deals with Einstein’s time because that is the phenomenon. It is the aspect we can observe and grasp in our static consciousness. Therefore, for us, velocity is defined as the derivative of position. But in truth this is a limitation of our perception, not a claim about reality in itself (the noumenon). Here Bergson is right, and this is the domain of philosophers, not physicists. This may also be why Einsteinian time in relativity varies with the frame of reference: each observer perceives it differently, since it belongs to the phenomenal aspect. By contrast, Bergsonian time is objective. According to Horwitz and colleagues in the article cited below, this time is the interval in relativity (usually denoted by the Greek letter τ), which is invariant—i.e., does not change among frames. In my terms, this is the noumenon of time. Returning to column 519 on Assaf Inbari, I can now say that Parmenidean time is scientific (Einsteinian) time—the phenomenon—and Heraclitean time is Bergsonian time—the noumenon.

Conclusion: Back to Art

In the previous column I noted the concatenation of a series of frames in text or painting into a dynamic film, and a single painting (a frame) that succeeds in expressing dynamism by focusing on one moment (freezing it) out of a dynamic whole. We can now say that the art of Bak and others is to represent the noumenon, which usually eludes us, and to set it tangibly before our eyes. This allows us to see Bergson time through Einstein time. When one sees a single painting and succeeds in extracting from it a dynamic sense, this means we are operating in the mode of an ideal movie camera (and thus there is no need for a derivative—i.e., for more than one frame—to apprehend that there is a dynamic process before us).

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