Zeno's Arrow Pardox and Contradiction - Politics Forum.org | PoFo

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I'm looking for some feedback in considering this Zeno paradox in part because it seems of particular interest to Marxists that claim an some level of affinity with dialectics. This doesn't have a clear goal, but just wanting to put it out and see what comes of it, trying to work through it as it was something that came up unexpectedly in a past discussion and left me overstretching myself into the unknown for a lack of education with certain concepts around motion.

Here is a summary of the Arrow Paradox explaining how it seems to make motion incomprehensible.
Spoiler: show
The arrow cannot move. To do so requires that it be in one place equal to itself during one part of an instant and another during another. Also, it would occupy a space larger than itself in order for it to have room to move.(4),(5)


Everything at a place equal to itself is at rest.

A flying arrow is always at a place equal to itself at every instant in its flight.

RP1 & RP2 => RC1

A flying arrow is at rest at every instant in its flight.

That which is at rest at every instant does not move.

RC1 & RP3 => RC2

A flying arrow does not move.
While the above rendition of the argument suffers from the fallacy of composition, it is possible to render the argument in a form not subject to this fallacy. This can be done as follows:



An instant is an indivisible minimal element of time.


Something is at rest (instantaneously) iff it is in its place (one place equal to itself) in one instant and it is in the same place (equal to itself) in different instants (remains at rest).


Something moves iff it is not at rest.

RD2 & RD3 => RC2

Something moves iff either (A) it is not in one place (equal to itself) in one instant or (B) it is in different places (equal to itself) in different instants.
I will present the two disjuncts as separate cases.

Case 1:

(A) That which moves is not in one place (equal to itself) in one instant.

It would appear that this case could be disposed of immediately by noting that it seems to contradict RP2 directly. -- RP2 could be interpreted that everything is always in a place equal to itself. -- However, it is instructive to analyze more deeply. We can consider the 'not' as applying to "one place" or alternatively to "one instant". Let us first consider the 'not' applied to "one place". "Not one place" becomes "different places".
RI42 RC2 => RC22
RC22 Something moves iff it is at different places in the same instant. (An instant has more than one place.)
Although this interpretation is practically inconceivable to us, it is the interpretation intended by the argument. But we can think of it as like the blurred photograph of something in motion. The object is apparently at (many) different places (equal to itself) at the "instant" the photograph was taken.


If something is at different places during the same instant then it is not at one place equal to itself.

RP4 => RC3

If something is at one place equal to itself then it is not at two different places during the same instant.

RP2 & RC3 => RC4

An arrow in flight is not at different places during the same instant.

RC22 & RC4 => RC5

An arrow in flight does not move.
The "also" clause has more the form of an "otherwise" clause.


Something cannot occupy a space smaller than itself.

If something is not at a place equal to itself then it occupies a space either smaller than or larger than itself.

RP5 & RP6 => RC7

If something is not at a place equal to itself then it occupies a space larger than itself.

RP4 & RC6 => RC7

If something is at different places during the same instant then it occupies a space larger than itself.

RC22 & RC7 => RC8

If something moves then it occupies a space larger than itself. (An arrow must occupy a space larger than itself if it is to move.)
RC22, however is also in direct conflict with a widely held premiss.

RP7 Nothing can be in two different places (equal to itself) during the same (one) instant.
This case concludes that the arrow cannot move during an instant. We are left with case 2.

Case 2:

(B) That which moves is at different places (equal to itself) in different instants.

RI41 RC2 => RC21
RC21 Something moves if it is at different places (equal to itself) in different instants. (This is our usual understanding of motion.)
We are considering whether an arrow can be in motion in an instant. By the above case, something could move only if it were in different places in different instants. Therefore, for it to move in the one instant under consideration, that instant would have to have two parts which were also instants. It would be these "sub-instants" in which the arrow were at different places. But, by RD1, an instant is indivisible; so, it has no such parts which are instants. Consequently, at each instant it is not possible for the arrow to be at different places in different instants.

In either case the arrow cannot move. Consequently the logical disjunction of the two cases also yields an unmoving arrow.

The physics of the relation between position and velocity has some interesting structural consequences. The Heisenberg Uncertainty Principle states that the position and the momentum (velocity) of an object cannot be simultaneously measured to any degree of accuracy; accuracy in the measurement of one is lost at the expense of accuracy in the measurement of the other.(16) A homely macroscopic analogy illustrates this principle.

Take a photograph of an object in motion. The length of time the shutter is open (the reciprocal of the shutter speed) can be used in conjunction with the amount of blur in the image to estimate the speed of the object. The longer the shutter is open the longer the blur and the more accurately the speed can be measured. But the longer the blur is, the less accurately one is able to determine the position of the object. Conversely, the sharper the picture is, the more accurate knowledge of the position of the object will be, but the more uncertain knowledge of its velocity will be.

In Zeno's thought experiment a very sharp view, namely, the arrow being "in its place", is taken; this leaves no blur at all to use in determining the velocity. The mental shutter speed would have to be infinite to obtain an indivisible instant -- we are left with an instant with zero duration. Since motion is measured by determining the ratio of distance traveled to the time duration, Zeno's thought experiment leaves zero (length blur) divided by zero (length duration) for the computation of velocity. And zero divided by zero is undefined. One has perfect information about the position but no information about the velocity. While it is true that a stopped object leaves no blur, it seems fallacious to assume that velocity is zero when one sees no blur. And any blur at all does have the immediate consequence that the object is indeed occupying a space larger than itself.

It has been argued that an object always takes up the space it occupies. It can also be argued that an object in motion always takes up less space than it occupies. Relativity theory holds that an object in motion is contracted in the direction of motion. The shortened length, X', can be calculated in terms of the at-rest length, X, and the velocity, V, using the Lorentz contraction equation. That equation is X'=X(1-V2/C2)1/2, where 'C' is the speed of light.(17) The faster an object is moving, that is, the larger V is, the smaller X' is. If an instant is indivisible and is the same "size" regardless of whether the object is at rest or in motion, then the moving, and hence contracted, object takes up less space; it has room to rattle around in the same sized instant which immobilizes the object at rest. But according to relativity theory objects in motion experience a time dilation effect. That can be interpreted to mean that the size of an instant is increased or "stretched".(18) This gives the moving object even more room. Also according to relativity theory faster moving objects are contracted more. The greater the contraction, the relatively greater the room to move -- and hence the greater the speed possible.

Consider the possibility that an instant is indivisible in the sense that it has no duration. An instant without duration is not consistent with our usual definition of velocity. Velocity is the ratio of distance traversed to the duration in which the traversal occurs. An instant with no duration would be just the temporal coordinate of an object. Taken together with its position coordinate, the result forms the event coordinates of a particular space-time point. It is not possible to determine the velocity of an object on the basis of a single event. At least two events are required. Even our notion of velocity at a point depends upon more than one event. We define the velocity at a point as the instantaneous rate of change of position with respect to time. An instantaneous rate of change, from the differential calculus, involves taking the limit of the ratio of the change in position (distance) to the change in time (duration) between two points. That limit may be some definite quantity, but all computations not at the limit require an extended duration, and no computation is possible at the limit (the denominator would be zero, and division by zero is forbidden). If an instant has no duration, then no velocity within the instant is possible. Velocity at an instant is not determinate without reference to the context of the instant, that is, events external to the instant.

The renditions of the argument as I have presented them significantly reflect our modern view of motion. The Arrow as originally presented speaks only to motion within an instant. The definition of motion I proposed, a modern one, in conjunction with the attending argument concludes, validly, that motion is a phenomenon that is "trans-instantaneous". If instants are infinitely divisible, as the divisionists presume, then any moving object exists in different places (equal to itself) at different instants, as close as you like. On the other hand, If instants are atomic, as the atomists presume, then any moving object exists in different places (equal to itself) at different instants, and the motion is as described more fully in the discussion of "uniform" velocity on page 72 below. But in both cases the concept of moving, as we understand it, cannot apply "within" an instant. Our concept of motion, which is based upon infinite divisibility, requires that we think of objects as moving continuously. If we think of instants as intervals (durations), that means we think of the object as crossing each interval continuously until it enters the next instant, crossing that into its successor, etc. But this view is not consistent with atomism, as the Arrow shows.

Why is this paradox of relevance, well it seems there is a charge laid against formal logic of having certain limits which it'll inevitable end up in contradiction.
An interesting example of how dialectical and formal logic can be seen as antagonistic is in Zeno’s arrow paradox. For Zeno since, at any discrete moment in time, the arrow is at rest, the arrow is at rest at every discrete moment throughout its flight. The arrow is both moving and always at rest: a paradox. For Hegel, the problem is with halting the arrow in its flight in order to “grasp” it at every distinct moment, much as we do when we capture a “piece” of data. It is clearly impossible to build up motion from a collection of moments at rest. However, if you start from the fact of motion, it is obvious that every moment of rest is merely a convenience, a way to “grasp” the arrow for analysis, but does not represent the truth of the arrow in motion. From a socio-political point of view, whenever we try to “grasp” a concept, institution, or phenomenon, we have to hold it still, violating the fact of its motion (i.e. how it changes over time); we also have to make it distinct (A and not not-A) by shearing it of all its relationships. Both of these operations do violence to the complexity and reality of the phenomenon under analysis. It is to try to resist this violence that dialectical logic is still important.

The idea for dialectical as I understand it, following Hegel, is that a contradiction is meant to be negated and raised to a higher conception free of the initial contradiction instead of the Kantian response.
The Kantian solution, namely, through the so-called transcendental ideality of the world of perception, has no other result than to make the so-called conflict into something subjective, in which of course it remains still the same illusion, that is, is as unresolved, as before. Its genuine solution can only be this: two opposed determinations which belong necessarily to one and the same Notion cannot be valid each on its own in its one-sidedness; on the contrary,they are true only as sublated, only in the unity of their Notion.

- Source

This has played out particularly in certain debates between logicians of the formal sort and dialectical camp.
Spoiler: show
The principle of continuity leads to the conclusion that during a certain timeinterval a changing object has contradictory attributes[682].

Schaff’s third and final argument for the rejection of the ontological principle of non-contradiction is Zeno’s time-honoured argument against motion. It was Hegel who in modern times revived Zeno’s paradoxes and turned them to his own advantage, namely to prove that motion is a ‘living contradiction’. Engels adopted Hegel’s use of Zeno’s arguments and Plekhanov made it prominent in his own presentation of dialectics. He also re-established the chain of ideas leading from Zeno through Hegel to Engels and passed it on to the Marxist-Leninist philosophy of our own times. Motion, wrote Plekhanov, is a ‘contradiction in action’, the moving body presents itself as an ‘irrefutable argument’ in favour of the ‘logic of contradiction’ [683].

Bertrand Russell observed a long time ago that Zeno’s paradoxes have inspired practically all the theories concerned with the concepts of time, space, infinity, and continuity, and that they have acted as a powerful stimulus to the development of these concepts from Zeno’s times to our own day. Marxist-Leninist philosophy is not interested in the theoretical aspect of Zeno’s paradoxes. The interest of Marxist-Leninist philosophy in Zeno’s arguments has been invariably limited to a single point, namely, to their usefulness in showing that contradictions sensu stricto exist ‘objectively’ in the phenomena of Nature.

Of all Zeno’s arguments against motion Schaff considered only one, that of the ‘arrow in flight’. In his opinion, it is not a paradox but a flawless argument[684].

It proves either that from the assumption ‘the arrow is in flight’ follows the conclusion ‘the arrow is at rest’, or that a moving body is at the same time in motion and at rest, it is at a given point and it is not there. In both cases the premiss that something moves entails a contradiction. We thus face the following alternative: either to deny the fact that motion is real, as Zeno did, or to accept Zeno’s argument and the reality of motion. The latter course implies the conclusion that there are contradictions sensu stricto in Nature. Something must be given up, either the reality of motion or the laws of traditional logic. A Marxist-Leninist accepts the evidence of experience and rejects the laws of traditional logic ‘Motion is a contradiction, a unity of contradictions’. Thus, to use Engels’ words, he recognises that contradictions are ‘objectively present in things and processes themselves’, and, as it were, assume a ‘corporeal form’ [685].
Thus, however, the contradiction allegedly inherent in mechanical motion remains as the only mainstay left to support the contention that the law of noncontradiction is a relic of the metaphysical mode of thought, to be abandoned at the dialectical stage of the development of mankind. The question as to whether Zeno’s analysis of the ‘arrow in flight’ can be maintained acquires crucial importance.

Ajdukiewicz subjected this matter to a thorough examination. His discussion of Zeno’s paradoxes, in many respects novel, deserves close attention in view of the role his examination has played in the evolution of Marxist-Leninist philosophy in Poland[706].

Ajdukiewicz examined two possible interpretations of the flying arrow argument. Schaff seems to have used both of them in different places or in support of each other. The first of these interpretations starts with the assumption that the arrow is in flight and reaches the conclusion that if it is in flight then it is at rest. Since

(p) : p ⊃ ∼ p . ⊃ ∼ p,

the assumption must be rejected as self-contradictory. The argument proceeds by the following stages.

If the arrow is in flight during an interval T, then for every instant t of the interval T there is a position x in which the arrow is to be found (A). If there is a position in which the arrow is to be found at every instant t of its flight (T), then the arrow remains at rest throughout its flight (B). This implication makes use of the definition of rest by which a body is considered to be at rest during the interval T if there is a position in which it is to be found at every instant t of the interval T. From (A) and (B) follows the conclusion: if the arrow is in flight throughout T, then the arrow is at rest throughout T. The assumption that the arrow is in flight implies that the arrow is not in flight.

The inference seems to be a substitution of the following theorem of the propositional calculus

p ⊃ q . q ⊃ r : ⊃ . p ⊃ r.

This, however, is not the case. The sentences to be substituted for q in p ⊃ q and in q ⊃ r are not the same but two different propositions. The consequent of the first premiss is, ‘for every instant t of the interval T there is a position x in which the arrow is to be found’, and the antecedent of the second premiss runs, ‘there is a position in which the arrow is to be found at every instant t of its flight (T)’. Symbolically expressed, the consequent in p ⊃ q has the form

(t) (∃x) φxt (q1),

and the antecedent in q ⊃ r

(∃x) (t) φxt (q2).

The inference under discussion is, therefore, a substitution of a different theorem of the propositional calculus, namely of

p ⊃ q . s ⊃ r : ⊃ . p ⊃ r,

which is a valid formula if and only if q ⊃ s. But this is not so in our case, since q1 does not imply q2 (though q2 does imply q1.). The rules which govern the use of quantifiers do not allow the inference

(t) [(∃x) φxt] . ⊃ . (∃x) [(t) φxt] (C).

A different example might make it clearer. The premiss ‘for every x there exists such a y that x < y’ does not imply ‘there is such a y that for every x:x < y’. Similarly, from the fact that for every man there is a man who is his father does not follow that there is a man who is the father of every man. Since the antecedent of (C) is true and the consequent is false, (C) is not a valid formula. The inference on which Zeno’s argument in its first interpretation is based is clearly fallacious[707].

The second, more common interpretation can be reduced to the following inference. If a body is in a definite position x at an instant t, then this body is at rest in x at the instant t (D). If the arrow in flight is in a definite position x at every instant of its flight, then the arrow is at rest at every instant of its flight (E). If the arrow in flight is at rest at every instant of its flight, then the arrow is at rest throughout its flight (F)[708].

The fallacy of Zeno’s argument is twofold. The first depends on the ambiguity of the connective ‘is’. The connective ‘is’ in the expression ‘the arrow is in the position x’ may mean the same as ‘the arrow remains at x’ (a). The connective ‘is’ may be also used, however, in the most general sense which does not specify the kind of relation between the arrow and its position, to mean ‘it passes x’ or ‘leaves x behind’ or ‘reaches x’ (b). If in the premiss (D) the first ‘is’ is used in the sense (b), the consequent of the premiss does not follow from the antecedent and the whole inference is destroyed. If, however, the connective ‘is’ in the antecedent of the premiss (D) has the meaning (a), it becomes a false statement, because a moving body certainly does not remain anywhere. Thus, there is no such meaning of ‘is’ in which the premiss (D) is true and the whole inference valid[709].

With the above distinction of meanings that the connective ‘is’ may have, Plekhanov’s problem does not offer any difficulty. If we disagree with the thesis that ‘a body in motion is at a given point, and at the same time it is not there’, Plekhanov wrote, we will be ‘forced to proclaim with Zeno that motion is merely an illusion of the senses’ [710]. This is exactly the contention which made of the ‘arrow in flight’ the main argument for the existence of contradictions in the phenomena of Nature. The antecedent of Plekhanov’s thesis is a perfectly true, though trivial proposition, if the first ‘is’ carries the meaning (b), and the second the meaning (a). We can agree with it without committing ourselves to the existence of ‘corporeal contradictions’. On the other hand, if ‘is’ is differently interpreted, Plekhanov’s antecedent becomes a false statement. We can disagree with it without being forced to admit that motion is illusory.

In order to know whether a body is in motion or at rest at an instant t, we must always consider what happens to this body at the instant earlier and later than t, i.e. we must consider a time interval that contains t [711]. Only then can we give exact definitions of what is motion, continuous motion, and rest. Zeno’s argument goes astray at this point. If the arrow is where it is at a given instant, it does not follow therefrom that the arrow is then at rest. We cannot know whether it is in flight or at rest without considering its position at earlier and later instants. Where it is now does not presume where it was before or where it will be after, i.e. whether it is or is not in flight. There are obvious excuses for Zeno’s paralogism; no physical definition of rest and motion was available at his time. Zeno cannot be blamed either for having made the assumption that there are consecutive instants and points. There are no such excuses for errors of the same kind when they are committed today[712].

To show that the alternative ‘either motion is illusory or it involves contradiction’ is invalid, since both its constituents are false, it is sufficient to apply the modern concept of mathematical continuity to Zeno’s paradoxes. Motion consists in the occupation by the moving body B of different places at different times. When throughout an interval, however short it might be, different times are correlated with different places, B is in motion. Similarly, B is at rest, when throughout an arbitrary interval different times are correlated with the same place. This prompted Bertrand Russell to say that Weierstrass ‘by strictly banishing all infinitesimal, has at last shown . . . . . . that the arrow, at every moment of its flight, is truly at rest’ [713]. By ‘being at rest’ Russell understands occupying a place equal to itself’. On this understanding Zeno was right in pointing out that an arrow in flight is at each instant where it is, irrespective of whether it does or does not fly. When the modern concept of continuity is applied, this does not imply that the arrow in flight is not at different places at different times throughout the interval of its flight.

But to the above I wonder if it misses the point or whether dialectical are indeed engaging in sophistry.
In regards specifically to Ajdukiewicz's treatment of Zeno's arrow paradox specifically, Henri Wald writes
Spoiler: show
p. 120
For fear of dialectics, B. Russel considered Heraclitus' famous aphorism: "We dive and do not dive in the same river" as mystical 68. However, he approves of Plato's formulation: "You cannot dive twice in the same river since new waters keep running over you" 69. In the latter formulation, the Heraclitean dialectical judgement: "The river that we dive in is concomitantly the same yet another" is reduced to the level of an elementary judgement, that no longer annoys the old logic.

The same is done by Polish logician K. Ajdukiewicz, who thinks that thinking may reflect the contradictory character of motion only be elementary logic. "For hundreds of years already the student in mathematics has known to point out Zeno's mistake in his argumentation. This mistake is to actually admit that the sum of the infinite time intervals ... would not be finite. With Zeno, the Sum t/2 + t/4 + t/8 + t/12 . . .cannot have a finite value. The elementary theory of the infinite geometrical sequences teaches us that the sum here is finite and has exactly the value of t" 70.

While Zeno raises the problem of the quality of motion, Ajdukiewicz measures it quantitative side. From Zeno's point of view, who refutes the very essential existence of motion, Ajdukiewicz measures only its appearance. The french philosopher Victor Bochard's reply to some mathematicians claiming that Zeno's aporiae have long been solved, is fully applicable to Ajdukiewicz, too " . . . to compute the exact moment when Achilles will reach the turtle. . . to answer the question when? while you are being asked how?" 71.

p. 203
Unfortunately, the Polish logician Kazimietz Ajdukiewicz also thinks that way. In his opinion, Zeno's famous aporia on movement was due to the fact that the famous Greek philosopher did not know that the sum of infinite geometric sequences is finite 35. Ajdukiewicz is angry with Marxists because they still believe in the contradictory character of movement long after mathematics has solved this problem. However, mathematics measured the quantative aspects of movement, whereas Zeno referred to the very quality of movement. Zeno refuted the essence of movement, and mathematicians measured only its appearance. Zeno asked himself how Achilles could reach the tortoise, whereas mathematicians computed when Achilles could reach it. Mathematicians solved a problem, other than that set by Zeno, to say nothing of the fact that the solution provided was also highly contradictory: the infinite geometric sequences have a finite sum.

Marxistm has already answered a similar "objection" meant to set metaphysics in place of dialectics. "This objection is unjust: (1) it describes the outcome of movement, not movement itself; (2) it does not show, it does not involve in itself the possibility of movement; (3) it shows movement as a sum, as a sequence of states of rest, viz, it has not removed the dialectical contradiction, it only covered it, hid it, attenuated it 36, Lenin replies.

The above point seems to not be perhaps unique to Henri Wald.
Spoiler: show
Why mathematical solutions fail. These and other attempts at resolving Zeno’s paradoxes may make perfect mathematical sense and yield equations that are of great use in that domain. Nevertheless, in metaphysical terms they do not even scratch the surface of the problem which was at the heart of Zeno’s formulation of his paradoxes: the impossibility to conceptualise the passage from One to Many.

With its manipulation of the unit mathematics finds “ways out” of the immobility of the arrow, condemned, according to Zeno, by the self-identity of its position at any moment, never to accomplish the transition from rest to motion. But the point, quite generally put, is that using Zeno’s rules of the game, the unit cannot be manipulated, and furthermore a manipulation of the unit does not resolve the problem of the passage from one to many, from the unit to concrete plurality. Zeno’s rules of the game were the acceptance of the Parmenidean prohibition19 that only one or the identical being can be thought of whereas the many of becoming as non-being yet or non-being anymore is unthinkable.

Only being is real. Only the identical with itself is thinkable. The "way out" of this imperative is the position of the pluralists who denied reality to the identity of one altogether and declared that in fact the process of becoming is real. In this way they did not have the problem of how to attain the multiplicity starting from the identity because they simply did not start from it, as they privileged life over logic. But this alternative position that privileges the reality of becoming over that of Being does not offer a way out of Zeno’s paradoxes for it simply dismisses the rules in which they are generated. These two positions, in fact, even though historically associated, are incommensurable.

Zeno’s target, then, seems more likely to be the Pythagorean pretence to get the many of the Universe by multiplication or addition of the unit. As Kathleen Freeman writes regarding this matter:

Zeno’s attack was on the idea of the Many, that is, of multiplication.....multiplication in itself is useless....It is useless because you are bound to start with either a Nothing or an Infinite, and by its means you get only what you start with, either a nothing or an Infinite.20

In other words Zeno argued that One (a non divisible) is one and can never become many and that Many (a divisible) will always be a quantity and, therefore, can never be exhausted by division in order to make of it a One. If you accept this logic, you are hooked and you can easily see how this assumption hinders the conceptualisation of change and movement, when this is intended as a passage from one to many, from the identical of a resting position to the concrete plurality of movement. When, in other words, one tries to find a way out of the simple logic of the identity in whose framework Zeno’s paradoxes arise. Many is either empirically or phenomenally given (experienced) or it cannot be conceptualised as a passage from the identical or being in which we think any existent, to the many of movement or change. If you think in terms of being what you are left with is always a new being, without so being able to capture the whom through which a certain being becomes a new different being.

But if you think in terms of a given change, you never conceptualise the identity of being as you are at another level, that of life rather than logic. In this sense we must also stress that Zeno’s paradoxes do not add anything to the Parmenidean prohibition to think only of the identity. One is One and cannot be Many. If we want to think logically, we can only think the identical, because identity is the form of our thought21: our thought can only be identical with itself when it thinks, that is it cannot think two things at the same time22. The response of the pluralists and all those who embraced a similar philosophical creed (see in more recent times Hegel and Bergson23) was to refuse to think of the existent as being, but to think of it as becoming. This as I said, though, was not a solution to Zeno’s paradoxes as it simply embraces a new “logic”, the logic of becoming that denies the identity. But if you acknowledge the logic in which Zeno’s paradoxes arose, you cannot but accept them as a description of an impasse that is constitutional to our thought.

This seems to be in accordance with one interpretation of Marx's sense of motion expressed by Bertell Ollman.
This is not to say that dialectical thinkers recognize the existence of change and interaction, while non-dialectical thinkers do not. That would be foolish. Everyone recognizes that everything in the world changes, somehow and to some degree, and that the same holds true for interaction. The problem is how to think adequately about them, how to capture them in thought. How, in other words, can we think about change and interaction so as not to miss or distort the real changes and interactions that we know, in a general way at least, are there (with all the implications this has for how to study them and to communicate what we find to others)? This is the key problem addressed by dialectics, this is what all dialectics is about, and it is in helping to resolve this problem that Marx turns to the process of abstraction.
Even today few are able to think about the changes they know to be happening in ways that don't distort—usually by underplaying—what is actually going on. From the titles of so many works in the social sciences it would appear that a good deal of effort is being directed to studying change of one kind or another. But what is actually taken as "change" in most of these works? It is not the continuous evolution and alteration that goes on in their subject matter, the social equivalent of the flowing water in Heraclitus' river. Rather, almost invariably, it is a comparison of two or more differentiated states in the development of the object or condition or group under examination. As the sociologist, James Coleman, who defends this approach, admits, "The concept of change in science is a rather special one, for it does not immediately follow from our sense impressions . . . It is based on a comparison, or difference between two sense impressions, and simultaneously a comparison of the times at which the sense impressions occurred." Why? Because, according to Coleman, "the concept of change must, as any concept, itself reflect a state of an object at a point in time" (Coleman, 1968, 429). Consequently, a study of the changes in the political thinking of the American electorate, for example, gets translated into an account of how people voted (or responded to opinion polls) in 1956, 1960, 1964, etc., and the differences found in a comparison of these static moments is what is called "change." It is not simply, and legitimately, that the one, the difference between the moments, gets taken as an indication of or evidence for the other, the process; rather, it stands in for the process itself.

In contrast to this approach, Marx set out to abstract things, in his words, "as they really are and happen," making how they happen part of what they are (Marx and Engels, 1964, 57). Hence, capital (or labor, money, etc.) is not only how capital appears and functions, but also how it develops; or rather, how it develops, its real history, is also part of what it is. It is also in this sense that Marx could deny that nature and history "are two separate things" (Marx and Engels, 1964, 57). In the view which currently dominates the social sciences, things exist and undergo change. The two are logically distinct. History is something that happens to things; it is not part of their nature. Hence, the difficulty of examining change in subjects from which it has been removed at the start. Whereas Marx, as he tells us, abstracts "every historical social form as in fluid movement, and therefore takes into account its transient nature not less than its momentary existence" (My emphasis) (Marx, 1958, 20).

Which points me to a differing ontology in which time is perhaps treated as an independent thing external to objects that acts upon them.
Time as Abstraction
2. A theory which takes on itself the task of explaining away change by what is abstracted from change, namely the time and the different values of the property that changes, will get into the kind of paradoxes as those presented by Zeno. No matter how you try to connect those abstractions mechanically from outside, you can never succeed to construct change from them,as those are not parts which are there by themselves merely connected in notion of change. For a particular change – a movement which can be described by the s=v*t formula, that mathematical description is merely an abstraction of how the potential measurements in some given framework happen, but the theory about time and values separate from change, about time and position separate from movement, will try to say that what is first is t and x. That there are ontologically existing positions, and ontologically existent times. And in the moment t1, it is in position x1; in moment t2 in position x2 etc… And in such theory there is no movement, just sequence of position in sequence of times, and there the movement appears as a merely nominal concept.

It is also imagined that the difficulty of Zeno’s paradoxes can be overcome by adding infinitesimals to the story. But often it is the “bad” notion of infinitesimals which is used, the one where the infinitesimals are something ontologically primary – again the abstract moments of time are imagined, but now which are infinitely close, i.e. where the distance between two moments is smaller then any given “distance”, but yet it is not zero. But this infinitesimals understood this way are yet another contradiction. And while it is true that calculus gives good way to solve the Zeno’s paradoxes, it is not through this contradictory interpretation of infinitesimals connected to the contradictory notion of change as aggregate of moments.

The mind left within the confusion created by combining those two notions (or interpretations), each contradictory in itself, is hoping that there is something about putting two things which mind can’t comprehend together, and through that – the magic of change comes about. And what the mind does in such situation is to blame the reality and our limited power of understanding for this weird state of affairs, as if the best we can do is to catch the pieces which reality throws to us. And while our minds are really to blame, it is not because our comprehension is limited, but for wanting to make our abstractions the basis of reality, instead of seeing them for what they are.

Many of the above themes come together in Graham Priest’s inconsistent account of motion in In Contradiction (1987). Priest sets up the opposing consistent account of change as what he calls the cinematic view of change. This is the view that an object in motion does no more than simply occupy different points of space at different times, like a succession of stills in a film only continuously connected. He attributes the view to Russell and Hume. It is an extrinsic view of change, in the sense that change is seen as a matter of a relation to states at nearby instants of time. The best-worked-out version of this view is the usual mathematical description of change of position by a suitable function of time; and then motion as velocity, that is rate of change of position, is given by the first derivative, which is a relation to nearby intervals.

To me it seems that time doesn't have ontological primacy but things in themselves are seen in a state of flux.
And there is probably good reasons to emphasize the primacy of things being in flux rather than time as an external thing.
In order to examine Dharmakı¯rti’s account of change in light of Priest’s inconsistent theory, we need to revisit Dharmakı¯rti’s notion of momentariness. Here I follow Georges Dreyfus’ authoritative reconstruction of Dharmakı¯rti’s philosophy from a Tibetan point of view.20

Dharmakı¯rti follows Vasubandhu’s view of momentariness as well as most of Vasubandhu’s metaphysics and ontology, in his Abhidharmakos´a.21 According to Vasubandhu,

Destruction of things is spontaneous. Things perish by themselves, because it is their nature to perish. Since they perish by themselves, they perish as they are produced. Since they perish by themselves, they are momentary.22

According to Dreyfus, Dharmakı¯rti elaborates on this ‘‘process view’’ of ontology in two ways: the argument from disintegration and the inference from existence.23 Dharmakı¯rti presents the argument from disintegration as a refutation of Nya¯ya ontology, according to which the term ‘‘disintegration’’ describes ‘‘the state of an already disintegrated thing.’’24 This Nya¯ya view of disintegration posits substances that undergo disintegration in dependence on causes of destruction (vina¯s´ahetu). Against this view, Dharmakı¯rti proposes a dilemma: is the disintegration of a substance an event that the substance undergoes or is it of the substance itself? If the former, it is difficult to see how all qualities disintegrate as soon as they are produced. If the latter, the substance itself must bring about its own disintegration. This means that one can infer the effect, that is, disintegration, from the cause, that is, substance. But for Dharmakı¯rti an inference from cause to effect is a fallacy. Hence, one cannot posit substances separate from their disintegration. Instead, we must think of the term ‘‘disintegration’’ as referring to the process of disintegration. What this means, by using Mortensen’s terminology, is that each spatiotemporal point is not occupied by a substance undergoing disintegration but by a process of disintegration itself.25

Dharmakı¯rti’s inference from existence goes as follows:

[T]hings truly exist insofar as they are able to perform a function. To function is to be capable of producing an effect, a faculty possible only if the object is constantly [ceasing to exist].26 A static object is not acting on anything else nor is it being acted upon. Therefore, that something exists shows that it is momentary.27

What this means is that if something exists, then it exists only momentarily. Hence, Mortensen is correct in saying that for Dharmakı¯rti every existing thing occupies only a spatiotemporal point

Nonetheless, Mortensen seems to underestimate the importance of Dharmakı¯rti’s process view of reality. For Dharmakı¯rti, if something exists, it occupies only a spatiotemporal point. But what exists is not really a substance that undergoes change. Rather, it is the process, or the change, itself that is at a spatiotemporal point. Once we understand Dharmakı¯rti’s view of momentariness in its full extent, it becomes unclear whether or not Dharmakı¯rti has enough resources to reject Priest’s argument for the inconsistent theory of change. This is what I will show below.

Dharmakı¯rti versus Priest

As we saw above, for Dharmakı¯rti there is no distinction between the object and the state in which the object is: the object is said to occupy a spatiotemporal point only to the extent that it is in the process of disintegration or cessation. Now, this view of reality rings a bell. It is a very similar view to Priest’s, if not the same, that a changing object must be in a state of flux at each spatiotemporal point. I examine in this section whether or not Dharmakı¯rti can resist the temptation of Priest and reject an inconsistent theory of change.

Which I assume is amicable with the idea that whilst in a mathematical formulate there is perhaps not sense of future, in reality we have entropy to always press us 'forward.
Entropy (arrow of time)

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