Essay Five: Motion Isn't Contradictory

[This Essay should be read in conjunction with Essays Four and Eight Parts One and Two.]

Preface

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As is the case with all my Essays, nothing here should be read as an attack either on Historical Materialism [HM] -- a theory I fully accept --, or, indeed, on revolutionary socialism. I remain as committed to the self-emancipation of the working class and the dictatorship of the proletariat as I was when I first became a revolutionary nearly thirty years ago. [The difference between Dialectical Materialism [DM] and HM, as I see it, is explained here.]

It is also worth pointing out that phrases like "ruling-class theory", "ruling-class view of reality", "ruling-class ideology" used at this site (in connection with Traditional Philosophy and DM) aren't meant to imply that all or even most members of various ruling-classes actually invented these ways of thinking or of seeing the world (although some of them did -- for example, Heraclitus, Plato, Cicero and Marcus Aurelius). They are meant to highlight theories (or "ruling ideas") that are conducive to, or which rationalise the interests of the various ruling-classes history has inflicted on humanity, whoever invents them. Up until recently this approach had almost invariably been promoted by thinkers who relied on ruling-class patronage, or who, in one capacity or another, helped run the system for the elite.

However, this will become the central topic of Parts Two and Three of Essay Twelve (when they are published); until then, the reader is directed here, here, and here, for further details.

It is also worth pointing out that a good 50% of my case against DM has been relegated to the End Notes. Indeed, in this particular Essay, most of the supporting evidence and argument is to be found there. This has been done to allow the main body of the Essay to flow a little more smoothly. In many cases, I have added numerous qualifications, clarifications, and considerably more detail to what I have to say in the main body. In addition, I have raised several objections (some obvious, many not -- and some that will have occurred to the reader) to my own arguments -- which I have then answered. [I explain why I have adopted this tactic in Essay One.]

If readers skip this material, then my answers to any qualms or objections readers might have will be missed, as will my expanded comments and clarifications.

[Since I have been debating this theory with comrades for over 25 years, I have heard all the objections there are! Many of the more recent on-line debates are listed here.]

Update 07/03/14: I have just received a copy of Burger et al (1980), the existence of which I had been unaware until a few weeks ago. One of the contributors to this book [i.e., Hyman Cohen (Cohen (1980)] seems to have anticipated (and answered) one or two of the points I have raised in this Essay Unfortunately, Cohen's 'answers' fails miserably; I will attempt to explain why over the next few weeks.

Finally, anyone puzzled by the unremittingly hostile tone I have adopted toward DM/'Materialist Dialectics' in these Essays should perhaps read this first.

As of March 2014, this Essay is just under 68,000 words long; a much shorter summary of some of its main ideas can be found here.

The material presented below does not represent my final view of any of the issues raised; it is merely 'work in progress'.

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(1)  Introduction

(2)  Initial Problems

(a) "Asserted" By Whom?

(b) "Solved" In What Way? And By Whom?

(c) Yet More Vagueness

(d) Yet More Dogmatism?

(e) Motion 'Itself'

(3)  Do Contradictions Explain Motion? Or Merely Re-Describe It?

(a) The Problem Stated

(b) Are Contradictions Causes?

(c) 'Internal Contradictions' And Motion

(d) An Indistinct Note

(4)  Is Engels's Account Comprehensible?

(a) A Minimum Requirement

(b) An Initial Ambiguity

(c) First Attempt At Disambiguation

(d) Second Attempt At Disambiguation

(e) Fatal Ambiguity

(5)  The Classical Response To Zeno

(6)  Back to The Drawing Board

(a) The Devil In The Detail

(b) Space To Let

(7)  Further Problems

(a) Pick Your Contradiction

(b) Theatre Of The Absurd

(c) Samuel Beckett Eat Your Heart Out

(8)  No Word Is An Island -- Philosophers Ignore Ordinary Language

(a) For Whom The Bell Tolls

(b) Ordinary Language And Paradox

(c) Lack Of Imagination

(d) Ordinary Objects Regularly Do The Impossible

(9)  Dialectical Objects Do The Oddest Things

(a) Do They Move Or Simply Expand?

(b) Or Do They Concertina?

(c) Coordinates To The Rescue?

(10)  Everyday Miracles?

(11) Inferences From Language To The World

(a) Thought Experiment In Place of Scientific Experiment

(b) Metaphysical Con-Trick

(c) Exclusively Linguistic

(12) Dialectical 'Contradictions'

(13) Conclusion

(14) Notes

(15) References

Summary Of My Main Objections To Dialectical Materialism

Abbreviations Used At This Site

Return To The Main Index Page

Introduction

In this Essay, I aim to examine the role that 'contradictions' are supposed to play in explaining motion and change.¹

Several prominent DM-theorists have attempted to illustrate the allegedly contradictory nature of reality by appealing to a variety of examples, some of which are based on variations of Zeno's Paradoxes. For instance, in order to highlight the limitations of FL, Engels directed our attention to the 'contradictory' nature of motion, depicting it in the following way:²

"[S]o long as we consider things as at rest and lifeless, each one by itself, alongside and after each other, we do not run up against any contradictions in them. We find certain qualities which are partly common to, partly different from, and even contradictory to each other, but which in the last-mentioned case are distributed among different objects and therefore contain no contradiction within. Inside the limits of this sphere of observation we can get along on the basis of the usual, metaphysical mode of thought. But the position is quite different as soon as we consider things in their motion, their change, their life, their reciprocal influence on one another. Then we immediately become involved in contradictions. Motion itself is a contradiction: even simple mechanical change of position can only come about through a body being at one and the same moment of time both in one place and in another place, being in one and the same place and also not in it. And the continuous origination and simultaneous solution of this contradiction is precisely what motion is." [Engels (1976), p.152.]³

In common with other dialecticians -- indeed, as is well-known, he lifted these ideas from Hegel --, Engels here connects change with motion, and both with "contradictions" in nature and society.

However, before this passage is examined in detail, there are a number of serious problems it raises which need addressing since they influence the overall interpretation of the conclusions Engels reaches. Left unresolved they threaten to undermine it completely.

Initial Problems

There are in fact five general difficulties with the above passage:

(1) "Asserted" By Whom?

Engels's closing sentence is rather odd:

"Motion itself is a contradiction; even simple mechanical change of place can only come about through a body being both in one place and in another place at one and the same moment of time, being in one and the same place and also not in it. And the continual assertion and simultaneous solution of this contradiction is precisely what motion is." [Ibid. Bold emphasis added.]

Exactly who is supposed to do the "asserting", and who the "solving", here? Of course, it could be argued that these words were meant to be taken metaphorically. But, if that is so, what is the force of Engels's use of "precisely"?

Even more to the point: if Engels was speaking figuratively what has "assertion and simultaneous solution" got to do with motion? This isn't even a good metaphor!

Perhaps Engels intended to say that these phrases merely pertain to the description of motion? In that case, his conclusions were restricted to language about motion, not motion itself.⁴

(2) "Solved" In What way?

How exactly are contradictions "solved"? Are they like puzzles, riddles and mysteries? If they are, do they disappear once they have been "solved"? Puzzles and mysteries cease to be such when they have been resolved. Is this the same with these contradictions? If it is, do new contradictions immediately take their place? Is each "solved" contradiction then replaced by the 'same' contradiction, or by an entirely new one? How might we conform this? And, how do we know if there is only one contradiction present, or countless thousands, for each unit of time involved? If there is more than one contradiction, how are they all connected with any given body in motion? Does each contradiction arise and fall as that body moves? Or is there a single, extended contradiction smeared, or spread out, as it were, right across its entire trajectory? Is this 'extended contradiction' then perhaps this: that a moving body is "here and not here, in general", so to speak?

More puzzling still: Are these contradictions "solved" by some mind or other comprehending them first? If not, what sense can be given to the word "solved"? And, what precisely is there to understand in a contradiction so that a 'solution' is required in the first place, but which now mysteriously still helps propel the moving object further along (if it does)? On the other hand, if a 'solution' is required, how was this achieved before human beings evolved?

At first sight, Engels appears to be arguing that it is only our understanding of motion that is contradictory:

"[A]s soon as we consider things…then we…become involved in contradictions…." [Ibid., p.152. Bold emphasis added.]

This might help explain why the passage refers to the "continual assertion" of contradictions, since it is evident that only human beings can assert things. If so, it looks like Engels thought that human observers can't avoid "asserting" such contradictions whenever they attempt to describe motion, and that could itself be a result of their partial understanding of the 'absolute truth' about motion. On the other hand, this conundrum could be a fault of logic, or even language, both of which are said by some to be inadequate to the task. But, that would fail to explain how and why contradictions, upon being "asserted", are immediately "solved", and then promptly re-asserted again.

Anyway, but worse, this would mean that it is only human understanding (of motion) that is contradictory, not reality itself -- unless, of course, we are meant to assume that nature is Mind, or even that it is the 'self-development of Mind' that propels bodies along. But, the former alternative suggests that when reality is fully understood, all such contradictions should disappear. If so, this implies that motion might one day cease, all contradictions having been 'solved'. Moreover, if contradictions actually 'cause' motion, then their total resolution should, it seems, freeze nature in its entirety. Or, is it that motion will just stop being (or appearing to be) contradictory one day, but otherwise carry on as normal? Or does this mean that nature will just slow down as it is understood better (i.e., if what we know about motion and change becomes less and less contradictory)? Who can say? Certainly, in the 140 years since Engels wrote these enigmatic words, DM-fans have been more content merely repeating them than they have been posing these rather obvious questions, let alone answering them.

Admittedly, DM-theorists distinguish between subjective and objective dialectics -- the former relating to our (perhaps decreasingly) partial grasp of the nature of reality, the latter relating to processes in the 'objective world' independently of our will. But, it is still unclear how this helps answer the above questions. If the mind "solves" the contradictions involved in motion, wouldn't this mean that things actually stop moving? And wouldn't this indicate, too, that movement only seems to be contradictory because of the partial nature of knowledge, implying that motion isn't really contradictory? Plainly, that is because these subjective contradictions ought to disappear as knowledge grows, meaning that (in the limit) reality isn't 'contradictory', after all. In that case, it is only our 'one-sided knowledge' of nature that fools us into concluding otherwise.

Well, perhaps then this just means that we don't really understand such 'contradictions' to begin with? But, yet again, that would fail to explain why contradictions are promptly reasserted upon being "solved", nor is it at all clear how they could be solved if no one understands them, or if no one understands nature fully. More alarmingly, this might mean that the objects in question aren't really moving, as Zeno originally contended.

Why then does Engels declare the following?

"…the continual assertion and simultaneous solution of this contradiction is precisely what motion is…." [Ibid., p.152. Bold emphasis added.]

This seems to confirm the view that motion isn't really 'contradictory-in-itself', and that it is simply our 'one-sided' perspective on it that misleads us. After all, Engels tells us that the "continual assertion" and "solution" of this contradiction is "precisely what motion is". Why does Engels say that this reveals "precisely" what motion is, as opposed to arguing that it merely depicts what we subjectively think it is?

An appeal to "objective dialectics" can't help us comprehend what Engels meant here either, since neither assertions nor solutions occur in nature (apart, that is, from the intelligent beings who make or who provide them). And, if that is so, these non-objective assertions and solutions can't have been reflected in the mind of observers as part of an objective scientific theory, or as part of 'objective dialectics'. If assertions and solutions don't exist in the world independent of the minds involved, there would be nothing there (in the material world) for the minds of scientists and/or dialecticians to reflect.

And, if that is so, what has assertion and solution got to do with motion in the real world? And why did Engels think they were at all relevant?

So many questions -- so few answers...

(3) More Vagaries

"Motion itself is a contradiction; even simple mechanical change of place can only come about through a body being both in one place and in another place at one and the same moment of time, being in one and the same place and also not in it. And the continual assertion and simultaneous solution of this contradiction is precisely what motion is." [Ibid. Bold emphasis added.]

More specifically, in relation to moving bodies, it is pertinent to ask: How far apart are the two proposed "places" that a moving object is supposed to occupy while at the same time not occupying one of them? Is there a minimum distance involved? The answer can't be "It doesn't matter; any distance will do." That is because, as we will see, if a moving object is in two places at once, then it can't truly be said to be in the first of these before it is in the second (since it is in both at the same time). So, unless great care is taken specifying how far apart these two places are, this view of motion would have, say, an aeroplane landing at the same time as it took off! If any distance will do, then  the distance between the two airports involved is as good as any. [I return to this topic later in this Essay.]

Indifference in this respect would have you arriving at your destination at the same time as you left home!

Anyway, whatever the answer to that question turns out to be -- as is well known -- between any two locations there is a potentially infinite number of intermediary points (that is, unless we are prepared to impose an a priori limitation on nature and deny this).

Does a moving body, therefore, (1) occupy all of these intermediate points at once? Or (2) does it occupy each of them successively?

If the former is the case, does this imply that a moving object can be in an infinite number of places at the same time, and not just in two, as Engels said? On the other hand, if Engels is correct, and a moving body only occupies (at most) two places at once, wouldn't that suggest that motion is discontinuous? That is because, such an account seems to picture motion as peculiar a stop-go sort of affair, since a moving body would have to skip past (but not occupy, somehow) the potentially infinite number of intermediary locations between any two arbitrary places (the second of which it then occupies), if it is restricted to being in at most two of them at any one time. But, that itself appears to run contrary to the hypothesis that motion is continuous and therefore contradictory --, or, it does so at least in any straight-forward sense. It is surely the continuous nature of motion that poses such problems for a logic (i.e., FL) which is allegedly built on static, discontinuous points in space and time, this being the picture that traditional logic is supposed to have painted --, at least, according to dialecticians.

It could be argued that no matter how much we 'magnify' the trajectory of a moving body, it will still occupy two locations at once, being in one of these and not in it at the same time. But, that doesn't solve the problem, for if there is a potentially infinite number of intermediary places between any two locations, a moving body musty occupy more than two at once, contrary to what Engels seems to be saying:

"[A]s soon as we consider things in their motion, their change, their life, their reciprocal influence…[t]hen we immediately become involved in contradictions. Motion itself is a contradiction; even simple mechanical change of place can only come about through a body being both in one place and in another place at one and the same moment of time, being in one and the same place and also not in it. And the continual assertion and simultaneous solution of this contradiction is precisely what motion is." [Ibid. Bold emphasis added.]

Hence, between any two points P and Q -- located at, say, (X_P, Y_P, Z_P) and (X_Q, Y_Q, Z_Q), respectively -- that a moving object M occupies (at the same "moment in time", T₁),  there are the following intermediary points: (X₁, Y₁, Z₁), (X₂, Y₂, Z₂), (X₃, Y₃, Z₃),..., (X_i, Y_i, Z_i),..., (X_n, Y_n, Z_n) -- where n cam be arbitrarily large. And the same applies to (X₁, Y₁, Z₁) and (X₂, Y₂, Z₂), and so on.

So, if Engels is right, M must occupy not just P and Q at the same instant, it must occupy all these intermediary points too -- again, all at T₁. That can only mean that M is located in a potentially infinite number of places, all at the same "moment". It must therefore not only be in and not in P at T₁, it must be in and not in each of (X₁, Y₁, Z₁), (X₂, Y₂, Z₂), (X₃, Y₃, Z₃),..., (X_i, Y_i, Z_i),..., and (X_n, Y_n, Z_n) at T₁ (and it must be in all those intermediary points between (X₁, Y₁, Z₁) and (X₂, Y₂, Z₂), too), if it is also to be in Q at the same "moment".

And, what is worse: M must move through (or be in) all these points without time having advanced one instant!

It will have done all this in zero seconds!

M must therefore be moving with an infinite velocity between P and Q!

Unless, of course, we decide to re-define velocity so that it is no longer distance divided by time.

But, then, what is it?

[An appeal to the Calculus here -- or, rather, to a DM-interpretation of the Calculus -- would be to no avail, as we will see in Essay Seven Part One. See also this sub-section of the present Essay.]

However, as we will see later, this alternative (i.e., that a moving body occupies all the intermediate points between any two points, all at the same time) poses other serious problems for Engels's theory, over and above it implying that objects move with infinite velocities.

On a different tack, do these contradictions increase in number, or stay the same, if an object speeds up? Or, are the two points depicted by Engels (i.e., the "here" and the "not here") just further apart, in that case? That is, are the two points that M occupies, at the same moment, if it accelerates just further apart? But, if it occupies them at the same time, it can't have accelerated. That is because it hasn't moved from the first to the second, since it is in both at once. Hence it isn't easy to see how, in a DM-universe, moving bodies can accelerate (or even move!) if they are in these two locations at once.

[I am of course using "accelerated" here as it is employed in everyday speech, not as it is used in Physics or Applied Mathematics.]

Accelerated motion involves a body being in (or passing through) more places in a given time interval than had been the case before it accelerated. But, if M is in these two places at the same time, how can it pick up speed? [For some of the complexities involved here, see Note 18c.]

So many more questions; even fewer answers...

(4) Yet More A Priori Dogmatics?

Quite apart from all this, Engels's endeavour to provide an overtly linguistic/'conceptual' solution to the 'problem of motion' suggests that there is more than a hint of LIE in his account. And no wonder: he borrowed this approach from Hegel, an Idealist of the worst possible kind.

[LIE = Linguistic Idealism; this term is explained here.]

This 'conceptual' approach to motion is evident from the way that Engels's depiction of it depends on a 'one-sided' consideration of just a few of the concepts that apply in this area, expressed though by means of some rather  ordinary-looking words, the meaning of which Engels simply took for granted (more on this later). Based on thought alone, Engels imagined he was able to conclude what must be true of every moving body in the entire universe, for all of time, without exception. But, how could he possibly know all this with so little evidence (in fact, none at all, as we will also see) to rely on?

"Motion is the mode of existence of matter. Never anywhere has there been matter without motion, nor can there be…. Matter without motion is just as inconceivable as motion without matter. Motion is therefore as uncreatable and indestructible as matter itself; as the older philosophy (Descartes) expressed it, the quantity of motion existing in the world is always the same. Motion therefore cannot be created; it can only be transmitted….

"A motionless state of matter therefore proves to be one of the most empty and nonsensical of ideas…." [Ibid., p.74. Bold emphases alone added.]

Clearly, Engels possessed a truly remarkable skill: the ability to determine fundamental features of reality, valid for all of space and time, based on the alleged meanings of a few words/'concepts'. Indeed, Engels's claims about motion are all the more impressive when it is recalled that he hit upon them in abeyance of any supportive evidence -- let alone a significant body of evidence. As it turns out (and this will also be demonstrated below), evidence would have been unnecessary, anyway.

As we have already seen (in Essay Two), all that an aspiring dialectician like Engels needs to do in such circumstances is briefly 'reflect' on the supposed meaning of a few words/'concepts', and substantive truths about fundamental aspects of nature, valid for all of space and time, spring instantly to mind.

Or, to be more honest, all he/she has to do is read Hegel's 'Logic'. This is a core feature of ruling-class forms-of-thought, imported into the workers' movement by incautious non-workers like Engels. [On this, see Essay Nine Parts One and Two, Twelve Part One and Fourteen Part Two.]

The only 'evidence' that supports Engels's interpretation of motion is this highly compressed argument, or rather, 'thought experiment', which is itself based on a consideration of what a few innocent-looking words/'concepts' must mean. Pressed for a justification of this line of reasoning, all that Engels could possibly have offered by way of substantiation would have been a rather weak claim that this is what the word "motion" really means. Clearly, such a (imagined, but plausible) rejoinder gives the game away since it would reveal that substantive truths about motion had indeed been derived from the meaning of certain words, and nothing more.

[The significance of that observation will emerge in Essay Twelve Part One.]

As noted above, an appeal to evidence would be irrelevant, anyway. That is because the examination of countless moving objects would fail to confirm Engels's assertion that they occupy two places at once. This is so no matter what instruments or devices are used to carry out these hypothetical observations, and regardless of the extent of the magnification used to that end, or the level of microscopic detail enlisted in support. No observation could confirm that a moving object is in two places at once (except in the senses noted below), and in one of these and not in it at the same time. This, of course, explains why Engels offered no scientific evidence whatsoever in support of his belief in the contradictory nature of motion. And this picture hasn't altered in the intervening years -- indeed, no book or article on DM even so much as thinks to quote such evidence in support of this thesis --, and that situation isn't likely ever to change.⁵

It could be objected to this that if, say, a photograph were taken of a moving object, it would show by means of the recorded blur, perhaps, that such a body had occupied several places at once. In that case, therefore, there is, or could be, evidence in support of Engels's claims.

The problem with this is that no matter how fast the shutter speed, no camera (not even this one, or this) can record an instant in time, merely a temporal interval. Clearly, to verify the claim that a moving object occupies at least two places in the same instant, a physical recording of an instant would be required. Plainly, since instants (i.e., in the sense required) are mathematical fictions, it isn't possible to record them.

It could be countered that as we increase the shutter speed of a camera, any photographs taken will always show some blurring. This supports the contention, then, that moving objects are never located in one place at one time. Maybe so, but it still remains the case that no photograph can catch an instant, and thus none can verify Engels's contention.

Again, it could be argued that it is reasonable to conclude that moving objects occupy two locations at the same moment from the above. Once more, since an instant in time is a fiction, it isn't reasonable to conclude this. Not even a mathematical limiting process could capture such ghostly 'entities' in the physical world, whatever else it might appear to achieve in theory. But, even it could, no camera (or radar device, or other piece of equipment) could record it. Hence, even if an appeal to mathematical limiting processes was both viable and/or available, it would be of no assistance. No experiment could conceivably substantiate any of the conclusions Engels reached about moving bodies.

And that explains why he and those who accept these ideas have to force this view of motion onto nature.

Hence, this thesis about moving bodies has not emerged from the facts, but has been imposed on them, in defiance of what Engels himself said:

"Finally, for me there could be no question of superimposing the laws of dialectics on nature but of discovering them in it and developing them from it." [Engels (1976), p.13. Bold emphasis added.]

"All three are developed by Hegel in his idealist fashion as mere laws of thought: the first, in the first part of his Logic, in the Doctrine of Being; the second fills the whole of the second and by far the most important part of his Logic, the Doctrine of Essence; finally the third figures as the fundamental law for the construction of the whole system. The mistake lies in the fact that these laws are foisted on nature and history as laws of thought, and not deduced from them. This is the source of the whole forced and often outrageous treatment; the universe, willy-nilly, is made out to be arranged in accordance with a system of thought which itself is only the product of a definite stage of evolution of human thought." [Engels (1954), p.62. Bold emphasis alone added.]

"We all agree that in every field of science, in natural and historical science, one must proceed from the given facts, in natural science therefore from the various material forms of motion of matter; that therefore in theoretical natural science too the interconnections are not to be built into the facts but to be discovered in them, and when discovered to be verified as far as possible by experiment.

"Just as little can it be a question of maintaining the dogmatic content of the Hegelian system as it was preached by the Berlin Hegelians of the older and younger line." [Ibid., p.47. Bold emphasis alone added.]

Of course, as noted above, part of the problem here is what the word "instant" means. [I am taking this word to mean the same as "moment in time", used by Engels.] So, it might be thought that this 'problem' could be solved by means of a suitable definition. However, even if this were possible, such an 'adjustment' would merely represent the adoption of a new convention, and would have no bearing at all on the nature of reality.^5a

As Trotsky argued:

"How should we really conceive the word 'moment'? If it is an infinitesimal interval of time, then a pound of sugar is subjected during the course of that 'moment' to inevitable changes. Or is the 'moment' a purely mathematical abstraction, that is, a zero of time? But everything exists in time; and existence itself is an uninterrupted process of transformation; time is consequently a fundamental element of existence. Thus the axiom 'A' is equal to 'A' signifies that a thing is equal to itself if it does not change, that is if it does not exist." [Trotsky (1971), p.64. Bold emphasis added.]

Unfortunately for Engels, if motion were to take place in one of these "moments", that would mean that it couldn't exist -– that is, not unless we are also prepared to reject the a priori conclusion Trotsky expressed in the above passage.

But, if motion actually takes place -- as it surely does -- then what are we to make of the claim that if something is moving it must be in at least two places in the same instant, when the latter do not exist (according to Trotsky)? Does this refute Trotsky, or Engels, or both? Is there even a straw-sized contradiction here for dialecticians to "grasp" to save their drowning theory?

Furthermore, an appeal to the abstract nature of some of the points made above can't rescue Engels, either. His analysis of motion is no less abstract itself! And, it can't have itself been derived by abstraction from all (or any) of the forms of motion hitherto experienced by either himself or humanity -- or even from a finite sub-set observed by scientists and/or philosophers down the ages. That is because Engels's thesis clearly appeals to things that, according to Trotsky, do not exist -- such as "moments" in time. And, even if they did exist, we couldn't experience or observe them, and hence we couldn't use them to confirm what Engels said. [Observations take place in time, and have a duration; "instants" do not.]

Worse still, we can't abstract from such instants in order to agree with Engels, either.⁶

Whichever way we turn we hit yet another non-dialectical brick wall.

Motion 'Itself'

To be sure, Engels promptly changed direction in the above passage, arguing that it is motion itself that is contradictory, not just our thoughts about it that are -- declaring that:

"Motion itself is a contradiction…." [Engels (1976), p.152. Emphasis added.]

In which case, it could be objected that Engels was actually arguing that our thoughts about motion are contradictory because motion itself is. That is, our theories depict the world more fully and truly when they reflect its contradictory nature, and that substantive claims about the universe are justified if and when our ideas capture reality more precisely (but, only if they have been tested in practice).

Unfortunately, if this response were correct, it would be inimical to DM, anyway, since that would mean DM-theory itself contains contradictions, which would imply it is a contradictory theory.⁷

[The disastrous implications that particular conclusion has for DM were outlined in Essay Seven, and Essay Eleven Part One.]

However, such a reply would give the game away, since it conforms an earlier accusation that this view has been imposed on nature because there is no way that Engels could know, or have known, that nature is contradictory in its entirety, and thus that all motion in the entire universe, for all of time, is as he says it is. The very best that Engels could claim is that our thoughts about motion are contradictory, and that this suggests that motion in nature might be, too.

The problem with this fall-back position is that (as will be apparent by the end of this Essay): our thoughts about motion aren't the least bit contradictory, either.^7a

Be this as it may, the above response fails anyway to neutralise the fatal consequences outlined earlier. That is because Engels's philosophical thesis, which was the result of an extrapolation from the meaning of a handful of words/'concepts' to fundamental aspects of reality, is openly Idealist (on this see Essay Two and Twelve Part One). Engels himself pointed this out:

"The general results of the investigation of the world are obtained at the end of this investigation, hence are not principles, points of departure, but results, conclusions. To construct the latter in one's head, take them as the basis from which to start, and then reconstruct the world from them in one's head is ideology, an ideology which tainted every species of materialism hitherto existing.... As Dühring proceeds from "principles" instead of facts he is an ideologist, and can screen his being one only by formulating his propositions in such general and vacuous terms that they appear axiomatic, flat. Moreover, nothing can be concluded from them; one can only read something into them...." [Marx and Engels (1987), Volume 25, p.597. Italic emphases in the original; bold emphases added.]

Compare these comments with those of George Novack:

"A consistent materialism cannot proceed from principles which are validated by appeal to abstract reason, intuition, self-evidence or some other subjective or purely theoretical source. Idealisms may do this. But the materialist philosophy has to be based upon evidence taken from objective material sources and verified by demonstration in practice...." [Novack (1965), p.17. Bold emphasis added.]

Worse still, and for reasons given above, not only can this 'theory' not be confirmed, its subject matter (i.e., the thesis that a moving body occupies and does not occupy the same place in the same instant, being in two places at once) resists all attempts to make sense of it, as we will see.

Substantive philosophical 'truths' like this (about motion) are ambitiously universal in intent, but are thoroughly parochial in origin. Indeed, their promulgators' epistemologically imperialist intentions (which stretch across all regions of space and time) remain stubbornly unmatched by any obvious capacity to satisfy such excessive philosophical ambitions with adequate material support, or any at all.

So, throughout history, traditional theorists (like Engels -- but more particularly, Hegel) have privileged speculation ahead even of a perfunctory search for supporting evidence. Indeed, they  assume that all of nature must be as their 'surgically enhanced' words seem to them to depict it.

However, if this approach to Super-Truth were valid, it would mean that the universe possessed these features simply because of the idiosyncrasies of Indo-European Grammar -- the language group in which most of this hyper-bold talk has been concocted.

(5) Explanation Or Re-Description?

The Problem Stated

Perhaps even worse still: It isn't easy to see how the 'contradictory' nature of motion could in any way explain it, nor is it easy to see how it could form part of a wider scientific account of anything at all. At best, this way of characterising motion simply re-describes it.

More specifically, it is difficult to see how one 'part' of a 'contradiction' is capable of exercising a causal influence over any other 'part', or indeed how one or both of these UOs (i.e., this "here" and that "not here") could make anything move.

[A more general objection to this way of seeing change is presented here.]

[UO = Unity of Opposites.]

As Engels depicts things, both 'parts' of this UO seem to appear together: a body is "here" and "not here" all at once, as it were:

"Motion itself is a contradiction; even simple mechanical change of place can only come about through a body being both in one place and in another place at one and the same moment of time, being in one and the same place and also not in it." [Engels (1976), p.152.]

In that case, it looks like questions concerning the proximate cause of motion (with the implied temporal concomitants such questions often motivate) can't be answered by anyone who prefers this way of depicting movement. The mere fact that a moving body is "here" does not appear to be capable of making it become "not here". Indeed, this alleged contradiction seems to lack any causal powers whatsoever, any capacity to make things happen. It isn't so much that the dialectical batteries have run down as it is that there don't appear to have been any supplied with the original item -- and it has no slot for them to fit into.

This probably explains why Engels didn't even attempt to construct a causal account of motion based on the contradiction he claims to have found there (and, as far as can be ascertained, no DM-theorist since has filled in the gaps). But, even if a DM/causal account were to be provided one day, it isn't easy to see how these alleged contradictions could explain motion; how does a being "here" and "not here" (all at once) explain why anything actually moves? What work do such contradictions do -- even if you believe in them?

It could be objected here that this radically misconstrues DM, for the counter-argument presented above misleadingly splits apart the assumed 'sections' of a contradiction when DM itself requires contradictions to be constituted by (or to be based upon) interpenetrated opposites. A dialectical contradiction is a relation, not a thing. Moreover, and contrary to the above, DM doesn't depict motion or change in such mechanical, causal terms. Dialecticians' various discussions of causation are specifically aimed at countering mechanistic and reductionist accounts like this.

Or, so a response might go.

However, even if this reply were acceptable, no attempt was made in Engels's work -- and, to my knowledge, none has been made anywhere else -- to explain how contradictions can have any effect on anything at all, anywhere, anyhow, and in whatever preferred causal or mediational/dialectical language they are expressed -- that is, other than figuratively. [More on this in Essay Eight Parts One and Two.] And, this is quite apart from the fact that this alleged contradiction (in motion) doesn't appear to be relational at all. What precisely is being related to what? What "relation" is this particular one meant to picture or reflect? Is a body related to itself as it moves? But, how would that make it move?

[The best attempt that I have so far seen from a Marxist Dialectician to explain the rationale behind this view of motion and change has been taken apart here.]

Of course, it could be argued that it is the relation among bodies that makes them move; that response will be examined below, and in more detail in Essay Eight Part Two.

Moreover, it is far from easy to see how a contradictions could exercise any sort of effect on anything at all unless it was translated into, or expressed somehow in, physical/material terms (which will be attempted below). At some point, bits of matter are going to have to be moved about the place. Now, this physically inconsequential word ("contradiction") doesn't seem to have the required physical presence -- the oomph, as it were -- to carry out menial tasks like this.⁸

[HM = Historical Materialism.]

Furthermore, if the volunteered DM-response above were correct (but see below), contradictions wouldn't appear to be of much help in explaining social change, let alone changes in nature. If no causal role is assignable to contradictions in DM (with respect to motion, or, indeed, with respect to anything at all), then they certainly can't serve in such a capacity in HM. The alleged contradictions in Capitalism, for example, wouldn't, therefore, make anything actually happen (they would, at best, be the result of other things happening (for which DM-theorists would no have no explanation, since these 'other things' wouldn't themselves have been caused by contradictions), or they'd be the result of certain specific social relations).

The cause or causes of social development would be totally obscure (given this (assumed) rejection of any causal role for such contradictions to play). In that case, we are forced to conclude that if there are any contradictions in reality, they must play some sort of causal role, at some level, in some form, otherwise dialecticians wouldn't be able to explain why anything actually happened in nature and society.

[Of course, that might be the real reason why they can't do this --, but they certainly do not see things this way, to state the obvious.]

Conversely, this could mean that if the development of class society is still to be accounted for in terms of the supposed contradiction between the forces and relations of production, contradictions could be dispensed with at no loss to HM, since (given the above response) contradictions would do no work in HM either, playing no causal role. In that case, the sooner they are pensioned-off the better. Attention could then be focussed on the genuinely causal nature of the above relations -- suitably phrased in historical and materialist terms. Naturally, this would involve a radical re-write of HM, abandoning much of the traditional Hermetically-inspired jargon, which has up until now only managed to stifle Marxist theory.

If so, this means that dialecticians need to specify -- as a matter of some urgency -- what, if anything, is so causal about the contradictions they seem to see everywhere, so that the latter can at least do some genuine work in HM. At present they do not appear to be part of the action; at best, they seem to be merely decorative.

On the other hand, the assignment of a causal role to contradictions in HM or DM -- so that they cease to be merely ornamental -- would generate insuperable difficulties for both theories, as we will soon see.

Are Contradictions Causes?

As hinted at above, even if it were possible to assign some sort of causal role to contradictions (albeit expressed in suitably acceptable dialectical language), it would still not help DM-theorists account for motion. That is because (according to Engels) motion allegedly involves a body being in one place and not in it, all the while being in two places at one and the same 'instant'/'moment'. The problem is: How does this actually explain motion causally -- or in any other sense? What exactly does it add to a scientific account of the same phenomenon? All it appears to offer is a paradoxically-worded re-description.

In order to make the last point clearer, it is worth pondering once again the answer to this question: Do contradictions cause motion (i.e., do they make it happen), or does motion merely reveal the presence of contradictions as it unfolds? On one reading of Engels's account, it looks like it is motion that causes (or creates) contradictions. Hence, according to this way of reading his exact words, something must be in motion first for it to bring about contradictory, simultaneous occupancy and non-occupancy of successive locations. But, as we will soon see, this would mean that one or both of the following hypotheticals would have to be true:

(1)  If contradictions didn't exist, motion could still take place.

(2)  If motion ceased, contradictions would still remain.

(1) The relevance of the first of these is underlined by the fact that unless motion was already underway, a contradiction could not be inferred.

At the very least, this option prompts a further question: Which came first -- movement or contradiction? The answer to this might be why Engels spoke about contradictions being "solved", and then "re-asserted", since, on that basis, it looks like motion causes contradictions, not the other way round.

Of course, it could be argued that these two go hand-in-hand; so it no more makes sense to ask which came first, movement or contradictions than it would to ask: "Which came first, counting or numbers?"

But, as we will see later on in this Essay, there are (in fact) examples of motion in the real world where no contradiction is implied, directly or indirectly. So, perhaps this is the case here, too?^8a

(2) The second option above follows from the simple observation that a stationary body can occupy two places at once, and it can be in one place and not in it at the same time. [Examples of both are given below.]

In that case, (2) suggests that contradictions aren't a sufficient cause of motion, whereas (1) indicates they aren't even necessary.

Moreover, and with respect to (1), once more, Engels himself appears to have reasoned from his understanding of what motion is to its contradictory implications. In that case, it looks as if there is no causal role for contradictions to play with respect to motion itself, as far as Engels saw things -- that is, there seems to be no way that they could make anything move. At best, they appear to be conceptually derivative, not causative; they depend on motion, not the other way round. Hence, as things now stand, it looks like things first of all move, and only then do contradictions emerge -- and even then this just applies to our depiction of motion.

If so, it might be correct to say that contradictions operate solely at a conceptual level -- they appear to have no part to play in the physical action, on the ground, as it were.

Given this modified view, it would seem that objects in the world just move, but they do not to do so because they become embroiled in literal contradictions.

[So, for example, moving bodies do not argue among themselves about the occupancy or non-occupancy of this or that particular "place" --, which would be the clear implication of the ordinary, literal use of the verb "to contradict". Nor do they become entangled in 'time-and-motion' wrangles about who or what was where, when, and why. Again, they would do this if literal contradictions (as opposed to a figurative, DM-extension to this word) were operative in such cases. (On this, see Note 1, and Essay Eight Parts One and Three.)]

In fact, given Engels's account of motion, it seems that it is we who derive these paradoxical conclusions in our attempt to depict something that just takes place (without any such fuss) in nature.

In other words, according to this interpretation of Engels's views, it looks like the 'fault' lies in us, not in things.

However, this way of depicting motion is clearly unacceptable to DM-theorists; they insist that we must begin with material reality not with a description of it. From there, according to them, we must postulate only those contradictions that really exist in nature or society -- based perhaps on their reflection in human thought, confirmed in practice. Clearly, human beings study motion and its attendant contradictions using the conceptual resources they have to hand, which might not always be up to the job. Or so a counter-claim might go.

But, even this response is no help. That is because there seems to be nothing in reality that thought could latch onto, or reflect -- and hence, nothing for anyone to abstract from, or to, and then test in practice -- that even remotely resembles the contradictions postulated by dialecticians.

[Why this is so occupies the latter three quarters of this Essay. Also, see here.]

In relation to Engels's account of motion, as will soon emerge, there is no clearly specifiable set of possibilities -- or even actualities -- with which his description could conceivably correspond. In fact, his words turn out not to be a depiction of the physical world in any shape or form. That isn't because he got the details wrong, or because he failed to capture nature accurately enough --, nor yet because nature is too complicated for us to describe -- it is because his words fail to be a description in the first place. Hence, Engels's 'description' of motion is not just empty, it isn't even a description!

Again, it could be objected that the above analysis is misguided since it compartmentalises reality, distorting the account of nature given in DM.

In response to this it is also worth pointing out that we do not have to divide the 'parts' of a contradiction one from another (or from other relevant aspects of reality) to make the above argument work.

If each and every contradiction postulated by dialecticians (whether derived from "really existing material forces", or not) is given a sufficiently complex, dialectical background (interconnected within the Totality, required by the theory, verified in practice, etc., etc.), that still would not amount to an explanation of the causal or "mediated" links that are required. A widening of the domain (to the entire Totality if need be) cannot suddenly provide an explanation of how the simultaneous presence and absence of an object in one and the same place could possibly make it move -- or even how it could account for motion in any way at all.

An appeal to forces here would be to no avail, either -- as will be demonstrated in detail in Essay Eight Part Two. Unless forces are anthropomorphised, they too cannot account for movement and change in DM-terms. [That cryptic comment will also be explained in Essay Eight Parts One and Two.]

Furthermore, but the alleged reflection of contradictions 'in the mind', which might be thought capable of providing the 'conceptual connection' that supposedly exists between a cause and its effects (or between various mediated items in the Totality), cannot create a genuine connection if there are none already there in reality for it to reflect. Contradictions must have some sort of material basis if they are to be reflected in thought; they cannot just be conceptual. And yet, what material form do they take?

Unless sense can be given to the idea that contradictions are capable of connecting things in the required way -- in reality and not just 'in the mind' --, in order to provide some sort of grist for the DM-causal/mediational mill to grind away at, a DM-style reflection would advance the explanation of motion not one inch.

Even assuming it could be shown that contradictions do in fact represent a material relation between objects and/or processes -- which have been abstracted from (or read into) the phenomena (in an as yet unspecified way!) -- they still couldn't account for motion. That is because this would simply amount to a re-description of the phenomena, once more. We still await the explanatory punch-line: how do contradictions make things move? What is the material point to this Hegelian myth?

If, though, it is now claimed that such a causal (mediational) link between events must to be postulated (i.e., it is just assumed to exist to make the theory work), then that would merely provide a conceptual link between the said events, once more -- and such it would remain until the physical details were filled in. Without the latter, the contradictory nature of motion would remain at best a conceptual, but not yet a material aspect of reality.

[That outcome should surprise no one given the Idealist origin of DM-use of "contradiction".]

If, on the other hand, it is claimed that the mere presence of the said conceptual connection indicates that such causal links must exist in reality -- that is, if the complex reflection theory of knowledge is assumed to be true (wherein the human mind acquires knowledge actively, in practice etc.) --, then that would still not explain how contradictions could actually cause motion. How do contradictions succeed in moving things about the place? Here, the dialectical spade is not just turned, it snaps in two.

Clearly, the above difficulties will only be resolved at some point if a clear explanation is given as to how contradictions can make things move -– or, at least, until it is shown how and in what way the above objections are misguided.

However, as should now seem plain, the role that contradictions supposedly play in motion is not helped by an account that depicts them (1) As the product, not the cause of motion (making them derivative), or (2) As the result of human reflection on the nature of motion (implying they are merely conceptual, and are thus Ideal).

Hitherto, DM-theorists have been content merely to label certain states-of-affairs "contradictory" without apparently giving any thought to the lack of explanatory role this empty ceremony assumes in their theory. Why call anything "contradictory" (and claim so much for the use of this term) if no account can be given of how this actually explains why anything changes or moves?

'Internal Contradictions' And Motion

At this point, it could be argued that the above objections are all irrelevant since DM-theorists are committed to the thesis that motion and change are caused by internal contradictions; the above account seems to be obsessed with external causes.⁹

Unfortunately, in connection with motion, there do not appear to be any internal contradictions capable of impelling objects along. No one supposes (it is to be hoped!) that an internal contradiction works like some sort of metaphysical motor, humming away inside a moving object, powering it on its way!^9a And there do not seem to be any 'struggles' taking place within moving bodies that impel them onward (perhaps in the way that a drunken brawl might make a train carriage wobble from side to side, only worse) --, and this would be so even if it were true that all bodies are in fact UOs. No matter how intense this supposed internal battle becomes, a 'metaphysical boxing match' of this sort seems incapable of generating self-propulsion.

Lenin's "demand", therefore, looks rather empty:

"Dialectics requires an all-round consideration of relationships in their concrete development…. Dialectical logic demands that we go further…. [It] requires that an object should be taken in development, in 'self-movement' (as Hegel sometimes puts it)…." [Lenin (1921), p.93. Bold emphases added. This entire topic is examined in great detail in Essay Eight Parts One and Two.]

"

Furthermore, there do not appear to be any identifiable contradictions situated at the leading edge of a moving body 'dragging' it along, just as there seem to be none at the back 'pushing'.

Worse still: both of these scenarios (even if they were remotely plausible) would clearly involve the creation of kinetic energy out of thin air.

In that case, with regard to individual bodies, motion can't be an example of change through "internal contradictions".

It could be replied that since locomotion and development in a system are the result of forces acting on bodies/processes, the contradictory nature of motion can easily be accounted for on the basis of a network of internal, systematically-opposed forces. This would then make the unit within which contradictions (and thus motion) occur the whole, not the part, which seems to be the assumption underlying the comments made in previous few paragraphs.

Naturally, that response would make a mockery of the claim that all objects change through self-development, or that they barrel along because they are self-motivated. Just as it would make a mockery of Lenin's contrast between a mechanical, 'external' account of movement  and change, and a dialectical account. On this modified, 'theory', no object would be self-motivated -- never mind what Lenin demanded. -- it would be moved by forces internal to the system of which it is a part, but external to each object in that system.

"Dialectics requires an all-round consideration of relationships in their concrete development…. Dialectical logic demands that we go further…. [It] requires that an object should be taken in development, in 'self-movement' (as Hegel sometimes puts it)…." [Ibid., Bold emphases added.]

"The identity of opposites...is the recognition...of the contradictory, mutually exclusive, opposite tendencies in all phenomena and processes of nature (including mind and society). The condition for the knowledge of all processes of the world in their 'self-movement,' in their spontaneous development, in their real life, is the knowledge of them as a unity of opposites. Development is the 'struggle' of opposites. The two basic (or two possible? Or two historically observable?) conceptions of development (evolution) are: development as decrease and increase, as repetition, and development as a unity of opposites (the division of a unity into mutually exclusive opposites and their reciprocal relation).

"In the first conception of motion, self-movement, its driving force, its source, its motive, remains in the shade (or this source is made external -- God, subject, etc.). In the second conception the chief attention is directed precisely to knowledge of the source of 'self'- movement.

"The first conception is lifeless, pale and dry. The second is living. The second alone furnishes the key to the 'self-movement' of everything existing; it alone furnishes the key to 'leaps,' to the 'break in continuity,' to the 'transformation into the opposite,' to the destruction of the old and the emergence of the new.

"The unity (coincidence, identity, equal action) of opposites is conditional, temporary, transitory, relative. The struggle of mutually exclusive opposites is absolute, just as development and motion are absolute...." [Lenin (1961), pp.357-58. Italic emphases in the original; bold emphases added. Quotation marks altered to conform to the conventions adopted at this site.]

However, even if systematically-opposed forces could somehow be interpreted as contradictions -- or if they could at least be regarded as constituting them -- that would still fail to show how internal contradictions could explain motion (or, rather, a change in motion), or even how they could initiate it. Nor would it account for the contradictory nature of motion itself; at best, all this would do is appeal to the allegedly contradictory nature of the system of forces that supposedly produced or changed any motion in the system. The fact that a moving body appears to be in at least two places at once (and hence contradictory in itself while moving) is in no way connected to whatever allegedly initiated that motion, or with whatever now maintains it (if anything does) -- at least not obviously so. Certainly, dialecticians have yet to connect contradictory forces themselves with the alleged fact that moving bodies appear to be in two places at once, in and not in at least one of them at the same time. Nor is it easy to see how this might be done.

Hence, whether or not it is true that movement is caused/mediated by a disequilibrium within a system of incipient forces ('internal' or otherwise), that still wouldn't affect the alleged fact that once moving, a body appears to do contradictory things. Even given the truth of such an 'internalist', or even 'externalist', account of contradictions and forces, the fact that a body is in two places at once is a consequence of this setup. But, the "in two places at once" (etc.) descriptor (or its physical correlate) doesn't also cause motion in addition to the forces at work in the system. Indeed, while forces might cause motion (or, rather, cause a change in motion), the alleged contradictory nature of the movement that results from this has no part to play in the action.

So, even if the 'internalist'/'externalist' picture were correct, Engels's analysis of motion would still amount to nothing more than a re-description of it; it would still be the case that motion makes bodies do allegedly contradictory things, not the other way round. Hence, the contradictions that Engels highlights are still derivative, and not at all explanatory.

It is worth re-emphasising this point: even if opposing forces could explain contradictory motion (which thesis is demolished in Essay Eight Part Two, anyway), the nature of the connection between the paradoxical states that moving bodies appear to display and forces has still to be established. All that the addition of opposing forces has achieved is to account for the origin of one contradiction (motion) in terms of another (oppositional forces). The contradictory nature of motion itself is still locked in the descriptive mode -- it does no work. Whether or not forces can explain motion (or even changes in motion) is not being questioned here, yet. Even supposing they could, the contradictions Engels supposedly saw in moving bodies remain descriptive. We are still owed an explanation as to why a moving body being "here and not here at the same time" and "in two places at once", accounts for its motion as opposed to merely re-describing it.

Of course, on this view, motion (or, indeed, change in motion) would be causally related to forces, but this just divorces the latter from the contradictory behaviour of moving bodies (a point Engels himself seems to have conceded -- on that, see Note 10). So, even if it were the case that opposing forces caused motion (or changed it), this still would provide no useful role for the observation that motion is itself contradictory. As far as DM is concerned (that is, on the basis one particular interpretation that appears to be inconsistent with what Engels himself said about forces -- again, see Note 10), what seems to be important is the alleged fact that opposing forces are contradictory; the other notion (about the contradictory nature of motion) still appears to be redundant; it serves no obvious purpose, and plays no role in the action.¹⁰

[As will be argued at length in Essay Eight Part Two, the appeal to oppositional forces to explain contradictions (and/or contradictory totalities) is no less misguided. There, it will be demonstrated in extensive detail that not only is there no conceivable interpretation of opposing forces that could account for contradictions (in FL or DL), there is no viable literal or figurative way of depicting contradictions as forces.]

[DL = Dialectical Logic; FL = Formal Logic.]

Of course, even more revealing is the fact that in classical Physics forces are supposed to change the motion of bodies; this means that the idea that something has to maintain movement (whether it is contradictory or not) depends on an obsolete Aristotelian theory of movement. If so, the fact that contradictions can't supply a causal explanation of motion is all to the good, for if the allegedly contradictory nature of motion caused and maintained movement, much of post-Aristotelian (Newtonian) mechanics would have to be binned.¹¹

But, then again, if such 'contradictions' don't and can't explain motion (i.e., they do not change or initiate it), why make such a fuss about them?

Well, despite the above, it could be objected that this whole discussion seriously misunderstands the nature and role of contradictions in dialectics. As John Rees points out:

"[These] are not simply intellectual tools but real material processes…. They are not…a substitute for the difficult empirical task of tracing the development of real contradictions, not a suprahistorical master key whose only advantage is to turn up when no real historical knowledge is available." [Rees (1998), pp.8-9.]

Hence, it could be argued that the problem with the above criticisms is that they substitute an abstract analysis for one that should be focus on real material forces.

This objection is considered in detail elsewhere at this site (here, here, here, and here), where Rees's and other dialecticians' epistemological and methodological claims are examined at length, alongside a consideration of the "real material contradictions" to which most DM-theorists appeal to illustrate their theory -- as well as the spurious claim that dialecticians do not use their theory as a "master key" to unlock reality, when they clearly do. [On that, see Essay Two.]

The claim will also be revived here and here (but, more specifically here, here and here) that material contradictions cannot account for change, since they are locked in the descriptive mode (and a confused mode, at that).

An Indistinct Note

However, one further possibility hasn't yet been examined: What if it is entirely unclear what Engels was trying to say in the passage under consideration? Indeed, what if it could be shown that he was in fact saying nothing at all comprehensible?

In that case, it would be completely beside the point whether or not there are any genuine examples of "material contradictions" in nature (at least, not as Engels saw them). Well, no more than there would be any point in Christians, for example, trying to locate the actual Trinity somewhere in outer space. The problem here lies not so much with the search itself (in that it might be too difficult, or would take too long), but with the nature and description of something that could be looked for. If we are given nothing comprehensible to search for, plainly no search can begin.

[As noted in Essay Six, you can look, for example, for your keys if you do not know where they are, but not if you do not know what they are.]

But, is there any substance to these claims?

The next few sections aim to show that there is -- and plenty more than enough.

Is Engels's Account Comprehensible?

A Minimum Requirement

Before an empirical investigation into the real material cause of motion can begin, we need to be clear precisely what it is we are being asked to examine. As things turns out, it isn't possible to determine what Engels was trying to say when he wrote the following about motion:

"[A]s soon as we consider things in their motion, their change, their life, their reciprocal influence…[t]hen we immediately become involved in contradictions. Motion itself is a contradiction; even simple mechanical change of place can only come about through a body being both in one place and in another place at one and the same moment of time, being in one and the same place and also not in it. And the continual assertion and simultaneous solution of this contradiction is precisely what motion is." [Engels (1976), p.152.]

In order to substantiate these allegations, several further ambiguities in Engels's account will need to be addressed first.

An Initial Ambiguity

Engels tells us that a body must be:

"[B]oth in one place and in another place at one and the same moment of time, being in one and the same place and also not in it." [Ibid., p.152.]

Here, he appears to be claiming two separate things that do not immediately look equivalent:

L1: Motion involves a body being in one place and in another place at the same time.

L2: Motion involves a body being in one and the same place and not in it.

L1 asserts that a moving body must be in two places at once, whereas L2 says that it must both be in one place and not in it, while leaving it unresolved whether it is in a second place at the same or some later time -- or even whether it could be in more than two places at once. To be sure, it could be argued that it is implicit in what Engels said that this occurs in the "same moment of time"; however, I am trying to cover every conceivable possibility, and it is certainly possible that he did not. [The significance of these comments will emerge as the Essay unfolds.]

It is also far from easy to see how a moving body can be "in one place and not in it", and yet still be in two places at once. If moving object M isn't located at X  -- if it is "not in X" --, then it can't also be located at X (contrary to what Engels asserts). On the other hand, if M is located at X, then it can't also not be at X! Otherwise, Engels's can't mean by "not" what the rest of us mean by that word. But, what did he mean?

Moreover, if M is in two places at once -- say in X and Y at the same time --, then it can't be in Y but must be in Y and another place -- otherwise it will be stationary at Y.

[We will return to these initial problems later in order to see if there is any way around them.]

Be this as it may, it is important to be clear what Engels means here because L1 is actually compatible with the relevant body being at rest! This can be seen if we consider a clear example: where an extended body is motionless relative to an inertial frame. Such a body could be at rest and in at least two places at once. Indeed, unless that body were itself a mathematical point, or discontinuous in some way, it would occupy the entire space between at least two distinct spatial locations (i.e., it would occupy a finite volume interval). But since all real bodies are extended in this way, the mathematical point option seems irrelevant, here. [Anyway, it will be considered below.]

A commonplace example of this sort of situation would be where, say, a train is at rest relative to a platform. Here, the train would be in countless places at once, but still stationary with respect to some inertial frame. [There are countless examples of this everyday phenomenon.]

[In this and subsequent paragraphs I will endeavour to illustrate the alleged ambiguities in Engels's account by an appeal to everyday situations (for obvious materialist reasons). However, these can all be translated into a more rigorous form using vector algebra and/or set theory. In the last case considered below, just such a translation will be given to substantiate that particular claim.]

Unfortunately, even this ambiguous case could involve a further equivocation regarding the meaning of the word "place" -- the import of which Engels clearly took for granted. As seems plain, "place" could either mean the general location of a body (roughly identical with that body's own topological shape, equal in volume to that body --, or, on some views very slightly larger than its volume, so that the body in question can fit 'inside' its containing volume interval). Alternatively, it could involve the use of a system of precise spatial coordinates (which would, naturally, achieve something similar), perhaps pinpointing its centre of mass, and using that to locate the body, etc.

Of course, as noted above, Engels might have been referring to the motion of mathematical points, or point masses. But, even if he were, it would still leave unresolved the question of the allegedly contradictory nature of the motion of gross material bodies, and how the former relates to the latter.

It is Engels's depiction of motion that is unclear; because of that, I will concentrate on ordinary material bodies. Anyway, since DM-theorists hold that their theory can account for motion in the real world, the motion of mathematical points -- even where literal sense can be made of them and of the idea that they can move (if such points do not exist in physical space, they can hardly be said to move) will not in general be entered into here.

L2 itself involves further ambiguities that similarly fail to distinguish moving from motionless bodies. Thus, a body could be located within an extended region of space and yet not be totally inside it. In that sense it would be both in and not in that place at once, and it could still be motionless with respect to some inertial frame.

Here the equivocation would centre on the word "in". However, it could be objected that "in" has been replaced here by with "totally inside". Even so, it is worth pointing out that Engels's actual words imply that this is a legitimate interpretation of what he said:

L2: Motion involves a body being in one and the same place and not in it.

If a body is in and not in a certain place it can't be totally in that place. So, a mundane example of this ambiguity is where, say, a 15 cm long pencil is sitting in a pocket that is only 10 cm deep. In that case, the pencil would be in, but not entirely in, the pocket -- that is, it would be both in and not in the pocket at the same time, but still at rest with respect to some inertial frame. L2 certainly allows for such a situation. Engels's use of the word "in" and the rest of what he said carries this ambiguity.

Hence, it seems that Engels's words are compatible with a body being motionless relative to some inertial frame. And this is still the case even when L1 and L2 are combined, as Engels intended they should:

L3: Motion involves a body being in one place and in another place at the same time, and being in one and the same place and not in it.

An example of L3-type -- but apparently contradictory -- 'lack of motion' would involve a situation where, say, a car is parked half in, half out of a garage. Here the car is in one and the same place and not in it (in and not in the garage), and it is in two places at once (in the garage and in the grounds of the house), even while it is at rest relative to a suitable inertial frame.

In which case, the alleged contradiction Engels is interested in can't be the result of motion; it is in fact a consequence of the vagueness or the ambiguity of his description.

This can be seen from the further fact that objects at rest relative to some inertial frame can and do display the same apparent 'contradictions' as those that are in motion with respect to the same inertial frame. Naturally, if things at rest share the very same vague or ambiguous features (when they are expressed in language) as those that are in motion, Engels's description clearly fails to pick out what is unique to moving bodies.

This is not a good start.

At best, L3 simply depicts the necessary but not the sufficient conditions for motion. In that case, the alleged contradictory nature of L3 has nothing to do with movement actually occurring, since the same description could be true of bodies at rest, which share the same necessary conditions. As already noted, alleged paradoxes like this arise from the ambiguities implicit in the language Engels himself used -- and, as it turns out, language he misused. [This will be discussed in greater detail below.]

Nevertheless, in the next few sections several attempts will be made to remove and/or resolve these equivocations in order to ascertain what, if anything, Engels might have meant by the things he tried to say about moving bodies.

First Attempt At Disambiguation

As will also be demonstrated in Essay Six -- in relation to Trotsky's (and indirectly Hegel's) attempt to analyse the LOI --, Engels's account of motion is in fact far too vague to be of much use.^11a

[LOI = Law of Identity.]

I now propose the following disambiguation of Engels's comments about motion in order to determine if there is any sense at all to be made of what he concluded about moving bodies:

L5: A body B in motion involves change of place such that:

L6: B is at (X₁, Y₁, Z₁) at t₁ and at (X₂, Y₂, Z₂) at t₁.

L7: (X₁, Y₁, Z₁) is not the same place as (X₂, Y₂, Z₂).

[Where, (X_i, Y_i, Z_i) etc., are coordinate triples, and t₁ is a temporal variable.]

This opening set looks more promising. However, it is worth noting that this clarity has only emerged because of the introduction of the phrase "change of place", in L5. Unfortunately, if this expression does succeed in bringing out what Engels meant it would suggest that change explains motion, not the other way round. Perhaps this minor difficulty can be circumvented; I will leave that for others to decide.

[Still others, of course, might wonder exactly how the word "change" could be explicated (given this theory) without an appeal being made to a definition that involved the word "motion" (a definition, it is worth remembering, that has yet to be attempted by dialecticians -- Graham Priest excepted, of course). Naturally, the use of the latter term would not alter the truth of L5, but it would make it eminently circular.]

However, even if this 'niggle' were resolvable, the initial promise the above set of sentences seemed to offer soon evaporates when it is remembered that L5-L7 fail to rule out cases where an extended body might move at a later time, say t₂, but not at t₁. That is, B could still be stationary at t₁, and in two different places at once (because it is an extended object), and at rest with respect to some inertial frame, with the subsequent motion taking place at t₂, not at t₁ -- as we saw above with that car.

The significance of this observation is easily lost, but it revolves around the fact that Engels's account is compatible with an object being stationary at t₁, and it is no reply to be told that this object moved later, when we are still owed a description of motion that captures its necessary and sufficient conditions, not a promissory note that the said object will move at some later time. Anyway, attempts to capture the necessary and sufficient conditions of the future predicted motion of this object will only attract the same criticisms -- that is, if L5-L7 were merely replaced with propositions that just change the temporal variable to t₂, while no other adjustments are made. This is because, in that case, questions will only arise as to why this minor alteration is capable of turning L5-L7 into necessary and sufficient conditions when the use of t₁ failed to do this originally.

L6-L8 below attempt to fix this glitch.

The problem, it seems, lies with L5, since it fails to connect the motion it mentions with the same instant recorded in L6 and L7. Hence, the following emendations need to be made, it would seem:

L8: A body B in motion involves change of place only at t₁, such that:

L6: B is at (X₁, Y₁, Z₁) at t₁ and at (X₂, Y₂, Z₂) at t₁.

L7: (X₁, Y₁, Z₁) is not the same place as (X₂, Y₂, Z₂).

Of course, the same caveats could be applied to later instants, so that such a description captures the movement of the body in question along its entire trajectory. That would merely entail the use of "t_i" in the place of "t₁" in L8 and L6. This complication will be ignored here, since it doesn't seem to affect the points at issue.

Unfortunately, however, L6-L8 do not appear to imply a contradiction --, that is, not unless it is clear that B is no longer at (X₁, Y₁, Z₁) at t₁, since it is possible for a body to be in two places at once. For example, few would regard it as a contradictory feature of reality that a cake, say, could be in a box and in a supermarket all at once, and stationary with respect to some inertial frame all the while.

On the other hand, if a die-hard dialectician could be found who thought that that scenario was contradictory, he/she would need to explain to the rest of us exactly what this alleged contradiction amounted to, and how, in virtue of its being in two such places at once, for example, the cake involved was engaged in some sort of 'struggle', and against what it was 'struggling'! As we will see in Essay Seven, the dialectical classicists hold that objects turn into whatever their opposites are, that is, whatever they turn into whatever they are contradicted by, or are 'struggling' with. In this example, that would seem to involve such cakes 'struggling' with and then turning into the buildings that housed them! Since no one in their left mind could reasonably be expected to believe this, cakes in supermarkets can't be regarded as in anyway contradicting the bricks and mortar that surround them. Anyone who still thinks this is advised to seek professional help.

So, in order to rectify this, we need to replace L6 with L9, as follows:

L8: A body B in motion involves change of place only at t₁, such that:

L9: B is at (X₁, Y₁, Z₁) at t₁ and not at (X₁, Y₁, Z₁) at t₁, and B is at (X₂, Y₂, Z₂) at t₁.

L7: (X₁, Y₁, Z₁) is not the same place as (X₂, Y₂, Z₂).

Now, this set (henceforth called ℒ) certainly looks inconsistent. The question, though, is: Can all its constituent sentences be false at once? Only if we can rule out that eventuality is it possible to construct a contradiction from all and only elements of ℒ.

[At this point it is worth recalling that a set S of sentences is inconsistent just in case not all of its elements can be true at once. But a "contradiction" requires more than this. In the simplest case, the elements of a binary sub-set of sentences formed from elements of S are contradictory just in case (1) those elements are inconsistent and (2) they cannot also be false. In short, they cannot both be true and they cannot both be false. This salient fact is invariably overlooked by DM-theorists, which, naturally, leads them into confusing contradictions with inconsistencies and/or contraries -- and, in many cases, with a host of other unrelated things, too. (Any who object to the alleged 'pedantry' here should read this first and then think again.)]

The question is, therefore: Can all of the elements of ℒ be false at once? If they can, then it won't be possible to construct a contradiction from all and only elements of ℒ. I propose to resolve this question by considering each of ℒ's constituent sentences in turn, but in reverse order:

(1) L9 would be false if at least one of its conjuncts was false. But the first part of L9 ["B is at (X₁, Y₁, Z₁)"] could be false in several ways: for example, if B is at (X₃, Y₃, Z₃) at t₁.

[In fact L9 is an inconsistent sentence anyway, and hence it is false (either that, or it isn't a proposition to begin with (which is what I would maintain, anyway)^11b --, depending on which branch of the Philosophy of Logic one attends to). But since DL is based on the claim that inconsistent sentences can be true, I have ignored this alleged fact here because it would beg the question.]

(2) L8 is linked to L9 by means of a "such that" phrase, so the truth or falsehood of L8 is sensitive to the truth or falsehood of L9. Hence, when L9 is false, L8 is, too.

(3) L7 could be false if (X₁, Y₁, Z₁) was the same place as (X₂, Y₂, Z₂). This would make L9 false, as well.

In which case, it looks like we can imagine situations in which, while not all of L's elements could be true at once, all could be false at once. This means that it isn't possible to construct a contradiction from all and only elements of ℒ.

Knowledgeable readers will have noticed the illegitimate way in which the schematic sentences of ℒ (and others) have been interpreted here to derive this spurious result. The reason for this ploy (and what its implications are) will be commented upon presently.

Second Attempt At Disambiguation

From this point on it will be assumed that the difficulties with Engels's account noted in the previous section (whether or not they are legitimate) can be resolved, and that there exists some way of reading his words that implies a contradiction, and which succeeds in distinguishing moving from motionless bodies.

Perhaps the following will suffice:

L10: For some body b, at some time t, and for two places p and q, b is at p at t and not at p at t, and b is at q at t, and p is not the same place as q.

This looks pretty contradictory. With suitable conventions about the use of variables we could abbreviate L10 a little to yield this slightly neater version:

L11: For some b, for some t, for two places p and q, b is at p at t and not at p at t, and b is at q at t.

However, one point needs underlining here: none of the strictures dialecticians impose on the LOI must be allowed to stand if L11 is to work, otherwise we would lose the ability to talk about "the same body", "the same time" or "the same place". This would also affect the application of certain conventions governing the use of terms such as "same variable", "same meaning" and "same reference". Hence, if we are to depict the contradictory nature of motion successfully we are forced to accept as valid the application of the LOI to the use of the same words and the same variables ranging over temporal instants (but, as a rule of language, not a 'logical truth' -- why this is so is explained in Essay Twelve Part One). Since protracted examples of motion take place over very long time periods, we can't appeal to the relative stability of language to fix the reference of these variables (or that of their ordinary language counterparts), if the LOI is not applicable in all cases.

[MFL = Modern Formal Logic; LOI = Law of Identity.]

But, if the LOI is rejected then Engels's description would become irredeemably vague. Many of the 'spurious' objections rehearsed toward the end of the previous section (in relation ℒ) depended on ignoring some or all of these conventions; as a result those objections were entirely illegitimate. Of course, that ploy itself was aimed at highlighting this very point: the use of variables in FL is based on conventions that DM-theorists must themselves observe (in ordinary discourse, and/or in logic) if Engels's analysis of motion is to be rendered comprehensible, but which conventions vitiate their own criticisms of the LOI! Naturally, it is a moot point which horn of this dilemma they will want to "grasp": either accept the Hegelian criticisms of the LOI and sink Engels's analysis of motion, or accept Engels's account and abandon Hegel's criticism of the LOI.

It could be objected that the above comments represent a caricature of the criticisms that dialecticians make of the LOI. The relative stability of both material bodies and linguistic expressions permits them to talk about such things as the "same body", the "same word", the "same variable", and so on. Moreover, dialecticians do not flatly deny the LOI, they just claim that it is true only within certain limits. In addition, they hold that objects and processes in change possess "identity-in-difference".

These responses are considered in detail in Essay Six (the relative stability argument, for example, is neutralised here); 'dialectical contradictions' themselves are analysed in Note 1 and here.

Of course, hard-nosed dialecticians might choose to ignore MFL altogether. That is, of course, their right. But they would then find it rather difficult to say what Engels actually meant in the quoted passage above. [Anyway, that bolt hole will be blocked later on in this Essay.]

Unfortunately however, even as it stands, and despite the foregoing (that is, if the contentious claims made above about the LOI and MFL are indeed misconceived, and are thus withdrawn), L11 would still fail to be a logical contradiction, and that is because of several more annoying ambiguities.

In fact, this new batch of vagaries turns out to be far more intractable than the relatively minor ones considered so far.

A Fatal Ambiguity

This latest set of equivocations revolves around the supposed reference of the "t" variable in L11:

L11: For some b, for some t, for two places p and q, b is at p at t and not at p at t, and b is at q at t.

It is always possible to argue that L11 really amounts to the following:

L12: For some b, during interval T, and for two 'instants' t₁ and t₂ [where both t₁ and t₂ belong to T, such that t₂ > t₁], and for two places p and q, b is at p at t₁, but not at p at t₂, and b is at q at t₂.

[In the above, t₁ and t₂ are themselves taken to be sets of nested sub-intervals, which can be put into an isomorphism with suitably chosen intervals of real numbers; hence the 'scare' quotes around the word "instant" in L12. (Incidentally, t₂ > t₁ means that t₂ is later than t₁.)]

Clearly, the implication here is that the unanalysed variable "t" in L11 actually picks out a time interval T (as opposed to an instant in time) -- brought out in L12 -- during which the supposed movement takes place. This would licence a finer-grained discrimination among T's sub-intervals (i.e., t₁ and t₂) during which this occurs.¹² Two possible translations of L12 in less formal language might read as follows:

L12a: A body b, observed over the course of a second, is located at point p in the first millisecond, and is located at q a millisecond later.

L12b: A body b, observed over the course of a millisecond, is located at point p in the first nanosecond, and is located at q a nanosecond later.

And so on…

Indeed, this is how motion is normally conceived: as change of place in time -- i.e., with time having advanced while it occurs. If this weren't so (i.e., if L12 is rejected), then L11 would imply that the supposed change of place must have occurred outside of time -- or, worse, that it happened independently of the passage of time --, which is either incomprehensible (as even Trotsky would have admitted), or it would imply that, for parts of their trajectory, moving objects (no matter of how low their speed) moved with an infinite velocity! [This was pointed out earlier.]

And yet, how else are we to understand Engels's claim that a moving body is actually in two places at once? On that basis, a moving body would move from one place to the next outside of time -- that is, with time having advanced not one instant. In that case, a moving body would be in one place at one instant, and it would move to another place with no lapse of time; such motion would thus take place outside of time. But, according to Trotsky, that sort of motion wouldn't exist, for it would not have taken place in time.

Indeed, we would now have no right to say that such a body was in the first of these Engelsian locations before it was in the second. [That is because "before" implies an earlier time, which has just been ruled out.] By a suitable induction clause, along the entire trajectory of a body's motion it would not, therefore, be possible to say that a moving body was at the beginning of a journey before it was at the end!

[The reasons for saying this will be detailed below, but the latter conclusion depends on the argument presented here, which should be read first. Trotsky's worries about instants are examined below, and in Essay Six. The contrary idea that if a body is located at a point at an instant, it must be stationary, is also examined below.]

Despite this it would seem that this latest difficulty can only be neutralised by means of the adoption of an implausible stipulation to the effect that whereas time isn't composed of an infinite series of embedded sub-intervals -- characterised by suitably defined nested sets of real numbers --, location is.

[Naturally, such a stipulation would have to reject Trotsky's strictures on events taking place only in time.]

This would further mean that while we may divide the position a body occupies as it moves along as finely as we wish -- so that no matter to what extent we slice or magnify a body's location, we would always be able to distinguish two contiguous (or proximate) points allowing us to say that a moving body was in both of these places at the same time --, while we can do that with respect to location, we can't do the same with respect to time.

Clearly, this is an inconsistent approach to the divisibility of time and space -- wherein we are allowed to divide one of these (space) as much as we like, while the same is disallowed of the other (time).

[It could even be argued that this is where the alleged 'contradiction' originally arose -- it was introduced into this 'problem' right at the start by this inconsistent (implicit) assumption, so no wonder it emerged at a later point -- no puns intended.]

This protocol might at first sight seem to neutralise an earlier objection (i.e., that even though a moving body might be in two places, we could always set up a one-one relation between the latter and two separate instants in time, because time and space can be represented as equally fine-grained), but, plainly, it only achieves this by stipulating (without any justification) that the successful mapping of places onto nested intervals of real numbers (to give them the required density and continuity) is denied of temporal intervals.

The Classical Response

So, there seem to be three distinct possibilities with these two separate variables (i.e., concerning location and time):

(1) Both time and place are infinitely divisible.

(2) Infinite divisibility is true of location only.

(3) Infinite divisibility is true of either but not both (i.e., it is true of time but not place, or it is true of place but not time).

Naturally, these aren't the only alternatives, but they seem to be the only ones that are relevant to matters in hand.

Of course, one particular classical response to this dilemma ran along the lines that the infinite divisibility of time and place implies that an allegedly moving body is in fact at rest at some point. Hence, if we could specify a time at which an object was located at some point, and only that point at that time, it must be at rest at that point at that time. [This seems to be how Zeno at least argued.]

Nevertheless, it seemed equally clear to others that moving bodies can't be depicted in this way, and that motion must be an 'intrinsic' (or even an 'inherent' property) of moving bodies (that is, we can't depict moving bodies in a way that would imply they are stationary), so that at all times a moving body must be in motion, allowing it to be in and not in any given location at one and the same time. [This seems to be Hegel's view of the matter -- but good luck to anyone trying to find anything that clear in anything he wrote about this topic!]

If so, one or more of the above options must be rejected. To that end, it seems that for the latter set of individuals 1) and 3) must be dropped, leaving only 2):

(2) Infinite divisibility is true of location only.

However, it is worth pointing out that the paradoxical conclusions classically associated with these three alternatives only arise if other, less well appreciated (often implicit) assumptions are either left out of the picture or are totally ignored -- i.e., in addition to those alluded to above concerning the continuity of space and the (assumed) discrete nature of time. As it turns out, the precise form taken by several of these suppressed and unacknowledged premisses depends on what view is taken of the allegedly 'real' meaning of the words like "motion" and "place".

In Essay Three Part One and Essay Twelve Part One, it was argued that philosophical 'problems' of this sort only arise when ordinary words are twisted beyond recognition (which view, incidentally, was endorsed by Marx), and the new conventions for the use of such terms that emerge as a result are then misinterpreted as super-empirical truths, not conventions at all.

In short, the 'classical' approach only gets off the ground if linguistic conventions/stipulations, and/or rules, are mistakenly viewed as Super-Scientific, mega-empirical propositions, or as matters of fact.

That is, this approach mistakes an implicit decision to use words in a novel way as a fundamental truth about reality itself. It misconstrues the medium for the content.

Indeed, this is precisely how theorists (in ancient Greece) began to misread the products of social relations (conventions/rules) as if they were the real relations between things, or even as those things themselves (thus fetishising language). Because of this they imagined they could 'derive' Super-theses like these from the 'philosophical' jargon they had invented.

As a result of this 'wrong turn' (although there were clear ideological motives for taking it -- on this, see below and here), Traditional Philosophers reasoned that the word "motion", for example, implied there was some sort of 'problem' (or 'contradiction', or 'paradox'), which needed to be resolved. Few, if any, questioned the original distortion/fetishisation that had been inflicted on ordinary words (for motion, place and change), which linguistic deformation had artificially created such 'difficulties'.

That is because, of course, these thinkers came from those sections of society that had been divorced from the world of collective labour and communal life, and whose theories reflected the ideal view of reality this privileged life-style encouraged. For the same reason it also arose from an ideologically-motivated denigration of the vernacular. [These allegations will be fully-documented in Essay Twelve (summary here).] So, in their view, if the world is ultimately Ideal, it would of course be quite 'safe' to infer Super-Scientific truths about it from language alone, as we saw George Novack point out earlier.

The fact that the classical 'paradox' of motion was based solely on a set of initial (surreptitious and, as it turns out, illegitimate and unacknowledged) false linguistic moves like this is confirmed by the further fact that the acceptance or rejection of one or more of the three options listed above (repeated below) can't be (and has never been) based on evidence of any sort. Severally or collectively, each of these alternatives is founded on a linguistic convention overtly or covertly accepted by all parties to this metaphysical con-trick, one that uses what is supposed to be the 'real' meaning of the word "motion" -- or, indeed, the 'real' meaning of any of the other terms associated with it (such as "place", "same", "time" and "instant").

Moreover, the choice of one or more of options (1) to (3) (as a way motivating a favoured 'solution' to this artificially-induced 'problem') also depends on the idea that even if the specification of the location of a body is in no way problematic (in that we can always declare that a moving body is in two places at once), the specification of time is.

Thus while the identification of point instants in time was seen to be the 'problem', the specification of points in space hardly raised a eyebrow. With respect to DM, this can be seen by the way that Trotsky, for example, failed to draw the same conclusions about locations in space that he drew about points in time.^12a

(1) Both time and place are infinitely divisible.

(2) Infinite divisibility is true of location only.

(3) Infinite divisibility is true of either (i.e., of time but not place, or of place but not time).

Nevertheless, these appear to be among the fundamental issues that have exercised philosophers for millennia -- and now dialecticians. In their case, however, the preferred 'solution' appears to rule out the possibility of a moving object being in two contiguous places at two different times. This means, therefore, that DM-theorists have implicitly opted for alternative (2):

(2) Infinite divisibility is true of location only.

[With the word "indefinite" perhaps replacing "infinite" here in some cases.]

As has already been noted, this choice was motivated by a surreptitious exclusion: the indefinite division of time was ruled out, while that of position wasn't.¹³

Finally, but more importantly, the traditional metaphysical 'solutions' that have been on offer for centuries were also based on the rejection of at least one implication of the ordinary understanding of motion, which is that moving bodies occupy different places at different times. This is such a mundane connotation of our every day grasp of certain kinds of motion that it seldom features in classical discussions, except perhaps where it is rejected out of hand as far too 'crude' to be worthy of consideration.

However, as we shall soon see (and again in several other Essays posted at this site), the protocols of ordinary language and common understanding are not so easily ignored, dismissed, or depreciated.

Back To The Drawing-Board

The Devil In The Detail

However, there are (and can be) no (a priori) empirical constraints on the length of time intervals. In fact, as was also noted above, Engels's account of motion was not (and could not have been) derived from observation, mediated via the naïve or the sophisticated version of the RTK. Nor could his idea of 'motion in general' -- nor, indeed, of 'abstract motion' -- have been materially-grounded, either.

[RTK = Reflection Theory of Knowledge.]

That is because human beings -- aided or not by the use of microscopes, computers, cameras or lasers -- do not possess powers of discrimination sufficiently fine-grained enough to allow the study of movement in the detail required, so that 'reflection' (or 'abstraction') could be presented with anything useable to work with, or upon, in order to decide what does or does not happen to moving bodies in an 'instant'.

And it is little use objecting that this or that 'must' be true of 'motion itself', for that would be to concede the fact that a 'must' like this had been derived from the meanings of a few words, which words, as we will soon see, are far less straight-forward than traditional theorists would have us believe.

It could be objected that the classical analysis of motion follows deductively from certain incontestable premises. There are only a handful of possibilities that the world could conceivably present to us; Engels's analysis, via Hegel, is based on one of these. So, what's the problem?

Once more, the problem is that the deducibility or otherwise of these conclusions depends on the use of several artificially modified/constrained words (such as "place", "move", "time", "moment", etc.), which have either been idiosyncratically, or narrowly (re-)defined, or which have had their meanings altered in other ways. In that case, nothing reliable can follow from them (as I hope to show later in this Essay). This was, of course, the point Marx was trying to make.

Even worse, not only does nothing follow from such distorted language (or abstract 'concepts'), it is impossible to give a clear sense (or any sense) to the classical account (nor, indeed, to more modern versions of it that depend upon the same defective tradition). In fact, as will be demonstrated in Essay Twelve Part One, all such accounts are non-sensical and incoherent; they not only do not say anything comprehensible about the world, they can't.

In that case, if humanity does in fact possess an 'abstract' idea of motion (but this will be contested below, and in other Essays posted at this site), it can't have been derived from 'reflection', nor could it have been based on anything found in the material world. And those observations become all the more apposite if this allegedly 'abstract' idea of motion itself originated from (a) The inequitable constraint mentioned above -- i.e., that which was arbitrarily imposed on the allowable length of temporal intervals, but excused of point locations in space --, for no good reason; and (b) A ruling-class view of reality.

In short, Engels's theory wasn't based on reflection (howsoever this 'process' is understood), or on evidence, or even on 'abstraction', but only on 'concepts' that are themselves the product of traditional/classical stipulations (or covert conventions) -- which were then imposed on reality inequitably!

Space To Let

Returning now to consider several earlier options:

L11: For some b, for some t, for two places p and q, b is at p at t and not at p at t, and b is at q at t.

L12: For some b, during interval T, and for two 'instants' t₁ and t₂ [where t₁ and t₂ belong to T, t₂ > t₁], and for two places p and q, b is at p at t₁, but not at p at t₂, and b is at q at t₂.

(2) Infinite divisibility is true of location only.

[L12a: A body b, observed over the course of a second, is located at point p in the first millisecond, and is located at q a millisecond later.

L12b: A body b, observed over the course of a millisecond, is located at point p in the first nanosecond, and is located at q a nanosecond later. And so on…]

However, if for some reason L12 were to be rejected as an alternative interpretation of L11 (that is, if the idea that time is continuous and indefinitely divisible is flatly denied (while this condition is asymmetrically allowed of space) -- i.e., if option (2) above is imposed on the phenomena) --, then there seems to be no consistent way of ruling out the following as yet another possible reading:

L13: For some b, for just one instant t, for three places p₁, p₂ and p₃, b is at p₁ at t, but not at p₂ at t, and b is at p₃ at t (where p₂ and p₃ are proper parts of p₁).

Here, a finer-grained discrimination of position (but not of time) means that L13 is not contradictory at all, since a body can be in two places at once whether it moves or not (as we have seen), with no implication that it both is and is not in any one of them.¹⁴

Translated, L13 could be read as follows:

L13a: A stationary body b, observed over the course of an instant, is at (X₁, Y₁, Z₁) and (X₃, Y₃, Z₃), but not at (X₂, Y₂, Z₂), where (X₃, Y₃, Z₃) and (X₂, Y₂, Z₂) are both located inside (X₁, Y₁, Z₁).

L13b: A moving body b, observed over the course of an instant, is at (X₁, Y₁, Z₁) and (X₃, Y₃, Z₃), but not at (X₂, Y₂, Z₂), where (X₃, Y₃, Z₃) and (X₂, Y₂, Z₂) are both located inside (X₁, Y₁, Z₁).^14a

[An obvious objection to the above is neutralised in Note 14a (link above).]

An everyday example of L13 might involve a case where a ship, say, enters port: here the ship could both be in the water and in the port at the same time (and hence simultaneously extended across several locations, and thus be in at least two places at once), and be moving, but with no implication that it is entirely in any one of these at one and the same instant, or that it is fully occupying any specific part at any moment, nor yet occupying every point in this finite region (so that it need not be in other areas of that port, for example, at that time). In the latter case, while it is still inside the said port it wouldn't be in, say, the dry dock (which is also part of that port), nor in the staff canteen, nor in a host of other places in that port, at that time.

Moreover, if the ship were stationary with respect to some inertial frame, the same possibilities would still apply. Here, this ship could be in one place and not in it (fully), and in two or more places at once, and stationary (or moving), and yet imply no contradiction. That is because this particular example employs a finer-grained division of place to compensate for the arbitrary imposition of the opposite convention on time.

In that case, the alleged contradiction vanishes once again.

[I have given a more technical version of this scenario in Note 15.]¹⁵

As pointed out above, L12 and/or L13 can only be rejected successfully by an ad hoc stipulation to the effect that while spatial location can be divided indefinitely, time may not.

But, even then we have just seen that Engels's claims still don't work!

In which case, of course, the allegedly contradictory nature of motion is at best an artefact of convention -- which only works by constraining the divisibility of time but not of place --; hence it isn't one based on any genuine features of reality.¹⁶

At worst, it is the product of a confused use of philosophical jargon.

Further Problems

Pick Your 'Contradiction'

It could be objected to all this that while it might not be possible to express the contradictory nature of motion in ordinary (or even technical) language,¹⁷ motion in the real world must nevertheless be contradictory.¹⁸ This might involve the acceptance of one or more of the following (but so far suppressed) assumptions:

L14: An object can't be in motion and at rest at one and the same time (in the same inertial frame).^18a

L15: If an object is located at a point it must be at rest at that point.^18b

L16: Hence, a moving body can't be located at a point, otherwise it wouldn't be moving, it would be at rest.^18b1

L17: Consequently, given L14, a moving body must both occupy and not occupy a point at one and the same instant.

In which case, it could be argued that L14-L16 (or their 'dialectical' equivalent) capture the rationale behind Engels's analysis of motion.

Indeed, if this weren't so, it would suggest that motion was either (a) impossible or (b) illusory --, or even (c) that it was a sort of 'stop-go' affair.^18c

As far as (c) is concerned, motion would be analogous to the way movement is depicted by, say, the use of film. Here, motion only appears to be continuous when it is in fact discontinuous, being composed of rapidly sequenced 'freeze frames', as it were. When played at a certain speed, this 'fools' the human eye into 'seeing' continuous movement. Given this 'quasi-static' view of motion, a 'moving' body (in the real world, not on film!) would occupy a point and be stationary at that point, and then occupy another point an instant later, and be stationary there too, and so on. Naturally, what the said object gets up to in between such locations at such times would be, on this view, somewhat mysterious. But, on its own, that wouldn't be enough to make this picture of motion false, no more than quantum discontinuities now invalidate QM -- that is, given the way that motion is depicted in Traditional Philosophy.¹⁹

[QM = Quantum Mechanics.]

Options (a) and (b) above are absurd and won't be considered in this Essay.

In order to reject this 'quasi-static' view (i.e., option (c)), consideration might be given to one or more of the following (each defined in relation to a suitable inertial frame, as necessary):

L18: If a body is located at a point it is at rest.

L19: If that body subsequently occupies another point, it must be at rest there, too.

L20: Hence, on this view (i.e., according to (c) above), motion is no more than successive point occupancy. This means that locomotion must be composed of either: (i) Successive states of instantaneous rest, or (ii) The sequential existence and non-existence of what only seem to be identical -- but which are in fact numerically different -- bodies at each of the said points, with that body falling into non-existence at the end of each moment of location/rest, followed by the subsequent entry into existence of a new, but seemingly identical body at the next moment, at the new point, giving only the impression of motion.

[Option (ii) would resemble the way that neon lights in a complex sign, say, can be turned on and off in sequence to create the illusion of motion. It seems Leibniz held a version of this theory.]

L21: L20(i) involves a body in discontinuous motion separated by periods of instantaneous rest. L20(ii) involves a body, or series of bodies, in discontinuous existence at contiguous locations.

L22: L20(ii) must be rejected as absurd.

L23: If L20(ii) is rejected then L20(i) implies that in between each successive point occupancy a body must pass through an indefinite (possibly infinite) number of intervening locations.

[Of course, this depends on the further assumption that there are an infinite, or a potentially infinite, number of points between any two points.]

L24: Hence, even on the assumption that motion is discontinuous, there will still be an indefinite number of such intermediate points that a moving object has to occupy while it is passing between the points at which it is said to be at rest in consecutive instants, but which intermediate locations the body must both occupy and leave at one and the same instant. In that case, that body can't be at rest at any of these intermediate points.

L25: Consequently, if motion takes place -- and it is either continuous or discontinuous -- a moving body must both be located and not be located in a given place at one and the same time, namely at these intermediate points, at least.

L26: Therefore, the assumption that a body is in motion only if it occupies and is at rest in successive locations at contiguous instants is false -- for even on that assumption, a body must violate this condition for an indefinite number of intermediate points between each successive instance of 'rest', at successive instants.

L27: Therefore, either motion is impossible or illusory (which is absurd), or motion can't be wholly discontinuous.

[It is possible to strengthen L27 by means of L27a, but that option will not be pursued further here:

L27a: Therefore, either motion is impossible or illusory (which is absurd), or motion can't be discontinuous.]^19a

However, it is worth noting that the above argument begins with the rejection of an apparent contradiction -- that which is expressed in L14 (restated here for ease of reference, but re-numbered L28, and hence very slightly altered) alongside its supposed contradictory, L29:

L28: A body can't be at rest and in motion at the same time in the same inertial frame.

L29: A body can be at rest and in motion at the same time in the same inertial frame.

Naturally, this depends on whether these are genuine contradictories; I will ignore that minor complication here. On the other hand, if they aren't even propositions, then they cannot be contradictories to begin with.

Nevertheless, I will assume they are propositions for the purposes of this argument; however, their status as propositions will be questioned in Essay Twelve Part One.²⁰

Hence, if these 'niggles' are ignored, L29 is true if L28 is false, and vice versa.

As is well-known, an analogous series of assumptions motivated Zeno to try to 'prove' that motion was either impossible or illusory. DM-theorists obviously reject Zeno's conclusion, but it seems they can only do that by accepting L28 (or its equivalent), and by rejecting L29, in order to derive their own contradiction expressed in L17, which was:

L17: A moving body must both occupy and not occupy a point at one and the same instant.

Plainly, if L28 were false (and L29 true) -- which would mean that a body could be moving and at rest at the same time --, L17 might not look quite so compelling. At any rate, it is clear that dialecticians have to reject one 'contradiction' (expressed in L29) in order to derive their own (in L17).

Now, when L17 is conjoined with L28 we obtain the following:

L17a: Since a body can't be at rest and moving at one and the same time in the same inertial frame, a moving body must both occupy and not occupy a point at one and the same time.

This seems to be the 'contradiction' that exercised Engels. If so, it is worth asking: Which one of the following 'contradictions' is it legitimate to accept or reject: L17 or L29?

L17: A moving body must both occupy and not occupy a point at one and the same instant.

L29: A body can be at rest and in motion at the same time in the same inertial frame.

Which of these 'contradictions' is the more absurd? If L29 is true, it looks like L17a cannot be derived in any obvious way from the sorts of considerations advanced in L14-L27. This would mean that Engels's analysis is defective -- always assuming, of course, that his 'argument' depends on such considerations and that some sense can be made of anything he said in this regard.

Nevertheless, it is clear from the way that the above argument has been constructed that L17a itself depends on the truth of L28 (repeated here again for ease of reference):

L28: A body can't be at rest and in motion at the same time in the same inertial frame.

L17a: Since a body can't be at rest and moving at one and the same time in the same inertial frame, a moving body must both occupy and not occupy a point at one and the same time.

That is because L14-L27 began with the assumed truth of L14 -- or, its equivalent, L28. The reverse implication doesn't appear to hold. This means that L28 does not seem to pre-suppose the truth of the conclusion drawn in L17a, whereas the conclusion drawn in L17a looks like it depends on L28. This in turn suggests that L28 might be the more fundamental of the two.

[L14: An object can't be in motion and at rest at one and the same time (in the same inertial frame).]

Be that as it may, L28 is itself false if L29 is true:

L28: A body can't be at rest and in motion at the same time in the same inertial frame.

L29: A body can be at rest and in motion at the same time in the same inertial frame.

Unfortunately, L29 is a familiar truth! An object can be at rest with respect to one inertial frame, and yet be in motion with respect to another. The wording of L29 doesn't rule this out. In order to eliminate this new difficulty, therefore, L29 must be modified; perhaps in the following manner:

L30: With respect to the same inertial frame and the same instant in time, a body can be at rest and in motion.

[L30 'contradicts' L30a:

L30a: With respect to the same inertial frame and the same instant in time, a body can't be at rest and in motion.]

L30 now certainly looks 'contradictory' (especially if "at rest" is taken to mean "not in motion with respect to the same inertial frame").

Nevertheless, it was the rejection of L30 (or its equivalent) that led to the derivation of L17a. Hence, if L30 is always false (i.e., if L30a is always true), it looks like L28 must always be true, too (given certain other assumptions, and if worded appropriately).

L17a: Since a body can't be at rest and moving at one and the same time in the same inertial frame, a moving body must both occupy and not occupy a point at one and the same time.

Consequently, if we deny that a body can be at rest and moving at the same time (in the manner indicated above), Engels's conclusion does appear to follow! This much seems reasonably clear.

Unfortunately, however, the following line of argument also shows that the derivation of L17a from the rejection of L30 isn't inevitable, and that Engels's conclusion doesn't automatically follow:

L31: A body can't be at rest and in motion with respect to the same inertial frame at the same time.

L32: If a body is wholly located at a point it can't be located wholly at any other point in the same reference frame at the same time.

L33: But, a moving body must be located wholly at two points at the same time, otherwise it would be at rest.

L34: Since L33 is impossible (by L32), motion can't take place. Hence, by L31, and despite appearances to the contrary, all bodies are at rest.

Of course, L34 is somewhat analogous to the conclusion Zeno himself drew, and it flatly contradicts experience. It is therefore unacceptable -- that is, if we allow experience to decide. But, L31-L34 demonstrate that L17a doesn't have to follow from the rejection of L30, even if the alternative outcome proves unpalatable.

It is now clear that the refusal to accept the 'contradiction' contained in L30 can lead to two distinct 'contradictory' conclusions. One of them is inconsistent with experience (i.e., the latter half of L34, i.e., L34b), while the other is self-contradictory (i.e., L17a):

L17a: Since a body can't be at rest and moving at one and the same time in the same inertial frame, a moving body must both occupy and not occupy a point at one and the same time.

L34b: Despite appearances to the contrary, all bodies are at rest.

Naturally, which one of these two outcomes proves to be the least unacceptable will depend on other priorities. If it is felt that experience is unreliable, L34b might be preferable. On the other hand, if contradictions are regarded as fundamental features of reality, and appearances are held to be deceptive, or unreliable, L17a might well be chosen. It is also worth noting, however, that neither option is empirically verifiable; in fact they both transcend any conceivable body of evidence and all possible experience.²¹

Nevertheless, given the fact that dialecticians also believe that appearances contradict underlying 'essences' they are the last ones who can legitimately appeal to experience to refute Zeno-esque conclusions like L34b. In fact, if the DM-thesis that underlying 'essences' 'contradict' appearances is itself true, then, since it appears to be the case that there are moving bodies, in 'essence' the opposite must be the case! Hence, if appearances 'contradict' reality it seems that, essentially, no bodies move, and Zeno was right after all!

Putting this annoying corollary to one side for now, it is worth emphasising that both halves of these two 'derivations' rely on the sorts of ambiguities encountered earlier with respect to L1-L13 (alongside several others analysed below). Aprioristic 'arguments' like these only seem to work because they are shot-through with equivocation and distortion; indeed, this is partly why the above two conclusions finally descend into absurdity and incoherence -- as we are about to discover.

Theatre Of The Absurd

The absurdity in L34b is quite plain for all to see and needn't detain us any longer. However, the ludicrous nature of L17a isn't perhaps quite so obvious. It may nevertheless be made more explicit by means of the following argument:

L35: Motion implies that a body is in one place and not in it at the same time; that it is in one place and in another at the same instant.

L36: Let A be in motion and at (X₁, Y₁, Z₁), at t₁.

L37: L35 implies that A is also at some other point -- say, (X₂, Y₂, Z₂), at t₁.

L38: But, L35 also implies that A is at (X₂, Y₂, Z₂) and at another place at t₁; hence it is also at (X₃, Y₃, Z₃), at t₁.

L39: Again, L35 implies that A is at (X₃, Y₃, Z₃) and at another place at t₁; hence also at (X₄, Y₄, Z₄), at t₁.

L40: Once more, L35 implies that A is at (X₄, Y₄, Z₄) and at another place at t₁; hence also at (X₅, Y₅, Z₅), at t₁.

By n successive applications of L35 it is possible to show that, as a result of the 'contradictory' nature of motion, A must be everywhere in its trajectory if it is anywhere, and all at t₁!²²

But, that is even more absurd than L34b!

L34b: Despite appearances to the contrary, all bodies are at rest.

The only way to avoid such an outlandish conclusion would be to maintain that L35 implies that a moving body is in no more than two places at once. But even this wouldn't help, for if a body is moving and in the second of those two places, it can't then be in motion at this second location -- unless, that is, it were in a third place at the very same time (by L15 and L35). Once again, just as soon as a body is located in any one place it is at rest there, given this way of viewing things. The proposed dialectical derivation outlined above required that very assumption, repeated here:

L15: If an object is located at a point it must be at rest at that point.

L35: Motion implies that a body is in one place and not in it at the same time; that it is in one place and in another at the same instant.

Without L15 (and hence L35), Engels's conclusions wouldn't follow; so on this view, if a body is moving, it has to occupy at least two points at once, or it will be at rest. But, that is just what creates this latest problem: if that body is located at that second point, it must be at rest there, unless it is also located at a third point at the same time.

This itself follows from L17 (now encapsulated in L17b):

L17: A moving body must both occupy and not occupy a point at one and the same instant.

L17b: A moving object must occupy at least two places at once.

Of course, it could be argued that L17b is in fact true of the scenario depicted in L35-L40 -- the said body does occupy at least two places at once namely (X₁, Y₁, Z₁) and (X₂, Y₂, Z₂). In that case, the above objection is misconceived.

Or, so it might be maintained.

The above objection would indeed be misconceived if Engels had managed to show that a body can only be in at most two (but not in at least two) places at once, which he not only failed to do, he couldn't do:

L17c: A moving object must occupy at most two places at once.

That is because, between any two points there is a third point, and if the body is in (X₁, Y₁, Z₁) and (X₂, Y₂, Z₂), at t₁ then it must also be in any point between (X₁, Y₁, Z₁) and (X₂, Y₂, Z₂), at t₁ --, say (X_k, Y_k, Z_k). But, as soon as that is admitted, there seems to be no way to avoid the conclusion drawn above: if a moving body is anywhere it is everywhere, at the same time.

[And that is why the question was posed earlier about the precise distance between the points at/in which Engels says a body performs such 'contradictory' marvels.]

On the other hand, the combination here of an "at least two places at once" with and an "at most two places at once" would amount to an "exactly two places at once".

L17d: A moving object must occupy exactly two places at once.

L15: If an object is located at a point it must be at rest at that point.

Any attempt made by DM-theorists to restrict a moving body to the occupancy of exactly two places at once would work only if that body came to rest at the second of those two points! L15 says quite clearly that if a body is located at a point (even if this is the second of these two points), it must be at rest at that point. In that case, the above escape route will only work if DM-theorists reject their own characterisation of motion, which was partially captured by L15. [This option also falls foul of the intermediate points objection, above.]

In that case, if L15 still stands, then at the second of these two proposed DM-points (say, (X₂, Y₂, Z₂)), a moving body must still be moving, and hence in and not in that second point at the same instant, too.

It is worth underling this conclusion: if a body is located at a second point (say, (X₂, Y₂, Z₂)) at t₁, it will be at rest there at t₁, contrary to the assumption that it is moving. Conversely, if it is still in motion at t₁, it must be elsewhere also at t₁, and so on. Otherwise, the condition that a moving body must be both in a place and not in it at the very same instant will have to be abandoned. So, DM-theorists can't afford to accept L17d.

Consequently, the unacceptable outcome --, which holds that as a result of the 'contradictory' nature of motion, a moving body must be everywhere along its trajectory, if it is anywhere, at the same instant -- still follows.

Again, it could be objected that when body A is in the second place at the same instant, a new instant in time could begin. So, while A is in (X₂, Y₂, Z₂) at t₁, a new instant, say t₂, would start.

To be sure, this ad hoc amendment avoids the disastrous implications recorded above. However, it only succeeds in doing so by introducing several new difficulties of its own, for this would mean that A would be in (X₂, Y₂, Z₂) at t₁ and at t₂, which would plainly entail that A was located in the same place at two different times, and that in turn would mean that it was stationary at that point!

It could be objected, once more, that A-like objects occupy two places at once, namely (X₁, Y₁, Z₁) and (X₂, Y₂, Z₂), so the above argument is defective. Indeed, this is why the above 'derivation' -- i.e., the one that show that a moving body must be everywhere along its trajectory, if it is anywhere, at the same instant -- can't work. We can perhaps clarify this objection by means of the following:

L38: L35 also implies that A is at (X₂, Y₂, Z₂) and at another place at t₁, hence it is also at (X₃, Y₃, Z₃) at t₁.

[L35: Motion implies that a body is in one place and not in it at the same time; that it is in one place and in another at the same instant.]

The idea here is that if we select, pair-wise, any two points that a body occupies in any order (either (X₁, Y₁, Z₁) and (X₂, Y₂, Z₂), or (X₁, Y₁, Z₁) and (X₃, Y₃, Z₃), or (X₁, Y₁, Z₁) and (X_n, Y_n, Z_n), and so on), then L17c will still be satisfied:

L17c: A moving object must occupy at most two places at once.

Unfortunately, this seemingly promising escape route turns into yet another cul-de-sac.

Here is why:

It was maintained that Engels just needed a body to be in any two places at once. But, the third place above -- (X₃, Y₃, Z₃) -- isn't implied by his description of the 'contradiction' involved. L38 only works by ignoring the fact that the other place that A is in is precisely (X₁, Y₁, Z₁); so, it can't be in (X₃, Y₃, Z₃) at that time --, or it doesn't have to be, which is all that is needed. So, when A is both in (X₁, Y₁, Z₁) and (X₂, Y₂, Z₂), and (X₁, Y₁, Z₁) and (X₃, Y₃, Z₃), and so on, it can't be in at most two places at once, since it is in this case in more than two. So, the use of "and" scuppers this line of defence.

It could be objected that the above response only works because an "and" has been substituted for an "or". The original response in fact argued as follows:

R1: If we select pair-wise any two points a body occupies in any order (either (X₁, Y₁, Z₁) and (X₂, Y₂, Z₂), or (X₁, Y₁, Z₁) and (X₃, Y₃, Z₃), or (X₁, Y₁, Z₁) and (X_n, Y_n, Z_n), or..., and so on), then L17c will be satisfied.

But not:

R2: If we select pair-wise any two points a body occupies in any order (i.e., (X₁, Y₁, Z₁) and (X₂, Y₂, Z₂), and (X₁, Y₁, Z₁) and (X₃, Y₃, Z₃), and (X₁, Y₁, Z₁) and (X_n, Y_n, Z_n), and..., and so on), then L17c will be satisfied.

Unfortunately, once more, this reply just catapults us back to an earlier untenable position, criticised above, as follows:

"This is because, between any two points there is a third point, and if the body is in (X₁, Y₁, Z₁) and (X₂, Y₂, Z₂), at t₁ then it must also be in any point between (X₁, Y₁, Z₁) and (X₂, Y₂, Z₂), at t₁ --, say (X_k, Y_k, Z_k). Once that is admitted there seems to be no way to forestall the conclusion drawn above that if the body is anywhere it is everywhere at the same time."

In that case, the reply encapsulated in L38/R1 fails. So, if a body is in (X₁, Y₁, Z₁) and (X₂, Y₂, Z₂) at t₁, it must also be in at least one of the intermediate points, say, (X_k, Y_k, Z_k), also at t₁. In that case, R2 is still a valid objection.

In order to see this, a few of the subscripts in R2 need only be altered, as follows:

R3: If we select pair-wise any two points a body occupies in any order (i.e., (X₁, Y₁, Z₁) and (X₂, Y₂, Z₂), and (X₁, Y₁, Z₁) and (X_k, Y_k, Z_k), and (X₁, Y₁, Z₁) and (X_i, Y_i, Z_i), and so on), then L17c will not be satisfied.

It is surely philosophically irrelevant whether we label such points with iterative letters (i.e., "k" or "i") or with numerals ("1", "2" or "3"). [Recall, the points labelled with iterative letters (i.e., "k" or "i") are intermediate points.]

In which case, R3 implies that if a body is in, say, (X₁, Y₁, Z₁) and (X₂, Y₂, Z₂), at t₁, it must also be in at least one of the intermediate points, say, (X_k, Y_k, Z_k), at the same moment.

R3 thus implies that L17c is false.

L17c: A moving object must occupy at most two places at once.

Since there is a potentially infinite number of points between any two points, there is no way that L17c can be true.

Moreover, it is also worth asking of L38: Is A at (X₂, Y₂, Z₂), at  t₁? If it is, then it must be elsewhere at the same time, or it will be stationary, once more. So much is agreed upon. In that case, the only way to stop the absurd induction (i.e., that which derived the conclusion that if a moving body is anywhere it must be everywhere) would be to argue as follows:

L38a: L35 also implies that A is at (X₂, Y₂, Z₂) and at another place at t₁, hence it is also at (X₁, Y₁, Z₁), at t₁, but not at (X₃, Y₃, Z₃), at t₁.

[L38: L35 also implies that A is at (X₂, Y₂, Z₂) and at another place at t₁, hence it is also at (X₃, Y₃, Z₃), at t₁.]

However, this 'straw', once clutched, has unfortunate consequences that desperate dialecticians might want to think about before they claw at it too eagerly:

L38b: If A is at (X₂, Y₂, Z₂) and (X₁, Y₁, Z₁), at t₁, but not at (X₃, Y₃, Z₃), at t₁, then it must be at (X₃, Y₃, Z₃), at t₂.

L38c: If so, A will be at two places -- (X₂, Y₂, Z₂) and (X₃, Y₃, Z₃) -- at different times (i.e., (X₂, Y₂, Z₂), at t₁, and (X₃, Y₃, Z₃), at t₂).

L38d: In that case, between these two locations (i.e., (X₂, Y₂, Z₂) and (X₃, Y₃, Z₃)), the motion of A will cease to be contradictory -- since it will not now be in these two places at the same time, but in these two places at different times.

Hence, dialecticians can only escape from the absurd consequence that their theory implies that a moving object is everywhere at the same time by abandoning their belief in the contradictory nature of motion at an indefinite number of locations intermediate in its transit -- for example, right after it leaves the first two places it occupied in its journey!

It now looks like DM-theorists can only counter the above criticisms, and maintain their view that motion is 'contradictory', if they are prepared to impose several more ad hoc stipulations on nature (of the sort mentioned above, none of which seem to work anyway).

But, as we have seen several times already, such a response would be fatal to DM since it would undermine their belief that reality itself is contradictory (rather than just the things we say about it that are), all the while confirming the suspicion that it is only certain ways of representing nature that appear to be contradictory -- which "ways of representing nature", incidentally, still await clarification.

This option would, of course, mean that this part of DM (at least) is thoroughly conventional, and thus entirely subjective -- and still defective!

As we will see throughout this site, the source of these (and similar) 'problems' lies in the repeated attempt made by dialecticians (and metaphysicians alike) to state 'necessary truths' (i.e., a priori 'theses') about reality. Such theses are based solely on an extrapolation from the supposed meaning of a few specially-selected words to fundamental truths about the universe, valid for all of space and time. Clearly, with respect to Engels's analysis of motion, this predicament is further compounded by an attempt to circumvent several fundamental conventions found in our use of ordinary language --, such as those expressed by the LOC and the LOI. [I will endeavour to substantiate these claims below, and in detail in Essay Twelve Part One.]

[LOC = Law of Non-contradiction; LOI = Law of Identity; FL = Formal Logic; DL = Dialectical Logic.]

Finally, it could be argued that the above criticisms beg the question, since dialecticians do not doubt the application of principles drawn from FL -- such as the LOC --, they merely point to their limitations when confronted with change and motion. That rather odd claim is neutralised in Essay Four and Essay Eight Parts One and Three.

Suffice it to say here that dialecticians themselves have yet to account for motion in anything like a comprehensible form -- or even depict it accurately! So, whether or not it is correct to say that FL can account for motion and change, it is now quite clear that DL itself can't.

Even more annoying: in Essay Four we saw that, contrary to what dialecticians constantly tell us, FL can quite easily cope with motion and change.

Yet Another Absurd Dialectical Consequence

Another, perhaps less well appreciated consequence of this view of motion and change -- which, if anything, is even more absurd than the one outline above --, is the following:

If Engels were correct (in his characterisation of motion and change), we would have no right to say that a moving body was in the first of these 'Engelsian locations' before it was in the second.

L3: Motion involves a body being in one place and in another place at the same time, and being in one and the same place and not in it.

That is because such a body, according to Engels, is in both places at once. Now, if the conclusions in the previous section are valid (that is, if dialectical objects are anywhere in their trajectories, they are everywhere all at once), then it follows that no moving body can be said to be anywhere before it is anywhere else in its entire journey! That is because such bodies are everywhere all at once. If so they can't be anywhere first and then later somewhere else.

In the dialectical universe, therefore, when it come to motion and change, there is no before and no after!^22a

In that case, according to this 'scientific theory', concerning the entire trajectory of a body's motion, it would be impossible to say it was at the beginning of its journey before it was at the end! In fact, it would be at the end of its journey at the same time as it sets off! So, while you might foolishly think, for example, that you have to board an aeroplane (in order to go on your holidays) before you disembark at your destination, this 'path-breaking' theory tells us you are sadly mistaken: you must get on the plane at the very same time as you get off it at the 'end'!

And the same applies to the 'Big Bang'. While we might think that this event took place billions of years ago, we are surely mistaken if this 'super-scientific' theory is correct. That is because any two events in the entire history of the universe must have taken place at the same instant, by the above argument. Naturally, this means that while you are reading this, the 'Big Bang' is in fact still taking place!^22b

To be sure, this is absurd, but that's Diabolical Logic for you!

No Word Is An Island

And Therefore Never Send To Know For Whom The Bell Tolls; It Tolls For DM

Several of the points raised above require further elaboration -- in the course of which we will discover once again that Engels was in fact saying nothing at all intelligible.

As we have seen, Engels asserted the following:

"[A]s soon as we consider things in their motion, their change, their life, their reciprocal influence…[t]hen we immediately become involved in contradictions. Motion itself is a contradiction; even simple mechanical change of place can only come about through a body being both in one place and in another place at one and the same moment of time, being in one and the same place and also not in it. And the continual assertion and simultaneous solution of this contradiction is precisely what motion is." [Engels (1976), p.152.]

However, in doing so, he was clearly appealing to what he regarded as the established, inter-subjective meaning of terms like "motion", "change", "place", "moment", and "time". This can be seen from the fact that he didn't even think to define or explain what he meant by these words. Ordinarily, that in itself wouldn't be a problem, since we understand phrases like these perfectly well in our day-to-day affairs without recourse to definitions and the like. But, in specialised areas of study (especially those that seek to revise or correct the way we see things), a sloppy approach to theory is unacceptable. Indeed, such cavalier attitudes to ordinary language have a tendency to backfire on those foolish enough to adopt them. Again, this is especially true of those who attempt to press the vernacular into service way beyond its remit.

The ability to think one's way around such linguistic conundrums with ease is supposedly what dialecticians mean by "grasping a contradiction". This seems to imply that when confronted with the many 'contradictions' that nature allegedly throws our way, dialecticians merely have to "grasp" them, and all is well. This neat trick then 'allows' these 'serial graspers' to ignore the internal contradictions this approach introduces into their own theory. [More on that here and here.]

However, as we will see in Essay Seven (and here), DM-theorists are highly selective over which 'contradictions' they choose to "grasp", and which they blame on defective or on competitive theories. Hence, when dialecticians "grasp" the 'contradictions' they claim to see in motion and change, they attribute these to nature itself, and fail to blame this on Hegel's logical incompetence, or on Engels's lack of clarity, or both.

On the other hand, when contradictions appear in rival theories, these become a handy excuse for rejecting them. In this way, they tell us that by rejecting or 'resolving' such contradictions science can advance. But, if science advances by rejecting or 'resolving' contradictions in and between theories, then, plainly, the science of kinematics can't advance unless this 'dialectical contradiction' is also resolved (as, indeed, it will be by the end of this Essay, except this will be achieved by dissolving it). However, just as soon as that has been done, dialecticians will surely have to abandon their belief in the 'contradictory' nature of motion, or risk holding up the progress of science.

This self-inflicted quandary I have called "The Dialecticians' Dilemma".

As seems obvious, 'dialectically' clutching at a 'contradiction' doesn't make it disappear. As DM-theorists view things, motion is still 'contradictory', whether or not anyone else sees things this way. Hence, the significance of "grasping a contradiction" appears to be this: Anything that might ordinarily seem puzzling or paradoxical suddenly stops bothering dialecticians (that is, if it ever did). But, this move only works if it is accepted as a fact that this is the way the world actually is. If so, and on this basis, DM-theorists seem to think they can cease worrying about the contradictions their approach introduces right at the heart of their own theory. They accept the fact that even though nature is deeply perplexing, a pair of well-adjusted DM-spectacles allows the world to be viewed in the right way (where "viewed" in fact means "Ignore what you can't explain", and then "Accuse critics of not understanding dialectics").

Despite the spin, this nevertheless implies that it is impossible to explain what it could possibly mean for something to be in two different places at once (save in the ambiguous manner described earlier in this Essay, and again below). If that is so, the dialectical 'analysis' of motion is of little use to anyone, least of all to dialecticians, now that it is plain that not even they can explain motion, merely re-describe it in perplexing ways. All that Engels's 'analysis' seems to have achieved, therefore, is stop dialecticians worrying about their defective theory, leaving motion, as they see it, still a 'paradox'.

In that case, if there is a rational solution to this 'paradox' (if we but knew what it was), it is no good asking dialecticians to search for it. They gave up on that score the moment they leafed through Hegel's 'Logic', and began "grasping" 'contradictions'.

Left to them, this branch of Physics would simply grind to a halt.²³

Ordinary Language And Paradox

However, Engels did at least make an attempt to use everyday terms in his endeavour to show that they were not all they seemed to be. Or, rather, that when considered 'dialectically', the vernacular reveals more about reality than might otherwise have been apparent -- especially to those who are mesmerised by 'commonsense', or perhaps those who have been bamboozled by that inner fifth-columnist, the "abstract understanding".

Nevertheless, anyone who disagrees with the 'dialectical' conclusion Engels drew would no doubt be reminded that these few words -- or the 'concepts' they supposedly represent -- clearly and unambiguously imply the 'contradictions' that Engels and Hegel said they did. In that case, defenders of this view of things could claim that these two had simply made several implicit 'contradictions' explicit.

Intentionally or not, by arguing this way Engels succeeded in situating his paradoxical theses in an ancient metaphysical tradition stretching back as far as Zeno, Parmenides and Heraclitus -– a tradition which ordinary working people had no hand in building, but which is (demonstrably) based on ruling-class forms-of-thought and on a distortion of the vernacular, the only language that links humanity directly with the material world, as Marx himself pointed out:

"The philosophers have only to dissolve their language into the ordinary language, from which it is abstracted, in order to recognise it, as the distorted language of the actual world, and to realise that neither thoughts nor language in themselves form a realm of their own, that they are only manifestations of actual life." [Marx and Engels (1970), p.118. Bold emphasis added.]

Indeed, Engels's approach began to falter as soon as he attempted to squeeze some metaphysical juice out of such desiccated philosophical raisins; that is, when he tried to extract 'paradoxical' conclusions from a few rather innocent looking words.

Naturally, only by those who have already capitulated to the view that reality is fundamentally 'contradictory' will agree with the conclusions Engels reached. Others, however, might be forgiven for remaining sceptical --, particularly those who (not unreasonably) think that Engels's 'solution' is far more puzzling than the original 'problem'. Indeed, if the nature of motion is problematic, calling it "contradictory", while making no attempt to explain how this actually accounts for anything, is no help at all.

If these alleged 'contradictions' do no work (as was argued above), then their presence here is, at best, unhelpful. That is because we can now see that these 'contradictions' are the direct product of an over-active imagination, compounded by a naive acceptance of the Idealist gobbledygook Hegel inflicted on humanity.

In that case, Engels's 'analysis' is an obstacle to understanding, which will, of course, need removing if science is to advance.

Lack Of Imagination

In fact, Engels failed to consider other far more likely possibilities; indeed, it looks like it never occurred to him that his 'contradictory' conclusions might not follow if he had instead given consideration to the full range of words/ and/or meanings available to us in ordinary language. To be sure, these are easily accessed by those determined to use the vernacular with far greater consistency, honesty and sensitivity than Engels, Hegel or Zeno ever managed.²⁴

Engels clearly wanted to make a specific point about the paradoxical implications of a handful of seemingly innocent ordinary words. As we will see, he did this by unwittingly altering their everyday use while imagining that the meanings of several other ordinary terms that are normally associated with them remained unaffected.

In doing this he wasn't, of course, alone. Semantic sleights-of-hand like this have been practiced time and again throughout the history of Traditional Philosophy; and they are still being played. Even careful philosophers often fail to notice that their own work involves what can only be called "piecemeal selectivity" in the use of certain words. Indeed, they have invariably assumed it is possible to tinker around with several specially chosen expressions while the meaning of any words normally associated with them remain the same. Selectivity like this is, alas, double-edged. In fact, these associated words -- whose meanings Engels simply also took for granted -- prove to be equally (if not more) problematic than those he finally latched onto. As we are about to see, this unexpected turn of events will not only vitiate Engels's 'analysis' of motion, it will fatally undermine every single classic account, too.

If, for example, an ordinary word like "motion" possess 'contradictory' implications -- according to Hegel and Engels --, then perhaps other terms they failed to consider might have analogously paradoxical connotations, given this perverse way of viewing language. What about the word "place", for instance? What if it turns out to be just as 'problematic'? In such circumstances, could we continue to accept the validity of Engels's conclusions about "motion" if the interplay between these two intimately connected words is more complex than he imagined, and an alteration to one changed the other?

More pointedly: What if certain senses of the word "place" neutralise Engels's interpretation of the word "move"?

Clearly, Engels's argument relies on the meaning of "place" remaining fixed while he tinkered around with "move". But, if "place" itself has no set meaning, then any conclusions based on the supposition that it has will automatically come under suspicion. Worse still, any argument based on one aspect of the ordinary meaning of "place", which undercuts the 'philosophical' sense of "motion", must be subject to even greater doubt. That is because, if the sense of the latter is compromised by the slippery nature of the former (or vice versa), the meaning of neither can emerge unscathed, in view of their intimate connection.

In fact, as we are about to see, this in-built complexity has the salutary effect of deflating the philosophically grandiose conclusions Engels (and others) thought he (they) could derive from a handful of mundane-looking words when he (they) used a non-standard application of "motion" with what he (they) took to be a standard use of "place", and vice versa.

Ordinary Objects Regularly Do The Impossible

Many of the ambiguities noted above (in relation to Engels's analysis of "motion") actually depend on systematic vagueness in the meaning of the word "place" and its cognates. Even when translated into the precise language of coordinate algebra/geometry, the meaning of this particular word doesn't become much clearer.

[Of course, this is not to criticise the vernacular; imprecision is one of its strengths. Nor is it to malign mathematics! However, when ordinary words are imported into Philosophy, where it is almost invariably assumed they have a single unique (or 'essential') meaning, problems invariably arise.]

In fact, as it turns out, there is no such thing as the meaning of the word "place" -- or of "move".

This lack of clarity carries over into our use of technical terms associated with either word; the application of coordinate systems, for example, requires the use of rules, none of which is self-interpreting. [The point of that comment will be explained presently.]

Nevertheless, it is relatively easy to show (by means of the sort of selective linguistic 'adjustment' beloved of metaphysicians, but applied in contexts they generally fail to consider) that ordinary objects and people are quite capable of doing the metaphysically impossible. The flexibility built into everyday language actually 'enables' the mundane to do the magical, and on an alarmingly regular basis. Such everyday 'prodigies' do not normally bother us -- well, not until some bright spark tries to do a little 'philosophising' with them.^24a

If the ordinary word "place" is now employed in one or more of its usual senses, it is easy to show that much of what Engels had to say about motion becomes either false or uninteresting. Otherwise, we should be forced to concede that ordinary people and objects can behave in extraordinary -- if not miraculous -- ways.

Consider, therefore, the following example:

L41: The strikers refused to leave their place of work and busied themselves building another barricade.

Assuming that the reference of "place" is clear from the context (that it is, say, a factory), L41 depicts objects moving while they remain in the same place -- contrary to what Engels said (or implied) was possible. Indeed, if this sort of motion is interpreted metaphysically, it would involve ordinary workers doing the impossible -- moving while staying still!

Of course, it could be objected here that L41 is a highly contentious example, and not at all the sort of thing that Engels (or other metaphysicians) had in mind by their use of the word "place".

But, Engels didn't tell us what he meant by this term; he simply assumed we'd understand his use of it. If, however, it is now claimed that he didn't mean by "place" a sort of vague "general location" (like the factory used in the above example), then that would confirm the point being made in this part of the Essay: Engels didn't say what he meant by "place" since there was nothing he could have said that wouldn't have ruined his entire argument. Tinker around with the word "place" and the meaning of "motion" can't fail to be compromised (as noted above). This can be seen by considering the following highly informal 'argument':

L42: Nothing that moves can stay in the same place.

L43: If anything stays in the same place, it can't move.

L44: A factory is one place in which workers work.

L45: Workers move about in factories.

L46: Any worker who moves can't stay in the same place (by L42, contraposed).

L47: Hence, if workers move they can't do so in factories (by L44 and L45).

L48: But, some workers stay in factories while they work; hence, while there they can't move (by L43).

L49: Therefore, workers work and do not work in factories, or they move and they do not move.

As soon as one meaning of "place" is altered (as it is in L44), one sense of "move" is automatically affected (as in L45), and vice versa (in both L47 and L48). In one sense of "place", things can't move (in another sense of "move") while staying in one place (in yet another sense of "place"). But, in another sense of both they can, and what is more, they can typically do both. Failure to notice this produces 'contradictions' to order, everywhere (as in L49).

Even so, who believes that workers work and do not work in factories? Or that they move and do not move while staying in the same place?

Perhaps only those who "understand" dialectics...?

Dialectical Objects Do The Oddest Things

Do They Move Or Simply Expand?

Clearly, Engels's 'theory' of motion has to be able to take account of ordinary objects if it is to apply to the real world and not just to abstractions, or to physically meaningless mathematical 'points'. But, this is just what his 'theory' can't do, as we are about to see.

It could now be objected to the comments made in previous few sections that if "place" is defined precisely (without altering the meaning of "move") it would be possible to understand what Engels and Hegel were trying to say. In that case, it could be argued that if "place" is defined by the use of precise spatial coordinates (henceforth, SCs), Engels's account of motion would become viable again.

Or, so some might think.

Of course, the problem here is that in the example above (concerning those contradictory mobile/stationary workers), if we try to refine the meaning of the word "place" a little more precisely, it will come to mean something like "finite (but imprecise) three-dimensional region of space large enough to contain the required object". Well, plainly, in that sense things can and do move about while they remain in the same region (i.e., "place") -- since, by default, any object occupies such a region as it moves -- that is, it must always occupy a three-dimensional region of space large enough to contain it as it moves; it certainly doesn't occupy a larger or a smaller space (unless it expands/contracts)! Moreover, objects occupy finite regions as they move in relation to each other (or they wouldn't be able to move).

Hence, if defined this way, moving objects always occupy the same space, and hence they don't move! That is, if they always stay in the same space, they can't move. As we have seen, objects always occupy the same space, even as they move. So, they both move and don't move! Plainly we need to be more precise.

So, of the many options there are before us, the following seem to be relevant to the point at hand:

(1) If an object always occupies the same space (which fits it like a glove, as it moves), then it can't actually move!

(2) If it occupies a larger space as it moves, it must expand.

(3) If it moves about in the same region of space (such as a factory), it still can't move!

(4) If it successively occupies spaces equal to its own volume as it moves, the situation is even worse, as we will soon see.

Now, if the 'regions' mentioned above are constrained too much, nothing would be able to move -- this is Option (1). Put each worker in a tightly-fitting steel box that exactly fits him or her and watch all locomotion grind to a halt.

Put that worker in a region of space, and he/she still won't be able to move -- this is Option (3). That is because if we define motion as successive occupancy of regions of space, then this worker can't move since he/she is always in the same region.

The difficulty here is plainly one of relaxing the required region an object occupies sufficiently enough to allow it to move from one place to another without stopping it moving altogether (that is, preventing Option (3) from undermining Option (4)), all the while providing an account that accommodates the movement of medium-sized objects in the real world. But, once this has been done the above difficulties soon re-appear, for it is quite clear that objects still move while staying in the same place -- if the place allowed is big enough for them to do just that!

Indeed, this fact probably accounts for most (if not all) of the locomotion in the entire universe. Clearly, and in the limit, if anything moves in nature it must remain in the same place, i.e., in the universe! Of course, that would be to relax the definition of "same place" far too much. But, how can we tighten the definition so that objects aren't put in straight-jackets once more? (i.e., Option (1), again).

At first sight, the above objection (concerning a precise enough definition of "place") seems reasonable enough. Engels clearly meant something a little more precise than a vague or general sort of location (like a factory). But, what?

It might seem that Engels's argument can be rescued if tighter protocols for "place" are prescribed --, perhaps those involving a reference to "a (zero volume mathematical) point, in three-dimensional space, located by the use of precise SCs". But, this option would embroil Engels's account in far more intractable problems. That is because such an account would plainly relate to mathematical point locations, or the movement of mathematical points themselves -- and we saw earlier that that was a non-starter.

Clearly, things cannot move about in such points -- but this has nothing to do with the supposed nature of reality. These 'entities' do not (and could not) exist in nature for them to contain anything. That is because such points aren't containers. They have no volume and are made of nothing. If this weren't the case, they'd not be mathematical points, they'd be regions.

As noted above, if Engels meant something like this in his use of "place", his account would fail to explain or accommodate the movement of gross material bodies in nature, for the latter do not occupy mathematical points.

Moreover, it is no use appealing to larger numbers/sets of such points located by SCs; no material body can occupy an arbitrary number of points, since points aren't containers.

Perhaps we could define a region (or a finite volume interval) by the use of SCs? Maybe so, but this would merely introduces another classical conundrum (which is a variation on several of Zeno's other paradoxes): how a region (or a volume interval) can be composed of points that have no volume. Even an infinite number of zero volume mathematical points adds up to zero. Now, there are those who think this conundrum has a solution (just as there are those who think it doesn't), but it would seem reasonably clear that the difficulties surrounding Engels's 'theory' are not likely to be helped by importing several more from another set of Paradoxes.

Be this as it may, it is far more likely that Engels's use of the word "place" implies an implicit reference to a finite three-dimensional volume interval (whose limits could be defined by the use of well-understood rules in Real and Complex Analysis, Vector Calculus, Coordinate Algebra and Differential Geometry, etc.).

Clearly, such volume intervals must be large enough to hold (even temporarily) a given material object. If so, this use of the phrase "volume interval" would in principle be no different from the earlier use of "place" to depict the movement of those workers! If they can move about in locations big enough to contain them, and who remain in the same place while doing so, Engels's moving objects can do so, too -- except they would now have a more precise "place"/region in which to do it.

However, and alas, this sense of "place" is no use at all, for when such workers move, they will, by definition, stay in the same place! [Option (3), again!]

Naturally, the only way to avoid this latest difficulty would be to argue that the location of any object must be a region of space (i.e., volume interval) equal to that object's own volume. This is in effect one of the classical definitions. In that case, as the said object moves, its own exact volume interval would move with it, too; the latter would follow each moving object around more faithfully than its own shadow, and more doggedly than a world-champion bloodhound. But, plainly, if that were the case, it would mean that objects would still move while staying in the same place -- since, plainly, any object always occupies a space equal to its own volume, which would, on this view, travel everywhere with it, like a sort of metaphysical glove. [Option (1), again!]

As seems plain: if that is so, we now have two problems where once there was just one, for we should have to explain not only how bodies move, but how it is also possible for volume intervals to move so that they can faithfully shadow the objects they contain!

However, and far worse: in that case, not only would we have to explain how locations (i.e., volume intervals) are themselves capable of moving, we would also have to explain what on earth they could possibly move into!

What sort of ghostly regions of space could we appeal to, to allow regions of space themselves to move into them?

Even worse still: these 'moving volume intervals' must also occupy volumes equal to their own volume, if they are to move (given this 'tighter' way of characterising motion). And, if they do that, then these new 'extra' locations containing the volume intervals themselves must now act as secondary 'metaphysical mittens', as it were, to the original 'ontological gloves'. Metaphorically speaking, this theory, if it took such a turn, would be moving backwards, since an infinite regress would soon confront us, as spatial mittens inside containing gloves, inside holding gauntlets, piled up alarmingly to account for each successive spatial container, and how it could possibly move. As seems reasonably clear, we would only be able to account for locomotion this way if each moving object were situated at the centre of some sort of 'metaphysical onion', each with a potentially infinite number of 'skins'! [Iterated Option (1)!]

It could be countered that even though objects occupy spaces equal to their own volumes, as they move along they then proceed to occupy successive spaces of this sort (located in the surrounding region, for example), all of which are of precisely the right volume to contain the moving object that now occupies them, and which can be located/defined precisely. On this revised scenario, moving objects will leave their old locations behind as they barrelled along.

This now brings us to a consideration of Option (2) and/or Option (4) -- now modified to (4a) --, from earlier:

(2) If (an object) occupies a larger space as it moves, it must expand.

(4a) An object successively occupies spaces (or volume intervals) equal to its own volume as it moves.

I will reject (2) as absurd. If anyone wants to defend it, they are welcome to all the headaches it will bring them its train.

Considering, now, Option (4a):

Even if (4a) were a correct interpretation of what Engels meant, and it was a viable option (and sense could be made of these new, and accommodating locations without re-duplicating the very same problem, noted in the previous few paragraphs), no DM-theorist could afford to appeal to such successive volume intervals. That is because dialecticians claim that moving bodies occupy at least two such "places" at the same time, being in one of them and not in it at the same moment. Clearly, if motion were defined in such terms (that is, if it were characterised as involving objects successively occupying spaces equal to their own volumes), then moving objects would occupy at least two of these volume intervals at once.

In that case, 'dialectical objects' would not so much move as stretch or expand! [Modified Option (2)!]

To see this more clearly, it might be useful to examine the above argument more closely.

If the centre of mass (COM) of a 'dialectically moving' object, D, were located at, say, (X_k, Y_k, Z_k) and (X_k+1, Y_k+1, Z_k+1), at the same time (to satisfy the requirement that moving bodies occupy at least two such "places" at the same time, being in one of them and not in it), it would have to occupy a space larger than its own volume while doing so.

Let us call such a space "S", and let the volume interval containing (X_k, Y_k, Z_k) and (X_k+1, Y_k+1, Z_k+1) be "δV₁", leaving it open for the time being whether S and δV₁ are the same or are different. Thus, if the COM of D is in two such spaces (i.e., (X_k, Y_k, Z_k) and (X_k+1, Y_k+1, Z_k+1)) at once, D would plainly be in S, and would occupy δV₁. But, once again, that would mean that D would move while remaining in the same space -- i.e., it would remain inside S, or inside δV₁ (whichever is preferred), as its COM moved from (X_k, Y_k, Z_k) to (X_k+1, Y_k+1, Z_k+1), in the same instant. [Option (3), again!]

[Except, we can't speak of a 'dialectal object' moving from one point to the next since that would imply it was in the first before it was in the second, and that it was in the second after it was in the first. But, as we have seen, if such an object is in both places at the same time, there can be no "before" or no "after", here.]

Now, the only way to avoid the conclusion that D moves while occupying the same space S and/or δV₁ --, and hence that it appears to stay still while it moves, just like the 'mobile/stationary' workers we encountered earlier -- would be to argue that such spaces remain where they are while D moves into successively new locations, or new spaces. This seems to be the import of Option (4a).

But, as D moves it still occupies δV₁, only we would now have to argue that as it does so it also moves into a new δV each time, say, δV₂ -- except that δV₂ must also contain (X_k+1, Y_k+1, Z_k+1) and (X_k+2, Y_k+2, Z_k+2) -- otherwise it would not be a new containing volume interval that satisfied the requirement that moving bodies occupy at least two such "places" at the same time, being in one of them and not in it.

Plainly, all objects have to occupy some volume interval or other at all times (or they would 'disappear'). However, in D's case it has to do this while also occupying new volume intervals at the same time as it moves along (otherwise, as we saw, it would move while being in the same place, and thus that it didn't move!).

So, if D occupies only one S or only one δV at once, it would be at rest in either. [Options (1) and (3).] Hence, it must occupy at least two of these at the same time (if, that is, we accept the 'dialectical' view of motion).

If so, the only apparent way of avoiding the conclusion that D-like objects move while staying still is to argue that they occupy two successive Ss, or two successive δVs (perhaps these are partially 'overlapping', perhaps not), at once. Unfortunately, this would now mean that D-like objects would have to occupy a volume/volume interval bigger than either of S or δV at once, and hence: they must expand or stretch.

It could be objected that two successive δVs would contain (X_k, Y_k, Z_k) and (X_k+1, Y_k+1, Z_k+1) -- that is, δV₁ would contain (X_k, Y_k, Z_k) and δV₂ would contains (X_k+1, Y_k+1, Z_k+1) --, so the above objection is misguided. Maybe so, but the point is that dialectical objects must occupy two δVs at once, and if that is so, both δVs must contain (X_k, Y_k, Z_k) and (X_k+1, Y_k+1, Z_k+1), jointly or severally, otherwise such objects couldn't occupy two spaces (two δVs) at the same time.

But, if that is so, and D isn't stationary while it occupies δV₂, and as we saw above in an analogous context, it must also occupy δV₃ at the same time, and so on. Successive applications of this argument would have D occupying bigger and bigger volume intervals (i.e., δV₁ + δV₂ + δV₃ + δV₄ +...,+ δV_n), all at the same time. In the limit, D could fill the entire universe (or, at least, the entire volume interval encompassing its own trajectory), all at the same time -- if it moves and if Hegel is to be believed!

There thus seems to be no way to depict the motion of D-like objects that prevents them from either moving while staying still, or, from expanding alarmingly like some sort of metaphysical Puffer Fish.^24b

Figure One: At Last! An Organism That 'Understands' Dialectics

Either way, Engels's theory finds itself in yet another Hermetic Hole.

The reader should now be able to see for herself what mystical mayhem is introduced into our reasoning by this cavalier use of (contradictory) metaphysical language. When one sense of "move" is altered, one sense of "place" can't remain the same, nor vice versa.

Of course, no one believes the above ridiculous conclusions, but there appears to be no way to avoid them using the radically defective and hopelessly meagre conceptual and/or logical resources DL supplies its unfortunate victims.

[DL = Dialectical Logic; SC = Spatial Coordinate.]

Or Do They Concertina?

On the other hand, it seems that 'Dialectical objects' must concertina as they move.

Consider a simple body B made of 3 connected parts: P₁, P₂ and P₃, all arranged in the same line, so that there are no gaps between them. Let B move such that at t₁, the centre of the leading edge of P₁ is at (X₁, Y₁, Z₁), the centre of the leading edge of P₂ is at (X₂, Y₂, Z₂), and centre of the leading edge of P₃ is at (X₃, Y₃, Z₃). Let us also assume that the centre of the leading edge of P₃ now moves to (X₄, Y₄, Z₄). Finally, let us assume that the distances between each of (X₁, Y₁, Z₁), (X₂, Y₂, Z₂), (X₃, Y₃, Z₃) and (X₄, Y₄, Z₄) are all the same.

Now, if all moving objects occupy two places at once, and if B moves in a line parallel to the line joining the centre of the leading edge of P₁ to the centre of the leading edge of P₃, then the centre of the leading edge of P₁ must occupy (X₁, Y₁, Z₁) and (X₂, Y₂, Z₂), the centre of the leading edge of P₂ must occupy (X₂, Y₂, Z₂) and (X₃, Y₃, Z₃), and the centre of the leading edge of P₃ must occupy (X₃, Y₃, Z₃) and (X₄, Y₄, Z₄), at the same time. In effect, B would concertina as it moved, with the front end of, say, P₁ crushing or penetrating the back end of P₂, and overlapping it right up to its own leading edge -- in effect, wiping P₂ out!

[This result, of course, depends on the answer to an earlier question: How far apart are the two places a moving body occupies that Engels referred to? If this is left indeterminate, then any length will do. Even then, if a specific length is decided upon, we could make the distance between the above parts equal to that length, and the same result will still follow. (That is done below, anyway.)]

It can't be the case that the trailing edge of P₂ will leave (X₂, Y₂, Z₂) just before the leading edge of P₁ reaches it, since, as we have already seen, there is no before and now after here, since all such motion must take place at the same time for it to constitute a 'dialectical contradiction'.

Now, it could be objected that P₁ and P₂ for example will occupy the space between (X₁, Y₁, Z₁) and (X₂, Y₂, Z₂) and (X₂, Y₂, Z₂) and (X₃, Y₃, Z₃), respectively, as B moves, but this isn't possible. That is because there are no gaps between any of the three parts of this object for any of those parts to move into. So, if B were, say, a single carriage train, then P₁ would comprise the rear section of that carriage, P₂ the middle third, and P₃ the front end. This view of motion would therefore have these parts of this single carriage crushing the one in front. [I have considered the option that there is such a gap, below.]

[At a later date, I will add diagrams to make this and the other examples considered here a little clearer to the reader.]

Again, it might be argued that the structural properties of B (intermolecular forces, etc.) will prevent this from happening. That is undeniable, but this response also has the unfortunate consequence that while B may be in two places at once as it moves, none of its parts could be! And that in turn would imply that while B was racing along, none of its parts would be moving -- since they are not allowed to be in two places at once!

Similar, if not worse problems afflict a 'dialectical object' undergoing circular or more complex forms of movement, such as helical or spiral motion.

So, consider a rotating disc, D, of negligible thickness divided by a diameter line, T, into two equal semi-circular sectors, S₁ and S₂. If we set the centre of this disc as the origin, then we can set the leading edge of S₁ so that it lies along T. In addition, let there be a point p₁ on T, r units from the centre, with co-ordinates (r, θ₁). Let the leading edge of S₂ also lie along T, and let there be a point p₂, r units from the centre, with co-ordinates (r, θ₂). Plainly, this means that p₁ and p₂ lie on the same trajectory, a circular path r units from D. [I have used polar co-ordinates in two-dimensions here to simplify this example.]

Now, if all moving objects occupy two places at once, and if B rotates clockwise, then the leading edge of S₁ must pass through both p₁ and p₂, and the leading edge of S₂ must pass through both p₂ and p₁, at the same time. But this is even worse than the 'dialectically linear movement' considered above, since, in this case, either (1) D will totally disappear -- as both of its sectors occupy the same semi-circle that the other one occupied -- or, (2) Both of these sectors must stretch to cover the entire disc, ramming into the back of one another as they did so, compressing each into a region with zero area!

Now it might be possible to defend this picture of dialectal objects as they smash into one another by arguing that the above scenarios are heavily biased. For example, in the linear case above, while the centre of the leading edge of P₃ might move to (X₄, Y₄, Z₄), the distance between (X₃, Y₃, Z₃) and (X₄, Y₄, Z₄) need not be equal to that between (X₁, Y₁, Z₁) and (X₂, Y₂, Z₂) and (X₂, Y₂, Z₂) and (X₃, Y₃, Z₃). Let us say, therefore, that the distance between (X₃, Y₃, Z₃) and (X₄, Y₄, Z₄) is δL. In this case, therefore, B will move forward δL units, as will each of its parts.

This would have the effect on S₁ such that it would no longer move to (X₂, Y₂, Z₂), but to some intermediate point (X_1+δx, Y_1+δy, Z_1+δz), with the same sort of thing happening to the other leading edges. The same would also happen to the trailing edge of S₂, which, let us say was at (X_i, Y_i, Z_i), at t₁. Now, the trailing edge of S₂ and the leading edge of S₁ can't occupy the same space, as should seem obvious; so let us say that the distance between (X₁, Y₁, Z₁) and (X_i, Y_i, Z_i) can be made as small as we like -- let us stipulate that this is δS (where it is left open whether or not δS > δL). In that case, there would be a gap, δS, between at least two of the parts.

Hence, the trailing edge of S₂ would move to (X_i+δx, Y_i+δy, Z_i+δz) while the leading edge of S₁ moves to (X_1+δx, Y_1+δy, Z_1+δz). Plainly, these are not the same points. If so, S₁ won't smash into the back of S₂ as imagined above. And, the same sort of conclusion can be drawn in connection with the rotating disc, too.

Unfortunately, this reply fails, too. That is because the centre of the leading edge of S₁ has to occupy two places at once, if Engels and Hegel are to be believed. So, the centre of the leading edge of S₁ has to occupy (X₁, Y₁, Z₁) and (X_1+δx, Y_1+δy, Z_1+δz), and the centre of the trailing edge of S₂ has to occupy (X_i, Y_i, Z_i) and (X_i+δx, Y_i+δy, Z_i+δz), at the same time. Now, if δS is zero, then (X_1+δx, Y_1+δy, Z_1+δz) will lie beyond (X_i, Y_i, Z_i), which means that the leading edge of S₁ will smash into the back of S₂. The same will happen if δS < δL. On the other hand, if δS > δL then a larger and larger gap will open up between S₁ and S₂, which will widen all the more as B continues to move. So, B will either (1) Begin to fragment, or it will (2) Concertina, as it moves. The same will happen to the disc.^24b1

So, this 'theory' is still stuck in a Deep Dialectical Ditch.

Coordinates To The Rescue?

Despite this, it could be argued that if the ordinary word "place" is so vague then it should be replaced with more precise concepts; those defined in terms of SCs, once more. But, as the following argument shows, that would be another backward move (no pun intended!):

L50: A place can be defined by the use of SCs.

L51: SCs are composed of ordered real number 3-tuples (i.e., number triples, defined precisely -- see L52) in R³.^24c

L52: However, when written correctly, the elements in such 3-tuples must occupy their assigned places (by the ordering rules). Consider then the following ordered triplet: <x₁, y₁, z₁>. Each element in such an SC must be written precisely this way, with x_i, y_i and z_i (etc.) all in their correct places.

L53: But, the situating of such elements can't itself be defined by exact SCs, otherwise an infinite regress will ensue.

L54: Consequently, this latter sense of "place" (i.e., that which underlies the ordering rules for SCs) can't be defined (without circularity) by means of SCs.

This means that the definition of "place" by means of SCs is itself dependent on a perfectly ordinary meaning of "place", and, further, that the latter sense of "place" must already be understood if a co-ordinate system is to be set-up correctly.

Therefore, the ordinary word "place" can't be defined without circularity by means of a coordinate system.