Essay Six: Trotsky And Hegel -- How To Misconstrue The 'Law Of Identity', And How That Law Is No Enemy Of Change


Ideally, this Essay should be read in conjunction with Essays Four and Five.


Technical Preliminaries


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As is the case with all my work, 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 thirty-five years ago.


The difference between Dialectical Materialism [DM] and HM, as I see it, is explained here.


It is worth pointing that phrases like "ruling-class theory", "ruling-class view of reality", "ruling-class ideology" (etc.) used at this site (in connection with Traditional Philosophy and DM), aren't meant to suggest 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 intended 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 dogmatic approach to knowledge had almost invariably been promoted by thinkers who either relied on ruling-class patronage, or who, in one capacity or another, helped run the system for the elite.**


However, that 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 more details.


[**Exactly how this applies to DM will, of course, be explained in the other Essays published at this site (especially here, here, and here). In addition to the three links in the previous paragraph, I have summarised the argument (but this time for the benefit of absolute beginners!) here.]


It is also worth adding out that a good 50% of my case against DM has been relegated to the End Notes. This has been done in order to allow the main body of the Essay to flow a little more smoothly. This means that if readers want to appreciate fully my case against DM, they will need to consult this material. In many cases, I have qualified or amplified what I have to say in these Notes, and I have also added extra supporting evidence. I have also raised numerous objections (some obvious, many not -- and some that will have occurred to the reader) to my own ideas, which I have then proceeded to answer. [I explain why I have adopted this tactic in Essay One.]


If readers skip this material, then my answers to any objections they might have to my arguments will be missed, as will the extra detail and evidence.


[Since I have been debating this theory with comrades for over 30 years, I have heard all the objections there are! Many of the more recent debates are listed here.]


Several readers have complained about the number of links I have added to these Essays because they say it makes them very difficult to read. Of course, DM-supporters can hardly lodge that complaint since they believe everything is interconnected, and that must surely apply even to Essays that attempt to debunk that very idea. However, to those who find such links do make these Essays difficult to read I say this: ignore them -- unless you want to access further supporting evidence and argument for a particular point, or a certain topic fires your interest.


Others wonder why I have linked to familiar subjects and issues that are part of common knowledge (such as the names of recent Presidents of the USA, UK Prime Ministers, the names of rivers and mountains, the titles of popular films, or certain words that are in common usage). I have done so for the following reason: my Essays are read all over the world and by people from all 'walks of life', so I can't assume that topics which are part of common knowledge in 'the west' are equally well-known across the planet -- or, indeed, by those who haven't had the benefit of the sort of education that is generally available in the 'advanced economies', or any at all. Many of my readers also struggle with English, so any help I can give them I will continue to provide.


Finally on this specific topic, several of the aforementioned links connect to web-pages that regularly change their URLs, or which vanish from the Internet altogether. While I try to update them when it becomes apparent that they have changed or have disappeared I can't possibly keep on top of this all the time. I would greatly appreciate it, therefore, if readers informed me of any dead links they happen to notice.


In general, links to 'Haloscan' no longer seem to work, so readers needn't tell me about them! Links to RevForum, RevLeft, Socialist Unity and The North Star also appear to have died.




As of May 2024, this Essay is just over 71,000 words long; a summary of some of its main ideas can be accessed here.


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


[Latest Update: 04/05/24.]


Quick Links


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(1) Introduction


(2) Trotsky On Identity


(3) Ironically, "Identical" Is Not Identical With "Equal"


(a) A Mistake Most Dialecticians Make


(b) Ordinary Language Thwarts Dialectics


(c) "Equal" And "Identical" Not Identical


(d) Mathematical Equality Vs Mathematical Identity


(e) Trotsky Changes The Subject


(4) Trotsky's Argument Dissected


(a) Precisely What Is Trotsky Denying?


(b) Trotsky Has This Base Covered -- Or Has He?


(5) Bags Of Sugar Actually Refute Trotsky


(a) Trotsky's Answer


(b) Mere Guesswork On Trotsky's Part?


(c) Trotsky's Instructions -- Followed Exactly And To The Letter?


(d) Yet Another Misidentification


(e) 'Norms Of Tolerance' Refute Trotsky, Too


(f) Wrong Anyway


(g) Identical A Priori Tactics


(h) Super-Science From Mere Words


(i)  Physicists Discover Identical Particles!


(j)  Changeless Sub-Atomic Particles


(k) An Everyday Example Of Absolute Identity


(6) Trotsky Uses Identity To Criticise Identity


(a) Same Moment


(b) A Turn To The Concrete


(c) Incomprehensible? Or Just Trivial?


(7) Did Trotsky Understand Identity?


(a) How Can Anyone Learn What Identity Is If There Is None?


(b) The Sting In The Tail


(c) 'Approximate' And 'Abstract' Identity


(d) Plato, Hegel, Trotsky And The Concept Of 'Abstract Identity'


(e) Dialectical Dilemma


(f)  Identity Schmidentity


(g) Trotsky's Exact Words Dialectically Implode


(h) Trotsky's Attack Unequal To The Task


(i)  Materially-Induced Dialectical Misery


(8) The Knock-Out Blow


(9) Traditional Logic Versus Modern Formal Logic


(a) Traditional Logic Defective?


(b) Dialectical Logic Superior To Formal Logic?


(c) Formal Logic: A Fragmented And Static View Of Reality?


(10) Notes


(11) References


Summary Of My Main Objections To Dialectical Materialism


Abbreviations Used At This Site


Return To The Main Index Page


Contact Me




Few other areas of FL cause dialecticians more problems than the LOI. For many it is the bête noir of "formal thinking". However, this Essay aims to show that not only have dialecticians misconstrued this so-called 'Law', the vast majority have in fact attacked the wrong target!


[FL = Formal Logic; LOI = Law of Identity; DM = Dialectical Materialism/Materialist, depending on the context.]


Hegel's Logic is the immediate source of these errors for it is there that we find Hegel applying his quirky 'reasoning powers' to something that is not, as it turns out, inimical to change. Identity is no more a threat to change than difference is to stability.


Nevertheless, the main thrust of my criticisms of Hegel's 'analysis' of this 'law' will appear in Essay Twelve. The objections I have raised here against the highly repetitious and misguided criticisms that Marxist Dialecticians level against the LOI also indirectly apply to his work.


Since these Essays have been written from within the Trotskyist tradition, and because Trotsky's comments on this 'law' are far more influential on active revolutionaries than Hegel's (few of whom have read or studied his work), it makes sense to begin with his widely quoted remarks.


Trotsky On Identity


In his debate with Burnham, Trotsky rehearsed several arguments aimed at exposing what he took to be serious limitations of the LOI, criticisms he had lifted directly or indirectly from Hegel that have resurfaced almost verbatim in the writings of Dialectical Trotskyists ever since.1 The motivation for Trotsky's analysis was his belief that FL deals only with 'static and lifeless concepts', rendering it incapable of grasping the dynamism found in concrete reality. Remarkably, Trotsky nowhere attempted to substantiate these sweeping allegations; in fact there is no evidence that he consulted a single logic text that had been written in the previous 100 years.2 Clearly, he didn't think that this failure to check his facts disqualified him from passing informed opinion on the subject. By the same token, therefore, we may suppose him an expert in High Energy Physics -- and perhaps even brain surgery!


This rather damning criticism applies equally well to the vast majority of Trotsky's epigones –- to say nothing of DM-theorists in general -- few of whom show any sign of ever having consulted a single logic text (ancient or modern), saving, of course, those two badly misnamed books written by Hegel: the Shorter Logic and the Science of Logic (i.e., Hegel (1975, 1999)).


[AFL = Aristotelian Formal Logic; MFL = Modern Formal Logic.]


Most of the criticisms DM-theorists level against FL in general have been examined in Essay Four, where they were shown to be based on a serious misunderstanding even of AFL, let alone MFL. This is hardly surprising given the allegations advanced in the previous paragraph.


Be this as it may, in this Essay I plan to focus on Trotsky's criticisms of the LOI, which DM-theorists -- at least those in the Trotskyist wing of Marxism -- generally regard as definitive. John Rees, for example, outlined one key issue as follows:


"[In FL] things are defined statically, according to certain fixed properties -– colour, weight, size, and so on. This is denoted by the expression 'A is equal to A'." [Rees (1998), p.272.]


Trotsky's own argument, however, was this:


"The Aristotelian logic of the simple syllogism starts from the proposition that 'A' is equal to 'A'. This postulate is accepted as an axiom for a multitude of practical human actions and elementary generalisations. But in reality 'A' is not equal to 'A'. This is easy to prove if we observe these two letters under a lens -– they are quite different from each other. But, one can object, the question is not the size or the form of the letters, since they are only symbols for equal quantities, for instance, a pound of sugar. The objection is beside the point; in reality a pound of sugar is never equal to a pound of sugar -– a more delicate scale always discloses a difference. Again one can object: but a pound of sugar is equal to itself. Neither is true (sic) -– all bodies change uninterruptedly in size, weight, colour etc. They are never equal to themselves. A sophist will respond that a pound of sugar is equal to itself at 'any given moment'…. 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), pp.63-64.]2a


One puzzling fact about this passage -- which it shares with the many references made to this 'law' in other DM-writings -- is that it ignores classical versions of the LOI, none of which Trotsky, Rees or other dialecticians ever bother to reference, let alone quote.3


[Except when I am directly quoting DM-sources, where I use capital letters in that way in this Essay I will highlight them in bold to distinguish them from the ordinary use of capital letters.]


So, there appear to be at least a dozen substantive points Trotsky was making, here:


T1: AFL begins with "A is equal to A".


T2: This "postulate" applies quite well in most practical situations. But, in reality "A is not equal to A".


T3: Close inspection under a lens, for example, will show that any chosen letter "A" is not exactly the same as any other letter "A".


T4: A similar observation applies if these letters stand for material objects, like a pound bag of sugar.


T5: Any two weighings of seemingly equal bags of sugar will always reveal minor differences.


T6: It is no use arguing that all bodies are equal to themselves since they all undergo constant change; so they are never equal to themselves.


T7: The sophistical response -- that objects are momentarily equal to themselves -- is based on an abstract conception of time.


T8: If a moment in time is an interval, then any object will undergo inevitable change in that interval.


T9: If it isn't an interval, it must be a mathematical abstraction, a "zero of time".


T10: Everything exists in time and existence is an "uninterrupted process of transformation"; time is a "fundamental element of existence".


T11: "A is equal to A" implies that objects are equal to themselves if they don't change.


T12: Objects that don't change, don't exist.


Trotsky nowhere backs any of these up with evidence (or, none that isn't itself based on further thought experiments), but that seemingly fatal defect rarely seems to bother dialecticians. In earlier Essays, we saw why DM-theorists airily brush aside the need to substantiate their theses with anything that remotely resembles proof: if the universe is governed at every level by DL, a simple 'thought experiment' is all the 'evidence' a dialectician really needs.


[DL = Dialectical Logic.]


Naturally, only consistent materialists will object at this point.


As George Novack argued:


"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 emphases added.]


Lest anyone object to the above, it is worth pointing out that what little 'evidence' Trotsky and/or his epigones have offered in support of their hyper-bold claims -- about every single object in the entire universe, for all of time -- will be examined below.


However, Trotsky's quasi-Hegelian observations were based on a serious 'misunderstanding' even of AFL -- a defect seriously compromised by an ironically appropriate mis-identification of the LOI, further compounded by the invocation of an abstract metaphysical doctrine of his own.


[Of course, there are many other serious weaknesses in Trotsky's argument, but they are merely consequences of the above.]


Ironically, "Identical" Is Not Identical With "Equal"


A Mistake That Applies Equally To Most Dialecticians


Trotsky's initial characterisation of the LOI is itself rather strange. His paraphrase of it went as follows:


S1: A is equal to A.4


But, as an accurate depiction of identity, S1 isn't even close -- not least because it omits mention of the word "identity"! Contrast S1 with the following far less inaccurate -- but simplified -- version of the same 'law':


S2: A is identical to A.


But, why have generations of dialecticians studiously avoided formulations of the LOI like S2 in favour of those that appear to be about something entirely different? [No irony intended.] Why did Trotsky prefer S1 to S2?


Clearly, his use of "equal" in S1 meant he was actually attacking the principle of equality -- not the LOI. Naturally, this means that Trotsky's criticisms of the LOI were misconceived from the start!


However, when confronted with the above claims, DM-apologists tend to respond, "So what? What's the difference between the two?" As will be appreciated, that reply is itself problematic -- not the least because it reveals that these individuals have an equally insecure grasp of the issues involved. [Irony intended.]




(i) If there is no difference between the two, then they are identical, which means that we would now have at least one genuine example of the LOI on which all could agree, namely this -- that equality is identical with identity!


(ii) If they are different after all, Trotsky clearly attacked the wrong target.


Now, when challenged with this dilemma, dialecticians either ignore it, or they appeal to the "It's just abstract" defence, accompanied, or not, with the by-now-clichéd accusation of "pedantry".


As we will soon see, this retreat is itself a step back too far, since there is a clear difference between abstract equality and abstract identity, too, which dialecticians have also failed to notice. So, abstract or concrete, the two notions aren't the same. [Anyway, 'Abstract', or 'Absolute', Identity will be dealt with later.]


Furthermore, as we discovered in Essay Three Parts One and Two, dialecticians have an insecure grasp of the nature of abstraction, and are largely content to be told what to think on this score by notorious Idealists -- like Aristotle, Spinoza and Hegel.


As we will also find out, our grasp of words that seek to depict or criticise the nature of 'abstractions' depends on the employment of very real, material correlates in this world. For example, the above objections have themselves to be committed to paper, typed on to a computer screen, or propagated through the air as sound waves. In which case, it becomes pertinent to ask whether sentences containing the word "identical" make exactly the same point as those containing the word "equal". If they do, then Trotsky's criticisms of this 'law' can't apply to any material embodiment of his ideas --, since, if they did, we would once again have an employment of this 'law' in the material world which undermines all he had to say about it -- for here we would have some very real, material sentences that would be identical in content -- if "equal" were the same as "identical".


On the other hand, of course, if they don't make exactly the same point, then, once more, Trotsky attacked the wrong target.


Finally, the fact that dialecticians -- who are supposed to be using 'cutting edge' science, philosophy, and 'logic' -- failed to notice this serious error, and continue to ignore it no matter how many times it is brought to their attention, seriously undermines their credibility. Indeed, these major interpretive blunders fatally compromise the claim that DM is a science to begin with, let alone a philosophical theory that merits serious attention.


Ordinary Language Once Again Thwarts Dialectical Casuistry


Our comprehension of words for identity, sameness, equality and difference clearly revolves around the use of certain expressions in ordinary life, whatever technical modifications we might subsequently want to introduce, and for whatever reason we might do that. But, the ordinary use of terms like "equal", "identical", "same" and "different" is itself highly complex, even though that isn't the impression one gets from reading Trotsky's comments (or, indeed, those of his epigones). Nor is it the impression one forms when reading Hegel, either.


[More on this later. The importance of ordinary language was highlighted in Essays Four and Twelve Part One (as well as the rest of Essay Twelve, summary here). It was also stressed by Marx himself.]


It could be objected that these two principles (i.e., equality and identity) are approximately identical, to such an extent that any difference between them can be ignored. However, as we will see, that isn't even remotely correct; these two concepts are radically different. But, even if it were the case that they are approximately identical, that would still be of no help. Unless we possessed a clear idea of what would count as absolute identity between these two, we would be in no position to declare they have only approximated to that ideal. An approximation only makes sense if we know with what it is that it approximates; but, for us to know that, we would have to know what it would be for the LOI to apply absolutely in this case so that we could say why this is merely an approximation. [There is more on this below.]


Whatever one thinks of the limitations (or otherwise) of the vernacular, unless we begin with an accurate and representative view of the use of such terms in ordinary language we stand in real danger of making fundamental mistakes in more complex or technical areas. As we will see, that is exactly what undermines the criticisms DM-theorists make of the LOI.


It would be a mistake, therefore, to think "equal", "identical", "same" (and related terms) all mean the same (no pun intended). But, because of his cavalier attitude to the vernacular, Trotsky either ignored, or was oblivious to, the conceptual space ordinary language opens up for its users in this regard, a flexibility that allows them to make complex and intricate allusions to identity, equality, similarity, difference, and much more besides, with relative ease.


Consider just a few examples: not only can two or more things be equal and not identical, they can be identical without being equal. For instance, two or more forces can be equal and opposite (or equal and not opposite), yet still fail to be identical. [If they were identical they couldn't be opposite.]


Again, two individual sportsmen/women could be identically the same player. For example, in cricket, they could be "opening bat", "first slip", or "wicket keeper", at different times in the same game or at the same time in different games, while being unequal in many other respects.


Not only that, but identically the same man or woman could occupy, say, two different official, semi-official or work-related posts at the same time, but have unequal powers in each (e.g., NN could be a Unison rep at the same time as being the Treasurer of her local branch of Stop the War Coalition (STWC)). In that case we could say that "The Unison rep is identical to the STWC Treasurer", and, since NN is both of these at once, slow or even change wouldn't affect this identity statement (unless, of course, she resigns from one or both, or dies).5


Furthermore, two or more things can be the same even if they aren't at all alike: for example, two copies of identically the same book (e.g., Das Kapital) in radically different languages (say, English and Chinese) are easily recognisable as the same book even if they are totally dissimilar. Minor differences between the two are irrelevant here. Only a fool would try claim that a copy of Das Kapital wasn't in fact a copy of that book because of a differently coloured cover, for example. So, while these books may not be identically the same physical object, they are identically the same work by Marx. Indeed, countless different readers can now access the same works of Marx's right across the planet at the Marxist Internet Archive. Despite the fact that they might access his work using different browsers, screen resolutions or text magnification, few would claim that these facts prevent readers accessing identically the same work.


This indicates that our application of identity criteria in different areas of discourse change depending on the substantival terms and the circumstances involved.5a In addition, this shows that in the vernacular there is no such thing as the meaning of any of our terms for identity, sameness or difference --, which further implies that Hegel and other dialecticians focussed their attention on an entirely spurious target.


To continue, two totally different things can be equal. For example, two non-identical athletes who cross the winning line together would both be equal joint-winners of the Gold medal, say. Two women at the front of two different queues in the same or different Post Office(s) would both be equally first in line. Two idiots who shout "Fire!" at the same time in a cinema are equally to blame for the ensuing panic. A bus or a train could be equally acceptable to a weary traveller as a means of transport. Two delivery men who carry a packing case up three flights of stairs will be equally responsible for delivering it. Two punters could equally share a lottery prize because they completed the same winning ticket together, and both chose identically the same numbers. Two comrades could sell equal numbers of different revolutionary papers on separate paper sales weeks apart. Instances like these are easy to multiply. No doubt two or more readers could imagine equally apposite (but non-identical) examples of their own to make identically the same point.5b


[The following material used to be in Note 5b.]




It could be argued that these examples of identity aren't in fact examples of strict identity since all the items listed will change in small ways, as will their relation to countless other local, or even remote, objects. In that case, absolutely nothing in nature will be identical to itself from moment to moment.


This objection has been partially defused in Note 5, and will be completely laid to rest below (herehere, and here), as well as the closing part of this Essay.


However, it is worth pointing out that the examples given above were listed in order to show that abstract, and what we might call, for want of a better phrase, material identity, aren't the same as abstract, and what we might also call, material equality, and that ordinary language (but not the obscure jargon philosophers have concocted) is our best guide to what we mean by identity, sameness, equality and difference.


[The primacy of ordinary language is taken for granted at this site; in Essay Twelve this stance will be defended in depth (summary here); but see also here.]


Now, the other objection (i.e., that an alteration to a body's relational properties changes that body) itself depends on the truth of several other DM-theses -- for example, DM-Holism and the doctrine of "internal relations". Since these are taken apart in Essays Three Part Three and Eleven Parts One and Two, no more will be said about that topic in this Essay. [However, on this, see here.]


Nevertheless, a few points need to be made about the doctrine of universal change.


Naturally, it would be perverse to deny that many things change; not only is this a given in our use of ordinary language and our common understanding (a highly truncated list of ordinary words we have for change has been posted here), it is a familiar feature of everyday life and highly confirmed by science.


However, even if the evidence we now have were to be multiplied by several million orders of magnitude (i.e., by a factor of, say, 102000000 -- or more), that would still fail to be enough to justify the sort of mad dog Heracliteanism we find in DM-texts:


"[A]ll bodies change uninterruptedly in size, weight, colour etc. They are never equal to themselves.... [E]verything exists in time; and existence itself is an uninterrupted process of transformation…. For concepts there also exists 'tolerance' which is established not by formal logic…, but by the dialectical logic issuing from the axiom that everything is always changing….... Dialectical thinking analyses all things and phenomena in their continuous change…. Dialectics…teaches us to combine syllogisms in such a way as to bring our understanding closer to the eternally changing reality." [Trotsky (1971), pp.64-66. Italic emphases added. Paragraphs merged.]


"Dialectics…prevails throughout nature…. [T]he motion through opposites which asserts itself everywhere in nature, and which by the continual conflict of the opposites…determines the life of nature." [Engels (1954), p.211. Italic emphases added.]


"[Among the elements of dialectics are the following:]…internally contradictory tendencies…in this [totality]…and unity of opposites…. [E]ach thing…is connected with every other…[this involves] not only the unity of opposites, but the transitions of every determination, quality, feature, side, property into every other…. In brief, dialectics can be defined as the doctrine of the unity of opposites. This embodies the essence of dialectics….


"The splitting of the whole and the cognition of its contradictory parts…is the essence (one of the 'essentials', one of the principal, if not the principal, characteristic features) of dialectics…. The identity of opposites…is the recognition…of the contradictory, mutually exclusive, opposite tendencies in all phenomena and processes of nature…. Development is the 'struggle' of opposites.… The unity…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.221-22, 357-58. Emphases in the original; some paragraphs merged.]


"According to Hegel, dialectics is the principle of all life…. [M]an has two qualities: first being alive, and secondly of also being mortal. But on closer examination it turns out that life itself bears in itself the germ of death, and that in general any phenomenon is contradictory, in the sense that it develops out of itself the elements which, sooner or later, will put an end to its existence and will transform it into its opposite. Everything flows, everything changes; and there is no force capable of holding back this constant flux, or arresting its eternal movement. There is no force capable of resisting the dialectics of phenomena….


"At a particular moment a moving body is at a particular spot, but at the same time it is outside it as well because, if it were only in that spot, it would, at least for that moment, become motionless. Every motion is a dialectical process, a living contradiction, and as there is not a single phenomenon of nature in explaining which we do not have in the long run to appeal to motion, we have to agree with Hegel, who said that dialectics is the soul of any scientific cognition. And this applies not only to cognition of nature….


"And so every phenomenon, by the action of those same forces which condition its existence, sooner or later, but inevitably, is transformed into its own opposite…. When you apply the dialectical method to the study of phenomena, you need to remember that forms change eternally in consequence of the 'higher development of their content'…. In the words of Engels, Hegel's merit consists in the fact that he was the first to regard all phenomena from the point of view of their development, from the point of view of their origin and destruction…." [Plekhanov (1956), pp.74-77, 88, 163. Bold emphases alone added. Some paragraphs merged.]


"'All is flux, nothing is stationary,' said the ancient thinker from Ephesus. The combinations we call objects are in a state of constant and more or less rapid change…. [M]otion does not only make objects…, it is constantly changing them. It is for this reason that the logic of motion (the 'logic of contradiction') never relinquishes its rights over the objects created by motion….


"With Hegel, thinking progresses in consequence of the uncovering and resolution of the contradictions inclosed (sic) in concepts. According to our doctrine…the contradictions embodied in concepts are merely reflections, translations into the language of thought, of those contradictions that are embodied in phenomena owing to the contradictory nature of their common basis, i.e., motion…. [T]he overwhelming majority of phenomena that come within the compass of the natural and the social sciences are among 'objects' of this kind…[:ones in which there is a coincidence of opposites]. Diametrically opposite phenomena are united in the simplest globule of protoplasm, and the life of the most undeveloped society…." [Plekhanov (1908), pp.93-96. Bold emphases alone added. Some paragraphs merged.]


"There are two possible ways of regarding everything in nature and in society; in the eyes of some everything is constantly at rest, immutable…. To others, however, it appears that there is nothing unchanging in nature or in society…. This second point of view is called the dynamic point of view…; the former point of view is called static. Which is the correct position?... Even a hasty glance at nature will at once convince us that there is nothing immutable about it….


"Evidently…there is nothing immutable and rigid in the universe…. Matter in motion: such is the stuff of this world…. This dynamic point of view is also called the dialectic (sic) point of view…. The world being in constant motion, we must consider phenomena in their mutual relations, and not as isolated cases. All portions of the universe are actually related to each other and exert an influence on each other…. All things in the universe are connected with an indissoluble bond; nothing exists as an isolated object, independent of its surroundings….


"In the first place, therefore, the dialectic (sic) method of interpretation demands that all phenomena be considered in their indissoluble relations; in the second place, that they be considered in their state of motion…. Since everything in the world is in a state of change, and indissolubly connected with everything else, we must draw the necessary conclusions for the social sciences….


"The basis of all things is therefore the law of change, the law of constant motion. Two philosophers particularly (the ancient Heraclitus and the modern Hegel…) formulated this law of change, but they did not stop there. They also set up the question of the manner in which the process operates. The answer they discovered was that changes are produced by constant internal contradictions, internal struggle. Thus, Heraclitus declared: 'Conflict is the mother of all happenings,' while Hegel said: 'Contradiction is the power that moves things.'


"...As we already know that all things change, all things are 'in flux', it is certain that such an absolute state of rest cannot possibly exist. We must therefore reject a condition in which there is no 'contradiction between opposing and colliding forces' no disturbance of equilibrium, but only an absolute immutability….


"In other words, the world consists of forces, acting many ways, opposing each other. These forces are balanced for a moment in exceptional cases only. We then have a state of 'rest', i.e., their actual 'conflict' is concealed. But if we change only one of these forces, immediately the 'internal contradictions' will be revealed, equilibrium will be disturbed, and if a new equilibrium is again established, it will be on a new basis, i.e., with a new combination of forces, etc. It follows that the 'conflict,' the 'contradiction,' i.e., the antagonism of forces acting in various directions, determines the motion of the system…." [Bukharin (1925), pp.63-67, 72-74. Bold emphases added. Some paragraphs merged.]


[Several more quotations from the dialectical classics and other DM-texts that echo the same sentiments can be found here. (No irony intended.)]


Unfortunately, the evidence supporting these hyper-bold claims is conspicuous by its absence, as we discovered in Essay Two. Despite this, the authority of assorted Idealists and Mystics (like Heraclitus and Hegel) seems to be sufficient for DM-theorists.


However, the uncontroversial admission (above) that change is a widespread phenomenon, throughout nature and society, doesn't amount to any sort of concession to DM, since 'dialectical change' is supposed to be caused by, or is the result of, 'internal contradictions'. Now, that doctrine has been demolished in Essay Eight Parts One and Two. In Essay Eleven Part One, Heraclitean change has also been destructively criticised.


So, non-dialecticians can agree with dialecticians on the reality of change; where they differ is over the cause of change: 'internal contradictions', and perhaps also the dogmatic claim that everything in reality constantly changes in every respect, always.


It is worth underlining here that the denial of universal change doesn't imply that everything is changeless; just that some things might be (and probably are) changeless, and some things maybe aren't. [On this see Note 11 and Note 12.] Plainly, this is an empirical question which can't be settled by appealing to the 'authority' of a handful Idealist Philosophers and Hermetic Mystics -- nor, indeed, on the basis of a set of highly repetitive and dogmatic assertions.


Nevertheless, let us suppose for the moment that some object, B, possesses the following properties, qualities or relations: B1, B2, B3,..., Bn.


According to several of the above dialectical worthies, all of these properties, qualities and relations must change all the time (into what they change we are kept in the dark don't -- other than that we are informed they change into their 'opposites' -- i.e., into not (B1, B2, B3,..., Bn), or perhaps (B1*, B2*, B3*,..., Bn*), where each term with an attached asterisk is the 'opposite' of the 'same' term without one).


[However, even that possibility is closed off in Essays Part Three and Eleven Part One. Henceforth, to save on needless repetition, I will simply call these properties, qualities or relations, "properties".]


Nevertheless, while B changes it is still identical with itself. In order to see this, let us suppose that when each property, Bi, changes, it becomes, say, Bi*, in the first instance, and then Bi** in the next, and so on. But, at any moment, B's identity will be given by its set of properties (if we must view identity traditionally). So, in the first case, for example, B will have changed into {B1*, B2*, B3*,..., Bn*}. Hence, even though B has changed, it retains its changed identity. In which case, as long as B exists it is identical to itself (albeit, its changed self). Consequently, viewed this way, identity is no enemy of change.


[Dialecticians often appeal to the existence of UOs to defuse this sort of objection; that topic has been examined in Essays Seven and Eight Parts One and Two.]


[UO = Unity of Opposites/Unities of Opposites, depending on the context.]


Of course, the above scenario (which is called Maximal Heracliteanism (or MAH) in Essay Eleven Part One -- link below) might not be the interpretation of change that most dialecticians would want to adopt (even though the DM-classicists quoted above seem to be sold on it). If so, they should pause for thought before finally deciding. That is because, if just one of the properties possessed by B -- say, Bk -- remains the same, even for a few nanoseconds, then the LOI must apply to it, and the dialectical game is up, for here we would have something that remained the same, and is identical with itself, even if only momentarily.


By way of contrast, the maximalist option (i.e., MAH -- again, this is explained at the following link) has even worse consequences for DM; these have been set out in detail in Essay Eleven Part One.


Either way, Heraclitus is no friend of dialectics -- or, if he is, he is also its worst enemy.


A nice unity of opposites there, one feels!




Of course, only those with the same opinion about this as Marx, who take their philosophical cue from ordinary language, will be impressed with the examples given earlier. On the other hand, since ordinary language is the means of communication invented, preserved and maintained by ordinary workers (as they interface with one another and the material world), only those who prefer for non-materialist language -- for instance, those with an inexplicable fondness for the obscure terminology concocted by Philosophers, or even worse, the jargon cooked up by Hegel -- over the materially-grounded vernacular will have reason to cavil. Annoyingly, those in that category be doing so for identically the same ideologically-compromised reasons.


[On that see, Essay Twelve (summary here).]


Such is the cunning of ordinary discourse.


[The following material used to be in Note 6.]




The use of ordinary words for identity and difference is reassuringly varied and can sometimes be bewilderingly complex. Consider the following (greatly shortened) list of examples:


E1: The same letter can appear in the same word in different places, and in a different word in the same place (e.g., "t" can appear first and fourth in both "trite" and "trot"). A different letter can appear in the same word, in the same or in a different place (e.g., if "chien" and "dog" are counted as the same word in different languages, "c" appears in the first place in the French word, and "d" in the same (i.e., the first) place in the English word, and different letter, "d" and "g", appear in the same English word). Moreover, the same word, in the same or different place in the same or different sentence, can mean the same or different things. For instance, consider Chomsky's example: "Pretty little girls' school"; the word "pretty" can be taken in several ways, depending on how the whole phrase is read, as can each of these sub-phrases: "Pretty little", "little girls'" and "Pretty little girls'", to name but three), and different words, in the same or different places can mean the same or different things (as in, "The striker hit the scab" and "The scab was hit by the striker" (where the same words mean the same in different places in two different sentences with the same sense, but the two sentences with the same words mean two different things); and "The striker hit the ball", where the same word could mean different things (i.e., "striker" could mean a player on a Football (soccer) field or someone engaged in a strike). Furthermore, the same word can mean different things at one and the same time to two different people (e.g., if one of them reads it as a code word, on one occasion), and different things to the same person at different times (if, say, their facility with the language concerned improves). Naturally, permutations like this can be knitted together endlessly to form complex identity/equality sentences that we can all understand, given the right level of concentration. For example, the same word could mean different things to the same person at different times, but the same thing in different places, while it could mean the same thing at the same time or at different times in the same place or in different places to the same or different people (etc., etc.).


E2: The same numeral can appear in the same place in the same number in different places at the same or different times (e.g., the figure "9" in a mathematics book, or on a bank statement), or in the same place in different numbers (as in 191 and 1911). Not only that, identically the same numeral can appear in the same number in different places, where it will have a different mode of signification (e.g., in 2500, 2450 and 2445; here the same numeral, "5", means something different in each case, or in 191 and 1911, where the "9" appears in the same place (i.e., second from the left) but means something different in each case, or where it appears in different places (in the tens column and in the hundreds column) but could mean the same (i.e., if the "9" in 191 stood for 90 ten cent coins, and the "9" in 1911 stood for 900 one cent coins)). And the very same number can change in other ways even while it stays the same -- for example, the numeral, "5", will stay the same but will signify something different as other numerals are added in. So, in 50, the 5 stands for five tens. If another zero is added (to yield 500), the very same 5 will now mean five hundreds, and so on. [The reader is to imagine all this being typed in one go, say, into a calculator) so that the very same figure "5" on the screen changes while it remains the same as zeros are added.] Furthermore, the same numeral can appear in the same sign in the same place and mean something different, depending on how it is read (e.g., the numeral "1" in "10" could mean "one" written in the tens column, or it could mean "one" written in the unitary power of two column in binary code, with the first "one" signifying "ten" and the second indicating "two"). Or the very same "2" on a clock face could signify 2am or 2pm. Or think of the way that "1" can mean something different if it occurs in the same place in 01/02; in the first sense it could mean the 1st of February (if read by a UK citizen), in the second, the 2nd of January (if read by a US citizen). So, in the last few cases, the very same thing could be identical in certain respects while being different or unequal in others. Examples are easy to multiply. The same points (or different ones) can be made about the same (or different) musical notes, dance steps, gestures, works of art, signs, signals, symbols and noises.


E3: The same day of the week occurs in the same place in different weeks, and for 24 hours on the same day in the same week. And it can occur in the same place in different weeks belonging to the same or different months. The reader, no doubt, can supply his/her own complex permutations as the temporal vocabulary employed is changed -- as in: same/different second, minute, hour, fortnight, year, decade, century, millennium, geological age, eon…


E4: The same book can appear in different libraries in the same place, or in different libraries in different places, and a different book can appear in the same or different libraries in the same or different places. The same copy of The New York Times can be read by different people in the same place at the same time, or in different places at the same time, or in the same place at different times -- and it can be read by the same person in different places at the same or different times, and so on. The same can happen with TV programmes, films, photographs, music scores, works of art, e-mails, computer games, texts, printed adverts, and plays.


E5: The same worker could join the same strike at different times, or different strikes at the same time (if he/she has two jobs and both are in dispute). And different workers could join the same or different strikes at the same or different times in the same or different places. And the same strike could spread to different places, involving different workers at the same or different times. The same or different cheques could be made valueless if the same Bank goes bust, and the same person could be made an orphan at the same time if both their parents are killed in the same or different accidents at the same or different times.


E6: The same element in the periodic table can appear in different parts of the universe at the same or different times, and in the same or different compounds at the same or different times. The same geodesic can be traversed by different particles, at the same or different times. The same inertial frame can contain the same or different objects at the same or different times, and different inertial frames can contain the same or different objects at the same or different times. The same (or different) considerations apply to packing cases, bags, holes, tins of beans, garages, flats, sheds, houses, cars, taxis, trains, aeroplanes, ships, buses, submarines, rockets...


And the same or different DM-fan could take exception to the same or different example(s) above.


Try expressing any of that in Hegel-speak!


But it's a doddle in the vernacular.


We needn't concentrate, either, on examples that some might still consider "abstract"; two (physical) ink marks on a page (two letter "A"s, again) which aren't even identical in shape or size (i.e., "a" and "A") could be identically positioned between other non-identical letters. So, in "pat" and "PAT", each letter "A" is sandwiched identically between two other non-identical letters (i.e., both are in the middle). Large or small physical differences between these letters, and any other incidental changes they might undergo (which don't affect their relative position) -- such as a change of colour on your screen, or on the page -- won't alter the fact that they are identically placed between two other letters. Indeed, the spacing of these letters could be grossly unequal, but that wouldn't affect the fact that these letters are placed identically in the middle.


[To be sure, the gap between the letters might be different, but that wouldn't alter the fact that both are identically placed in the middle. And by "middle" is meant "having one letter either side", not "located at or near the geometric centre".]


Now, the position of ink marks on a page (or even those electronically produced as pixels on your screen) isn't abstract, it is eminently material --, so much so that one or both can be obliterated by the non-dialectical use either of some Tipp-ex or the delete key.


And deletion isn't the removal of an abstraction.


[Alternatively, just try deleting an abstraction!]


As noted elsewhere, ordinary and technical or even semi-technical languages have seemingly limitless capacities for allowing their users to express complex and subtle differences in meaning way beyond that permitted by the obscure and lifeless language Hegel inflicted on his readers. This shouldn't surprise anyone; ordinary, technical and semi-technical languages have been created over countless centuries by working people and/or scientists in their interaction with the world and with one another. These systems of communication reflect our species' complex inter-relationship that each individual, class or group has with others, and with changing reality -- and which contain our best guide to identity, sameness and difference, and much else besides.


In contrast, Hegel's opaque, jargon-bound language reflects alienated, ruling-class forms-of-thought and material interests -- cobbled-together as part of a dubious, class-compromised intellectual tradition that stretches back well over 2500 years --, and which (in Hegel's case) was invented by an individual who, in his theoretical activity, was far more concerned with his relation to the world of ideas than he was with his interaction with ordinary human beings, objects and processes in the material and social world. Small wonder then that his ideas can't cope with changing reality.




Clearly, Trotsky and Hegel created serious problems for themselves when they erected an insecure 'logical' edifice on such an insubstantial linguistic base. This predicament was further compounded by their choice of an extremely narrow range of examples when compared with the countless available to them (and to ordinary speakers -- on that see above), which permit talk of equality, sameness, identity and difference with ease.


Equally annoyingly: Traditional Philosophers have in general done exactly the same (irony intended).6


"Equal" And "Identical" Not Identical


As will no doubt be apparent to any competent user of language, "equal" and "identical" aren't synonymous. Several examples given above illustrate this fact; the distinction can also be seen if "equal" is substituted for "identical" in either of the following sentences:


S3: NN and NM are identical twins.


S4: The money that the victim of the racial assault received was equal to that stolen in the assault.


The use of "equal" in S3 would make it meaningless (viz., "NN and NM are equal twins"), and the presence of "identical" in S4 would change its sense entirely:


S4a: The money that the victim of the racial assault received was identical to that stolen in the assault.


Clearly, the implication of S4a is that the very same notes and coins were returned, whereas S4 itself would be true if the money the victim received was merely the same value as the money taken (perhaps presented to her in cheque form).


Mathematical Equality Vs Mathematical Identity


Moreover, we needn't restrict our attention to ordinary sentences (even though Trotsky himself did); the above distinction is found in mathematics. Consider the following:


S5: x2 - x - 42 = 0 x = 7, or x = -6.


S6: cos3θ + sinθ º 4sinθcos2θ.


[In S6, "º" is the sign for identity or equivalence.]


Nobody confuses "=" with "º" in mathematics. Moreover, in S5, just because x = 7 or x = -6, that doesn't mean x is identical with either -- otherwise it could never stand for another number (as it does in, say, x2 + x - 56 = 0 x = -7, or x = 8) -- and, of course, if it couldn't stand for a different number, it wouldn't be called a variable.


Worse still: two or more identicals can be equal to, but different from, the same identical. For instance, while 0 = 0, it is also true that 0 + 0 = 0, and 0 x 0 = 0 -- even though it is also true that neither 0 + 0 nor 0 x 0 are identical to 0, nor to one another. Even worse, some things can change even while they remain the same. For example, it is relatively easy to transform 1/√n into √n/n -- as follows: 1/√n x √n/√n º √n/n. But, 1/√n doesn't even look like √n/n, although the two are identical: 1/√n º √n/n. So, here we have change with no change!


In which case, equality and identity don't prevent change, nor do they even imply that things can't change -- at least, not in mathematics.7


In MFL (i.e., outside of mathematics), the distinction between these two is even more pronounced. The "=" sign is used as a relational expression (which is legitimately flanked only by Proper Names and other singular terms, such as Definite Descriptions), whereas "º" is a truth-functional operator (and can be flanked only by propositions or clauses).


Of course, these distinctions aren't the same as those we find in ordinary language (no irony intended), nor yet those in Traditional Philosophy -- more on that below.


["Truth-functional" is technical term that expresses a logical link between propositions, the alteration of which changes the conditions under which they are true or false. For those not too familiar with MFL, I have posted a simplified explanation of these and other terms, here.]


[MFL = Modern Formal Logic.]


It could be objected once again that these examples are all abstract; in which case, the reader is re-directed to my earlier response to this objection.


So, the question returns: why did Trotsky make a claim about equality when he was trying to discuss identity? The fact that he ignored all of the classical formulations of the LOI (such as Leibniz's) only compounds the problem.8


Perhaps this was an oversight? But, this glaring omission -- coupled with Trotsky's subsequent and rather odd digression over bags of sugar and eye-glasses, as well as his failure to consider the wider use of words for identity in the language of everyday life -- tends to suggest that he didn't really understand the very thing he was criticizing: identity.8a


It therefore looks like Trotsky tried to undermine the LOI by appealing to a principle (equality) that wasn't identical with it (irony intended).


Trotsky Changes The Subject


One answer to this 'puzzle' might lie in the fact the change of subject recorded in S1 allowed Trotsky to go on to raise what turn out to be largely irrelevant claims points about things like bags of sugar. Because the latter involve objects that can be measured (as opposed to their being counted), the interpretation of the "A"s in S1 as quantities of sugar heavily biased Trotsky's criticism -- it allowed him to focus his attention on one particular aspect of equality that isn't necessarily connected with identity.


S1: A is equal to A.


For example, one and the same bag of sugar could be 'self-identical' and equal to itself in weight even while it was unequal in weight to a second seemingly identical bag. [How this is possible will become clear as the argument unfolds.] And, two different bags of sugar could be equal in weight (even if only momentarily), as far as our most sensitive instruments are able to detect. Not only that, two separate bags could both have their weights changing in exactly the way Trotsky described (no irony intended); the first bag could have its weight falling, the second rising. At some point, therefore, their two weights could momentarily be identical. How could these possibilities be ruled out?


Furthermore, in two separate piles, bag B in pile one, and bag C in pile two, could be the heaviest in their respective heaps. In that case, each bag would be equally the heaviest in their respective groupings while still being non-identical in weight with one another. No doubt the reader can imagine other cases Trotsky failed to consider.


Clearly, Trotsky's analysis blurred these clear distinctions -– those, incidentally, that are easily drawn in ordinary language (indeed, as they have been here), and which are readily understood, even by working-class children.


More importantly, Trotsky clearly failed to notice that even though objects might vary in weight, they could still be identical in number. Indeed, as is patently obvious, any object is identical to itself in number (and each is identical in number with any other single object, too). Moreover, close inspection over an extended period of time will fail to reveal any relevant difference with respect to their number, even if other aspects of the said object(s) changed markedly (such as their weight). Trotsky overlooked this obvious counter-example to his claim that things can't remain the same while they change: in at least this sense most do.


Of course, it could be objected here that not only do some things divide as they change, others merge together; in such cases, their number wouldn't be identical from moment to moment. That is undeniable. However, descriptions of divisions and mergers depend on the said objects being identifiable first, which process clearly depends on the application of the LOI. If we can't count objects before or after they divide or merge, we are surely in no position to judge that they have changed in this respect. Since counting depends on identification under a given general term (so that we can say we have, for instance, 2 bags of sugar -- or one amoeba, then two), that aspect of this objection itself depends on an application of the LOI as a rule of language.8b


Anyway, the above comments still apply to objects that don't divide or merge. Plainly, there are uses of numerical identity that aren't susceptible to this objection (concerning objects that divide or merge). For example, if we consider, say, the number of volumes of Das Kapital, it is clear that there are just as many volumes today as there were 100 years ago (viz., three -- that is, if we don't count Theories of Surplus Value; six if we do). Even though the number of copies of Das Kapital has increased markedly over the years, and most copies will have changed markedly in the meantime, the number of volumes of Das Kapital remains steadfastly fixed on three (or six). Hence, the following statements are true:


L1: The number of volumes of Das Kapital in the year 1900 is identical to the number of volumes of Das Kapital in 2019 (namely, three (or six)).


L2: The number of volumes of Das Kapital on any one day in 2021 is identical to the number of volumes of Das Kapital on the same day in 2021 (namely, three (or six)).


L3: A is identical to A.


In L1, we have identity over time and in L2 identity at any moment in time.


But, even though the "A"s in L3 stand for "The number of volumes of Das Kapital on any one day in 2021" (when interpreted as they are in L2), it is clear that it isn't possible to map the same "A"s consistently onto anything analogous in L1. That is because the first "A" would have to stand for "The number of volumes of Das Kapital in the year 1900", the second for "the number of volumes of Das Kapital in 2019", which phrases are clearly not typographically identical, even though they are both part of sentences that express a simple rule we have for identity.


This demonstrates that Trotsky's narrow interpretation of the variable letter "A"s (in L3 or S1) fails to capture the much wider use of words we have for identity in ordinary language -- some of which were considered above, and several more will be examined below. Even so, both L1 and L2 surely count as further counter-examples to Trotsky's ill-considered charges against the LOI.


It is also worth recalling that the volumes of Das Kapital are just as material as bags of sugar.


Again, it could be objected that number is an abstract property of objects, making the above points irrelevant. But, according to Lenin, anything that enjoys objective existence external to the mind is material:


"[T]he sole 'property' of matter with whose recognition philosophical; materialism is bound up is the property of being an objective reality, of existing outside our mind." [Lenin (1972), p.311.]


Well, the three volumes of Das Kapital surely exist just as objectively "outside the mind" as do pound bags of sugar. Moreover, if Trotsky is allowed to refer to the measurable properties of bags of sugar -- such as their weight, which will also be recorded by the use of several numbers --, critics of the above can't consistently object to a similar appeal to their countable properties.


[Anyway, 'Abstract Identity' will be examined below.]


In addition, consider the following perfectly normal examples of the use of words associated with identity:


L4: The number of months of the year is identical to the number of Apostles.


L5: The number of elements lighter than Helium is identical to the number of authors of The Proletarian Revolution and the Renegade Kautsky.


L6: The Morning Star is identical to the Evening Star.


L7: The population of the United Kingdom at noon on any day in January 2021, is identical to a whole number somewhere between 50 and 80 million.


L8: The point of all these counterexamples is identical in each case: to refute Trotsky's criticisms of the LOI.


L9: The stance of the majority Trotskyists is identical to that of Marx on the following issue: The emancipation of the working class is an act of the workers themselves.


L10: The editor of International Socialism in 2009 is identical to the author of A People's History Of The World.


L11: Mount Godwin-Austen is identical to K2.9


Once again, sentences like these can be multiplied indefinitely. As a highly competent user of language, Trotsky can't have been unaware of this. So why did he feign ignorance? Was his analysis of this 'concept' biased by an extremely narrow focus on a specific philosophical use of words for identity -- derived from that notorious Idealist, Hegel -– and one that didn't match their application in ordinary language?


As we will see, these rapidly-forming suspicions aren't easy to dismiss.


Trotsky's Argument


Precisely What Is Trotsky Denying?


However, returning to Trotsky's actual argument:


"In reality 'A' is not equal to 'A'. This is easy to prove if we observe these two letters under a lens -– they are quite different to each other…." [Trotsky (1971), p.63.]


But, not even a lens-wielding Trotsky would consider making the same point (no irony intended) in relation to the following legitimate example of the use of the "=" sign:


S7: 2(x + 1) = 2x + 2.


[S1: A is equal to A.


S1a: A is equal to itself.]


[I am concerned here with Trotsky's use of "=", not with a more proper use of "º".]


But, if not, why not? In S7, the two sides of the equation don't even look similar (with or without the aid of a magnifying glass!), quite unlike the two "A"s in S1. Despite that, few would question the fact that the left-hand side of S7 is still equal to the right-, for all x. In which case, this use of "=" isn't susceptible to Trotsky's 'microscope argument'. That suggests this particular point about the "A"s in S1 was equally misguided (irony not intended, again).


It could be objected that S7 is an 'abstract' example, which exempts it from such criticism. But, Trotsky's point about the two "A"s in S1 is no less 'abstract'. Initially for Trotsky, the "A"s in S1 were merely letters. And yet, if the symbols in S7 were to be interpreted in the same light, his lens-inspired criticism would make no sense. Who in their left mind would use a magnifying glass to check whether 2(x + 1) is exactly equal to 2x + 2 in form? And who would ever employ S7-type sentences in mathematics if the use of an equal sign was only legitimate when the symbols on either side of it had to be identical in shape, or microscopically indistinguishable under a lens? When employing such sentences we surely advert to the rule they express, not the physical form of the letters they happen to assume. Hence, despite the fact that the symbols appearing in S7 look totally different to the naked eye, no one would question their role in expressing a simple algebraic rule.


In that case, why did Trotsky use such a crass argument against the expression of a logico-linguistic rule in S1? If mathematicians were to scrutinise each other's work in the same crude way, they would surely dispense with what we now call "proof", and resort to inspecting alleged 'proofs' with magnifying glasses. Mathematical advancement would then depend, not on proof, but on proof-reading!


Trotsky Has This Base Covered -- Or Has He?


Some might object and claim that Trotsky had anticipated this point when he said:


"[Concerning] the proposition to 'A' is equal to 'A'[:] This postulate is accepted as an axiom for a multitude of practical human actions and elementary generalisations. But in reality 'A' is not equal to 'A'. This is easy to prove if we observe these two letters under a lens -– they are quite different from each other." [Trotsky (1971), p.63.]


Hence, it could be argued that even though mathematicians deal with "abstract concepts", the symbols they use to express them are constrained by limitations imposed on anyone operating in this world. In that case, since no two symbols could be absolutely identical, Trotsky's point remains valid -- or so this argument might go.


However, in the vast majority of cases in mathematics symbols like "=" and "º" occur between symbols that don't even look remotely the same. Several examples were given above. Anyone who doubts this should consult a handful of mathematics texts (of any level of difficulty equal to or above Intermediate Standard). There they will find few examples of schematic sentences like S1, but countless like S5 or S6.


S5: x2 - x - 42 = 0 x = 7, or x = -6.


S6: cos3θ + sinθ º 4sinθcos2θ.


Trotsky's analysis thus fails completely to account for this use of symbols. In fact, not only are mathematicians not really interested in "approximate equality", the notion of "abstract identity" -- if any sense can be made of it -- is itself parasitic on ordinary identity, or on a surreptitious material application of the LOI (as a rule, not as a truth), as we shall soon discover.


Bags Of Sugar In Fact Refute Trotsky


Trotsky's Answer


[In what follows, I am assuming the sugar in question isn't loose, but is held in container or a bag of some sort while it is being weighed -- clearly that would reduce any effect the environment had on the sugar itself, etc. This minor adjustment does not, I think, affect the points Trotsky wanted to make. If anyone thinks differently, please contact me and explain why you disagree.]


Again, some readers might still think that Trotsky had anticipated these relatively minor quibbles, since he went on to consider a possible response that the two "A"s in S1 might really be "symbols for equal quantities, for instance, a pound of sugar".


S1: A is equal to A.


S1a: A is equal to itself.


In response, he pointed out that in the real world a pound of sugar is never equal to a pound of sugar, since any apparent equality will vanish upon closer examination:


"But, one can object, the question is not the size or the form of the letters, since they are only symbols for equal quantities, for instance, a pound of sugar. The objection is beside the point; in reality a pound of sugar is never equal to a pound of sugar -– a more delicate scale always discloses a difference. Again one can object: but a pound of sugar is equal to itself. Neither is true (sic) -– all bodies change uninterruptedly in size, weight, colour etc. They are never equal to themselves. A sophist will respond that a pound of sugar is equal to itself at 'any given moment'…. 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." [Trotsky (1971), p.64.]


Mere Guesswork?


The problem with this is that Trotsky was clearly guessing here. He had no way of knowing for sure that greater accuracy in weighing would always reveal detectable differences.


Indeed, there are several possibilities that he failed to consider. For example, the weighing scales themselves could alter slightly, which could be the real cause of the inferred change in the weight of the sugar. Indeed, the weighing scales could alter to such an extent that they compensated for the change in the weight of the sugar, so that in the end no overall difference was detectable. How could Trotsky rule either of these out? Plainly, he couldn't do so if constant change -- including that experienced by instruments -- is a central postulate of dialectics. How could he be so sure that the hypothetical differences he says must exist between these bags (or between a bag and itself) weren't artefacts of the machines themselves -- or of some other ambient cause -- as opposed to these being genuine phenomena representing actual changes in the weight of the sugar? For all he knew the sugar itself could remain the same for a few seconds (or minutes), with any apparent change that had been detected being the result of other incipient factors. In fact, as seems clear, Trotsky could only be 100% confident that any subsequently detectable differences were always and only the result of changes to the sugar itself because of an a priori stipulation to that effect. And, as seems plain, a stipulation is different from an imposition on nature in name alone.10


Of course, that doesn't mean Trotsky was wrong in this case. No doubt if a series of identical experiments -- note the use of the highlighted word here -- were conducted, differences would be detected. But, given Trotsky's stated views on change he would have had no way of knowing whether any of them were a result of changes in the scales, the sugar, the eyesight of the observer, the relative strength of the surrounding gravitational field, or a combination of one or more of these --, or, indeed, whether or not they were attributable other proximate causes.


Some might think the above considerations are irrelevant; if things change, who cares what causes it? But, Trotsky is here appealing to the results of an experiment -- one that he clearly didn't carry out himself -- to substantiate a hyper-bold claim about all objects everywhere in the entire universe, for all of time. It now turns out that because of that thesis itself, it might not be possible to confirm what he said. If so, we are still owed an explanation as to why Trotsky thought it correct to say everything changes all the time, when this thesis can't actually be verified. And this isn't just because many of the above complications could cancel each other out, or mask a temporary lack of change in other things, it is because we don't have access to the vast majority of regions of space and time, and never will!10a


And, as we are about to find out, any attempt to plug the gaps in Trotsky's argument only succeed in punching even bigger holes in it.


Trotsky's Instructions -- Followed Exactly And To The Letter?


As we have seen, Trotsky's argument partly revolves around the accuracy of measuring a pound bag of sugar (etc.). However, he then rather boldly extrapolated from a few (theoretical) observations about local conditions connected with weighing that sugar to hyper-bold, general claims about all objects for all of time:


"[A]ll bodies change uninterruptedly in size, weight, colour etc. They are never equal to themselves…. [T]he 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…. For concepts there also exists 'tolerance' which is established not by formal logic…, but by the dialectical logic issuing from the axiom that everything is always changing…. Hegel in his Logic established a series of laws: change of quantity into quality, development through contradiction, conflict and form, interruption of continuity, change of possibility into inevitability, etc…. All this demonstrates, in passing, that our methods of thought, both formal logic and the dialectic, are not arbitrary constructions of our reason but rather expressions of the actual inter-relationships in nature itself. In this sense the universe is permeated with 'unconscious' dialectics." [Trotsky (1971), pp.64-66; 107. Emphases added.]


But: how could Trotsky possibly have known any of this? His only 'evidence' appears to have been this thought experiment!


Again, it could be objected that Trotsky was merely drawing out the consequences of accumulated human experience of change, deriving reasonable conclusions from thousands of years of our growing knowledge of the nature and society, much of it scientific.


In response to this (and in addition to the points made in this subsection) it might be worth trying to reconstruct the reasoning behind Trotsky's claims. Presumably he was arguing as follows:


T1: Anyone who performs a weighing experiment exactly as specified -- who repeats the same procedures -- will find that no two measurements are exactly the same.


But this imputed argument -- if it is indeed his -- would embroil Trotsky in having to invoke the LOI yet again, only now applied to his own instructions, and hence to the measurements required by anyone who wanted to test his claims (brought out by the highlighted words in T1). Clearly, these instructions would have to be carried out identically and to the letter each time if they were to count as a verification of Trotsky's predictions -- only now aimed at demonstrating that the LOI is defective after all, even when applied to instructions!


Of course, given Trotsky's strictures on the LOI, no two attempts to carry out any set of instructions would ever be the same. In that case, there would be no accurate way to test his or anyone else's predictions!


[Once more, an appeal to "approximately identical instructions" has been batted out of the park, below.]


On the other hand, if Trotsky had been faced with someone who claimed that at least two of their measurements were identical, he could only have responded in one or more of the following ways:


(1) Insisting that this experimenter must have been mistaken.


(2) Pointing out that the machines used weren't accurate enough.


(3) Maintaining that his instructions hadn't been carried out exactly and to the letter.


(4) Arguing that identically the same experiments hadn't been performed each time.


In other words, in the absence of a mistake (and if the same results were recorded on more accurate scales -- i.e., ruling out (1) and (2) above), Trotsky would only be able to criticise the above reported experimental verification of the LOI by an appeal to that very same 'law', but now applied to his own instructions! [I.e., options (3) and (4).]


Hence, in order to counter results that would disconfirm his predictions in the above manner Trotsky would have to argue that only those who followed his instructions identically and to the letter could disprove the LOI!


The irony here is quite plain: identically performed experiments are required to prove that nothing is identical with anything else -- including experiments!


To be sure, anyone who only roughly followed his instructions (who was, perhaps, content with a wishy-washy, "approximate-within-certain-limits" dialectical-sort-of-equality) would probably find that many (if not most) of their measurements gave identical results for the weight of these bags of sugar -- thus 'confirming' this 'law'!


In which case, Trotsky's predictions about such objects would end up being refuted by anyone who adopted the diluted DM-version of the LOI! Such experimenters would succeed in confirming the absolute form of the 'LOI by employing an approximate version of it!


A use of 'approximate identity' would show that two or more bags of sugar all weighed the same, and, indeed, that a bag of sugar weighed as much as itself.


Conversely, the more exactly an experimenter adhered to Trotsky's instructions, the more likely it would be that they detected non-identical weights.


In that case, they would succeed in disconfirming the absolute version of this 'law' by applying an exact copy of Trotsky's instructions!


So, by reverse irony, they would refute Trotsky in practice by doing exactly as he indicated, using the LOI applied to his instructions in order to disconfirm it when applied to bags of sugar!


In short: relying on evidence alone, Trotsky was certainly not justified in projecting his conclusions as far as he thought he could --, i.e., across the entire universe for all of time -- not least because there is no evidence that he performed these experiments himself. And anyone who did, or who might perform them would be caught in the above practical refutation of his criticisms of that 'law' whenever they performed such experiments exactly as he instructed.


[It is worth adding that the development of science has at least suggested a much more cautious approach should be taken to theory and evidence. On that, see Note 11 and Note 12, below.]


Hence, Trotsky's claim that all objects, everywhere, change all the time, if extrapolated beyond the aforementioned conventions and scientific facts, would transform the LOI into a metaphysical truth -- that is, into a 'fundamental truth' supposedly valid for all of space and time.


But, there could be no body of evidence large enough to support an extrapolation as bold as this -- or, at least, none that wasn't also based on those very same conventions relating to identically performed experiments and the use of ordinary words for identity. Extrapolation beyond these -- by means of them -- to universal theses that are applicable everywhere and everywhen would convert them into universal and changeless truths -- the very thing Trotsky affected to disavow.


In that case, Trotsky would have to appeal (explicitly or implicitly) to the LOI as a universal truth in order to justify his general conclusions about everything in existence behaving exactly as he said it would --, for example, with every human being measuring objects identically throughout all of human history, in order to show that no one could actually do this, because of his criticism of a 'law' they would have to have used in order to do just that!


So, any evidence (either from the past or the present) to which DM-theorists might appeal in order to undermine this 'law' would automatically call into question the methods by means of which that evidence had been collected, processed and checked. Without the LOI applied as a rule of language, or as a rule guiding practice, no one would be able repeat the same experiments to verify, or refute, earlier results, or even check measurements and hence confirm the accuracy of Trotsky's predictions.


Nor would they be able to learn to use the very same theories that others had used, or appeal to the very same 'law', or its alleged refutation (in the way that, say, Trotsky and Hegel claimed to have done).


Without this 'law' applied as a rule of language, or one that helped guide experiment, there would be no conceptual space within which science or ordinary practice could develop --, and thus no reliable data, no settled theories --, for anyone to begin even to think about confirming the DM-hypothesis of universal change.


In which case, there could be no science or philosophy that questioned the application of this 'law' as a rule of language, or of practice, while still hoping to remain viable.


That, of course, helps explain the source of the difficulties highlighted above (in connection with the postulated refutation of Trotsky's predictions about the weight of a bag of sugar); it also reveals why Hegel got into such a mess over his attempt to half accept and half reject this 'law', and why this entire topic became such a puzzle to him and his epigones. If this 'law' is treated as a metaphysical truth (which has generally been the approach adopted by Traditional Theorists) -- i.e., as a 'necessary truth' --, then its falsehood becomes impossible to state (as we have just seen, and as we will continue to see throughout this Essay), at least in comprehensible language, or in language that doesn't implicitly rely on this very 'law' to state anything at all.10b


Of course, the 'truth' of this 'law' doesn't seem (to some) to be at all trivial. If it is viewed (traditionally) as a 'law' that is said to depict, or determine, the fundamental nature of everything in existence, then it appears (to Hegelians and DM-fans) to rule out change. But even then, this 'defect' proves impossible to express in any language that doesn't also rely on this very 'law', only now operating as a rule governing the use of words to make that very point! (Again, as we have just seen.)


Alternatively, if this 'law' is viewed rightly as a rule of language, or of practice (without which humanity couldn't have developed a single coherent idea), then these pseudo-problems over 'identity' simply vanish. On that basis, the LOI isn't a universal truth because of what Hegel or anyone else said about it; it isn't a truth to begin with! That is because it is a rule (and a rather badly stated rule about our use of words for identity, sameness or difference, too), it can't be true or false, only practical or impractical, useful or useless.


In fact, as we have seen, it is impractical -- or, rather, it is the height of folly -- even to attempt to reject this rule.


In that case, without a clear idea of how to use words for identity (etc.), it would be impossible even for DM-theorists to begin to wonder whether or not our words were approximately stable from moment-to-moment, or if they altered in other alarming ways. If the LOI is rejected (as a rule of language, or of practice) -- or if it is held to be an 'approximate truth' -- then all of the above points will have their place.


For example, in order to be able to say whether or not something was true (or 'partially true'), we would need to know how to use the word "truth" in the same way from moment-to-moment (why that is so will become clearer as this argument proceeds), just as we would also have to know the same with respect to our other words for identity, sameness and difference. Without some notion as to what counts as identically the same employment in all these cases, we couldn't even begin to say what would constitute an approximation to anything whatsoever, or, indeed, in what way something fell short of a standard that presupposed the applicability of words we have for identity (etc.).


[Incidentally, I have employed "know how to use" a certain word in a pragmatic sense; there is no suggestion that we apply language according to some theory we supposedly hold or follow. In fact, I have borrowed this locution from Gilbert Ryle. On that, see here.]


Now, this fundamentally important point sailed right over Hegel's head, and it seems that that is also the case with Engels, Lenin, Trotsky and other DM-theorists. They regarded the LOI as a truth (even if it was one that the "abstract understanding" employed uncritically, or which was valid "only within certain limits"). But, it can't be a truth (for the reasons outlined above); our use of words for identity express rules constitutive our ability to utter sentences that we can even so much as begin to regard as true or false (should we so choose).


[Incidentally, the above isn't a 'transcendental argument', merely a reminder that we have no choice but to use words to express our thoughts, and we may only do certain things with such words because of contingent facts about our social history and our physical constitution. On this, see here and here, where I enter into this topic in more detail.]


So, this 'law', applied as a rule of language (or of practice), has to be employed even to make the point that there are such things as approximate truths (since anyone doing this would have to at least use the word "truth" in a consistent way to make that very point!); in which case, this 'law' can neither be true nor false itself.


And this is what makes the comments of dialecticians in this area valueless, just as it is also why their ideas collapse so readily into incoherence.


All this shows why an appeal to human experience since the beginning of time, on its own, is irrelevant. Empirical evidence can't be used to attack the LOI without that attack implicitly employing that very same 'law' as a rule of language, or as a practical rule applied to experiments, the use of instruments, Trotsky's own writings, and those of relevant scientists (etc.). But, when the LOI is deployed in this manner in an attempt to expose its alleged empirical limitations (à la Trotsky) -- or to reveal its supposed theoretical short-comings (à la Hegel) --, that attempt itself will self-destruct. For if that 'law' (applied as a rule) is unsafe, then so are the methods used gather any evidence aimed at questioning it, and so are the arguments used to undermine it.


Which is indeed what we have seen.


[More on this in Note 15, below. See also Note 13.]


Naturally, a grudging acceptance of the above (linguistic and socially-sanctioned) conventions -- on the following lines "Ok, so it's a rule, but that doesn't show that objects in the real world obey it; in fact, they don't, they change all the time" -- would have the opposite effect. It would involve dialecticians using criteria that delineate the conditions required for the performance of identical, but real experiments in this world (etc.) in order to confirm, for example, Trotsky's point about a bag of sugar -- which would undermine their own ideal 'thought experiments' aimed at revealing the alleged deficiencies of the LOI.


Of course, it is up to such individuals whether or not they prefer ideal 'thought experiments' to the identically performed, real experiments carried out in this world to test their ideas, which, once they had been performed to the letter, would, in that very act, refute such ill-considered 'thought experiments' - again as e have just seen!


Now, DM-theorists might sincerely believe that all objects constantly change, but that is all this will ever remain: a mere belief, an act of faith. There could be no conceivable body of evidence in favour of this leap of faith that wasn't itself dependent on conventions of measurement, counting and comparing, which can't themselves be subject to Trotsky's (or Hegel's) strictures. And, as we will soon see, that is why both Hegel and Trotsky ended up having to use this 'law' (implicitly) in their futile endeavour to undermine it, and why they both wound up, in practice, refuting their own criticisms of it.


Beyond this, the idea that reality is in the grip of a universal 'Heraclitean Flux' is supported by nothing more than an unfounded extrapolation from a few badly-worded 'thought experiments', themselves based on a laughably superficial understanding of a seriously mis-identified 'law'.


[Having said that, this doesn't commit the present writer to the opposite view that nothing changes! Once again, this is an empirical matter to be settled by evidence, not a priori stipulation or the musings of a confused Christian mystic.]


Yet Another Misidentification


All this, of course, is quite independent of the fact that Trotsky seems to have confused the LOI with something else completely different (no irony intended):


"Every worker knows that it is impossible to make two completely equal objects. In the elaboration of a bearing-brass into cone bearings, a certain deviation is allowed for the cones which should not, however, go beyond certain limits…. By observing the norms of tolerance, the cones are considered as being equal. ('A' is equal to 'A')…. Every individual is a dialectician to some extent or other, in most cases, unconsciously." [Trotsky (1971), pp.65, 106.]


From this it is clear that Trotsky misconstrued his own version of the LOI! If he had wanted to direct our attention to the lack of identity between two different objects (the two "cone bearings" in the above passage) he should have used the following schema:


W1: A is equal to B.


But not:


W2: A is equal to A.


In the quotation above, Trotsky referred to the manufacture of "cone bearings" in his argument against the unrestricted application of his own simplified version of the LOI. Here, he was clearly interpreting the two "A"s in W2 as standing for different (even if somewhat similar) "cone bearings", that is, he was in fact employing W1. Naturally, this throws into serious doubt Trotsky's ability to spot even when something is or isn't an instance of his own garbled version of the LOI!


Some might regard that as unfair. Surely, Trotsky's point was to argue that just as cone bearings look very similar (but are nevertheless distinct), the two "A"s in W2 are equally similar but distinguishable (in some way). So, he was right to use W1.


This objection has some force -- but, fortunately, not much. That is because Trotsky began with the following assertion: 


W3: Every worker knows that it is impossible to make two completely equal objects.


[Trotsky's point about two letter "A"s in "A is equal to A" being non-identical themselves was tackled earlier in this Essay.]


The idea seems to be that workers often (invariably?) realise that the LOI is of limited applicability when they make things. However, even if that were correct, Trotsky's main point would be irrelevant. His avowed target had been the LOI ("A is equal to A", not "A is equal to B"), since he hoped to show that workers in their practical activity implicitly or explicitly reject that 'law', or, at least, that they are aware of its limitations. In order to do this, he advanced the claim that workers in general know that it is impossible to make two objects exactly alike. But, one of his criticisms of the LOI was that all objects change continually and hence they are never equal to themselves. Now, even if we accept Trotsky's version of the LOI, it doesn't refer to two separate objects being the same; in its classical form (and sometimes even in Trotsky's version) it is manifestly about an object's alleged relation to itself.11


If, on the other hand, Trotsky had written:


W4: Every worker knows that it is impossible to make an object completely equal to itself,


the absurdity of what he was saying would have been clear to all. No worker (or anyone else for that matter) would entertain such a crazy idea.


W1: A is equal to B.


However, in W1, Trotsky's point is completely different; there he was arguing that different objects aren't identical, and that workers know this. In this particular case, he wasn't saying that any one specific object isn't self-identical, but that of any two objects, not only can workers see that they aren't the same, they also know they can't make two that are identical. He didn't say that workers are aware that they can't make one object the same as itself. But, that is precisely what Trotsky needed to show, that no worker believes that one object can be made the same as itself -- i.e., that it is impossible to make one that is self-identical. He manifestly failed to do this.


In any case, Trotsky's point (in W3) can't even be derived from his own criticism of the LOI. W3 isn't even a DM-thesis! And, that is quite independent of whether or not workers conclude everything he said they should or would. As seems clear, it isn't relevant to claim that workers are automatic dialecticians because they assent to a banal truth that isn't actually part of DM. It isn't a DM-thesis that two objects are different, only that no object is self-identical. What is wanted here is an example taken from DM that workers could assent to before they were persuaded to do so by a fast-talking Dialectical Missionary. What we actually have is a truism that any card-carrying member of the ruling-class could accept; even George W Bush knows that two apples aren't one apple!


W3: Every worker knows that it is impossible to make two completely equal objects.


Despite this, it could be argued that Trotsky's point is that all workers are aware of change, since they know that the same machine, for example, produces seemingly alike but different objects.


If that is what Trotsky meant then it is certainly unexceptionable, but it isn't what he said. And, even if he had said it, it wouldn't have distinguished a DM-description of reality from one available to anyone using ordinary language, or, indeed, anyone cognizant of 'bourgeois' science. And, we can go further: no sane Capitalist believes that all commodities (or even any two of them) are identical or that things don't change.


[I have covered this topic extensively in Essay Nine Part One; readers are directed there for more details.]


'Norms Of Tolerance' Refute Trotsky, too


But, what about this comment?


"Every worker knows that it is impossible to make two completely equal objects. In the elaboration of a bearing-brass into cone bearings, a certain deviation is allowed for the cones which should not, however, go beyond certain limits…. By observing the norms of tolerance, the cones are considered as being equal. ('A' is equal to 'A')…. Every individual is a dialectician to some extent or other, in most cases, unconsciously." [Ibid., pp.65, 106. Bold emphasis added.]


In the above, it is clear that Trotsky failed to notice that the alleged limitation to which he referred (concerning the making of two identical items) doesn't appear to affect the judgement of whoever is responsible for applying the said "norms of tolerance". According to Trotsky's own description, workers are at least able to determine what constitutes the same application of these norms to different cone bearings. But, that surely means that these hypothetical workers would have to employ a norm that encapsulated the dread LOI in order to apply it equally, or identically, between cases. That is, they would have to know (in practice) what constituted an identical application of these norms each time, since an approximate application of those norms to two very similar cones might very well pass them off as identical!


Hence, in order for a worker to do what Trotsky says, they would have to know precisely what constitutes the correct application of the same norm to at least two different cone bearings. Even if these workers rejected the LOI (which is doubtful), or, indeed, recognised its supposed limitations, they would still have to use a norm expressing that 'law' in order to be able to agree with Trotsky that it fails to apply to cone bearings!


In short, they could only concur with Trotsky after completing a practical refutation of what he declared they all implicitly knew!


Wrong Anyway


Despite this, what Trotsky actually said is patently incorrect. His comments clearly ruled out the possibility that (i) two different objects could become the same, that (ii) a worker could make two distinct objects into one and the same thing, and that (iii) workers know this. In fact, ordinary language and common experience allows for both eventualities -- (i) and (ii) -- about which workers would be well aware, anyway.


Examples of two things becoming one include the following:


(1) Two streams can flow into the same river.


(2) Two items of cloth can be combined in the same garment.


(3) Two cricketers or baseball players can become the same fielder (at the same time in different matches, or at different times in the same match), or two soldiers or union officials could be promoted to the same rank (with similar provisos).


(4) Two scabs could become the same target of the one brick; or two bricks could form part of the same defence against a police attack.


(5) Two workers could form the same small picket in the same or different strikes.


(6) Two copies of The Daily Mail could become the lining of the same pigsty -- but, only after suitable apologies had been extended to the pigs, of course.


Examples of two items being made into one include the following:


(1) Two rivets can be made into the same seal between two plates of metal.


(2) Two buckets of paint can be mixed to form the same colour (i.e., green and red making brown).


(3) Two wooden posts can form the same support in a mine.


(4) Two ropes can form the same towline.


(5) Two plastic pipes can combine into the same outlet.


(6) Two miscounted Widgets can become the same excuse for a strike.


(7) Two sentences can form the same paragraph of the same or different strike leaflets.


(8) Two (or more) of the above can form the same excuse for dialecticians to ignore them.


Of course, if we are no longer restricted to considering only two items then it would be possible to multiply the above examples indefinitely. For instance, one hundred thousand workers could form the same revolutionary column, or two million people could form the same march against the war in Iraq. Or even: two thousand police officers could constitute the same panic-stricken retreat from either of the former.



Figure One: London February 15th 2003 -- Two Million 'Unconscious' Anti-Dialecticians?


It could be objected here that these 'counter-examples' beg the question since, if Trotsky were right about the defects of the LOI, none of the above would be genuine identity statements.


However, as argued earlier, our ordinary use of words for identity (i.e., "the same as", "exact", "similar", "identical", "not different", "precisely", etc.) is highly complex. It is far more sophisticated than Trotsky and Hegel imagined in their 'theoretical' deliberations -- although in their everyday speech they couldn't have been unaware of this fact; they would have used sentences employing terms like the above countless times throughout their lives.


The vernacular -- which, it is worth reminding ourselves yet again, is derived from everyday, material practice -- allows for the expression of all manner of complex identities; the lists given above outline only a few of these. [There are many more of the same here -- no irony intended.]


Anyone who failed recognise these as examples of sameness and identity (etc.) would be deemed not to understand their own language (since they would be incapable of recognising, using or comprehending the same words from that language in the same way as the rest of us); indeed, they might in some circumstances become a danger to themselves. In which case, they would hardly be in a position to criticise the 'law' that supposedly operates behind such words, or between such objects and processes.


Indeed, the employment of such words in contexts like those outlined above tells us more about their meaning than could be gleaned by reading the same comments in Hegel's work an indefinite number of times (irony intended). His extremely narrow, metaphysical use of less than a handful of our words for identity and change shares nothing with their ordinary employment; as such, his theses are not only devoid of sense, they are incoherent. [Why that is so is explained in Essay Twelve Part One.]


Some might object that many of the above examples fall foul of the point Heraclitus made about constant change -- for example, that it is impossible to step into the same river twice:


"It is not possible to step twice into the same river according to Heraclitus, or to come into contact twice with a mortal being in the same state. (Plutarch)" [Quoted from here.]


It is in fact impossible to make sense of this odd statement. Had Heraclitus said that it wasn't possible to step into the same body of flowing water twice, he might have had a point. Even so, it is quite easy to step into the same river, and many times, too! Indeed, without that capacity, not even Heraclitus could test his own 'theory' (or even imagine such a test in his 'mind's eye'), for he wouldn't be able to recognise anything as "the same river" to test it on, even in theory!


In fact Heraclitus got into a terrible mess over the criteria of identity for mass nouns and count nouns. [I have dealt with these terms elsewhere in this Essay.]


Consider the following conversation:


R1: NN -- "I swam in the Thames this morning, and in a different river in the afternoon."


R2: NM -- "In which river did you swim in the afternoon, then?"


R3: NN -- "The Thames!"


R4: NM -- "But, I thought you said you had swum in a different river in the afternoon!"


R5: NN -- "Ah, but according to Heraclitus, it isn't possible to swim in the same river twice."


R6: NM -- "In that case, you have just refuted him, since you swam in the Thames twice!"


R7: NN -- "Not at all. The water I swam in was different in the afternoon."


R8: NM -- "But, it was still H2O, wasn't it?"


R9: NN -- "I don't get your point."


R10: NM -- "You said the water was different in the afternoon. In that case, was it chemically different -- not perhaps H2O? If it wasn't H2O, you didn't swim in water the second time! Perhaps you meant the water was different in some other way?"


R11: NN -- "The water molecules were the same, obviously, but it was just a different body of water molecules in the afternoon."


R12: NM -- "How does that make it a different river? Surely our concept of a river involves the fact that it flows. If it didn't, it wouldn't even be a river! So, just because a different body of water molecules flows along it every day, doesn't mean it's a different river."


R13: NN -- "I see your point. Perhaps Heraclitus meant that!"


R14: NM -- "If so, it is possible to step into the same river twice, or, indeed, a hundred times."


As noted above, Heraclitus had clearly confused criteria of identity we have for count nouns (same river, same molecule, etc.) with those we have for mass nouns (same water).


So, to return to one of the examples given earlier (slightly modified):


(1a) Two streams can flow into the Thames.


Even if the water molecules were different, or the body of water were changing all the time, (1a) refers to the same river -- the Thames -- not some other river -- say, the Severn.


In which case, the point Heraclitus wished to make (if any sense can be made of it, that is) doesn't count against the perfectly ordinary examples given above. If we take account of the vernacular and our ordinary ways of speaking -- and ignore the confused things Idealist Philosophers (like Heraclitus and Hegel) have to say --, it is quite clear that two or more things can be, or can become, or can be made to become, identical.


Taking account of ordinary ways of speaking is, after all, what Marx himself enjoined upon us:


"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 alone added.]


[I will return to the theme of constant change in Essay Eleven Part One, where I aim to show that this particular doctrine (in general) makes no sense, and not just when it is applied to rivers or bags of sugar. Readers are directed there for more details.]


It could be objected that none of us use our words in the same way, so the above observations are misguided. However, that objection itself is (at least) based on that very objector using the words I have employed in the 'offending' paragraphs above in the same way as I have just done. If that objector's words aren't being used in the same way, then that objection itself is misguided since it would be aimed at the wrong target; manifestly they wouldn't be aimed at what I had said, but at something else (the nature of which would be inaccessible to anyone, and probably not even the objector her/himself). And the same can be said of any other general attempt to advance the same objection -- since it can't be the same objection if the words it contains are employed differently.


Of course, this isn't to deny that we sometimes use words in different ways (or that the meaning of words change over time), but this can't be typical otherwise communication would be impossible. Moreover, anyone who rejected this simple point would, plainly, fail to get their point across, having used words differently! Alternatively, if they managed to get their point across, they would have used their words in the same way as the rest of us, undermining the very point the wanted to make!


If, on the other hand, these examples fail to tell us what our words for identity (etc.) mean -- if they are defective or inadequate in some way -- then even those who criticise their use must fail to grasp what they themselves are criticising (i.e., the ordinary use of the word "identity" and its cognates), since they won't be able to put into words what constitutes the same use either of that word or its associated terms. [The reasons for saying this are explained in more detail in Note 19.]


As this Essay will show, it is in fact impossible to decide what (if anything) Trotsky actually meant by his criticism of the LOI. All this suggests that the above examples represent a far more legitimate use of words for identity than the seriously limited selection employed by Hegel, Trotsky and his latter-day clones. Hence, as far as ordinary language is concerned, it is easy to speak about making two or more things exactly the same -- which is all that us non-Idealists need.


It is certainly all that workers need.


Identical A Priori Tactics


In Essay Five we saw how Engels had irresponsibly extrapolated from a sketchy thought experiment about moving bodies -– 'complemented' by an idiosyncratic understanding of ordinary words like "move", "place" and "contradiction" -- to universal truths supposedly valid everywhere for all of time. Here, we see Trotsky doing something similar based on his idiosyncratic interpretation of a seriously restricted set of words for identity -- in fact only "equal"! From this he also attempted to derive several substantive theses about every object and process in existence, also valid for all of space and time, deriving Super-scientific truths from a superficial and misguided 'conceptual analysis' of what he assumed were the meanings of words like "identical", "change", "equality", "time", "moment", and "measure". Just like Engels, he based these universal conclusions on an alarmingly narrow evidential base -- a set of words none of which turned out to be about identity --, 'supported' by a 'thought experiment' about bags of sugar, which (as will soon become clear) ends up undermining his own ambitious conclusions !


And this is supposed to be cutting-edge philosophy?


yet More A Priori Super-Science


Even when the serious difficulties that have already been considered are put to one side, it is clear that Trotsky's analysis is deeply flawed for other reasons. This can be seen if consideration is give to the rejoinder Trotsky himself advanced (in S9) to a hypothetical objection (recorded in S8):


S8: A pound of sugar is equal to itself.


S9(a): All bodies change uninterruptedly. (b) They are never equal to themselves.


Unfortunately, Trotsky failed to say how he knew that both halves of S9 were true. In fact, only if he were a semi-divine being could he possibly know that all bodies are never equal to themselves. Even if he had access to every single observation humans beings have made of bodies in recorded history, that would still fail to substantiate the hyper-bold claim recorded in S9 since those observations only amount to a vanishingly small fraction of all the bodies there are, have ever been, or will ever be. Neither could what he asserted have been based on scientific evidence because that body of data is equivocal, at best. For example, since the 'quantum revolution', it is now thought that several sub-atomic particles are equal to themselves for unimaginably long periods of time, and possibly forever. Protons, for instance, have an estimated half-life in excess of 1033 years, which is approximately 1020 times longer than the currently accepted age of the Universe. During that time they don't change (as far as we know), and as such they are surely equal to themselves.12 Indeed, proton decay has yet to be observed, so it might turn out that protons do not change at all, ever. And that isn't the only example. [On this, see the next subsection]


Physicists Discover Identical Objects


[This used to form part of Note 11.]


Recall what Trotsky asserted:


"Every worker knows that it is impossible to make two completely equal objects. In the elaboration of a bearing-brass into cone bearings, a certain deviation is allowed for the cones which should not, however, go beyond certain limits…. By observing the norms of tolerance, the cones are considered as being equal. ('A' is equal to 'A')…. Every individual is a dialectician to some extent or other, in most cases, unconsciously." [Trotsky (1971), pp.65, 106.]


However, contrary to what Trotsky said, it is very easy to make two identical objects -- indeed, every single one of us does so when we throw a light switch. Here is some material devoted to this topic taken from another Essay posted at this site:


Physicists tell us that every photon, for example, is identical to every other photon. This how Steven French puts things:


"It should be emphasised, first of all, that quantal particles are indistinguishable in a much stronger sense than classical particles. It is not just that two or more electrons, say, possess all intrinsic properties in common but that -- on the standard understanding -- no measurement whatsoever could in principle determine which one is which. If the non-intrinsic, state-dependent properties are identified with all the monadic or relational properties which can be expressed in terms of physical magnitudes associated with self-adjoint operators that can be defined for the particles, then it can be shown that two bosons or two fermions in a joint symmetric or anti-symmetric state respectively have the same monadic properties and the same relational properties one to another. [French and Redhead (1988); see also Butterfield (1993).] This has immediate implications for Leibniz's Principle of the Identity of Indiscernibles which, expressed crudely, insists that two things which are indiscernible, must be, in fact, identical." [French (2019). {This is in fact a quote from an earlier version of French (2019).} Bold emphases and links added. Referencing altered to conform with the conventions adopted at this site.]


The above passage has now been replaced by the following in the new, 2019, edition of this article:


"Now, of course, both quantum and classical objects of the same kind -- such as electrons, say -- are indistinguishable in the sense that they possess all intrinsic properties -- charge, spin, rest mass etc. -- in common. However, quantum objects are indistinguishable in a much stronger sense in that it is not just that two or more electrons possess the same intrinsic properties but that -- on the standard understanding -- no measurement whatsoever could in principle determine which one is which. If the non-intrinsic, state-dependent properties are identified with all the monadic or relational properties which can be expressed in terms of physical magnitudes standardly associated with self-adjoint operators that can be defined for the objects, then it can be shown that two bosons or two fermions in a joint symmetric or anti-symmetric state respectively have the same monadic properties and the same relational properties one to another (French and Redhead 1988; see also Butterfield 1993). This has immediate implications for the Principle of the Identity of Indiscernibles which, expressed crudely, insists that two things which are indiscernible, must be, in fact, identical." [Ibid. Bold added.]


Of course, French offers his own solution to this difficulty, but it isn't one that challenges the identity of quantal particles, just their lack of individuality. And, Nobel Laureate, Paul Dirac, made a similar point this way:


"If a system in atomic physics contains a number of particles of the same kind, e.g., a number of electrons, the particles are absolutely indistinguishable. No observable change is made when two of them are interchanged…." [Dirac (1967), p.307.]


However, one might wonder how anyone could possibly know two particles had been interchanged if they are indistinguishable. On the other hand, Pure Mathematician that he was, Dirac might merely be making a theoretical point on a par with the following: "If we swap one number in this equation for another (identical) number, no change will be observed: 2 + 3 = 5". We can see this perhaps more clearly with this example: "Two plus Three equals Five" is mathematically indistinguishable from "2 + 3 = 5" even though "2" and "Two", for instance, are plainly different.


In that case, every time a worker turns on a light, he or she makes or generates countless trillion identical objects per second -- which must mean that they are "unconscious" anti-dialecticians, if we apply the same sort of reasoning here as Trotsky.


Naturally, contentious claims like these can only be neutralised by an a priori stipulation that every photon in existence (past, present and future) must be non-identical -- despite what scientists tell us and in abeyance of the impossibly large (finite) amount of data that would be needed to support such a cosmically ambitious claim. At this point, perhaps, even hardnosed dialecticians might be able to see in this a blatant attempt to impose DM on reality.


Some might want to argue that photons don't occupy the same spatio-temporal co-ordinates, and so can't be absolutely identical, but this certainly isn't how Trotsky (or even Hegel) argued. When confronted with two letters "A"s on the page, Trotsky didn't ague that they occupied different spatio-temporal co-ordinates and hence aren't identical, he appealed to their assumed physical differences. But, photons are physically indistinguishable according to the above physicists.


Independently of this, photons refute Trotsky's claim that we can't make two identical objects (which must be located at spatio-temporal co-ordinates), for here we can.


Despite this, hardcore DM-fans might still want to argue that the above spatio-temporal objection means that photons aren't identical, but this objection is based on a certain definition of identity forced on nature in defiance of the claim that DM-supporters never do this.


[A recent discussion of these issues can be found in Brading and Castellani (2003), and Castellani (1998). An even more recent discussion can be found in Saunders (2006) (this links to a PDF), and particularly French and Krause (2006). See also Hilborn and Yuca (2002), Ladyman and Bigaj (2010), and the Wikipedia entry here.]


It could further be objected that Trotsky would surely have been unaware of developments in Physics that took place after he died, but, as the references given above show, these facts were largely true of classical particles, too; quantal particles merely present a more extreme form of strict identity. And, Lenin it was who reminded us that science is ever revisable; hence, no dialectician who agrees with Lenin could consistently rule out the possibility that scientists would one day discover identical particles -- as indeed they have.


Even so, Trotsky was quite happy to impose this view on nature before all (or most of) the evidence had been collected, in defiance of what he said elsewhere:


"The dialectic does not liberate the investigator from painstaking study of the facts, quite the contrary: it requires it." [Trotsky (1986), p.92. Bold emphasis added]


"Dialectics and materialism are the basic elements in the Marxist cognition of the world. But this does not mean at all that they can be applied to any sphere of knowledge, like an ever ready master key. Dialectics cannot be imposed on facts; it has to be deduced from facts, from their nature and development…." [Trotsky (1973), p.233. Bold emphasis added.]


And the facts tell us that photons are identical and in all likelihood, changeless.


I now turn to that topic, changeless identical sub-atomic particles.


Changeless Sub-Atomic Particles


[This used to form part of Note 12.]


The following material was published in another Essay at this site:


In fact, the half-life of a proton is reckoned to be in excess of 1033 years -- i.e., one followed by thirty-three zeros! Estimates vary, but this is approximately 1020 (i.e., one followed by twenty zeros) times longer than the age of the known universe, if current theory is correct. Experimental evidence suggests its half-life is probably longer even than this. Predicted proton decay hasn't actually been observed yet. Apparently, electrons are even less 'dialectical'. In that case, there could in fact be more changeless objects in nature than there are changeable. The point is, of course, that this is an empirical matter, not -- as Engels, Plekhanov, Lenin, Trotsky and Mao seem to have thought -- an a priori truth based on the musings of an Idealist who died over 2500 years ago (i.e., Heraclitus).


As far as protons are concerned, we are told the following:


"Along with neutrons, protons make up the nucleus, held together by the strong force. The proton is a baryon and is considered to be composed of two up quarks and one down quark.


"It has long been considered to be a stable particle, but recent developments of grand unification models have suggested that it might decay with a half-life of about 1032 years. Experiments are underway to see if such decays can be detected. Decay of the proton would violate the conservation of baryon number, and in doing so would be the only known process in nature which does so." [Quoted from here.]


Wikipedia adds:


"In particle physics, proton decay is a hypothetical form of radioactive decay in which the proton decays into lighter subatomic particles, usually a neutral pion and a positron. Proton decay has not been observed. There is currently no evidence that proton decay occurs." [Accessed 29/07/2015.]


One Professor of theoretical physicist had this to say:


"The only known stable particles in nature are the electron (and anti-electron), the lightest of the three types of neutrinos (and its anti-particle), and the photon and (presumed) graviton (which are their own anti-particles). The presumed graviton, too, is stable. The other neutrinos, the proton, and many atomic nuclei (and their anti-particles.... I'm going to stop mentioning the anti-stuff, it goes without saying) are probably not stable but are very, very, very long-lived. Protons, for instance, are so long-lived that at most a minuscule fraction of them have decayed since the Big Bang, so for all practical purposes they are probably stable. The other rather long-lived particle is the neutron, which when on its own, outside an atomic nucleus, lives just 15 minutes or so. But neutrons inside many atomic nuclei can live far longer than the age of the universe; such nuclei provide them with a stable home. Finally, I should add that if dark matter is made from particles, then those particles, too, must be stable or very, very long-lived.


"Why are these particles stable? It turns out that our world imposes some rules on particle behaviour, ones not visible to us in the physics of waves and vibrations that we encounter in daily life, that prevent some particles from decaying, either rapidly or at all. The fundamental rules are 'conservation laws', statements that certain quantities in the universe never change in any physical process. (These quantities include energy, momentum, electric charge, and a few others.) There are also some approximate conservation laws, stating that certain quantities only change very rarely. Conservation laws do not appear from nowhere, imposed out of thin air by theorists; they are related to other properties of the world. For example, if the laws of nature do not change over time, then it follows (thanks to a theorem of the mathematician Emmy Noether) that energy is conserved. Meanwhile, the stability of the matter out of which we are made provides strong tests of these conservation laws, as we'll see.


"Combining these laws with the properties of particles leads to a set of simple rules that determine when particles simply cannot decay, or when they can at most decay very rarely. And these rules are (almost) entirely sufficient to explain the stability of the particles out of which we are made, and those that we interact with most often." [Matthew Strassler, quoted from here; accessed 19/02/2014. Quotation marks altered to conform with the conventions adopted at the site. Spelling adjusted to agree with UK English; several links added. Italic emphases in the original. The professor then proceeds to spell out those rules, which I have omitted.]


Another Physics site also added this comment (in October 2014):


"Protons live a long time but perhaps not forever. Several theories predict that protons can decay, and a handful of experiments have tried to detect such an event. The Super-Kamiokande experiment in Japan has the longest track record in the search for proton decay, and its researchers have now published a new lower bound on the proton lifetime that is 2.5 times greater than their previous bound. The proton's observed stability places constraints on certain extensions of the standard model of particle physics.


"Proton decay is an expected outcome of most grand unified theories, or GUTs, which meld together the three main particle forces -- strong, weak, and electromagnetism -- at high energy. A certain class of GUTs, for example, predicts that a proton should decay into a positron and π meson with a lifetime of about 1031 years, which means roughly 1 decay per year in a sample of 1031 protons. Experiments have already ruled this possibility out.


"Other GUTs that incorporate supersymmetry (SUSY), a hypothetical model that assumes all particles have a partner with different spin, predict that the proton decays into a K meson and a neutrino with a lifetime of less than a few times 1034 years. The Super-Kamiokande collaboration has looked for signs of this decay in a 50,000-ton tank of water surrounded by detectors. If one of the many protons in the tank were to decay, the K meson's decay products (muons, π mesons) would be detectable. The researchers simulated such proton decay events but found no matches in data spanning 17 years. From this, they conclude that the proton lifetime for this SUSY-inspired decay pathway is greater than 5.9 x 1033 years." [Quoted from here, accessed 23/08/2015; bold emphasis and links added.]


Update, 09/08/21: The following report, from Nature Reviews Physics, is dated January 8, 2021:


"Does the proton decay? The proton could decay into a positron and a pion and if so the predictions of some beyond standard model grand unified theories would be confirmed. Since the 1980s underground experiments have been looking for signs of such decay. No proton decay has yet been spotted, but one of the experiments, the Kamioka Nucleon Decay Experiment...detected neutrinos from the supernova SN1987a instead. Its successor, the Super-Kamiokande, set the best current lower-limit for proton lifetime: 1.6 x 1034 years. Some grand unified theories predict proton lifetimes up to around 1036 years so there is still room for surprises. Upcoming neutrino experiments also have proton decay measurements planned and are expected to push the limits on the proton lifetime. These measurements will take place at Deep Underground Neutrino Experiment (DUNE) currently under construction [at the] Jiangmen Underground Neutrino Observatory (JUNO) which is almost completed and Hyper-Kamiokande whose construction received the green light last year." [Quoted from here; accessed 09/08/2021. One link and bold emphasis added.]


Much of the universe, it seems, doesn't 'understand' dialectics, either.


Even Peter Mason, an otherwise enthusiastic champion of DM, had to admit the following (while criticising fellow Trotskyists, Woods and Grant):


"Although dialectics certainly suggests that science will find a time limit beyond which protons will decay in some way, and teams of scientists are testing to find that limit -- nevertheless the proton is stable over very long periods. A twelve-year experiment, started in 1989, suggested that the proton has a lifetime of at least ten million billion billion billion years (1034 years -- a one with 34 zeros after it). It does not ceaselessly change, as Woods asserts." [Mason (2012), p.14.]


Of course, it could be objected that particles such as protons (i.e., hadrons) are composed of even more fundamental particles, which do enjoy a contradictory life of their own 'inside' each host 'particle'; their interactions would therefore mean that apparently changeless protons are in fact changing 'internally' all the time. But, this response simply pushes the problem further back, for these other, more fundamental particles (i.e., quarks --, in the case of protons, two "up" and one "down" quark), are themselves changeless, as far as is known. Moreover, these quarks certainly have no 'internal contradictions' of their own to worry about. Moreover, since protons are baryons -- i.e., they are composed of three quarks --, it isn't easy to see how their inner lives are in any way 'contradictory' (with three terms?). That is quite apart from the additional fact that DM-theorists have yet to establish that every single proton has a ceaselessly changing internal life of its own. Have they looked inside every single one?


Even more difficult to account for dialectically are electrons and photons (which are leptons and gauge bosons respectively), since they have no known internal structure (but see below). Unless acted upon externally, their 'lifespan' is, so we are told, infinite; hence, if they change, it isn't because of any 'internal contradictions'. [Although there are reports that electrons have been, or can, split.]


An appeal to antiquarks here, to save the dialectical day, would be to no avail, either. That is because quarks don't struggle with and then turn into antiquarks, nor vice versa, which is what the Dialectical Holy Books tell us should happen to all such 'opposites'. [On that, see here.] When they interact they annihilate one another. Furthermore, DM-theorists equivocate over the meaning of "internal". Sometimes they mean by this word, "logically internal", while at others they mean, "spatially internal". The latter interpretation would be fatal to DM (on that see here); the former is far too confused to make much sense of, anyway.


Someone might object that these particles are always moving, and hence changing. Again, the evidence that every single particle in the universe -- past, present and future -- is/was/will always be moving has yet to be produced. But, even if they are/were/will be always moving, that would simply mean that the distance between particles was changing, not the particles themselves.


It seems, therefore, that the picture of reality painted by dialecticians is more Jackson Pollock than it is Van Eyck.



Figure Two: The 'DM-View' Of Nature



Figure Three: Genuine Science 'Pictures' The World Rather More Precisely


[On protons, see here and here; on electrons, here; leptons, here; photons, here.]


Moreover, every electron is identical with every other electron, and the same applies to other elementary particles (such as photons).


[On this topic in general, cf., Perkins (2000), French (2019), French and Krause (2006) -- but more specifically Saunders (2006) (this links to a PDF). See also Dieks and Versteegh (2007/2008), Ladyman and Bigaj (2010), and Caulton and Butterfield (2012), and above.]


In fact there appear to be two schools at work here; those who hold that all such particles are identical and indiscernible (rather like the dollars or pounds in your bank account, not the dollars or pounds in your wallet or pocket), and those who claim they are identical but discernible. [On this, see the above references as well as Muller and Seevinick (2009), and Muller and Saunders (2008).]


"Discernible" appears to mean different things to different philosophers, too (no pun intended). In practice it seems to imply that while such objects are identical, nevertheless they aren't!


Naturally, dialecticians might want to object to the above on the lines that electrons, for example, aren't really particles --, or that they are probability waves or perturbations in the field, or that they are this or they are that. Perhaps so, but, once again, whatever they are, they are identical with that, and they change equally quickly as they themselves do. And, if they change, that can't be because of their 'internal contradictions' -- which DM-fans have yet to show they have.


[This comment puts paid to much of the confused ruminations on sub-atomic 'particles' found in, for example, Woods and Grant (1995/2007). More details on this will be posted in Essay Seven Part Two at a later date. On change through 'internal contradiction', see Essays Seven Part Three and Eight Parts One, Two and Three.]


Of course, the above considerations will only offend those who, for some odd reason, might prefer to foist dialectics on nature.


But, who on earth would want to do that?


Finally, it could be argued that since the relations between particles are always altering, even if certain particles are seemingly changeless, those particles will be changing all the time as a result.


This issue is discussed in more detail in Essays Six, Seven, Eight Part One, and Eleven Part Two. Suffice it to say here that:


(a) If this contention were correct, most of the elementary objects in the universe wouldn't be self-developing, but would be affected by external causes. In that case, this particular idea would support one strand of DM by torpedoing another. Indeed, it would also introduce into nature a "bad infinity" (as Hegel would have called it) as these external causes stretch off into the blue beyond. In that case, and once more, yet another a priori dialectical thesis will have been holed below the waterline. [On Hegel and the infinite, see Houlgate (2006).]


(b), there is no evidence that every particle in nature has an effect on every other, which supposedly changes it, or them. [So-called "Quantum Entanglement" is discussed here.]




(c) As noted earlier, even if this were correct, it would simply mean that the relations between these particles changed not the particles themselves.


If, on the other hand, change is defined in such a way that an alteration to the relations between objects also counted as a change to those objects themselves (these are often called "internal relations" by assorted Idealists and DM-fans alike) then, once more, that re-definition would be no different from an imposition onto nature of something that it might not have. If dialecticians have any evidence that there are indeed such "internal relations" in nature and society (or which affect everything in the universe in this way), then they need to produce it, or resist the temptation to keep advancing such claims.


Of course, there are DM-theorists (mainly of the HCD persuasion) who employ a handful of 'arguments' culled from the aforementioned Idealists, which are aimed at showing that such relations do indeed exist between objects and processes; but they would, wouldn't they? They are Idealists. They prefer 'conceptual arguments' over evidential proof any day of the week -- as George Novack pointed out. Indeed, the more they try to defend "internal relations" with a priori arguments like this, the more they confirm Novack's allegations.


Plainly, such bogus reasoning wouldn't be needed if DM-fans had any scientific evidence to back up these ancient, mystical dogmas.


It is, of course, a Hermetic doctrine that everything is interconnected, and is a union of opposites, as Magee noted:


"Another parallel between Hermeticism and Hegel is the doctrine of internal relations. For the Hermeticists, the cosmos is not a loosely connected, or to use Hegelian language, externally related set of particulars. Rather, everything in the cosmos is internally related, bound up with everything else.... This principle is most clearly expressed in the so-called Emerald Tablet of Hermes Trismegistus, which begins with the famous lines "As above, so below." This maxim became the central tenet of Western occultism, for it laid the basis for a doctrine of the unity of the cosmos through sympathies and correspondences between its various levels. The most important implication of this doctrine is the idea that man is the microcosm, in which the whole of the macrocosm is reflected.


"...The universe is an internally related whole pervaded by cosmic energies." [Magee (2008), p.13. More on this here, and here.]


Bertell Ollman's recent book is just the latest example of this mystical genre -- Ollman (2003). Ollman's work has been analysed in more detail in Essay Three Part Two.


["Internal relations" will be destructively criticised in Essay Four Part Two, when it is published.]


Moreover, we have already seen that ordinary language itself contains countless expression (simple and complex) that allow speakers to talk about objects and processes that not only can but do remain identical, sometimes over many years. Hence, neither human experience nor scientific theory agrees with Trotsky's hyper-ambitious claims.


An Everyday Example Of Absolute Identity


Consider the following scenario: two individuals, NN and NM, board the same train at the same time and remain on it for two hours. While these two are still on that train, whatever changes it experiences or undergoes as it speeds along (and this includes changes to all its contents and its relations with the entire universe in this two hours), this will always be true: NN and NM are on an absolutely identical train as each other.


More precisely:


A1: During temporal interval, T -- lasting m hours --, a set of predicates, P, is true of, or can be used to form true propositions about, (i) Train, L, (ii) All on board, and (iii) L and the relation its contents have with everything in the universe -- where P is comprised of the following elements: {P1, P2, P3,..., Pn}, and where n is indefinitely large (m ε ).


A2: NN and NM are both passengers on L during any sub-interval, tk, of T, of arbitrary length.


A3: During tk, a subset of P, namely {Pi..., Pk}, can be used to form true propositions about (i) L, (ii) All on board, and (iii) L and the relations its contents have with everything in the universe, including any changes that occur to one or both, during tk.


A4: So, during tk, NN and NM are on absolutely identically the same train as one another, since, while they are on L, the valid applicability of every element of {Pi,..., Pk} remains the case throughout.


Now, every element of {Pi,..., Pk} -- i.e., these predicates, or what they 'reflect' -- can change 'dialectically' all the time, but that won't alter the result, since any changed element of {Pi,..., Pk} will also be an element of that sub-set, by the above definitions.


So, here we have an example of absolute identity that would remain such even in a Heraclitean universe.


[I have tried to use DM-phraseology here wherever possible (e.g., "reflect", "dialectically", as well as "L and the relation its contents have with everything in the universe") so that this argument doesn't become needlessly controversial by the employment of too much non-DM terminology. Its use, however, doesn't imply I accept its validity; it is merely being employed in order to assist in its long-overdue demise.]


Finally, what applies to passengers on trains applies in like manner (irony intended) to those in or on ships, boats, aeroplanes, spacecraft, buses, taxis, rickshaws, bikes, cars and skateboards at the same time. Indeed, it applies in equal measure (more irony intended) to individuals in the same cinema, theatre, café, restaurant, hotel, hostel, house, office, hospital, waiting room, stadium, palace, mansion, igloo, cave, tree house, park, and graveyard, at the same time -- just as it applies to all of us on the same planet, at the same time. And if we go for broke, it applies to everything in the universe at the same time.


[I have added another detailed example of this sort of identity, concerning two climbers on the same mountain at the same time, below.]


So, far from being a rather rare occurrence (or even an impossibility according to DM-fans) absolute identity like this is as common as sand in the Sahara.


Of course, it could be objected that this sort of identity isn't the sort that interests dialecticians, so the above comments aren't just beside the point, they are irrelevant. But, as we will see as this Essay unfolds, we don't yet know what DM-theorists mean by the odd things they say about identity, so not even they may consistently advance that objection this side of saying for the very first time, with any clarity, what they do mean.


One thing is clear, neither Trotsky's nor Hegel's criticisms of the LOI are successful against the above sort of identity.


Using The LOI To Criticize The LOI


Same 'Moment'


However, even if Trotsky were right, and everything in the entire universe changed all the time, it would still be unclear what he was trying to say.


"But, one can object, the question is not the size or the form of the letters, since they are only symbols for equal quantities, for instance, a pound of sugar. The objection is beside the point; in reality a pound of sugar is never equal to a pound of sugar -– a more delicate scale always discloses a difference. Again one can object: but a pound of sugar is equal to itself. Neither is true (sic) -– all bodies change uninterruptedly in size, weight, colour etc. They are never equal to themselves. A sophist will respond that a pound of sugar is equal to itself at 'any given moment'…. 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." [Trotsky (1971), p.64. Bold emphasis added.]


For instance, it is far from certain what he had in mind when he asserted S9(b) (repeated below). Consider the following interpretations of S8 as possible targets of his:


S8: A pound of sugar is equal to itself.


S10: Let A1 be a pound of sugar at time, T1.


S11: Let A2 be a pound of sugar at time, T2. [T2 > T1.]


S12: S8 means A1 is equal to A1.


S13: S8 means A1 is equal to A2.


S1: A is equal to A.


S9(b): [All bodies] are never equal to themselves.


At first sight, it seems that Trotsky might have had S13 in his sights when he asserted S9(b), since it compares a pound bag of sugar with itself as it changes over time, which is perhaps the normal way of regarding change. But, S13 doesn't even look like a classical formulation of the LOI; nor does it look like Trotsky's own simplistic version (recorded in S1), either. It more closely resembles a quasi-empirical claim about the temporal continuity of material bodies. Clearly, if Trotsky had wanted to use S9(b) to refute S13, then S12 (which is surely a more likely target) would have been left unscathed. This suggests that S13 wasn't the interpretation that Trotsky had in mind. He must have read S8 as equivalent to (i.e., identical with -- note the irony here!) S12, which he plainly thought was refuted by S9(b):13


S8: A pound of sugar is equal to itself.


S12: S8 means A1 is equal to A1.


S9(b): [All bodies] are never equal to themselves.


If so, it is quite clear that Trotsky had to assume the truth of the LOI in order to declare it false; that is, he had to assume the LOI was reliable in order to try to show it was unreliable. Clearly, that means this 'demonstration' is radically defective, since Trotsky's 'analysis' only succeeded in undermining itself.


To see this more clearly it is worth making S9(b) a little more precise, perhaps along the following lines:


S14: For any object, A, at any time, t, A at t is not equal to A at t.


[I have avoided the construction: "For all objects, A, at any time, t, A at t is not equal to A at t" since it is a stylistic monstrosity. Not much hangs on the difference.]


Given the above caveat, S14 expresses the content of S9(b) a little more clearly; indeed, Trotsky himself employed a tensed ordinary language quantifier expression in S9(b) -- viz., "never".13a


"Again one can object: but a pound of sugar is equal to itself. Neither is true (sic) -– all bodies change uninterruptedly in size, weight, colour etc. They are never equal to themselves." [Ibid. Bold emphasis added.]


Unfortunately, this change of emphasis introduced a serious problem Trotsky failed to notice. This can be seen if we refer back to S1, S9 and S14:


S1: A is equal to A.


S9(a): All bodies change uninterruptedly. (b) They are never equal to themselves.


S14: For any object, A, at any time, t, A at t is not equal to A at t.


S9(b): [All bodies] are never equal to themselves.


Clearly, S9(b) -- when interpreted along the lines suggested by S14 -- implies that S1 must be rejected because:


S15: It is never true that A is equal to A.


However, S15 appears to imply the following:


S16: For any time, t, and any A, A at t is not equal to A at t.


But, this now transfers the emphasis onto the temporal aspects of identity, which underlines the points Trotsky himself tried to make about time and change:


"Again one can object: but a pound of sugar is equal to itself. Neither is true (sic) -– all bodies change uninterruptedly in size, weight, colour etc. They are never equal to themselves. A sophist will respond that a pound of sugar is equal to itself at 'any given moment'…. 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 emphases added.]14


It is here that Trotsky unwittingly introduced the serious difficulty alluded to above, one that is now connected with the identity of temporal instants. This is highlighted by his use of the phrase "that 'moment'" (which clearly means "that same moment", no matter how short it is), during which an object supposedly changes. But, in referring to time this way, the phrase "that moment" suggests that we can make sense of the following schematic sentence:


S17: For some instant, t, t is identical to t.


[S17 is just a more formal paraphrase of "that [same] 'moment'".]


If Trotsky did in fact rely on S17, it would be fatal to his argument since it claims that the instant in question is self-identical and hence subject to the LOI!


That being so, Trotsky's argument clearly requires something to remain the same (i.e., an instant in time during which an object changes) so that his objection to the LOI can gain some purchase. The serious problems Trotsky's analysis now faces become a little clearer just as soon it is realized that the following sentences follow from S16:


S16: For any time, t, and any A, A at t is not equal to A at t.


S18: There is a time, t1, and an A, such that A at t1 is not equal to A at t1.


S19: At one and the same instant, A is not equal to A.


Now, the phrase "instant in time" -- represented in S18 by the use of "t1" -- is just as legitimate a substitution instance as the phrase "pound bag of sugar" for the "A"s in S1. This can be seen if the following are compared (and if we wave for the time being the fact that "identical" isn't identical to "equal"):


S20: t1 is equal to t1.


S1: A is equal to A.


S17: For some t, t is identical to t.


[S20 of course is just a less specific version of S17, but it is nevertheless a legitimate version of S1.]


This means that Trotsky requires S20 to be true so that he can reject the LOI as false! As noted above, this implies that he needs the LOI to apply in an unrestricted sense to certain things (i.e., temporal instants) so that he can deny it of others (i.e., pound bags of sugar, or even distinguishable letter "A"s).


It could be objected here that Trotsky was merely criticising the LOI as it applies to objects and processes that change in time; his argument was certainly not about "temporal instants", which are abstractions anyway. Indeed, he was simply pointing out that no matter how narrow the time frame, change still takes place. That is abundantly clear from what he said:


"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.]


Or so a response might proceed.


Nevertheless, if "that moment" is interpreted exactly as Trotsky indicates, he still has to be able to refer to that very same moment during which the object in question changes; if "that moment" weren't the same, then his argument would plainly fall flat, for the assumed change in question would have taken place in a different moment. Clearly, no one would object to that.


Moreover the counter-claim that moments in time are abstractions won't work either; if bags of sugar can be weighed and found to vary, moments in time can be measured and shown to differ, too -- if Trotsky's other assertions about measuring things are taken as read. Temporal intervals are just as measurable as the weight of bags of sugar. It isn't easy to see how Trotsky could consistently criticize the LOI on the basis of the hypothetically differing weight of these bags without this spilling over into a general criticism of anything at all that is capable of being measured, such as time. If time is to be exempted from his critique of measurement, why not weight?14a


It seems, therefore, that Trotsky's argument relies on these measurable intervals staying the same while those measurable bags of sugar don't!


In fact, no matter how short the interval within which a given change is supposed to occur, Trotsky has to be able to refer to identically the same one to make his case. Without this his whole analysis collapses. So, whether or not these "moments" are extensionless temporal points, or extended intervals in time, Trotsky still has to employ the LOI so that he can assert that at least one of these is the very same moment during in which the assumed changes occur.


A Turn To The Concrete


However, even if the above serious problems were put to one side, and it is conceded that reference to identical temporal instants is in the end an unfair criticism of Trotsky, and S14 is examined without the quantifier switch (this links to a PDF) expressed in S16, the same conclusions follow (no irony intended).


Consider this putative substitution instance of S14 expressed in S21(a):


S14: For any object, A, at any time, t, A at t is not equal to A at t.


S21(a): There is an A and a time, t1, such that A at t1 is not equal to A at t2.


S21(b): There is an A and a time, t1, such that A at t1 is not equal to A at t1.


[S16: For any time, t, and any A, A at t is not equal to A at t.]


As the reader will no doubt have noticed, S21(a) is in fact an illegitimate substitution instance of S14 because the variable letters "t1" and "t2" aren't identical, which violates certain conventions governing the interpretation of bound variables in quantified contexts (in MFL).


[MFL = Modern Formal Logic.]


Now, Trotsky has to make use of something like S21(b) -- or its colloquial equivalent -- if his criticisms of the LOI are to succeed. This means that even if the objections noted above (about abstract "temporal instants" etc.) are either ignored or rejected, Trotsky's argument still has to rely on the identity of quantified tensed variables -- or their equivalent in ordinary language (such as "same moment") -- if it is to work. In short, he has to use the same material objects -- words or symbols, written on paper or reproduced on a screen -- to make his point.


In that case, his argument depends on either one or both of the following:


S20(a): Variable t1 is equal to (or identical with) variable t1.


S20(b): In referring to the same moment during which an object changes, critics of the LOI have to use the identical phrase "same moment", and they all have to mean exactly the same by it each time.


But, an appeal to the identity of symbols like these -- the linguistic expression of which must surely count as concrete and material -- would be just as good an instance of the application of the LOI as any Trotsky himself examined.15


Furthermore, Trotsky's analysis can't be seen as an 'immanent critique' -- i.e., as one undermining the LOI from within -- as it were. That is because his argument depends on this 'law' being absolutely correct while he is using it, and absolutely true after he has used it. If it is limited (or relativised) in any way, then he automatically loses the right to talk about the "same moment" or the "same interval" in which the alleged changes take place. Moreover, if the words he employed aren't identical in meaning as and when he used them, then his conclusions fail, too.


[Identity criteria for words (as opposed to letters) are considered in more detail below. Objections to the above, based on the 'relative stability' defence, have been neutralised in Note 15, and here.]


In that case, it is difficult to see how Trotsky's attack on the LOI can either proceed or succeed -- expressed in any language (technical, scientific or ordinary) -- without an implicit or explicit use of the very 'law' under scrutiny. In order to reject the LOI as it relates to objects, it now looks like Trotsky has to admit that it applies either to temporal instants or to tensed variable letters -- or, to ordinary words that give expression to either. In the latter two cases (i.e., variable letters and ordinary words), this will involve a use of very real material objects (written in ink or even pixels on your screen), just like those "A"s, to which Trotsky took exception.


Either way, Trotsky's analysis is now involved in intractable problems; identity criteria for temporal instants (even if these interpreted as discrete) are notoriously difficult to define -– even if you accept the LOI. They are far more problematic than identity criteria for material objects.16


And, of course, those governing the concrete employment of tensed variables are governed by convention. In which case, it looks like Trotsky has to appeal to the identity of tensed variables -- or to identical marks on the page, or ordinary words that have been identically applied -- if his argument is to work against the very same 'law' he used in his criticism of it!


Hence, in order to make his case, Trotsky had to ignore in practice what he had earlier concluded in theory, undermining what he said about the LOI by disregarding his own strictures against it.


So, if truth is confirmed in practice, Trotsky effectively scuppered his entire criticism of the LOI by having to apply, in practice, that very same 'law'.


It seems, therefore, that it isn't possible to attack this 'law' as it applies to any specific case without also appealing to its unrestricted application somewhere else.


Trotsky's Analysis -- Incomprehensible, Or Just Trivial?


On the other hand, if Trotsky had been aware of these problems and had still rejected the LOI (as it supposedly applies to temporal instants, tensed variables or even the application of ordinary words for identity), his criticisms would have either been (i) rendered far less grandiose than they might otherwise seem, or they would have (ii) collapsed into incomprehensibility.


As far as (i) is concerned, if reference to absolutely the same instant (or the same anything) is to be regarded as illegitimate (since it would clearly require yet another application of the LOI, as noted above), then Trotsky could only have meant one or more of the following:


S22(a): No object is identical with itself at a later time.


S22(b): No object is identical with itself at a different time.


[S13: S8 means A1 is equal to A2.]


[S21(a): There is an A and a time, t1 such that A at t1 is not equal to A at t2.]


But, both S22(a) and S22(b) express the banal truth expressed in S13 and S21(a), with which few would want to quarrel.


Clearly, therefore, S22(a) and S22(b) are almost certainly not what Trotsky had in mind; he surely wanted to argue that no object is self-identical at the same instant -- presumably because all objects are subject to their own internal struggles each of which is generated by a UO. Unfortunately, as we have just seen, without an (implicit, or even explicit) appeal to the LOI applied to temporal instants, tensed variables or their ordinary language equivalents, he can't assert this.


[UO = Unity of Opposites/Unities of Opposites, depending on the context; IED = Identity-in-Difference (i.e., 'Improvised Explanatory Device').]


On the other hand, option (ii) would apply whenever, say, each and every reference to the LOI was dialectically 'made and un-made', as it were, at the same time. That is, it would apply where "same" and "different" were said to "interpenetrate" one another, as in "same and non-same", or "the same and not the same" (using the IED ploy). This would then have Trotsky meaning one or other of the following:


S23(a): No object is self-identical at the same non-self-identical instant in time.


S23(b): No object is self-identical at the same and not the same self-identical instant in time.


But, as we have already seen with other dialectical principles, this threatens to explode in the following manner:


S23(c): No object is self-identical at the same and non-same non-self-identical instant in time.


S23(d): No object is self-identical at the same and not the same and not the same and not the same self-identical instant in time.


Where the word "same", as it appears in S23(a) and S23(b), has been replaced in italics by its (assumed) 'dialectical' meaning -- "same and non-same", or "same and not the same" -- in S23(c) and S23(d) respectively, in order to make explicit the radical confusion this option would create.


But, what could any of these possibly mean? What precisely is a "same and non-same non-self-identical instant in time"? Either it surreptitiously employs the LOI again by the use of the word "same", or it is meaningless.


[Of course, only those who reject the IED ploy, and hence those who repudiate classical DM, will object (successfully) at this point.]


Any who feel confident in their ability to explain what S23(c) could possibly mean should not be given the benefit of that considerable doubt until they have managed to do likewise with S23(d). But, what on earth could this mean: "the same and not the same and not the same and not the same self-identical instant in time"?


If now the same word "same" is given similar treatment in S23(d), it rapidly collapses into the following linguistic mess:


S23(e): No object is self-identical at the same and not the same and not the same and not the same and not the same and not the same and not the same and not the same self-identical instant in time.


As each "same" in S23(d) is replaced with its 'dialectical meaning': "same and not the same".


[As should seem clear, this process of 'dialectical' explication can be continued indefinitely. However, an "excessive tenderness" for my readers' eyesight and sanity prevents me from extending it any further.]


Of course, some might feel that the above is a ridiculous explication of "same" as that word is used in DL. Maybe so, but until dialecticians tell us what they do mean by their sloppy use of such words, it will have to do.


Moreover, it would be to no avail to argue that bags of sugar, for example, are the "same, yet different" (employing the IED-gambit, again) since Trotsky had already scuppered that response by declaring that all things are never the same:


"Again one can object: but a pound of sugar is equal to itself. Neither is true -- all bodies change uninterruptedly in size, weight, colour etc. They are never equal to themselves." [Trotsky (1971), p.64. Italic emphasis added.]


If objects and processes are never the same, they can't be "the same, yet different", they can only be "different, yet different". Of course, if they are indeed the "same, yet different" then it can't also be true that they are never the same, "never equal to themselves". Either way, any employment of this IED explodes it in the face of those reckless enough to try to use it.


It could be objected that Trotsky really meant that objects couldn't simply be the same, but that they are always the same yet different. However, we have seen in Essay Four that that option results in even greater confusion!


In that case, it is now clear that it is impossible to refute the LOI in the same crude way attempted by Trotsky (irony intended). As the above shows, this can't be done without using the LOI in any endeavour to do just that. Hence, anyone wishing to argue on exactly the same lines as Trotsky (or Hegel) will be forced to use the LOI twice -- first: by having to reproduce an identical copy of that argument; second: by using the LOI applied to tensed variable letters (or their equivalent in ordinary language) to establish their case.


The question now is: How might anyone who agrees with Trotsky accomplish the 'very same' task without falling into this double trap?


By means of semaphore, telepathy or Aldis Lamp?


Did Trotsky Understand Identity?


Learning About Identity In A World Where There Supposedly Isn't Any


As seems reasonably plain, in order to draw conclusions from a putative identity statement like S1, Trotsky must have been able to understand the words it contained -- even if he subsequently claimed that such propositions weren't strictly or always true. Clearly, therefore, Trotsky had to be able to comprehend the LOI before he could criticise it -- and the same goes for Hegel, too.17


S1: A is equal to A.


But, this raises another serious question: From where did Trotsky's understanding of the LOI originate? It isn't easy to see how it was possible for him to have grasped the concepts involved if he only ever encountered them in false sentences, or if their use was never absolutely legitimate, as was claimed in S9:


S9(a): All bodies change uninterruptedly. (b) They are never equal to themselves.


On the surface, it would seem impossible for anyone to learn what something meant if all they ever experienced weren't examples of the very thing they were being taught, or if they were all incorrect instances of it.18


The Sting In The Tail


Of course, it could be objected here that Trotsky's ability to understand everyday words for identity is irrelevant since the concepts they express are only valid "within certain limits"; even dialectical concepts only approach the truth "asymptotically".19


Hence, although Trotsky clearly knew how to use ordinary language, it could be argued that DL reveals that the vernacular is in fact ill-suited to depict change adequately. Indeed, the whole point of philosophical criticism is to demonstrate that everyday notions (which are perfectly legitimate in their own sphere of application) are incapable either of reflecting fundamental aspects of change or the fluid nature of reality. This has nothing to do with understanding or even with failing to understand anything.


Or so an objection might go.


However, this problem doesn't just affect the vernacular; the same considerations apply to technical and scientific languages. They would be unusable unless it was possible to specify the conditions under which their empirical propositions were true or the conditions under which they were false (should there be any). [Why this is so will be examined in Essay Twelve Part One. Those who might be tempted to refer to Hegel's criticisms of the LEM should read this, and then perhaps think again.]


But, if identity statements can't ever be true -- not just as a matter of fact, but of logic --, it would surely be impossible for anyone to comprehend them. Even assuming identity statements are only ever approximately true, no one would be able to grasp their content, if that were the case. That is because it would be impossible for anyone to comprehend in what way such ordinary identity statements could fall short of something (i.e., "absolute identity") that was never anywhere instantiated in reality; hence, no one would have the slightest idea what "approximate identity" actually approximated to.


This point isn't easy to see, so some elaboration is necessary.


Trotsky described objects and processes as follows:


"[A]ll bodies change uninterruptedly in size, weight, colour etc. They are never equal to themselves…. For concepts there also exists 'tolerance' which is established not by formal logic…, but by the dialectical logic issuing from the axiom that everything is always changing…. Hegel in his Logic established a series of laws: change of quantity into quality, development through contradiction, conflict and form, interruption of continuity, change of possibility into inevitability, etc…." [Trotsky (1971), pp.64-66. Bold emphases added.]


If, as a matter of logic, 'absolute identity' is nowhere instantiated in reality (which seems to be the significance of the use of the word "axiom", above; indeed, Trotsky is merely paraphrasing a bogus 'logical' argument Hegel cobbled-together), how would anyone be able to declare with any confidence that approximate identity failed to match this ideal by so much or so little? Worse still, if no one had any grasp at all of perfect identity, how could dialecticians be so sure that it is never instantiated in nature or society?


In such circumstances, exactly what is being ruled out?


It is no good, either, appealing to sentences like S1a to answer queries like this since all such sentences are material objects in their own right, and because of that are surely susceptible to Trotsky 'lens argument'. On that view, no material copy of S1 or S1a would identical to itself, or to any subsequent copy of it, and no copy of S1 or S1a would stay the same over time -- since "They [too] are never equal to themselves". In that case, S1a-type sentences can't actually inform us of exactly what Trotsky and/or Hegel were ruling out. Clearly, in order to do so, their content would have to remain the same, which possibility is something S9a, S9b and S1a exclude.


So, we have as yet no idea what 'absolute', or even 'abstract', identity could possibly be, since no material embodiment of it would fully express this 'concept', given this view.


S1: A is equal to A.


S1a: A is both equal to non-A and to non-non-A.


S9(a): All bodies change uninterruptedly. (b) They are never equal to themselves.


[Here, in this Hegelianesque 'proposition', the "non-non-A" refers to the negation of the negation of A, as new content 'emerges'. (The 'relative stability of words and meanings' response will be considered presently.) See also, Note 15.]


Again, that is because, if we now represent S1a by the propositional letter "P", we are instantly faced with serious problems, as S1a is applied to its own material embodiment -- the very material sentence expressed by P:


S1b: P is equal to non-P and to non-non-P.


And then, if we replace each "P" with what it is 'dialectically' identical with, we would have:


S1c: Non-P is equal to non-non-P and to non-non-non-non-P.


S1d: Non-non-P is equal to non-non-non-P and to non-non-non-non-non-non-non-non-P.


And so on, ad infinitem.


And, there is little point irritated DM-fans arguing that this is ridiculous, since the Hegelian rule (expressed in S1a) directly implies the bowl of logical spaghetti represented by S1d. Hence, only the rejection of S1a will prevent this logical bindweed from propagating itself.


Worse still, such material embodiments can't tell us what 'approximate identity' is even an approximation to.


And, if identity statements are never absolutely true (in this material world), how could anyone learn what the implications of accepting or rejecting sentences like S1 amounted to? Again, what exactly would they be accepting or rejecting?


[Note the significance of the italicised word, "exactly" above. No irony was intended, just a reminder that we can't evade or side-step the use of the LOI as a rule of language if we are to understand one another. If anyone knew exactly what they were rejecting by repudiating absolute identity, then they would have to use, and hence implicitly accept, the LOI (as a rule of language) in that very act of repudiation!]


Even to ask the question requires a use of the dread LOI. If no one has a clue what absolute identity is (and on this theory, just as soon as anyone claimed they did have just such a clue, that very idea would itself change, if DL were correct), the question naturally arises: What precisely is being rejected by the repudiation of sentences like S1?


S1: A is equal to A.


If, on the other hand, the LOI encapsulates what is in effect merely an empirical truth -- even if only an approximate one -- we would still be owed an explanation of what this empirical truth approximates to. And how could that be ascertained in this world if we never encounter the absolute limit of identity -- and worse: if we don't know what it is, even in theory, and (supposedly) can't express it in any known language?


It is no use, either, appealing to a more rigorous definition of this 'law' -- such as Leibniz's -- since any material embodiment of that 'law' must change even as it is being written, spoken or thought about, if Trotsky and Hegel are to be believed. In which case, if DL is correct, we couldn't even theorise a correct application of this 'law', or imagine an 'absolute' version of it so that we might then rule one or both out as 'empirically limited', or 'true only within certain limits'.


Questions like these apply equally well to any rejection of propositions expressing identity (as false, or as only partially/relatively true) -- just as they apply to the thoughts and/or words (which would also change just as soon as they were formed/thought) of those that raise doubts about the material application of this 'law' (again, as supposedly correct 'within certain limits'); the negation of an identity statement requires an understanding of what would make it true so that it might legitimately be rejected as false --, even if this is only to point out how limited or "one-sided" it is. But, if there is no way of saying what would make an identity statement true, its denial must lack a sense, for, once more, no one would have the slightest idea what was being ruled out.


Hence, the earlier question about what Trotsky did or didn't understand is relevant after all.


It could be argued that it is possible to say what would make an identity statement true; it is just that they are never absolutely true. However, as we will soon see, in order for anyone to be able to say that such statements are never true, the content of at least one identity statement would have to remain exactly the same over that same length of time, otherwise, anyone who criticised the absolute application of the LOI wouldn't be using, or appealing to, the same principle from moment-to-moment, day-to-day, or year-to-year. In this, it is quite easy to see there are several applications of the LOI (as a rule of language), and over many years, too.


Once more, any rejection of the LOI self-destructs, for if it isn't possible for at least one identity statement to remain exactly the same over a given length of time, then that will apply to any sentence a putative objector might think to use, too. In that case, what Hegel had to say about this 'law' must also have changed -- implying that we can't now access what he really meant to say. On the other hand, if what he had to say hasn't changed, then what Hegel had to say was false.


[Abstract identity will be discussed below.]


Again, it could be objected that identity statements are in fact both true and false (this would seem to be the implication of repudiating the LEM as it is believed to apply without qualification -- and it seems to be what underlies S1a). But, it is entirely unclear what even that could mean in relation to the LOI. It would either imply that material identity only approximates to absolute identity (since no one would then have a clue what absolute identity was, on this view), or if it is false then material identity would still only approximate to absolute identity, because, again on this view, identity propositions are merely, we are told, approximations. Either way, no identity statement would be absolutely true.


S1a: A is both equal to non-A and to non-non-A.


[LEM = Law of Excluded Middle.]


But, both of these amount to the same criticism of the LOI (i.e., that it only captures approximate identity in this world), and since we still have no idea what ordinary identity approximates to, this particular objection is itself completely empty. We still remain in the dark over exactly what is being ruled in or out (that is, we still have no idea what it is that approximate identity approximates to) -- with or without an appeal to the truth or falsity of the LEM.


So: even if the LOI were both true and false, and the LEM was as unsafe to use (unrestrictedly) as we have been told by dialecticians, what exactly would the ascriptions contained in any material example of the LOI be both true and false of? But, no answer to that can be given short of a clear grasp of absolute identity, which, alas, no one is supposed to have.


In that case, this new set of 'difficulties' is independent of whether the LEM is safe, unsafe, or both.


Again, it could be pointed out that we do have a clear idea what absolute identity amounts to; it is just that it doesn't apply to anything in reality. And the following is what we associate with it:


S1: A is equal to A.


We can surely have a clear idea of something even if it is never true, if it is only partially true, or if it is never instantiated in reality.


Or so an objection might go.


But, in that case, as pointed out above, if DL is valid, even that 'clear idea' will have to remain the same from moment-to-moment, year-to-year, generation-to-generation, and if that is itself so, then there is something in reality that instantiates an absolute application of the LOI, namely this 'clear idea'. On the other hand, if this 'clear idea' doesn't remain the same over many centuries, or even from moment to moment, then we wouldn't have a 'clear idea' of absolute identity!


Once again, it could be objected that propositions about identity could still be true and false, so that one option (i.e., the truth of identity statements) needn't necessarily mean that the other must be set in opposition to it. Maybe so, but if we still have no idea what absolute identity is then neither option is viable, whatever anyone does with it. In that case, if no one has a clue what the subject of both this denial and this affirmation amount to, if no one knows what the 'concept' of absolute identity is, then nothing true or false (or both, or neither) could be said of 'it'.


In order to see this, the reader should try to say something true or false about meskonators (over and above a handful of trivialities, perhaps relating to the spelling of that word, for instance).


["Meskonator" is a meaningless word; it is a pure invention.]


Plainly, if no one knows what a meskonator is, then no one could possibly know if anything approximated to 'it', or what was absolutely identical to, or totally different from, 'one of them'.


Of course, with identity itself, the situation is even worse, for the last sentence couldn't even be formulated coherently if "meskonator" is replaced by "identity", and no one knew what absolute identity was. The latter term would then be implicated in its own lack of meaning, as opposed to being implicated in that of another word.


To be sure, identity isn't an object -- but who says 'a' meskonator is? If we have no 'clear idea' of the nature of either of these, or, indeed, no idea at all, then we would be in no position to say anything coherent about them. Worse still, given the way that relational terms (such as "identical") are depicted in DM (i.e., as abstract objects -- we saw this in Essays Three Part One, Four Part One, and Eight Part Three), dialecticians are the last ones who may legitimately raise this particular objection.


However, the situation is even worse than the above might suggest. As hinted earlier, if dialecticians are correct, just as soon as anyone claimed they had an idea of absolute identity (via 'abstraction', or whatever), that idea itself (which must be embodied in some form or other in this changing material world, or in the central nervous system) must alter, and hence become different -- since all objects and processes "are never equal to themselves" (bold added). But in that case, how could anyone tell whether that idea had changed without being able to compare that altered idea of absolute identity with another idea of absolute identity that hadn't changed? Without that comparison, no one would be in any position to say whether their idea of absolute identity (a) coincided with absolute identity itself, (b) fell short of it, or (c) was totally different from it.


And, concentrating on words or concepts 'in the mind' -- contemplated or not by 'speculative reason', à la Hegel --, would be to no avail, either. If DM is correct then nothing remains the same: not words, not thoughts (whether these are, or aren't, about absolute identity, or even about this wishy-washy DM-approximation to it), not anything. Now, if that is true, then not even the last sentence could be securely grasped, for it, too, contains a phrase we have yet to comprehend: "absolute identity" -- since it would change just as soon as it was either perceived, conceptualised or vocalised.19a


Furthermore, if words or concepts actually remained the same (even if only temporarily), then Trotsky's (and Hegel's) criticisms of the LOI would be seriously compromised. Hence, an appeal to the 'relative' stability of words and concepts would be of little assistance to beleaguered DM-fans. If we have no idea what the original phrases "absolute identity" and "approximate identity" relate to (i.e., if we have as yet no clue how far short of, or how close to, the intended target they lie), then we would surely have no way of knowing if and when such words remained stable --, i.e., "absolutely identical", or even "approximately identical", with themselves over time. Hence, if the DM-view of the LOI is correct, 'relative stability' is itself parasitic on a concept we don't as yet comprehend: identity in any of its forms.


And, it is little use arguing that minor changes in words over short periods of time can be ignored, for on this view we would have no way of knowing by how far or how little each word has or hasn't altered if we still haven't a clue what absolute identity is so that we might compare such altered words with exemplars of them that haven't changed.


Some might want to argue that it is reasonable to assume our words remain the same from moment-to-moment; but, on this view, since we don't yet understand the word "same" (let alone "assume"), no one would know if anyone else was employing that word in the 'same' way as anyone else, or even in the 'same' way as they themselves had used it only a few seconds earlier!


Unless, of course, we grant our brains and our memories with an exemption certificate in this respect.


[There is more on this in the next sub-section.]


Once identity is questioned (at any level), all such comparisons and contrasts fail.


In short, there is no way out of the Dialectical Hole Hegel dug for all those who believe exactly the same about identity as him (irony intended).


'Abstract' And 'Approximate' Identity


In response, it could be objected that the above comments are completely misguided since Trotsky's argument was aimed at showing how no absolute sense could be made of the LOI when applied to objects and processes in material reality. He only needed to appeal to approximate identity in the first instance -- the sort of identity we meet in everyday life (from which the LOI has been abstracted) -- to underline the limitations of the Ideal or abstract version of LOI when it is confronted with concrete reality and change. The same perhaps could be said of Hegel (no irony intended).


[To save on needless repetition, whenever I refer to "absolute identity" I also mean "abstract identity" (as DM-theorist appear to understand both terms), and vice versa.]


To that end, therefore, the relative apparent stability of material objects allowed Trotsky to refer coherently to such things as the "same" instant, or the "same" object changing over time, and so on -- perhaps employing the IED argument, once more. This certainly didn't commit him to using the LOI in order to criticise it. Trotsky was obviously arguing dialectically, accessing everyday notions to show how they become contradictory when they are applied beyond the usual boundaries of commonsense.


Or, so the argument might go.


However, and once again, if sense is to be made of approximate identity, some grasp must surely be had of absolute identity so that a vague idea might be formed of what this watered-down version of it actually approximates to. If this is to be achieved by a retreat into the abstract then we are no further forward.


Indeed, if there is a problem about material (or even approximate) identity in this world, there must surely be an even more intractable one about the nature of abstract identity in an Ideal world. In the absence of a clear account of this abstract notion of identity, we would still have no idea what ordinary identity is approximating to. But, how might that be determined without another surreptitious appeal to the LOI?


Hence, we would surely need to know by how much or how little approximate identity was or wasn't absolutely identical to absolute identity itself before we could even begin to 'abstract' absolute identity into existence. Waving a phrase about (i.e., "abstract identity") in no way helps anyone understand what concrete identity is supposed to approximate to.


Consider this sentence:


A1: Approximate identity is approximately identical with absolute identity.


Until we understand absolute identity, any approximation to it will remain an empty notion.


Moreover, and more concretely: Just how is the sense of this abstract notion of identity fixed so that two or more references to it at different times might pick out identically the same target, as opposed to nearly the same target? How might even a latter day Hegel determine whether the notion he had formed one day of absolute identity was or was not absolutely (or even approximately) identical with the one he had accessed on another?


Consider further these sentences:


A2: Comrade NN means by "abstract identity" exactly the same as comrade NM.


A3: Comrade MM means by "abstract identity" approximately the same as comrade MN.


A4: Comrade NP means by "abstract identity" exactly the same as she did yesterday.


A5: Comrade PN means by "abstract identity" approximately the same as he did yesterday.


A6: Comrade PQ's memory of "abstract identity" is exactly the same as it was yesterday.


A7: Comrade NQ's memory of "abstract identity" is approximately the same as it was yesterday.


How could anyone determine what the word "same" meant in any of the above contexts before they knew what the intended goal was (i.e., identity itself)? For all anybody knew, an intentional target like this could be entirely different in the minds of two different abstractors, or even in the mind of one of them. It isn't as if either of them could check inside each other's skulls, or access their present or past thoughts to monitor their precision. But, how do dialecticians themselves lock on to identical ideas of abstract identity, those they supposedly share exactly the same with one another (or with Hegel, or with Trotsky, or with Lenin...) -- even before they had managed to determine what the supposed target of their words happens to be -- and, indeed, whether there is in fact one target here, or many -- or if there is in fact anything there at all? Is it just luck? Or, do they know something the rest of us don't?


Moreover, if memory were capable of accessing what each of us meant by "identity" --, say, even the day before (never mind several years ago) --, how might any of us ascertain if we recalled this word the same, or differently, each time? Or, the even same from moment-to-moment? Plainly, in order to do that we would have to have access to a concept of identity that didn't change -- that is, we would have to appeal to the only thing in the entire dialectical universe that had received a Developmental Exemption Certificate from the hand of Heraclitus himself: the word "identity" (or our 'idea' of it).


And we can't simply assume this is possible; if no one knows what absolute identity is, then an assumption to the effect that it means this, or it means that, would be about as useful as assuming that (a?) meskonator is, say, a new brand of deodorant. Nor can we appeal to ordinary language for assistance. If this theory undermines the vernacular -- which we have seen it does -- then it can't possibly help rescue the very thing that caused the problem.


To be sure, as language users we already know what our words for identity and difference mean -- that is, we know how to use them. This can be seen from the fact that few readers who have made it this far will have failed to grasp the import of the examples of the use of these terms in everyday contexts given earlier in this Essay. However, in these new and rarefied 'dialectical' contexts, where vagueness, obscurity and ceaseless change rule the day, ordinary terms fare rather badly. Even worse, the jargonised expressions dialecticians inherited from Hegel survive not at all. In fact, they commit hara-kiri, as we have just seen.


Plato, Hegel, Trotsky And The Concept Of 'Abstract Identity'


[This material used to appear in Note 20.]


The serious problems associated with abstractionism were rehearsed in Essay Three Parts One and Two. However, for present purposes, consideration need only be given to the following question: How could anyone possibly abstract into existence the -- or even a -- concept of absolute or abstract identity? [Again, I will merely refer to 'Absolute Identity', but this should be taken to include 'Abstract Identity'.] If absolute identity is nowhere to be found in reality, what is there for novice abstractionists to begin to work on? At least cats have the decency to exist as cats -- not 'approximate cats' -- in the natural world so that even a vague idea of the Abstract Cat could be assembled (so we are told). With such accommodating felines, abstractionists at least have something to get their metaphysical teeth into.


And, it isn't as if absolute identity is "what all forms of material identity have in common" (since according to DL they don't -- nothing in reality instantiates absolute identity, so nothing could possibly share it), as the Abstract Cat is supposed to be what all cats have in common. Absolute identity can't underlie all forms of approximate identity (otherwise they wouldn't be approximate versions of it!). This 'concept' isn't the 'general form of identity', either. If it were, then everything would instantiate it. There thus seems to be no grist for the 'dialectical mill' to grind away at, here.


It could be objected that mathematicians, for example, regularly abstract into existence points, lines and planes (and other such objects) that nowhere exist in reality. If so, what is the problem?


But, no mathematical point, line or plane shares any of its features with material points, lines or planes -- for example, points have no shape, circumference, diameter, thickness and they aren't made of anything -- so they can't have been abstracted from them, nor can 'abstract points line and planes' be applied to their supposed physical correlates as a general law, as we are told should be the case with 'abstract concepts' in 'dialectics'. [More details about this will be given in a later Essay. Until then, see here, but more specifically, here.] The same goes for other mathematical concepts and objects. What mathematicians actually do is to make use of a publicly accessible language (albeit one that has often been translated into their own symbolic code or shorthand); they then extend the terms already in use by analogy, metaphor, re-definition, etc. If this weren't so, they wouldn't be able to agree among themselves what their proofs actually demonstrated. That explains why mathematicians don't even think to scan each other's brains when they check new proofs; nor do they indulge in psychometric testing to assess the work of other theorists in their field. But, they would have to do this if they were dealing with abstractions. And it is no use appealing to a common mathematical language here, since, if abstractionism were true, the meaning of the terms in that language would be given by the abstract ideas in each individual mathematician's skull. In which case, no mathematician would mean the same by the words they use as any other in the field. Communication would be impossible. [This is quite apart from the fact that the entire abstractionist case grew out of bourgeois individualist epistemology -- on that, see here.]


[I have said more about mathematical abstraction and 'mental arithmetic', here and here.]


Furthermore, how would it be possible to abstract into existence, say, the Complex Plane, an Abelian Group or Hermite Polynomials? Or even, as noted earlier, a mathematical point? In material reality, what would there be for anyone to 'mentally slice' up in this regard? Worse still, how would anyone know if and when they had reached, or created, the right 'mental' target? Does a bell ring or a light go on when each lone abstractor hits the 'correct idea'?


Indeed, how would it be possible for dialectical abstractors to know when they, too, had succeeded in abstracting the right concept? How would they know whether their individual ideas of absolute identity are absolutely identical with 'absolute identity itself' -- or identical with the ideas abstracted into existence by fellow abstractors of the 'same' notion --, without already having a concept of absolute identity by means of which they could compare it, or them? Once again, approximate identity would be of no use here since that notion could only tell aspiring abstractors that they had approximated to a concept they hadn't yet derived or obtained. And, how might they become aware of their having fallen short even of that ideal target if they don't already possess it? It is little help knowing that abstractors have approximated to a 'who-knows-what?' sort-of-concept if no one knows what this 'who-knows-what?' sort of concept is, which they were supposedly approximating to.


As should seen clear, the word "identity" in the phrase "approximate identity" will as yet have no meaning, or no "determinate" meaning (on this view -- I am not expressing my own opinions here!) until that concept (i.e., 'identity itself') had been 'obtained'. And, how might anyone know they had managed to do that correctly without already knowing what absolute identity is so that they could determine if the concept they had 'obtained' is (or isn't) identical with its supposed target? In that case, the phrase "approximate identity" would be both meaningless and useless.


And, manifestly the word "identity" can't itself also mean "approximate identity", otherwise "approximate identity" would then have to become "approximate approximate identity". Not only does that odd phrase still use the problematic word, "identity", it has now been buried underneath another layer of approximations -- and, it is worth recalling, these are still approximations to 'who-knows-what?'


Once more, the phrase "approximate identity" must surely remain meaningless until absolute identity has itself been 'abstracted into existence'. But, given this theory, how would a single abstractor know what she or he was looking for if all they have to guide them is this permanently meaningless phrase?


To use an analogy: you can look for your keys if you don't know where they are, but not if you don't know what they are.


Try looking for, say, a meskonator (sic).


No luck?


Well, now try abstracting one, or the 'concept' of one!


Or, even an 'approximate meskonator', just for starters.


Still no luck?


Perhaps you don't 'understand' dialectics!


It could be objected that 'meskonators' don't already feature either in language or in the world, so this analogy misfires. However, the point of raising that analogy was to underline the fact that it isn't possible to abstract something if no one has a clue what any of the words that are supposedly being used to depict it, or search for it, actually mean. And since no one has a clue what absolute identity is (given the truth of this theory), it can't therefore be abstracted into existence -- and hence, neither can 'approximate identity'.


To be sure, we already use words for identity, and these feature in practices around which various rules for the use of this narrow vocabulary have grown over the centuries. But, it is our use of these words that tells us what identity is, not some 'law', nor yet some 'concept' of 'absolute identity' -– still less the obscure writings that Hegel conned his publishers into publishing. [On this, see Note 20a, below.]


In that case, howsoever it is conceived, the process of abstraction is surely surplus to requirements; in order to abstract absolute identity successfully into existence abstractors would already need to have access to that very concept in order to be able to decide if they had arrived at the right target. They will need to know whether or not any of the concepts they have lined-up in their thoughts ready to assume the legitimate title is the rightful heir, or is just a pretender to the throne. And, how are they going to do that without already having the 'concept' they were looking for?


Imagine for a moment that Abstractor NN was searching for a concept she called 'Absolute Identity' -- or, "AB", for short. Let us suppose she had managed to form a concept she called "CD" in her search for AB. The question would then be: Is CD absolutely identical to AB? How might NN decide without already having 'in her mind' the right concept, AB, so that she could decide if her concept was identical with it?


Now, vary the question slightly: Is CD approximately identical with AB? Again, how might NN decide if she wasn't already in possession of the concept, AB, but merely had an approximation to 'it' -- and when she can't even raise that specific question without already having AB?


So, even if there were such a thing as 'the process of abstraction', it would be of no use.


In fact, the same comments apply to the attempt to abstract into existence any concept whatsoever; in order to form it correctly, each concept would have to be known already.


So much was known at least to Plato.


It is worth pointing out here that this doesn't mean I agree with Plato, only that this way of regarding abstract ideas (or the names thereof) makes them resemble physical objects (or the names thereof) in the material world so that it makes sense to pose the conundrum: "If you are looking for something you must already know what it is. So, if you are looking for a concept, you must already have it. Otherwise you won't know what to look for."


This is a very short paraphrase of an argument in Plato's dialogue, Meno, and is known as "Meno's Paradox":


"Meno: And how will you enquire, Socrates, into that which you do not know? What will you put forth as the subject of enquiry? And if you find what you want, how will you ever know that this is the thing which you did not know?


"Socrates: I know, Meno, what you mean; but just see what a tiresome dispute you are introducing. You argue that a man cannot enquire either about that which he knows, or about that which he does not know; for if he knows, he has no need to enquire; and if not, he cannot; for he does not know the very subject about which he is to enquire. [Quoted from here. Capitals in the original.]


A recent translation renders this passage as follows:


"Meno: How will you look for it, Socrates, when you do not now at all what it is? How will you aim to search for something you do not know at all? If you should meet with it, how will you know that this is the thing that you did not know?


"Socrates: I know what you want to say, Meno. Do you realise what a debater's argument you are bringing up, that a man cannot search either for what he knows or for what he does not know? He cannot search for what he knows -- since he knows it, there is no need to search -- nor for what he does not know, for he does not know what to look for." [Plato (1997b), p.880; 80 d5-e5.]


The confusion of concepts with objects in this way creates insoluble 'problems' like this. Hegel's 'solution' was to invent concepts that 'self-develop' (predicated on a clear analogy with objects that plainly do develop). But, that is no solution, since we still don't know if they are the correct 'concepts'! And Hegel gave up the right to argue they were the right ones the moment he began to question the LOI.


[This topic is discussed in detail in Essay Three Parts One and Two, so the reader is directed there for more details. See also, Essay Four Part One (here, here and here) for my thoughts on the general confusion of concepts with objects, which crippled Traditional Philosophy for well over two thousand years.]


For Plato, Identity was a Form that exists in an ethereal world we inhabited before we were born, and which he claimed explained how we already possessed this concept by the time we were born -- but then only vaguely after that --, since we had been acquainted with it in this pre-existing state. The shock of birth all but wiped such concepts from our memories. The role of Philosophy was therefore to remind us of what we already know, if we but knew it!


In Plato's case, knowing the Forms was a type of recognition, somewhat like recalling the faces of old friends. So, knowledge for Plato was recollection, a recalling to mind of the 'faces of long lost concepts', as it were, reified (as quasi objects!) in Platonic Heaven, which all of us had encountered before birth. In that pre-existent state we all 'knew' the Forms because, as denizens of that Ideal World, we could see them, face-to-face, since they, too, were in fact Ideal Objects (or as 'Exemplars' somehow masquerading as Ideal Objects; Plato commentators are divided on this one!). It was a case of like recognising like -- a pre-existent soul recognising a fellow 'Ideal Object'.


The above dialogue continues:


"Meno: Well, Socrates, and is not the argument sound?


"Socrates: I think not.


"Meno: Why not?


"Socrates: I will tell you why: I have heard from certain wise men and women who spoke of things divine that --


"Meno: What did they say?


"Socrates: They spoke of a glorious truth, as I conceive.


"Meno: What was it? and who were they?


"Socrates: Some of them were priests and priestesses, who had studied how they might be able to give a reason of their profession: there have been poets also, who spoke of these things by inspiration, like Pindar, and many others who were inspired. And they say -- mark, now, and see whether their words are true -- they say that the soul of man is immortal, and at one time has an end, which is termed dying, and at another time is born again, but is never destroyed. And the moral is, that a man ought to live always in perfect holiness. 'For in the ninth year Persephone sends the souls of those from whom she has received the penalty of ancient crime back again from beneath into the light of the sun above, and these are they who become noble kings and mighty men and great in wisdom and are called saintly heroes in after ages.' The soul, then, as being immortal, and having been born again many times, and having seen all things that exist, whether in this world or in the world below, has knowledge of them all; and it is no wonder that she should be able to call to remembrance all that she ever knew about virtue, and about everything; for as all nature is akin, and the soul has learned all things; there is no difficulty in her eliciting or as men say learning, out of a single recollection all the rest, if a man is strenuous and does not faint; for all enquiry and all learning is but recollection. And therefore we ought not to listen to this sophistical argument about the impossibility of enquiry: for it will make us idle; and is sweet only to the sluggard; but the other saying will make us active and inquisitive.


"Meno: Yes, Socrates; but what do you mean by saying that we do not learn, and that what we call learning is only a process of recollection? Can you teach me how this is?


"Socrates: I told you, Meno, just now that you were a rogue, and now you ask whether I can teach you, when I am saying that there is no teaching, but only recollection; and thus you imagine that you will involve me in a contradiction." [Quoted from here. Links added.]


[There can be no generality associated with 'concepts' like this located in this Ideal World: they are 'abstract objects' (and were thus singular in form), reified in our 'previous presence'. The importance of that observation was underlined in Essay Three Part One. These comments will be connected with criticism of "Representationalism", another dominant ruling-class form-of-thought based on just this confusion (i.e., of concepts with objects), in Essay Thirteen Part Three.]


So, for Plato, knowledge is based on recollection.


[Indeed, even for Descartes, knowledge was based on a form of recollection; recollecting the forms (his "clear and distinct ideas") that 'God' had planted in us all. In fact, this is true of all forms of rationalist epistemology, even those that were propounded by erstwhile atheists. So, for Chomsky, grammar (or 'unbounded Merge') had been implanted in us all (i.e., in humanity) long before we were born, the result of a mega-mutation in the brain of one of our ancestors, a theory as miraculous as anything one finds in The Book of Genesis. So, we don't actually learn a language, we 'sort of remember it' when we are inducted into a speech community as infants. (I have criticised Chomsky's theory in Essay Thirteen Part Three. Readers are directed there for more details.)]


[On this, see Fine (2000, 2003b, 2007, 2018), Moravcsik (1971), and Scott (1999, 2007).]


Now, in modern English we don't really have a word for this sort of knowledge (i.e., knowing someone) -- "acquaintance" is the closest we have, but it is far too weak (for example, not all your acquaintances are your friends, and you are more than merely acquainted with the latter). We used to have two such terms in Old English (as they still do in German): "witten" and "kennen" (the latter word shows up in the song: "Do you ken John Peel" -- I owe this point to Peter Geach), and there are two in modern French: savoir and connaitre. [On this, see here and here.]


Knowing a Concept, or Form, was for Plato like knowing (connaitre) a friend. Hence, in this life you couldn't know that something was the correct Form unless you had already met it (and, presumably, seen it, 'face-to-face'); for him that was in this pre-existing state. Unfortunately, this just confused propositional knowledge (savoir) -- "NN knows that p" -- with personal knowledge (connaitre) -- "NN knows MM" --, conflating both as different varieties of acquaintance (this confusion crops up in Bertrand Russell's theory of 'Knowledge by Acquaintance', but it is implicit in all empiricist theories of knowledge -- a sort of obverse mirror image equivalent of rationalist theories, which depend on 'innate knowledge' -- they are based on the conflation of objects and concepts). So, this confusion has to a greater or lesser extent dogged Epistemology ever since Plato made it explicit (although from what he says, it seems it was implicit in Greek religion and poetry long before). Knowledge is thus viewed as a relation of some sort -- this is a relation we are said to have with the ideas, thoughts, 'representations', or 'qualia' we supposedly have in our heads. Alternatively, it is a relation we have with facts, propositions, statements, sentences, concepts, categories -- all in our heads, or in 'heaven' (in the 'Mind of God'), or even in this world somewhere, which we reflect in our heads. It resurfaces in this famous comment of Marx's:


"To Hegel, the life process of the human brain, i.e., the process of thinking, which, under the name of 'the Idea,' he even transforms into an independent subject, is the demiurgos of the real world, and the real world is only the external, phenomenal form of 'the Idea.' With me, on the contrary, the ideal is nothing else than the material world reflected by the human mind, and translated into forms of thought." [Marx (1976), p.102. Quotation marks altered to conform with the conventions adopted at this site. Link added.]


So, what had been for Hegel 'self-developing concepts in the mind' were now 'objects in the world reflected in our minds', just like ordinary objects are. It is the same confusion in reverse. All forms of 'knowledge by reflection' depend on this ancient conflation.


So, objects of knowledge, since Plato's day (at least), have been interpreted along these lines, that is, as entities referred to by Proper Names (abstract or even concrete), or other singular terms, which is why abstract particulars have become one of the most important factors in what is taken, by many, to be 'genuine' philosophical knowledge (and hence in DM, too -- again as we saw in Essay Three Part One). Here is why:


"The ideas of the ruling class are in every epoch the ruling ideas, i.e. the class which is the ruling material force of society, is at the same time its ruling intellectual force. The class which has the means of material production at its disposal, has control at the same time over the means of mental production, so that thereby, generally speaking, the ideas of those who lack the means of mental production are subject to it. The ruling ideas are nothing more than the ideal expression of the dominant material relationships, the dominant material relationships grasped as ideas; hence of the relationships which make the one class the ruling one, therefore, the ideas of its dominance. The individuals composing the ruling class possess among other things consciousness, and therefore think. Insofar, therefore, as they rule as a class and determine the extent and compass of an epoch, it is self-evident that they do this in its whole range, hence among other things rule also as thinkers, as producers of ideas, and regulate the production and distribution of the ideas of their age: thus their ideas are the ruling ideas of the epoch...." [Marx and Engels (1970), pp.64-65, quoted from here. Bold emphases added.]


[I have explained how and why these ideas serve the interests of the ruling-class in Essay Three Parts One and Two.]


This meant that knowledge was in fact a relationship between the 'Knower' and the 'Known' -- which pair of 'objects' now faced each other like two acquaintances -- just as it deflected attention away from the special nature of propositional (and hence scientific) knowledge. This in turn meant that both propositional and scientific knowledge were interpreted along similar lines, as a relation between the Knower and a peculiar sort of object, a proposition -- now objectified and/or reified (as an object -- either a sort of shadowy entity in extra-mental reality or as something lodged 'in the brain'). So, these days, such reified objects are called "facts", or "truthmakers", and knowledge becomes a relation between the 'Knower' and just such a fact ('in the head' or outside it). [On this, see also here.]


[Incidentally, this is also partly how the phrase "objective knowledge" entered philosophical currency. On that, see here. Notice, too, how this model keeps cropping up all over the place: everything has to be 'objectified' or given a Proper Name. This picture has held us captive for over two thousand years (to paraphrase Wittgenstein). Exactly why both Traditional-, and DM-theorists have fixated on this name-relation is explored in Essay Twelve (summary here).]


This ancient error also re-surfaces in Engels's confused musings about the alleged relationship between "thinking" and "being" -- in LF, for example -- where he too pictures knowledge as a relation between the knower and the known, objectifying both, just like Plato. This paradigm also lies behind his asymptote metaphor: absolute knowledge is pictured as an Ideal sort of object (or limit), toward which humanity is ever moving. This is in effect a sort of reverse Platonism: instead of being born into this world having (all but) forgotten the Forms, on this view, humanity is collectively struggling toward them, this prolonged dialectical meander subsequently(?) re-birthing humanity in a future Ideal State of 'Objective Knowledge', à la Hegel. So, DM is not so much 'upside down' Hegelianism, as back-to-front Platonism! Not so much Hegel 'on his feet' as Plato Moon Walking!


[LF = Ludwig Feuerbach And The End Of Classical German Philosophy; i.e., Engels (1888).]


So, the DM-Elect, gathered at some distant terminus, atop 'Epistemological Mount Olympus' at 'the end of time' -- that is, should any make it that far -- will all know (connaitre), by direct acquaintance, that they have arrived at Epistemological Nirvana because they will recognise it for the object it is. Failing that, a helpful Dialectical Guru will lift the scales from their eyes allowing 'dialectical enlightenment' to flood in. This means that DM is a form of Teleological Platonism; humanity will regain what it lost by their 'epistemological fall from grace' -- or, to use the buzz word, it will wave 'goodbye' to 'partial' and 'relative' knowledge, and ascend to the sunny uplands of 'absolute knowledge'.


[Admittedly, we have also (implicitly) been told that humanity will never attain to this Blessed State, but it is still the goal for any self-respecting DM-Truth-Seeker -- hence the term "Teleological Platonism".]


The other form of knowledge (i.e., savoir, or propositional knowledge) can't be squeezed into this bourgeois individualist straight-jacket, and more naturally harmonises with familiar social aspects of science. [That was one of the main topics of Essays Three Parts One and Two.]


[However, for Hegel, knowledge also appears to be a form of recollection -- on this see Magee (2008), pp.86-99. This ancient nostrum is thus no respecter of "genius". (There is a summary of this section of Magee's book, published at a right-wing site (note especially the précis of Chapter Three of that book), here.) See also, here.]


This then 'allows' Hegelians (and 'materialist' dialecticians) to talk about concepts 'developing' as knowledge grows, since for them propositional knowledge (savoir) has now been fused (à la Plato) with some sort of personal knowledge (connaitre) of concepts -- which also seem somehow to have developed in a 'Collective Mind' of some description, as 'objects' of a special kind, right before our eyes --, at least in so far as these 'concepts' or 'objects' have been illuminated by the light of 'speculative reason'. The 'objective' and the 'subjective' are then joined together in a grandiose version of the infamous 'Chymical Wedding of Christian Rosenkreutz' (which fantasy was highly influential on Hegel's thought -- also detailed in Magee's book, especially, pp.21-83; 223-257; the Introduction to that book has been posted here).


[Concerning the Hermetic and Rosicrucian background to all this (in addition to Magee (2008)), see Essay Four Part One, here.]


This also accounts for the popularity of abstraction among dialecticians, since they, too, have adopted this bourgeois individualist form of epistemology, even if they have stripped away much of the mystical garbage Hegel dumped all over it.


From such seemingly insignificant 'linguistic false turns', large metaphysical 'tumours' have grown. [More about this, too, in Essays Three Part Four (when it is published), and Twelve Part One.]


Now, as should seem obvious from other Essays at this site, I reject this entire way of speaking about knowledge. I am employing these notions here in order to help undermine yet another conservative thought-form that dialecticians have imported into our movement from Traditional Philosophy. It is certainly not the case that I think we need to know our concepts before we know them! But it is a very real paradox implied by DM-Epistemology.


Some might be tempted to appeal to the 'dialectical development' of concepts, but there is nothing here for dialecticians to begin with so that the DM-bandwagon can't even start to roll. And yet, even if it could begin to lumber on, there would be no way of knowing whether, of any two dialecticians, that they had developed the very same notion of absolute identity as any other DM-fan (without having a third notion of it (possibly the same, possibly different) with which to compare their results!). Plainly, in order to compare comrade NN's concept of Identity* with comrade NM's concept of Identity**, we would need to have access to absolute identity in order to decide whether or not they were approximately, or fully, identical --, as, indeed, would be the case with NN and MM.


[Asterisks were used in the previous paragraph to indicate that these two 'concepts of identity' might be totally different in each case -- and if DM-theorists are to be believed --, they will be different. Because of Heraclitean change, coupled with Hegel and Trotsky's criticism of the LOI, they can't be the same.]


Of course, if concepts have a sort of life of their own (as they seem to have in Hegel's scheme-of-things), there would be less of a problem (except perhaps one of accounting for their uncanny ability to regiment themselves, lock-step, in the minds of those they colonise, with no one to check on their progress). But, one would have thought materialists would pause at this point. In this material world, with respect to absolute identity, there is nothing for abstraction to begin with. According to Hegel and Trotsky nothing in material reality is absolutely identical with anything else (or even with itself!). That being the case, no concept of absolute identity could appear at the beginning of this process (applied as some sort of 'law of cognition'), or even emerge at the end, for reasons explored earlier.


Not only that, but there isn't even an identical terminus where all dialectical travellers must aim toward, let alone alight, which destination would help them decide whether or not one and all had abstracted the right concept(s) into existence, or even whether or not they had done any of it to very same concept all along -- or even to an approximate version as they themselves had entertained, or as any other dialectician wrestling with their own batch of individualistically processed 'concepts', had. And, as we have seen, the phrase "approximate identity" would be no use here since it contains a word ("identity") that must forever remain empty until the right objective had finally been reached. And how might that be decided upon?


Of course, the 'solution' to this Platonic/Hegelian quandary is to dissolve it. [That will attempted in Essay Twelve Part Six.]


In advance of that, the reader is encouraged to think of a concept, not as a 'mental construct', or as a rather strange sort of 'object' (in 'the mind', or anywhere else for that matter), but the use of which is an expression -- or better, the use of which depends on the exercise of a specific linguistic skill, which has had to be acquired through socialisation. This change of emphasis redirects attention away from a mysterious, hidden, inner world of 'abstracted objects' (and, indeed, away from socially-isolated, lone abstractors beloved of bourgeois ideology), back toward the public arena where such skills are acquired (in the open), and are subject to scrutiny and correction.


The above (very brief!) HM account of concepts relocates their source (which tradition would have it, used to be in a pre-existing world of Forms and Abstractions) and places it in a pre-existing linguistic community, which community socialises every single one of us in the above manner (albeit heavily distorted by class-division).


We can now see that the role of Philosophy isn't to inform us of the Super-scientific Truths of Traditional Philosophy, but to clear away the confusions created in our thought when we allow ourselves to stray away from the linguistic resources the above community has passed on to us -- when we allow "language to go on holiday", to paraphrase Wittgenstein. That is, when we forget to use the vernacular in the way we do in ordinary life. The key here is, of course, to return to using the resource with which we are all familiar, ordinary language; indeed, as Marx himself enjoined.


Hence, the role of Philosophy is still to remind us of what "we already know" (to paraphrase Wittgenstein, again), and this it does by reminding us, in this case, that we already know how to use ordinary words for identity, and that we only end up confusing ourselves when we forget this and begin (mis-)using words in the same way as Philosophers:


"Learning philosophy is really recollecting. We remember that we really use words in this way. The aspects of...language which are philosophically most important are hidden because of their simplicity and familiarity. [One is unable to notice something because it is always (openly) before one's eyes.]" [Wittgenstein (1993), p.179. Paragraphs merged.]


Dialectical Dilemma


As we also discovered in Essay Three Parts One and Two, theorists who rely on the existence of "abstract concepts" (in order to help justify the "objectivity" of human knowledge) find themselves caught in a serious dilemma at this point.




(1) They  admit that such concepts already exist (independently of us), toward which human knowledge advances or is approximating, or,


(2) They concede that it is we who construct such notions out of experience by a mysterious process of 'abstraction' (which still awaits explication).20


Option (2), it seems, underlies the proffered 'dialectical' response rehearsed above, reiterated by Lenin:


"Cognition is the eternal, endless approximation of thought to the object." [Lenin (1961), p.195.]


"Knowledge is the reflection of nature by man. But this is not simple, not an immediate, not a complete reflection, but the process of a series of abstractions, the formation and development of concepts, laws, etc., and these concepts, laws, etc., (thought, science = 'the logical Idea') embrace conditionally, approximately, the universal, law-governed character of eternally moving and developing nature.... Man cannot comprehend = reflect = mirror nature as a whole, in its completeness, its 'immediate totality,' he can only eternally come closer to this, creating abstractions, concepts, laws, a scientific picture of the world...." [Ibid., p.182. Bold emphasis alone added.]


On the other hand, (1) barely conceals its Platonic/mystical provenance, but which also appears to lie behind Engels's (hopelessly unclear) asymptote metaphor:


"'Fundamentally, we can know only the infinite.' In fact all real exhaustive knowledge consists solely in raising the individual thing in thought from individuality into particularity and from this into universality, in seeking and establishing the infinite in the finite, the eternal in the transitory…. All true knowledge of nature is knowledge of the eternal, the infinite, and essentially absolute…. The cognition of the infinite…can only take place in an infinite asymptotic progress." [Engels (1954), pp.234-35. Italic emphasis in the original; bold emphasis added.]


"The identity of thinking and being, to use Hegelian language, everywhere coincides with your example of the circle and the polygon. Or the two of them, the concept of a thing and its reality, run side by side like two asymptotes, always approaching each other but never meeting. This difference between the two is the very difference which prevents the concept from being directly and immediately reality and reality from being immediately its own concept. Because a concept has the essential nature of the concept and does not therefore prima facie directly coincide with reality, from which it had to be abstracted in the first place, it is nevertheless more than a fiction, unless you declare that all the results of thought are fictions because reality corresponds to them only very circuitously, and even then approaching it only asymptotically…. In other words, the unity of concept and phenomenon manifests itself as an essentially infinite process, and that is what it is, in this case as in all others." [Engels to Schmidt (12/03/1895), in Marx and Engels (1975), pp.457-58, and Marx and Engels (2004), pp.463-64. Bold emphasis added.]


Attentive readers will no doubt have noticed how Engels treats the object of knowledge and knowledge itself as "concept"-based:


"the concept of a thing and its reality, run side by side like two asymptotes, always approaching each other but never meeting. This difference between the two is the very difference which prevents the concept from being directly and immediately reality and reality from being immediately its own concept." [Ibid; bold added.]


While it isn't too obvious what Engels is trying to say here, it is nevertheless reasonably clear that he seems to think "reality" can be "its own concept", which can only mean that for him those concepts are actually independent of our knowledge of them, in that they are "reality". How would it be possible to approach them "asymptotically" if that weren't so? In which case, they must predate human existence in some form.


Of course, it would be unwise (and unfair) to place too much weight on these words of his -- an irredeemably unclear passage from an unpublished letter -- and read some form of Platonism into it. However, having said that, we have already seen (again, in Essay Three Part One), that Traditional Philosophers (including Hegel, and hence DM-classicists like Engels) regularly confused talk about talk with talk about the world -- that is, they ran together questions about how language works with what language is supposedly about. More specifically, they imported an ancient Greek and Medieval view of predicates both as (a) Linguistic expressions, and (b) What those expressions refer to in 'extra-mental reality' -- and they then proceeded to fuse them together (under Hegel's influence).


Briefly, here is the train-of-thought that led DM-theorists by the nose in that direction (covered in extensive detail in the above Essay):


The reconfiguration of predicate expressions as the Proper Names of 'Abstract Particulars' (by Ancient Greek Theorists) was greatly expanded and elaborated upon in the Middle Ages by Roman Catholic Philosophers by means of their newly-invented 'Identity Theory of Predication', This was used to 'reveal' that what we might at first sight take to be an ordinary predicative sentence (employing an otherwise innocent-looking copula -- the verb "to be" --, for instance, in "John is a man", Lenin's example) in fact expresses an identity relation between the predicate expression and the subject term, or indeed, what they supposedly 'reflected' extra-linguistically. So, "John is a man " became "John is identical with Manhood"; just as "John" refers to an extra-linguistic 'object', predicate expressions (like "a man") also refer to something extra-linguistic (traditionally this was one or more of the following: 'Platonic Forms', 'Universals', 'concepts', 'Ideas', 'abstractions'. Exactly where these were supposed to exist depended on how consistent an Idealist a given philosopher happened to be.


This slide helped motivate the above confusion of linguistic expressions with whatever they supposedly refer to extra-linguistically. Hegel swallowed that confusion whole:


"To define the subject as that of which something is said, and the predicate as what is said about it, is mere trifling. It gives no information about the distinction between the two. In point of thought, the subject is primarily the individual, and the predicate the universal. As the judgment receives further development, the subject ceases to be merely the immediate individual, and the predicate merely the abstract universal: the former acquires the additional significations of particular and universal, the latter the additional significations of particular and individual. Thus while the same names are given to the two terms of the judgment, their meaning passes through a series of changes." [Hegel (1975) §169, p234. Bold emphases added.]


"To say 'This rose is red' involves (in virtue of the copula 'is') the coincidence of subject and predicate. The rose however is a concrete thing, and so is not red only: it has also an odour, a specific form, and many other features not implied in the predicate red. The predicate on its part is an abstract universal, and does not apply to the rose alone. There are other flowers and other objects which are red too. The subject and predicate in the immediate judgment touch, as it were, only in a single point, but do not cover each other.... In pronouncing an action to be good, we frame a notional judgment. Here, as we at once perceive, there is a closer and a more intimate relation than in the immediate judgment. The predicate in the latter is some abstract quality which may or may not be applied to the subject. In the judgment of the notion the predicate is, as it were, the soul of the subject, by which the subject, as the body of this soul, is characterised through and through." [Ibid., p.237, §172. Bold emphases added.]


Plainly, for Hegel, predicate expressions don't just refer to 'universals', they are 'universals', and are 'abstract' into the bargain -- in which case, they thereby cease merely to be linguistic.


The rapid slither from there to the doctrine that predicates are extra-linguistic, extra-mental entities in their own right was further accelerated by the invention of the 'reflection theory of knowledge' by dialecticians. Predicate expression now somehow, in some as-yet-unexplained way, 'reflect' real material objects and processes in the world. So, as we also saw in Essay Three Part One, it then became all to easy for DM-theorists to run together linguistic expressions with what they supposedly referred to or 'reflected' extra-mentally. [I have entered into this in more detail, here.]


For the above dialecticians, predicate expressions now referred to 'abstractions'/'concepts' apprehended by, or cobbled-together in, 'the mind' (depending on which DM-theorist one attended to) -- and this was accomplished by the mysterious 'process of abstraction'. Unfortunately, that now presented DM-theorists with the knotty problem in the shape of the above dilemma, but with an added twist -- the problem of explaining where exactly these 'abstractions'/'concepts' are supposed to exist. If they exist only 'in the mind', what 'objectivity' do they possess? Alternatively, if they exist extra-mentally, what form do they take? And where exactly do they exist?


As we saw in Essay Three Parts One and Two, those questions remain unanswered to this day.


However, in order for these 'abstractions' to be able to deliver, or form part of, 'objective knowledge', they too had to be 'objective':


"Thought proceeding from the concrete to the abstract -- provided it is correct (NB)… -- does not get away from the truth but comes closer to it. The abstraction of matter, the law of nature, the abstraction of value, etc., in short all scientific (correct, serious, not absurd) abstractions reflect nature more deeply, truly and completely." [Lenin (1961), p.171. Bold emphasis alone added.]


"Knowledge is the reflection of nature by man. But this is not simple, not an immediate, not a complete reflection, but the process of a series of abstractions, the formation and development of concepts, laws, etc., and these concepts, laws, etc., (thought, science = 'the logical Idea') embrace conditionally, approximately, the universal, law-governed character of eternally moving and developing nature." [Ibid., p.182. Quotation marks altered to conform with the conventions adopted at this site.]


"Logical concepts are subjective so long as they remain 'abstract,' in their abstract form, but at the same time they express the Thing-in-themselves. Nature is both concrete and abstract, both phenomenon and essence, both moment and relation. Human concepts are subjective in their abstractness, separateness, but objective as a whole, in the process, in the sum-total, in the tendency, in the source." [Ibid., p.208. Bold emphases alone added.]


Here we see this confusion (between linguistic expressions and what they supposedly 'reflect' extra-linguistically) in all its glory, for nature is now said to be abstract, just as human concepts are supposed to be 'objective'! Of course, these ideas do not sit at all well with what Lenin had said about 'objectivity' a few years earlier:


"To be a materialist is to acknowledge objective truth, which is revealed to us by our sense-organs. To acknowledge objective truth, i.e., truth not dependent upon man and mankind, is, in one way or another, to recognise absolute truth." [Lenin (1972), p.148. Bold emphasis added.]


"Knowledge can be useful biologically, useful in human practice, useful for the preservation of life, for the preservation of the species, only when it reflects objective truth, truth which is independent of man." [Ibid., p.157. Bold emphasis added.]


So, it seems that if an 'abstraction' (or a 'concept') is to be 'objective', or if it is to contribute to 'objective truth', it must be independent of the human mind -- and if that is so, many of them must have pre-existed the evolution of the human species.


I can see no way of avoiding that conclusion.


[If anyone does, please let me know.]


We see this confusion (between linguistic expressions and what those expressions supposedly refer to) float to the surface in George Novack's truly awful book (about the 'Logic of Marxism'), and his bizarre 'analysis' of the by-now-familiar sentence, "John is a man":


"This law of identity of opposites, which so perplexes and horrifies addicts of formal logic, can be easily understood, not only when it is applied to actual processes of development and interrelations of events, but also when it is contrasted with the formal law of identity. It is logically true that A equals A, that John is John…. But it is far more profoundly true that A is also non-A. John is not simply John: John is a man. This correct proposition is not an affirmation of abstract identity, but an identification of opposites. The logical category or material class, mankind, with which John is one and the same is far more and other than John, the individual. Mankind is at the same time identical with, yet different from John." [Novack (1971), p.92. Bold emphases added.]


The supposed predicate "a man", which is a linguistic expression, has now been transformed into a "category", and, just as John is an 'extra-mental object', so is "mankind". So, the term, "a man", is now both a linguistic expression and an extra-linguistic entity in its own right. The two have now been run-together; which is the only reason the predicate can be inserted into an identity relation with John. Plainly, John can't be identical with a linguistic expression, but he can be said to be identical with another extra-linguistic entity (conjured into existence by this confusion, and by it alone).


[The "is" of predication (in "John is a man") has also been reconfigured as an "is" of identity, and that in turn has been transmogrified into an 'abstraction' -- "abstract identity". I have given several further examples of this confusion of linguistic expressions with extra-linguistic entities in the work of other DM-theorists in Essay Three Part One -- especially, here, here, and here. I have also subjected the above passage of Novack's to searching criticism, here.]


But, whichever horn of this dilemma dialecticians finally grasp (if they ever actually make their minds up, or are finally much clearer than Engels ever was), neither is welcome news to those who accept the 'dialectical analysis of Identity' -- Trotsky's or Hegel's version.


There are at least two main reasons for saying this about their take on identity:


(a) If there is an 'abstract concept' of absolute identity towards which our knowledge slowly edges then presumably that goal must remain the 'same' while it is being targeted. In that case, the LOI must apply to it (since, ex hypothesi, it doesn't change). But, no self-respecting DM-theorist can possibly agree with a Platonic view of abstract concepts like this, even though their 'asymptotic approach' metaphor, and what has been argued in the last few paragraphs suggests that they should. Admittedly, this is a controversial claim, but it may only be neutralised when it becomes clear what this metaphor implies -- more pointedly, when it becomes clear whether or not it means that as far as individual dialectical truth-seekers are concerned, there is indeed a goal (identical in the case of each dialectical pilgrim), which they have correctly targeted -- collectively or severally -- toward which they are all slowly gravitating. Naturally, a positive answer here would sink the dialectical analysis of the LOI, since it would (i) make plain that this metaphor implies that there is something unchanging called "absolute identity" that all dialectical truth-seekers are homing in on, and upon which all are already --, or about which they will be --, in equal agreement. On the other hand, a negative response would undermine DL equally quickly; if there is no such goal then (ii) approximate identity must approximate to nothing at all.


(b) Alternatively, if human knowledge is dialectically conditioned, and there are no abstract concepts that exist independently of our knowledge of them, then there would be no objective way to decide whether or not any two randomly selected dialecticians were aiming at the very same intentional target. Indeed, they would be hard-pressed to say what could count as the same goal in such circumstances (that is, without surreptitiously using the LOI in order to help them decide). But, if such dialectical truth-seekers haven't locked on to the very same target, then the second of the above conclusions (i.e., point (a(ii)) above) must surely apply. In that case, their search is, to put it bluntly, aimless. It is headed nowhere, since there is nowhere for it to head. On the other hand, if there is a way of delving into the minds of any two randomly selected dialectical abstractionists, and ask what it is that enables one or both to decide whether they are in pursuit of an identical goal, the first of the above points would then apply (i.e., (a(i))) -- for it would then be obvious that these truth-seekers had used the LOI to identify exactly the same intentional target, and, indeed, had done it with equal accuracy.


In fact, if anyone were to advocate, or even reject, one or other of the above options, they would still have to appeal to the very same 'law' in question in order to maintain the belief that approximate identity is more-or-less identical with abstract identity (whether or not "abstract identity" was understood Platonically, or as a quasi-Hegelian/'dialectical' construct/abstraction), or whether it isn't more-or-less identical with abstract identity. Of course, this concept (i.e., abstract identity) would have to remain locked, rock solid -- frozen in 'mental' or 'conceptual' space -- while it was being approximated to.


If that weren't so, this 'target' must surely have been misidentified by anyone foolish enough to blaze an intentional path toward such a mutating 'object'. As was argued in the last sub-section, unless dialecticians were able to specify under what conditions their notion of absolute (or abstract) identity doesn't change over time -- but remains absolutely self-identical in the minds of supporters and critics alike, and over many centuries, for them to be able to say with confidence that they are talking about the same thing as one another (or even the same thing as Hegel or Trotsky) --, any reference to 'it' by critic and believer alike would be entirely empty.


Otherwise, they should acknowledge their irresolvable differences, and cease their pointless blather.


And, even if the concept of 'abstract identity' were to change (as it is apprehended by one or all), then in order to express that very fact, some way must be found to declare that it was no longer absolutely identical with whatever it used to be absolutely identical with. In that case, access to an unchanging version of absolute identity would still be needed to classify any mutated version of absolute identity as a mutant. Without that, of course, they would lose the right to say that absolute identity has, or might have, changed.


Indeed, as should now seem obvious, we would need it not to change in order to say that it had!


[Once more, this just underlines the intimate connection there is in language between change and identity, contrary to what dialecticians will tell you -- if you were even bothered to listen.]


Moreover, an implicit reference to the LOI would have to be made as part of each and every claim that any randomly-selected dialectical abstractionist or truth-seeker had a concept of 'abstract identity' which was identical with that of any other, so that it could be said that they were referring to the same 'abstract concept' when making the 'same' point even about "approximate identity" --, and that would have to be the case even if they were disagreeing!


But, if these assumed ideas of "abstract identity" (or even of "approximate identity") weren't exactly the same, then agreement and disagreement over what they were talking about would be illusory, too. On the other hand, of course, if these ideas were 'absolutely identical' across two or more dialectical heads, then these criticisms of the LOI would plainly self-destruct.


Which explains the reference to hara-kiri, earlier.


Furthermore, if the presumptive subject of enquiry here were only 'roughly identical' in the minds of the many DM-fans sat round the dialectical table, not only would that fact be both untestable and unverifiable, it would mean that the topic of discussion would be indeterminate, too -- and for the same reason. Again, this would mean that any and all criticisms levelled against the LOI would have been misdirected, That is because not only would no one know exactly what "abstract identity" was so that it could be criticised equally the same -- and by the use of identical arguments -- by those who don't, shall we say, believe in the absolute validity of the LOI, no two critics of the LOI would be able to say that they had the very same thing in mind when they were even so much as pointing out its possible or actual limitations! Indeed, they couldn't even use the word "same" with any clear meaning in this or any other context --, and, annoyingly, that would be so for the same reason.


On the other hand, if it were now conceded that any two notions of strict (or even approximate) identity were exactly the same in the minds of any two intrepid dialectical abstractors, so that it could be said of one or both that they were talking about the very same thing, there would be no point in criticising the LOI, for it would be valid -- and admitted to be such -- at least here, by its severest critics.


Worse still, if were denied that anyone had an exact notion of strict identity (based on the claim that everyone holds only an 'approximate copy' of it), we should still want to know exactly what was being ruled out. In that case we (they) would have to have an idea of strict identity to be able to deny they (we) had any such idea!


Identity Schmidentity!


In order to underline this point, it might prove helpful to consider an analogy: let us suppose that someone introduced a word into the language -- say "schmidentity" -- but couldn't provide a single example of anything in reality that exhibited "schmidentity". If we were then told that certain things were only "approximately schmidentical" (or even "schmidentical within certain limits") we would still have no clear idea what this new word meant. If we don't know what "schmidentity" is, we certainly don't know what "approximate schmidentity" is. And calling this new 'concept' "abstract schmidentity", "absolute schmidentity" -- or even "relative schmidentity" -- would be equally futile.20a


It could be objected that the words we have for abstractions are merely extensions to words we already use in language, so the analogy with "schmidentity" is inapt. In that case, abstract or absolute identity would be an extrapolation from our ordinary words for identity, etc. [This point has in fact been covered in the last few subsections.]


Even so, let us change the example. Suppose someone introduced the phrase "absolute cat-hood" into the language, but couldn't say what an 'absolute cat' was, or by how much the average moggie differed from this quintessential feline. I think we would judge this person radically confused; we certainly wouldn't be inclined to adjust scientific, philosophical or ordinary language to accommodate them.


In that case, until we are told by how much or how little absolute or even abstract identity differs from our everyday words for identity, we would be wise, I think, to adopt the same policy.


And, even though there are abstract nouns in ordinary language, their mode of signification can't be assumed to be the same as that of any allegedly analogous, or even typographically identical, philosophical terms that have been thrown into the ring by Traditional Thinkers. Indeed, we can say more than this: in view of the argument in Essay Three Part One, since ordinary general terms retain their generality in everyday contexts -- whereas these philosophical monstrosities don't --, the two are definitely not identical (irony intended). Nor could one be an extension of the other if one of them destroys or undermines what is unique to the other -- in this case, an ability to express generality.


Of course, any dialecticians who disagree with this will have to abandon Trotsky's critique of the LOI, for if the two sorts of terms are identical, then the 'LOI' applies to them, at least.


[This commits me neither to a belief in absolute or abstract identity, but it does succeed in exposing the DM-'concept' as empty at best, or incoherent at worst.]


In that case, when dialecticians presume to tell us that a set of words in ordinary language connected with sameness and identity, which we all know how to use, doesn't mean what we usually take it to mean (and they then proceed to highlight its limitations, implying that our understanding of this set is defective to some extent, howsoever nuanced that turns out to be), then the onus would be on them to tell us what these dialecticians do mean by this set of words. Until they do, they might as well be talking about schmidentity.21


Again, it is little point referring to Hegel's criticisms of the LOI. As I have demonstrated here, he badly misconstrued this 'law', compounding his folly with a series of crass errors over the nature of predication, among other things.


Indeed, for all DM-fans know, they could very well be talking about schmidentity -- or, far more likely, nothing at all.


For example, how do they know that their 'notion of identity' isn't 'absolutely identical' with schmidentity? Or, indeed with nothing? The fact that I haven't defined "schmidentity" is no objection. They have yet to tell us what they mean by their use of words for identity. In fact, they mis-identify this word from the get-go -- and, what is even more interesting, they copied this exact misidentification from Hegel! [Irony intended.]


In which case, they probably are talking about nothing whatsoever.


Nevertheless, there are several other serious problems that confront the objection outlined at the beginning of the last subsection. What these are can be seen when we consider the exact words Trotsky himself used to criticise the LOI over 70 years ago (irony intended, once more).


Trotsky's Exact Words Dialectically Implode


The claim that our concepts are only approximately true -- if true itself -- would undermine DM more effectively than anything that has been argued at this site.


In order to see this, let us introduce the term "adequate" to describe the language belonging to, or used by, any given theory, but understood in the following manner:


S24: A language is adequate to a theory if, when expressed in that language, the empirical propositions of that theory can be deemed true (by appropriate means).


However, if it is impossible to develop a language adequate to a theory no matter what we do, then it would surely be impossible to grasp that theory's content, or determine what it implied, or was even about. In the case of identity and DM, this problem is particularly acute.


With respect to the matter-in-hand, this fatal defect can be underlined by a consideration of Trotsky's own words:


"The Aristotelian logic of the simple syllogism starts from the proposition that 'A' is equal to 'A'…. In reality 'A' is not equal to 'A'. This is easy to prove if we observe these two letters under a lens -– they are quite different to each other. But one can object, the question is not the size or the form of the letters, since they are only symbols for equal quantities, for instance, a pound of sugar. The objection is beside the point; in reality a pound of sugar is never equal to a pound of sugar -– a more delicate scale always discloses a difference. Again one can object: but a pound of sugar is equal to itself. Neither is true (sic) -- all bodies change uninterruptedly in size, weight, colour etc. They are never equal to themselves." [Trotsky (1971), pp.63-64.]


The question here is: Are the above words identical to the ones Trotsky actually wrote? Now, even though I have carefully copied the above quotation of Trotsky's words from my own copy of IDM (in fact I copied and pasted them from the on-line version), and checked them against the versions found in TAR and RIRE, as no doubt others have done with their own copies of these words --, which copies themselves were copies of successive generations of further copies of the originals --, despite all this, in one sense they aren't the very same words that Trotsky committed to paper over seventy years ago: they are just copies! So, in that sense they aren't exactly his words. Hence, if we count words as ink marks on the page, or collections of pixels on a screen, they are manifestly not physically identical with the originals. How could they be?


[IDM = In Defense of Marxism; i.e., Trotsky (1971); TAR = The Algebra of Revolution (i.e., Rees (1998); RIRE = Reason in Revolt (i.e., Woods and Grant (1995/2007); UO = Unity of Opposites/Unities of Opposites, depending on the context.]


On the other hand, in another perfectly ordinary sense of the term, the words quoted above are identical with the originals; they are Trotsky's words, and that is why his followers constantly quote them. We have criteria of identity for words (and also for ink marks), which Trotsky also had to rely on, or observe, in order to present his case against the LOI to anyone interested in reading it, and upon which his followers also depend. Otherwise he wouldn't have written them!


And, I am now about to use those words against him (and his epigones).


The fatal consequences mentioned above revolve around the import of the last few sentences above, and centre on the paradoxical conclusions that arise if we reject what they say -- as it seems DM-critics of the LOI must do to remain consistent with their own precepts.


For example, an acknowledgement that the above quoted words are identical with Trotsky's own would mean that anyone reading them now, and accepting his case against the LOI as valid, must have (implicitly) employed that very 'law' in order to criticise it! They would have to say (in effect) that in their copy these words are exactly the same as the ones Trotsky typed or penned all those years ago, which words now undermine an absolute application of the very 'law' they had just used to arrive at that conclusion!


Clearly, Trotsky and his epigones failed to take into account this perfectly ordinary sense of identity: anyone who reads Trotsky's words today has before them material objects (i.e., ink marks on the page or pixels on a screen) that are clearly identical with countless other such objects separated from one another in space and time (i.e., other ink marks on paper or pixels on a screen that represent still other copies of the very same article he wrote), by means of which an indefinite number of readers may access the very same ideas that Trotsky intended they should, but which marks aren't numerically, or materially, identical to the ones he originally penned. This is as clear a case of ordinary 'identity-in-difference' as one should wish to find, but one that doesn't commit us to a belief in those terminally obscure DM-UOs (in fact, there are none here) -- and, as will become plain, this is an example of 'identity-in-difference' that completely undermines Trotsky's criticism of the LOI.


In this case, countless manifestly (and optically) different material objects -- separated in space and time -- are nonetheless uncontroversially identical. That fact could only be denied by those who possess a defective copy of Trotsky's writings! Moreover, the admission of this fact does nothing to undermine anyone's belief in change.


[Annoyingly, that belief itself could remain the same even while agreement with Trotsky on the LOI changed as a result of the above argument!]


Of course, anyone who disagreed with the above conclusions would then perhaps be committed to the view that the words in their copy of Trotsky's writings weren't identical with those that Trotsky authored seventy or more years ago. Indeed, that idea might itself have been prompted by Trotsky's own writings and the message they conveyed, and which is what that message seems to express -- that is, that nothing is identical with itself, or with anything else, including those very words, or his message!


Someone could object: (α) the above considerations actually support Trotsky's case, for here we have several objects that we ordinarily call identical, which are manifestly not the same. So, our ordinary grasp of identity isn't as secure as some might imagine.


In response, it is worth pointing out that we count certain words, phrases and symbols as identical even though they use different letters, or none at all. For example, few of us would say most or even all of the following aren't the same word or symbol, or that they didn't mean the same:


B1: (i) Cat, CAT, Katze, gato, gatto, and chat; (ii) Trotsky and TROTSKY; (iii) colour and color; (iv) maths and math; (v) In Defence of Marxism and In Defense of Marxism; (vi) Das Kapital and Capital; (vii) vixen and female fox; (viii) one, eins, un, uno, een, en, and egy; (ix) 10 minus 9, 2 divided by 2, and 1; (x) 3 multiplied by 2, and 6; (xi) February 12th and 12th of February; (xii) 02/06/21 (in the USA) and 06/02/21 (in the UK), (xiii) Beta and β; (xiv) dog and chien; (xv) red and red; (xvi) red and red; (xvii) red and rot; (xviii) bold and bold; (xix) italic and italic; (xx) and and and; (xx) a short empty space on a page, and        ; (xxii) arrow, , , and ; (xxiii) 6 and upside down 9; (xxiv) half full and half empty; (xxv) non and uou, written upside down; (xxvi) dot and .; (xxvii) dash and —; (xxviii) aluminium and aluminum; (xxix) xxix and 29; (xxx) etc., and and so on...


That is because our criteria of identity for words, symbols and letters aren't in every case the same (no irony intended). With respect to words and symbols, a whole host of criteria apply; physical form is clearly not the only one (as the above objection -- (α) -- seems to assume). Hence, even though the physical form of the words and symbols used might be (totally) different, we nonetheless recognise most or all of them as identical.


Furthermore, it can't be the meaning of some or all the words and symbols in B1 that makes them identical, although this is certainly true in some cases. Hence, we would regard "schmidentity" and "Schmidentity" as identical words or inscriptions even though they have no meaning. That in turn can't be because they both lack a meaning, otherwise, on that basis, "schmidentity" and "meskonator" would have to be counted as the same word!


Of course, when meaning is introduced into the equation, the situation becomes even more problematic for dialecticians.


In that case, the meaning of Trotsky's words today must be identical to the meaning they had seventy odd years ago, even if their physical form isn't -- otherwise, it would suggest that what Trotsky intended to convey (e.g., that nothing is self-identical over time, for instance) must itself mean that that very message wasn't the same as the one he propagated all those years ago. That is because his message implies that not even messages are identical over time! If, therefore, a reader had actually accessed and read the identical message put out by Trotsky that urged them to come to the (same!) conclusion that even Trotsky's message must change over time, then they can't have understood that message accurately if they now agree with a corrupted or changed copy of it -- which original, for all they know, might support an absolute, unqualified belief in the LOI! And that possibility can't be ruled out if we accept the idea that:


"...all bodies change uninterruptedly in size, weight, colour etc. They are never equal to themselves..." [Ibid.]


In either case, such a critic wouldn't be able to ascertain exactly what Trotsky had written or meant, even while they feigned assent to the exact import of what he said in order to arrive at that sceptical conclusion -- and which had just prompted those very doubts about messages corrupting over time!


Alarmingly, that must mean that any such latter-day critic of the LOI must have access to Trotsky's precise thoughts by other means, over and above the physical text they will now have to concede can't be exactly the same as that which Trotsky originally penned. This alternative route to Trotsky's thoughts must therefore go beyond the confines of the physical document itself --, since the latter (on this view) exists now in this ever-changing world in corrupted form.


This 'alternative means of communication' could only be 'ethereal' or 'telepathic'. So, such Telepathic Trotskyists and Ethereal Epigones seem to be able to intuit Trotsky's exact meaning -- which now unfortunately prompts them to question any such exactitude!


But, even this would imply that while this 'ethereal' message (issuing somehow from Trotsky) was identical to the original he transmitted through the ether all those years ago -- enabling these very doubts about identity to be accessed exactly and with no loss of meaning over the years in such an esoteric  manner by contemporary recipients  -- the physical message wasn't!


Naturally, that would still commit such individuals to the validity of LOI -- only now applied to 'occult' messages and ethereal identities!


An Attack On The LOI Unequal To The Task


A moment's thought will confirm the fact that the idea that our concepts are somehow inadequate to the tasks we set them can't be correct, even if the LOI were as defective as DM-dreamers would have us believe. That is because, if our words are inadequate in some way then so were Trotsky's when he criticized the LOI, and so are those of anyone who now echoes such doubts. But, that in turn would mean all such attacks on language are defective in virtue of that very assault. Indeed, any words that expressed even an abstract disquiet about the adequacy of the vernacular would be inadequate to that task -- which means, plainly, that such inadequacies could never be adequately expressed!


Short of saying nothing at all, this impasse will always block the thoughts and ideas of anyone who thinks their words are permanently inadequate to any task set for them. Of course, that would imply that sceptical doubts about the adequacy of language (howsoever mildly expressed) must either be disingenuous or self-refuting. [Those who still harbour doubts should read this, and then perhaps think again.]


[And that includes anyone who tries to impugn ordinary language, howsoever nuanced and indirect it might be -- as we will see in Essay Twelve. Some thoughts on this are expressed in Note 19, below, others can be found in Essay Three Part Two, and by following the link at the end of the previous paragraph.]


Furthermore, if what Trotsky had intended to say about the limitations of the LOI were in principle impossible to express in any language, if even its physical embodiment wasn't identical with his thoughts as they were taking physical shape (i.e., when he was committing them to paper) -- because of his claim that all objects (including words forming 'in the mind', or those being committed to paper) are "never equal to themselves", let alone anything else --, then Trotsky himself couldn't have intended to mean anything by them! That is because there would have been nothing for him to have intended to have meant in such circumstances. A faded simulacrum of his own thoughts about the LOI would have been of no use even to Trotsky. Hence, not one nanosecond after being thought, Trotsky's very own words would be non-self-identical. On paper, or in the mind, they wouldn't be the same as those he had thought micro-seconds earlier, and he would then have no access to his own precise intentions, and for the same reason -- unless, that is, we are supposed to exempt memory from Heraclitean change, claiming perhaps that it is the only thing in the entire universe that isn't susceptible to the following:


"...all bodies change uninterruptedly in size, weight, colour etc. They are never equal to themselves..." [Ibid.]


If not, and the above strictures apply across the board, Trotsky's own thoughts would be forever lost -- as if they had never existed.


Hence, if what Trotsky said were correct then not even he could have affirmed, or confirmed, whether or not he was criticising the same 'law' from moment-to-moment -- without surreptitiously, or implicitly, appealing to the very same 'law' (as a rule of language), while he was attempting to do just that.


More generally: if it is impossible to specify what it is that is being attacked (on the basis that whatever is thought about 'it' is in principle not identical to what had just been thought about 'it' micro-seconds earlier), no intention to criticise 'it' can crystallise, for there would be no such 'it' to denigrate.


Once again, an appeal to 'approximate identity' might suggest itself here, but that would be to no avail. As was argued earlier, approximate identity is parasitic on concepts of identity not semantically-challenged in this Idealist manner. Hence, we would need some idea of what was being approximated to if the notion of 'approximate identity' is capable of doing any real work. But, ex hypothesi, that can't be accessed without an appeal being made to the validity of the LOI as a linguistic rule (but not a metaphysical truth) -- and, what is more, to the very same rule applied repeatedly. [Irony intended.]


Trotsky (or Hegel -- or, indeed, anyone who is in exact agreement with either or both of them), would be forced to use (implicitly or explicitly) such linguistic criteria to formulate their self-refuting Idealist notions about a 'law' they seriously misidentified to begin with!


Yet More Materially-Induced Dialectical Misery


Unfortunately, these fatal defects don't stop there: anyone who consults Trotsky's words today, and who agrees with his case against the LOI, owes us an explanation why, on the one hand, the "A"s in S1:


S1: A is equal to A


are subject to the following strictures:


"But in reality 'A' is not equal to 'A'. This is easy to prove if we observe these two letters under a lens -– they are quite different from each other...," [Trotsky (1971), p.63.]


when Trotsky's own words (written many years ago) aren't. Such a person would in effect be saying:


I agree that the abstract version of the LOI is defective because of what Trotsky's words say. That is because the marks on paper that I am now reading in IDM express exactly what Trotsky was thinking all those years ago; they convey the very same message he intended. And yet, in at least this respect the LOI must be correct for me to understand Trotsky, agree with him, and arrive at exactly the same conclusion he did, and which he intended his readers should. That allows me now to disagree with the LOI that I have just used in arriving at identically that result!


On the other hand, if my conclusions are only approximately the same as his, I must have some grasp of his exact intentions so that I may truthfully say with what my own opinions are in fact an approximation. And this I must know, for I assent to the idea that all identity statements are approximations because I agree with exactly what Trotsky says. Without his words I would still be under the illusion that identity was absolute -- er..., which it must be if I have just used the LOI to arrive at to this very point....


But, if the LOI is correct at least once, and the same person arrives at exactly the same conclusion as Trotsky -- only decades later -- then what Trotsky says in S9, for example, can't itself be true:


S9(a): All bodies change uninterruptedly. (b) They are never equal to themselves.


[And, of course, words are also "bodies".]


In that case, such a person could only agree with Trotsky's criticism of the LOI by appealing (implicitly) to the validity of the very same defective 'law' every time they accessed his words, drawing identical conclusions from it, many times over, throughout their no doubt very confused dialectical lives. Naturally, this would mean that anyone agreeing with Trotsky, who derived the same conclusion, would only be able to do so on condition that they then promptly disagreed with Trotsky in practice, implying that the exact opposite result was correct --, a result, which because of this, is not now identical with the one Trotsky had obtained!


This further implies that Trotsky's claims are right only if they aren't, and that if what he had intended to say were true, it would then become impossible for anyone (including Trotsky) to determine what it was he wanted to say, whether it was right or wrong, or whether it had been committed to paper accurately, transposed correctly between copies -- or even whether these questions are themselves defective or not --, and for the same reasons.


Once more: profound dialectical confusion has been exposed for all to see in actual practice.


This new (and ironic) dialectical inversion (whereby a rejection of the LOI depends on its successful application in practice by anyone trying to ascertain exactly what Trotsky, or Hegel, thought he/they wanted to say) just confirms how complex the conventions of ordinary language really are (to paraphrase Wittgenstein), and how it isn't possible to criticise those conventions without that attack itself falling apart for want of words with which it might be accomplished!


Clearly, this paradoxical result is a consequence of the cavalier attitude adopted by dialecticians (like Trotsky) toward ordinary language, seriously compounded by a direct assault on the LOI. It isn't possible to criticise that 'law' in this manner -- that is, by treating it as a putative truth.


Traditional Thinkers have always regarded the LOI a deep metaphysical truth about everything in existence -- that is, as a 'necessary truth'. However, as will be argued in Essay Twelve Part One, all such metaphysical theses are non-sensical and incoherent. Indeed, their denial is equally non-sensical (irony intended).22


Small wonder, then, that Trotsky's 'analysis' collapses so readily into incoherence.


The weaknesses of the LOI in fact lie elsewhere.23


The Anti-Dialectical Knock-Out Punch


Finally, it is worth noting that the fact that objects in the world undergo constant change can't in general be used to refute any of the above points since, no matter how fast anything changes, whatever it is identical with will change equally quickly.


In that case, the LOI is no enemy of change.


With that observation, much of 'materialist dialectics' falls apart.


[Further details are spelled out in Note 5b, and in Essay Eight Parts One, Two, and Three.]


Traditional Versus Modern FL


Traditional Logic Defective?


There is a serious, non-academic point at stake here. Traditional AFL, criticised by Trotsky, not only ignored complex inferences inexpressible in syllogisms, it totally failed to cope with relational expressions, quantifiers expressing multiple generality, internal and external negation and scope ambiguity. [This links to a PDF.] That was partly because of the way that quantifier expressions themselves had been interpreted by earlier logicians, who, with their slavish adherence to the traditional grammar of subject and predicate, helped cripple logic for over two thousand years. [On the origin of some of these confusions, see Barnes (2009).]


It is no exaggeration to say that much of Traditional Philosophy (i.e., Metaphysics) depends on antiquated and garbled logic like this. In which case, over two millennia of philosophical confusion -- including much that is found in Hegel -- largely derives from what is in effect sub-Aristotelian Logic.


Now, many of the 'difficulties' outlined in the last three Essays (i.e., Essays Four and Five, as well as this one) are a direct consequence of the crude way that quantifiers, relational expressions and tense operators had been interpreted (or ignored) by traditional and by dialectical logicians. In fact, progress toward unravelling pseudo-problems like this could only begin after Frege had completely re-laid the foundations of FL 130 years ago. As noted earlier, this salient fact has yet to register with most dialecticians -- no matter how many times they are apprised of it.


At first sight, considerations like these might appear to be dry, impractical and academic, of no interest to revolutionaries. However, if Marxists plan to use a radically flawed system (DL) in their endeavour to help change the world then this is relevant.24


Indeed, the long-term lack of success 'enjoyed' by Dialectical Marxism for the best part of a century suggests that DL has in fact been a millstone around its neck.


History has so far delivered an unambiguous verdict in this respect: DL has been tested in practice and found wanting.


DL Superior To FL?


Essay Four began by asking which one of the two rival logics (DL or FL) is the more adequate for use in science, and which could most easily accommodate change, identity and motion. It is now quite clear (from Essays Four to Seven Part Three) that DL is vastly inferior to FL in every single department; it is incapable of handling even the simplest forms of change, to say nothing of more complex developments -- or even of describing them! This is partly because (i) it relies on a garbled version of AFL, compounded by a confused metaphysic which has itself been hobbled by the impenetrable jargon invented by Hegel, and partly because (ii) its devotees unwisely undermine ordinary language.


In fact, DL is so fundamentally defective that it can't even cope with a simple bag of sugar -- let alone "long drawn out processes".


Small wonder then that it has seriously hindered the scientific development of Marxism.


FL And A Fragmented And Static View Of Reality


It might be felt at this stage that it is hardly surprising that the views expressed here and elsewhere at this site reach the conclusions they do since they depend on analysis -– that is, they are based on a fragmentary view of reality which splits the world and its contents into separate and un-mediated compartments. Naturally, when divorced from the whole, reality is going to appear paradoxical. Only against a wider background is it possible to comprehend the world correctly. In broader contexts, at different levels of abstraction and generality, the contradictory nature of objects and processes becomes much easier to appreciate, as indeed are the inadequacies of FL and the LOI.


This objection introduces the centrally important DM-concept: the "Totality". This terminally nebulous notion will be discussed in detail in Essay Eleven Part One. The idea that FL (ancient or modern) trades on a static view of the world was batted out of the park in Essay Four Part One.


Be this as it may, many DM-advocates direct our attention toward the 'three laws of dialectics' which Engels helped codify. These 'laws' they feel more accurately encapsulate the core ideas expressed in DM.


It is to this topic that I now turn.




1. As we will see, Hegel did at least mention identity in his critique of the LOI, even though it is clear from what he said that not only had he given this topic insufficient thought, he advanced superficial and erroneous claims about it. [A fuller consideration of Hegel's 'analysis' of identity can be found in Essays Twelve (summaries here, here and here), and Eight Part Three. I have also added additional thoughts on this topic to Appendix D of Essay Three Part One.]


Hegel's alleged denial of the LOI and the LOC are examined in Pippin (1978), Hanna (1986), and Hahn (2007). Even though these two authors struggle heroically to make Hegel comprehensible on this and other issues, it is difficult to tell whether they have succeeded or not, or both. [I will add some thoughts on these confused attempts to Essay Eight Part Three.]


It is also worth pointing out that the LOI was unknown to Aristotle.


[Unfortunately, since this Essay was originally published, the article at the above link has been changed; the one I referenced above can be found here. On this, see also here.]


After Another Edit: The Wikipedia article now attributes this 'law' to Aristotle's Metaphysics, but it is quite clear that it hasn't been stated there in its hackneyed form (i.e., A = A). Indeed, if anything, Aristotle derides this 'law':


"Now 'why a thing is itself' is a meaningless inquiry (for -- to give meaning to the question 'why' -- the fact or the existence of the thing must already be evident -- e.g., that the moon is eclipsed -- but the fact that a thing is itself is the single reason and the single cause to be given in answer to all such questions as why the man is man, or the musician musical, unless one were to answer 'because each thing is inseparable from itself, and its being one just meant this' this, however, is common to all things and is a short and easy way with the question)." [Metaphysics Book VII, Part 17. This can be found in Aristotle (1984b), p.1643. Bold added.]


Quoting this as an example of the use of, or as an allusion to, this 'law' would be to distort what Aristotle says; so I have added the following comment to the Wikipedia article (slightly edited):


And the quotation takes this 'law' out of context, for not only does Aristotle not mention 'identity', he specifically talks about predication (and since identity is a relation, he can't be talking about identity here):


"Let us state what, 'i.e. what kind of thing, substance should be said to be', taking once more another starting-point; for perhaps from this we shall get a clear view also of that substance which exists apart from sensible substances. Since, then, substance is a principle and a cause, let us pursue it from this starting-point. The 'why' is always sought in this form -- 'why does one thing attach to some other?' For to inquire why the musical man is a musical man, is either to inquire -- as we have said why the man is musical, or it is something else. Now 'why a thing is itself' is a meaningless inquiry (for -- to give meaning to the question 'why') the fact or the existence of the thing must already be evident -- e.g. that the moon is eclipsed -- but the fact that a thing is itself is the single reason and the single cause to be given in answer to all such questions as why the man is man, or the musician musical, unless one were to answer 'because each thing is inseparable from itself, and its being one just meant this'; this, however, is common to all things and is a short and easy way with the question). But we can inquire why man is an animal of such and such a nature. This, then, is plain, that we are not inquiring why he who is a man is a man. We are inquiring, then, 'why something is predicable of something' (that it is predicable must be clear; for if not, the inquiry is an inquiry into nothing). E.g. why does it thunder? This is the same as 'why is sound produced in the clouds?' Thus the inquiry is about the predication of one thing of another. And why are these things, i.e. bricks and stones, a house? Plainly we are seeking the cause. And this is the essence (to speak abstractly), which in some cases is the end, e.g. perhaps in the case of a house or a bed, and in some cases is the first mover; for this also is a cause. But while the efficient cause is sought in the case of genesis and destruction, the final cause is sought in the case of being also." [Ibid., bold emphasis added.]


So, I think the article needs amending.


Finally, since this 'law' is foreign to Aristotle, how can the author of this article say the following?


"The law of identity has deep impact on Aristotle's ethics as well. In order for a person to be morally praiseworthy or blameworthy for an action, he or she must be the same person before the act as during the act and after the act. Without the law of identity, Aristotle notes, there can be no responsibility for vice."


Personal identity isn't the same as the 'law of identity'.


However, I couldn't find in the Nicomachean Ethics anything like this reference to personal identity; so perhaps the author of this article will provide an exact reference?


In fact the Wikipedia article is highly misleading. For example, it attributes this phrase to Aristotle, which can't be found in Metaphysics VII Part 17: "a fixed constant nature of sensible things", and as such misrepresents what he was saying.


Update August 2011: I have just read Deborah Modrak's book on Aristotle (i.e., Modrak (2001)); this author devotes an entire section to Aristotle's views on identity -- i.e., pp.194-98. However, Modrak in fact concentrates on Aristotle's views on sameness; identity is conspicuous by its absence. Certainly, there is no mention of the LOI.


[Incidentally, I have added detailed comments on the only two other passages found in Aristotle's work (i.e., from Topics and from De Sophistici Elenchi), which could conceivably be recruited to the cause of saddling him with this Medieval 'Law', to Essay Four Part One.]


Nevertheless, the defects of the LOI lie elsewhere; these are outlined in Wittgenstein (1972), pp.97, 105-07, and Wittgenstein (1958), pp.84-85, 91, 111. [Cf., Glock (1996), pp.164-69.]


The best analysis of Wittgenstein's criticisms of identity can be found in White (1978). See also Marion (1998), pp.48-72 for an extended discussion. On identity in general, see Geach (1967, 1968, 1973, 1975, 1990), Griffin (1977), Noonan (1980, 1997, 2022) and Williams (1979, 1989, 1992). See also Deutsch (2018).


There is as yet no definitive account of Wittgenstein's use words for identity; however Roger White's long-awaited book should cast considerable light on this topic. [Cf., White (2006), and (forthcoming).]


There is an excellent summary in Glock (1996), pp.256-64, a more considered account in Diamond (1991) and in several articles published in Crary and Read (2000) -- for example, Conant (2000). See also Goldfarb (1997) and McGinn (1999).


Here are just a few examples of the extremely repetitive nature of this part of dialectics:


"[T]he first of [the universal Laws of Thought], the maxim of Identity, reads: Everything is identical with itself, A = A…." [Hegel (1975), p.167, §115.]


"In this remark, I will consider in more detail identity as the law of identity which is usually adduced as the first law of thought.


"This proposition in its positive expression A = A is, in the first instance, nothing more than the expression of an empty tautology." [Hegel (1999), p.413, §875.]


"Abstract Identity (a = a…) is likewise inapplicable in organic nature. The plant, the animal, every cell is at every moment of its life identical with itself and yet becoming distinct from itself….The law of identity in the old metaphysical sense is the fundamental law of the old outlook: a = a." [Engels (1954), pp.214-15.]


"The 'fundamental laws of thinking' are considered to be three in number: 1) The Law of Identity… [which] states that 'A is A' or A = A…." [Plekhanov (1908), p.89.]


"…Hegel elucidates the one-sidedness, the incorrectness of the 'law of identity' (A = A)…." [Lenin (1961), p.134.]


"Formal Logic starts from the proposition that A is always equal to A. We know that this law of identity contains some measure of truth…. Now…when we go to reality and look for evidence of the truth of the proposition: A equals A…we find that the opposite of this axiom is far closer to the truth." [Novack (1971), pp.32-33.]


"Formal Logic asserts: 'A is A'. Dialectical Logic is not saying 'A is not-A'…. It says: A is indeed A, but A is also not-A precisely so far as the proposition 'A is A' is not a tautology but has real content." [Lefebvre (1968), p.41.]


"The Law of identity is usually expressed in the form, A is A. That is, each thing is identical with itself." [Somerville (1946), p.183.]


"The Aristotelian conception of the laws basic to correct thinking may be stated as follows: 1. Law of Identity: Each existence is identical with itself. A is A…." [Somerville (1967), pp.44-45.]


"Classical, Aristotelian logic takes as its fundamental premise the Law of Identity, the statement that a thing is identical with itself. Expressed in a formula: A is A…. In Aristotle's formal logic A is A, and never non-A. In Hegel's dialectics A is A as well as non-A." [Baghavan (1987), pp.75-76.]


"The biggest contradiction of all lies in the fundamental premises of formal logic itself…. The basic laws…are:


1) The law of Identity ('A' = 'A')…." [Woods and Grant (1995/2007), pp.90-91. (This is found on p.95 of the 2nd edition.)]


"Dialectics, or the logic of motion, is distinct from formal or static logic. Formal logic is based on three fundamental laws:


"(a) The law of identity: A is equal to A; a thing is always equal to itself." [Mandel (1979), p.160.]


"The laws of logic are based on two main propositions. The first is that of identity or of self-conformity. The proposition very simply states: 'A is A,' that is, every concept is equal to itself. A man is a man; a hen is a hen; a potato is a potato. This proposition forms one basis of logic. The second main proposition is the law of contradiction, or as it is also called, the law of the excluded middle. This proposition states: 'A is either A or not A.' It cannot be both at the same time. For example: Whatever is black is black; it cannot at the same time be black and white. A thing -- to put it in general terms -- cannot at the same time be itself and its opposite. In practice it therefore follows that if I draw certain conclusions from a given starting point and contradictions arise, then there are errors in thinking or my starting point was wrong. If from some correct premises I come to the conclusion that 4 is the same as 5, then I deduce from the law of contradiction that my conclusion is false.


"So far all appears to be clear and certain. What can be a clearer law than that man is man, a rooster a rooster, that a thing is always the same thing? It even appears to be absolutely certain that a thing is either large or small; either black or white, that it cannot be both at the same time, that contradictions cannot exist in one and the same thing.


"Let us now consider the matter from the standpoint of a higher doctrine of thought, from the standpoint of dialectics. Let us take the first law which we have developed as the foundation of logic: A is A. A thing is always the same thing. Without testing this law, let us consider another one which we have already mentioned, the law of Heraclitus which says 'Everything is in flux,' or 'One cannot ascend the same river twice.' Can we say that the river is always the same? No, the law of Heraclitus says the opposite. The river is at no moment the same. It is always changing. Thus one cannot twice nor, more exactly, even once ascend the same river. In short: the law 'A is A' in the last analysis is valid only if I assume that the thing does not change. As soon as I consider the thing in its change, then A is always A and something else; A is at the same time not-A. And this in the last analysis holds for all things and events." [Thalheimer (1936), pp.88-89.]


"The first principle [the 'Law of Identity' -- RL] , then, declares that A is A, or to speak mathematically, every quantity is equal to itself. In plain English: a thing is what it is; no thing is what it is not....


"[The old logic] insists on its first, second and third law, on its identity, its law of contradiction and excluded third, which must be either straight or crooked, cold or warm and excludes all intermediary conceptions." [Dietzgen (1906), pp.386-89.]


"The central principle on which formal logic is built can be expressed in a simple formula that at first glance appears to be a self-evident truth 'A equals A'." [Conner (1992), p.22.]


"The basic principles of this Aristotelian or formal logic were the 'law of identity' and the 'law of non-contradiction'. The 'law of identity' stated, in symbolic terms, that A is equal to A, or an ounce of gold equals an ounce of gold, or, taking a unique object..., Leonardo da Vinci's Mona Lisa is equal to Leonardo da Vinci's Mona Lisa. The 'law of non-contradiction' stated that A cannot be equal to non-A, it makes no sense to say that an ounce of gold is not an ounce of gold or the Mona Lisa is not the Mona Lisa. On the basis of these apparently 'obvious' propositions a system of logic or sound reasoning was erected, exemplified by the syllogism." [Molyneux (2012), p.43. Quotation marks altered to conform with the conventions adopted at this site. Italic emphases in the original.]


"Dialectics is quite simply the logic of motion, or the logic of common sense to activists in the movement. We all know that things don't stand still, they change. But there is another form of logic which stands in contradiction to dialectics, which we call 'formal logic', which again is deeply embodied in capitalist society. It is perhaps necessary to begin by describing briefly what this method implies.


"Formal logic is based on what is known as the 'law of identity', which says that 'A' equals 'A' -- i.e. that things are what they are, and that they stand in definite relationships to each other. There are other derivative laws based on the law of identity; for example, if 'A' equals 'A', it follows that 'A' cannot equal 'B', nor 'C'....


"Whereas the formal logician will say that 'A' equals 'A', the dialectician will say that 'A' does not necessarily equal 'A'. Or to take a practical example that Trotsky uses in his writings, one pound of sugar will not be precisely equal to another pound of sugar. It is a good enough approximation if you want to buy sugar in a shop, but if you look at it more carefully you will see that it's actually wrong." [John Pickard, quoted from here.]


"The philosophical underpinning of this pursuit of knowledge was grounded in the empirical method, which guided the scientific inquiry into the interactions conceived of as external to these discrete and now well-defined entities. The law of identity was critical to the project: A thing is always equal to or identical with itself. Or stated in algebraic terms (sic): A equals A. One corollary of the idea that A is always identical to A is that A can never equal not-A." [Quoted from here; accessed 21/05/18.]


Examples like these can be multiplied almost indefinitely. Even though there are several minor differences in emphasis between them, the basic point of the above comments is reasonably clear: DM-theorists have fixated on a superficial form of the LOI, one they copy from each other generation after generation. Seldom do they bother to check that what they are criticising even remotely resembles anything taken from a logic text written in the last 120 years, or even one written by Aristotle!


In at least this respect, DM-authors are (ironically) identical. And, as we will see, in response to Hegel (here), identity statements aren't tautologies. Moreover, as we will see in this Essay, the LOI doesn't preclude change.


As Essay Four Part One demonstrated in detail, this tactic is part of a long and disreputable tradition religiously observed by 'Materialist Dialecticians': Define the basics of logic in a completely fanciful way, ridicule them, and then proclaim the superiority of DL over this straw man. Here is John Molyneux again:


"Marxist materialism is repeatedly attacked by the method of oversimplifying and caricaturing it to the point where it is obviously false...." [Molyneux (2012), p.36.]


And yet, this is precisely what DM-apologists do with respect to FL!


Novack's prize-winning entry in this respect, however, goes as follows:


"There are three fundamental laws of formal logic. First and most important is the law of identity. This law can be stated in various ways such as: A thing is always equal to or identical with itself. In algebraic terms: A is equal to A.… If a thing is always and under all conditions equal or identical with itself, it can never be unequal or different from itself. This conclusion follows logically and inevitably from the law of identity. If A always equals A, it can never equal non-A." [Novack (1971), p.20. Paragraphs merged. What this has to do with 'algebra' Novack annoyingly kept to himself.]


Clearly Novack failed to consider these counter-examples to his 'logical' conclusion:


N1: The number of volumes of Das Kapital is equal to the number of goals in a hat-trick.


N2: There were equal numbers of Union and non-Union members at the meeting last night.


N3: Although NN and MM have different disabilities they came equal first in the 100 metres final at the Para-Olympics, sharing the Gold Medal.


N4: Those two comrades sold equal numbers of papers on two different demonstrations last week.


N5: The author of Novack (1971) is identical to the comrade who penned the words in the last quotation above.


None of these suggests that the items they allude to can never change -- but, when they do change, anything identical to them will change equally quickly.


Otherwise, they weren't identical -- or, of course, they will cease to be identical with whatever they used to be!


Apart from Woods and Grant (1995/2007), none of the above theorists referred their readers to a single logic text (save those written by Hegel and other 'dialectical logicians'); worse still, not one of them bothered to quote Aristotle, even though many attribute this idea to him!


However, several of the above dialecticians at least mentioned the word "identity", but they then confused it with equality. Hence, most of the criticisms levelled against Trotsky in this Essay apply equally to them (no pun intended).


But, Woods and Grant go further:


"Firstly, let us note that the appearance of a necessary chain of reasoning, in which one step follows from another, is entirely illusory. The law of contradiction merely restates the law of identity in a negative form. The same is true of the law of the excluded middle. All we have is a repetition of the first line in different ways. The whole thing stands or falls on the basis of the law of identity ('A' = 'A'). At first sight this is incontrovertible, and, indeed, the source of all rational thought. It is the Holy of Holies of Logic, and not to be called into question. Yet called into question it was, and by one of the greatest minds of all time (sic).


"There is a story by Hans-Christian Andersen called The Emperor's New Suit of Clothes, in which a rather foolish emperor is sold a new suit by a swindler, which is supposed to be very beautiful, but invisible. The gullible emperor goes about in his fine new suit, which everyone agrees is exquisite, until one day a little boy points out that the emperor is, in fact, stark naked. Hegel performed a comparable service to philosophy in his critique of formal logic. Its defenders have never forgiven him for it." [Woods and Grant (1995/2007), p.91. (This can be found on pp.95-96 of the 2nd edition.) Quotation marks altered to conform with the conventions adopted at this site.]


This is typical hyperbole from Woods and Grant, who seem to think that modern logicians will be in any way fazed by the confusions that litter Hegel's badly misnamed book (on 'logic'). Indeed, the vast majority of them pay no more attention to Hegel, or Woods and Grant, than Woods and Grant themselves pay to the work of, say, Frédéric Bastiat (and probably for the same reason). Furthermore, based on Woods and Grant's execrable book -- even if any Logicians could be bothered to read it -- in this respect they aren't likely to change (irony intended).


Moreover, as we will see here, the LOC and the LOI aren't connected in the way that Woods and Grant clearly think they are. They have simply copied this idea off Hegel and Engels without bothering to check whether the 'negative form of the LOI' one implies the LOC. And, as we saw in Essay Four, neither AFL nor MFL is based on the LOI (although there are several systems of logic where the LOI plays a key role). In their haste to blame FL for everything but the Black Death and the demise of the Dinosaurs, Woods and Grant failed to acknowledge this salient fact -- always assuming they were even aware of it. [On this, also see my comments over at Wikipedia.]


[LOC = Law of Non-Contradiction; FL = Formal Logic; AFL = Aristotelian FL; MFL = Modern FL.]


2. There is evidence that Trotsky read and studied Hegel's Logic (see the Introduction to Trotsky (1986), and the reference to the Logic in Trotsky (1986) itself -- e.g., p.98). Nevertheless, even if questions about the accuracy of the title of Hegel's two books on 'logic' are put to one side, it is reasonably clear that apart from the Hegel's Logic Trotsky seems not to have consulted a single logic text before he began issuing ex cathedra pronouncements about it.


Having said that, Isaac Deutscher tells us that in the early 1930s, in preparation for a book he intended to write on Lenin, Trotsky "went back to classics of logic and dialectics, Aristotle and Descartes, but especially to Hegel" (Deutscher (1970), p.267). If so, Trotsky might have studied Aristotle's logical texts; if he did, it is plain that his view of Aristotle was heavily skewed by Hegel's misrepresentations and errors.


It is also possible that Jean van Heijenoort, a member of Trotsky's entourage, subsequently an expert logician, gave him some advice; but if he did, there is precious little evidence that much of it sank in. On this, see Van Heijenoort (1978), and Feferman (1993).


In this respect Trotsky wasn't alone; DM-theorists in general are only too happy to regale us with their home-spun ideas about FL, fables whose pristine simplicity hasn't been compromised by the arduous task of opening a single book devoted to AFL, let alone MFL.


Finally, those who think that AFL is based on the LOC (etc.) should consult Lear (1980), pp.98-114, where they will find a more balanced and scholarly account.


2a. Although I used part of the following quotation earlier, there is a passage in Engels's DN which closely resembles what Trotsky was saying:


"Abstract identity (a = a; and negatively, a cannot be simultaneously equal and unequal to a) is likewise inapplicable in organic nature. The plant, the animal, every cell is at every moment of its life identical with itself and yet becoming distinct from itself, by absorption and excretion of substances, by respiration, by cell formation and death of cells, by the process of circulation taking place, in short, by a sum of incessant molecular changes which make up life and the sum-total of whose results is evident to our eyes in the phases of life -- embryonic life, youth, sexual maturity, process of reproduction, old age, death. The further physiology develops, the more important for it become these incessant, infinitely small changes, and hence the more important for it also the consideration of difference within identity, and the old abstract standpoint of formal identity, that an organic being is to be treated as something simply identical with itself, as something constant, becomes out of date. [In the margin of the manuscript occurs the remark: 'Apart, moreover, from the evolution of species.'] Nevertheless, the mode of thought based thereon, together with its categories, persists. But even in inorganic nature identity as such is in reality non-existent. Every body is continually exposed to mechanical, physical, and chemical influences, which are always changing it and modifying its identity. Abstract identity, with its opposition to difference, is in place only in mathematics -- an abstract science which is concerned with creations of thought, even though they are reflections of reality -- and even there it is continually being sublated. Hegel, Enzyklopädie, I, p. 235. [This is a reference to Hegel (1975), pp.169-70, §117; see below -- RL.] The fact that identity contains difference within itself is expressed in every sentence, where the predicate is necessarily different from the subject; the lily is a plant, the rose, is red, where, either in the subject or in the predicate, there is something that is not covered by the predicate or the subject. Hegel, p.231. [This is a reference to Hegel (1975), pp.166-68, §115; see below -- RL.] That from the outset identity with itself requires difference from everything else as its complement, is self-evident.


"Continual change, i.e., sublation of abstract identity with itself, is also found in so-called inorganic nature. Geology is its history. On the surface, mechanical changes (denudation, frost), chemical changes (weathering); internally, mechanical changes (pressure), heat (volcanic), chemical (water, acids, binding substances); on a large scale – upheavals, earthquakes, etc. The slate of today is fundamentally different from the ooze from which it is formed, the chalk from the loose microscopic shells that compose it, even more so limestone, which indeed according to some is of purely organic origin, and sandstone from the loose sea sand, which again is derived from disintegrated granite, etc., not to speak of coal.


"The law of identity in the old metaphysical sense is the fundamental law of the old outlook: a = a. Each thing is equal to itself. Everything was permanent, the solar system, stars, organisms. This law has been refuted by natural science bit by bit in each separate case, but theoretically it still prevails and is still put forward by the supporters of the old in opposition to the new: a thing cannot simultaneously be itself and something else. And yet the fact that true, concrete identity includes difference, change, has recently been shown in detail by natural science (see above).


"Abstract identity, like all metaphysical categories, suffices for everyday use, where small dimensions or brief periods of time are in question; the limits within which it is usable differ in almost every case and are determined by the nature of the object; for a planetary system, where in ordinary astronomical calculation the ellipse can be taken as the basic form for practical purposes without error, they are much wider than for an insect that completes its metamorphosis in a few weeks. (Give other examples, e.g., alteration of species, which is reckoned in periods of thousands of years.) For natural science in its comprehensive role, however, even in each single branch, abstract identity is totally inadequate, and although on the whole it has now been abolished in practice, theoretically it still dominates people’s minds, and most natural scientists imagine that identity and difference are irreconcilable opposites, instead of one-sided poles which represent the truth only in their reciprocal action, in the inclusion of difference within identity." [Engels (1954), pp.214-16. Bold emphases alone added. Quotation marks altered to conform with the conventions adopted at this site.]


Which only serves to confirm the allegation made above that DM-theorists uncritically copy ideas off one another.


Here are the relevant sections of Hegel's Shorter Logic:


"Difference is first of all (1) immediate difference, i.e. Diversity or Variety. In Diversity the different things are each individually what they are, and unaffected by the relation in which they stand to each other. This relation is therefore external to them. In consequence of the various things being thus indifferent to the difference between them, it falls outside them into a third thing, the agent of Comparison. This external difference, as an identity of the objects related, is Likeness; as a non-identity of them, is Unlikeness.


"The gap which understanding allows to divide these characteristics is so great that although comparison has one and the same substratum for likeness and unlikeness, which are explained to be different aspects and points of view in it, still likeness by itself is the first of the elements alone, viz., identity, and unlikeness by itself is difference.


"Diversity has, like Identity, been transformed into a maxim: 'Everything is various or different': or 'There are no two things completely like each other'. Here Everything is put under a predicate, which is the reverse of the identity attributed to it in the first maxim: and therefore under a law contradicting the first. However, there is an explanation. As the diversity is supposed due only to external circumstances, anything taken per se is expected and understood always to be identical with itself, so that the second law need not interfere with the first. But, in that case, variety does not belong to the something or everything in question: it constitutes no intrinsic characteristic of the subject: and the second maxim on this showing does not admit of being stated at all. If, on the other hand, the something itself is, as the maxim says, diverse, it must be in virtue of its own proper character: but in this case the specific difference, and not variety as such, is what is intended. And this is the meaning of the maxim of Leibnitz (sic).


"When understanding sets itself to study Identity, it has already passed beyond it, and is looking at Difference in the shape of bare Variety. If we follow the so-called law of Identity, and say, The sea is the sea, The air is the air, The moon is the moon, these objects pass for having no bearing on one another. What we have before us therefore is not Identity, but Difference. We do not stop at this point, however, or regard things merely as different. We compare them one with another, and then discover the features of likeness and unlikeness. The work of the finite sciences lies to a great extent in the application of these categories, and the phrase 'scientific treatment' generally means no more than the method which has for its aim comparison of the objects under examination. This method has undoubtedly led to some important results; we may particularly mention the great advance of modern times in the provinces of comparative anatomy and comparative linguistics. But it is going too far to suppose that the comparative method can be employed with equal success in all branches of knowledge. Not -- and this must be emphasised -- can mere comparison ever ultimately satisfy the requirements of science. Its results are indeed indispensable, but they are still labours only preliminary to truly intelligent cognition.


"If it be the office of comparison to reduce existing differences to Identity, the science which most perfectly fulfils that end is mathematics. The reason of that is that quantitative difference is only the difference which is quite external. Thus, in geometry, a triangle and a quadrangle, figures qualitatively different, have this qualitative difference discounted by abstraction, and are equalised to one another in magnitude. It follows from what has been said formerly about mere Identity of understanding that, as has also been pointed out (s.99), neither philosophy nor the empirical sciences need envy this superiority of Mathematics.


"The story is told that when Leibnitz (sic) propounded the maxim of Variety, the cavaliers and ladies of the court, as they walked round the garden, made efforts to discover two leaves indistinguishable from each other, in order to confute the law stated by the philosopher. Their device was unquestionably a convenient method of dealing with metaphysics -- one which has not ceased to be fashionable. All the same, as regards the principle of Leibnitz (sic), difference must be understood to mean not an external and indifferent diversity merely, but difference essential. Hence the very nature of things implies that they must be different." [Hegel (1975), pp.169-70, §117.]


"The Essence lights up in itself or is mere reflection: and therefore is only self-relation, not as immediate but as reflected. And that reflex relation is self-identity.


"This identity becomes an Identity, in form only, or of the understanding, if it be held hard and fast, quite aloof from difference. Or, rather, abstraction is the imposition of this Identity of form, the transformation of something inherently concrete into this form of elementary simplicity. And this may be done in two ways. Either we may neglect a part of the multiple features which are found in the concrete thing (by what is called analysis) and select only one of them; or, neglecting their variety, we may concentrate the multiple character into one.


"If we associate Identity with the Absolute, making the Absolute the subject of a proposition, we get: The Absolute is what is identical with itself. However, true this proposition may be, it is doubtful whether it be meant in its truth: and therefore it is at least imperfect in the expression. For it is left undecided, whether it means the abstract Identity of understanding- abstract. that is, because contrasted with the other characteristics of Essence -- or the Identity which is inherently concrete. In the latter case, as will be seen, true identity is first discoverable in the Ground, and, with a higher truth, in the Notion. Even the word Absolute is often used to mean more than 'abstract'. Absolute space and absolute time, for example, is another way of saying abstract space and abstract time.


"When the principles of Essence are taken as essential principles of thought they become predicates of a presupposed subject, which, because they are essential, is 'everything'. The propositions thus arising have been stated as universal Laws of Thought. Thus the first of them, the maxim of Identity, reads: Everything is identical with itself, A = A: and negatively, A cannot at the same time be A and Not-A. This maxim, instead of being a true law of thought, is nothing but the law of abstract understanding. The propositional form itself contradicts it: for a proposition always promises a distinction between subject and predicate; while the present one does not fulfil what its form requires. But the Law is particularly set aside by the following so-called Laws of Thought, which make laws out of its opposite. It is asserted that the maxim of Identity, though it cannot be proved, regulates the procedure of every consciousness, and that experience shows it to be accepted as soon as its terms are apprehended. To this alleged experience of the logic books may be opposed the universal experience that no mind thinks or forms conceptions or speaks in accordance with this law, and that no existence of any kind whatever conforms to it.


"Utterances after the fashion of this pretended law (A planet is a planet; Magnetism is magnetism; Mind is Mind) are, as they deserve to be, reputed silly. That is certainly a matter of general experience. The logic which seriously propounds such laws and the scholastic world in which alone they are valid have long been discredited with practical common sense as well as with the philosophy of reason.


"Identity is, in the first place, the repetition of what we had earlier as Being, but as become, through supersession of its character of immediateness. It is therefore Being as Ideality. It is important to come to a proper understanding on the true meaning of Identity; and, for that purpose, we must especially guard against taking it as abstract identity, to the exclusion of all Difference. That is the touchstone for distinguishing all bad philosophy from what alone deserves the name of philosophy. Identity in its truth, as an Ideality of what immediately is, is a high category for our religious modes of mind as well as all other forms of thought and mental activity. The true knowledge of God, it may be said, begins when we know him as identity -- as absolute identity. To know so much is to see all the power and glory of the world sinks into nothing in God's presence, and subsists only as the reflection of his power and his glory. In the same way, Identity, as self-consciousness, is what distinguishes man from nature, particularly from the brutes which never reach the point of comprehending themselves as 'I'; that is, pure self-contained unity. So again, in connection with thought, the main thing is not to confuse the true Identity, which contains Being and its characteristics ideally transfigured in it, with an abstract Identity, identity of bare form. All the charges of narrowness, hardness, meaninglessness, which are so often directed against thought from the quarter of feeling and immediate perception rest on the perverse assumption that thought acts only as a faculty of abstract Identification.


"The Formal Logic itself confirms this assumption by laying down the supreme law of thought (so-called) which has been discussed above. If thinking were no more than an abstract Identity, we could not but own it to be a most futile and tedious business. No doubt the notion, and the idea too, are identical with themselves: but identical only in so far as they at the same time involve distinction." [Ibid., pp.166-68, §115. Bold emphasis added.]


There are similar, even longer and considerably more opaque passages in Hegel (1999).


Hegel's confused attempt to 'analyse' the LOI is examined in detail in Essays Eight Part Three and Twelve Parts Five and Six (not yet published).


3. That is, if they are even aware of them! On this, see Note 8, below.


4. In fact, Trotsky's version went as follows:


S1(a): 'A' is equal to 'A'.


However, since not much seems to hang on Trotsky's use of single quotation marks (over and above his rather odd reference to the microscopic examination of the letters in question), I have largely ignored them in what follows.


5. Change in, or to, NN is irrelevant here; that is because, howsoever much her two roles alter, since NN occupies both at the same time it will always be true that "The Unison rep is identical with the STWC Treasurer". The same point applies, mutatis mutandis, to the other examples listed in the main body of this Essay. [See also Note 5b, below.]


5a. A substantival term is a common noun that in general (but not always) admits of number (i.e., it has a plural form) --, e.g., three books, two people, five comrades -- and where they do admit of number, they are often called 'count nouns'. Their distinguishing mark centres on the criteria of identity we use in each case. The following is what Professor Lowe had to say about this topic:


"...[N]ot all general terms are common names -- for instance, adjectival or characterizing general terms such as 'red' and 'circular' are not, nor are abstract nouns such as 'redness' and 'circularity' (if indeed the latter are deemed to be general terms, for an alternative view is that they are singular terms referring to abstract individuals). The distinguishing feature of common names -- sometimes also called substantival or sortal general terms -- is that they have associated with them, as a component of their meaning, a criterion of identity for the individuals to which they apply (see Lowe 1989, Ch. 2). A criterion of identity for individuals of a kind K is a principle which determines, for any individuals x and y of kind K, whether or not x and y are one and the same K. Thus, the criterion of identity for cities tells us that Paris and London are different cities, since they occupy different locations; and the criterion of identity for rivers tells us that the Isis and the Thames are the same river, since they flow from the same source to the same mouth. Different kinds of individuals, denoted by different sortal terms, very often have different criteria of identity governing them -- and in some cases there is philosophical debate as to precisely what these criteria are (for example, in the case of persons). Credit is once more due to Frege for recognizing the important role that criteria of identity have to play in the semantics of sortal terms." [Lowe, Internet Resource [2].]


Count nouns are to be distinguished from other common nouns which don't admit of number, e.g., mass nouns (such as chalk, cabbage, meat, etc. -- it makes no sense to refer to "two meat", or "three chalk". Of course, these can be converted into count nouns, as in "two cabbages", "three chalks" or "four meats" (the latter two of which are in general short for "three pieces or sticks of chalk" or "two varieties, portions, slices, or cuts of meat"). However, some mass nouns are also substantivals, e.g., gold, lead, and plastic.


Many of the mistakes dialecticians make in relation to identity originate in their failure to notice the different logic that applies to these two sorts of nouns. [The same lack of attention to detail vitiates, for example, Heraclitus's comments about stepping into a river. I will say more about that later in this Essay. I am told that Greek makes no distinction between mass and count nouns; maybe so, but then no Greek would ask another to count chalk, rain, or wine.]


On this in general, see here and here.


However, as with most things in philosophical logic, things are never quite so simple. On this, see Geach (1968), pp.39-41 (this in fact links to the 3rd edition, so the pages are different: pp.63-64), and Lowe (1989, 2015). [The complex logic of mass nouns has proven to be particularly difficult for modern logicians to grapple with. Although much progress has been made over the last forty of fifty years, I am far from convinced there is a definite work on this topic. It might indeed be a mistake to look for a single logic of mass nouns.]


5b. The material that used to be here has now been moved the main body of the Essay.


6. Hegel's egregious logical blunders have been exposed here, here, and here (and they will be completely demolished in Essay Twelve Parts Five and Six -- summaries here and here).


The material that used to be here has been moved to the main body of this Essay.


7. However, as is the case with other philosophical 'problems', Trotsky is in good company; philosophers and modern logicians also succeed in confusing equality with identity. In fact, this is a still highly neglected area in the Philosophy of Logic. There are signs, though, that this might be beginning to change; on that, see Sanford (2005).


In fact, astute logicians have been aware of such complexities for years, but have been opposed by traditionalists reluctant to change (no irony intended). On this, see Geach (1967, 1973, 1975, 1990), Griffin (1977), and Noonan (1980, 1997, 2022). See also Deutsch (2018).


8. As pointed out above, DM-theorists' comments on FL are exceedingly repetitive and almost entirely garbled (Graham Priest's work being a notable exception in this regard). In stark contrast to their attempts to master other areas of knowledge -- for example, classical and modern economics, science, history, politics and current affairs --, when it comes to FL, dialecticians display little or no comprehension even of Elementary Logic. Indeed, their writings almost invariably contain highly superficial and inaccurate characterisations of what turn out to be obsolete forms of AFL. Not surprisingly, such 'logical straw men' are easy to knock over. [On this, see Essay Four, here and here.]


It is instructive to compare this dishonest approach to FL with the justifiable condemnation that the very same DM-theorists level at analogously distorted views of revolutionary socialism found in the writings of the vast majority of Marx's bourgeois critics. Here, again, is John Molyneux:


"Marxist materialism is repeatedly attacked by the method of oversimplifying and caricaturing it to the point where it is obviously false...." [Molyneux (2012), p.36.]


And yet this is precisely what he and other DM-fans regularly do when they attempt to summarise, discuss or criticise FL.


[For some reason, the words "sauce", "goose", and "gander" come to mind here.]


In this regard, it is quite clear that John Rees, for instance, can't possibly have checked a single logic text (other than those mis-titled works written by Hegel, perhaps) before he wrote what he did about FL and the LOI in TAR. As also seems to be the case with most other Marxist critics of FL, Rees appears to have confined his 'research' on this topic to reading only what previous dialecticians had written about it, simply copying what he found there. [Indeed, a supporter of this site tried to point this out to him at a large public meeting in London in 1990; that was clearly a waste of breath. I tried to point this out to him a few weeks ago on Twitter, and he promptly blocked me!]


In modern symbols, one form of Leibniz's Law is as follows:


[1] (x)(y)((x = y) º (Fx ® Fy)).


[1] is otherwise known as the Indiscernibility of Identicals. Translated, it roughly reads: "Any two objects are identical if and only if they share the same properties" -– or, "…whatever is true of one is true of the other." This particular 'Law' won't be defended here for reasons outlined in Note 1. Its translation into ordinary language isn't happy on any reading. That alone shows it isn't equivalent to the ordinary use of "equal to", "the very same as", "identical with", or even "numerically identical with". [For complications, see Gallois (2003).]


Nevertheless, it is important to note that the use of the "=" sign in [1] has been strengthened by the presence of the bi-conditional "º"; hence, it isn't identical to Trotsky's use of the former sign, either (no irony intended).


Contrast [1] with the following version of the same 'Law':


[2] (x)(y)((F)(Fx º Fy) ® (x = y)).


[2] is otherwise known as the Identity of Indiscernibles.


[Unfortunately, Internet Explorer 10 doesn't seem able to display all the symbols the above linked article uses; IE 11 appears to be able to cope with them, though. Edge also reproduces them correctly.]


Loosely translated [2] reads: "Any two objects that share every property in common are identical." However, this version requires quantification across properties, which is controversial.


One of Trotsky's mistakes was to suppose that this 'Law' is empirically testable -- where, for example, (i) he supposed that the truth of S1 [i.e., "A is equal to A"] could be checked directly by the use of an eyeglass, and where (ii) he referred to the weighing of bags of sugar. Trotsky clearly regarded the failure of S1 to pass such tests as sufficient grounds for rejecting this 'Law' (or, at least, an 'absolute' application of it, especially where change is involved). However, it didn't seem to occur to him that an empirical test of Leibniz's Law is wholly inappropriate; it would be just as misguided as an empirical test of, say, "a + b = b + a" (i.e., commutativity over addition) in Mathematics. Anyone who thought to check such a rule in the same manner as Trotsky (irony intended) would be regarded as hopelessly confused, and rightly so.


[On the other hand, when this 'Law' is regarded as the expression of a rule, the temptation to think it can be tested in such a crude manner vanishes. On this, see Note 10, below. Of course, it is possible to use this 'Law' to test whether or not two or more bodies are identical, but that is a different matter (no irony intended!).


Incidentally, on this issue it won't do to point out that certain operators in mathematics don't commute since no one in their right or left mind would empirically test these, either. Of course, tests might be performed to see whether or not certain systems in nature displayed commutativity -- or, rather, whether certain rules adequately coped with such systems (for example, in Matrix Mechanics) -- but no test would or could be run on the principle itself. And, despite what dialecticians say, the same goes for the LEM, as it has allegedly been used/refuted in/by QM. More on that later.]


[QM = Quantum Mechanics; LEM = Law of Excluded Middle.]


Compare the two versions of Identity outlined above with the following:


[3] j(y) º [(x)((x = y) & j(x))].


[3] appears in Griffin (1977), p.1, which also contains a strengthened version of Leibniz's Law:


[4] (x)(y)[(x = y) º (j)(j(x) º j(y))].


[4] roughly reads: "Any two objects are identical if and only if for any property, one has it if and only if the other also has it." Griffin gives several other versions of [1] and [2] above, (ibid., p.2).


Incidentally, [3] roughly says "Anything true of some object is equivalently true of any object identical to it." [In fact, I have used this version in the present Essay to show that identity is no enemy of change. Because this rule doesn't require quantification across properties, it is, in my view, preferable to both [2] and [4].]


["" is the universal quantifier, equivalent to "All", "Any" or "Every"; "" is the existential quantifier, equivalent to "Some" or "At least one"; "º" is the sign for logical equivalence, i.e., "If and only if"; "j" and "F" are predicate variable letters; "®" is the implication arrow, equivalent to "if...then"; "x" and "y" are "bound" quantifier variables. For more on this, see, for example, Priest (2000) and Tomassi (1999). On quantifiers, see Note 13a. However, for an important word of warning about variables, see here.]


8a. And the same could be said -- but, with slightly less justification -- about Hegel; at least he used the right word, even if it is clear that he failed to grasp the complexity of this logical 'law', just as he failed to do justice to the ordinary words we have for identity and difference, etc.


8b. Incidentally, this doesn't confuse identity with identification, it merely reminds us that mastery of the latter requires a certain level of proficiency concerning the vocabulary associated with the former.  


9. This is from The Guardian newspaper (Wednesday, 18/10/1995):


"K2 appeared over the 40 million years or so that India has been colliding with greater Asia. It was 'discovered' (i.e., by the British) and designated K2 (Karakoram Peak 2) in 1856. The peak was granted the name [Mount Godwin-Austen] in 1888, after its first surveyor, Col Henry Haversham Godwin-Austen (1834-1923). The previous title is now preferred as being less imperialistic. Ordinarily a mountain would revert to its local name, but K2 is so remote that it appears never to have gained one." [Letter from S. McDiarmid, reprinted in Notes and Queries. See also here.]


This is perhaps a more interesting example of ordinary ascriptions of identity than those considered by Frege (for instance, with respect to the Evening Star and the Morning Star, etc.), in Frege (1892). [On that, see, for example, here and here (the latter links to a PDF).]


From the same newspaper:


"In Nepal they call it Sagarmatha. To the people of Tibet, it is Chomolungma, though the ruling Chinese prefer the variant Qomolangma. When the British first began mapping India, they knew it as Peak B, then as Peak XV. But in 1865, to honour the surveyor-general of India who first mapped it, Peak XV was given the name Mount Everest. And Everest the mountain has remained throughout much of the rest of the world to this day.


"Now China is launching a fresh effort to outlaw the name Everest. Accusing British colonialists of 'raping the sacred mountain of Tibetans by giving it a false name', Chinese newspapers are calling on the world to 'respect Tibetans' by using the 50th anniversary of the first ascent next year to recognise the mountain henceforward as Qomolangma.


"At first sight, the proposal does not seem unreasonable. There are, after all, lofty precedents for such renaming. The highest point in Africa, the summit of Kilimanjaro, which was once known as Kaiser Wilhelm Spitze, properly became Uhuru Peak. The world's second-highest peak, once Mount Godwin-Austen to the British, has become K2 (ironically this also dates from imperial survey days). But doubts about the new proposal soon creep in. If Everest is unacceptable, why should the world not prefer the Nepalese name to the Chinese or Tibetan one? Who are the Chinese, of all people, to accuse others of raping Tibet? And how is the 'English language hegemonism' of which China complains worse than its Chinese language equivalent?


"We hold no great brief for the name Everest, though it has to be said that the word has a fine ring to it. But the answer is to live and let live. If people prefer Chomolungma, let them use that name. If others want to stick with Everest, let them do so too. We have no problem with diversity, though the Chinese may. In the end, the world's greatest mountain is surely more important than any name that mere mortals give to it." [The Guardian, 20/11/02. Italics added; quotation marks altered to conform with the conventions adopted at this site.]


So, here we have several perfectly ordinary identities: Everest is identical to Chomolungma, Qomolangma, Peak B and Peak XV; Kilimanjaro is identical to Kaiser Wilhelm Spitze, Mt Cook is identical with Aoraki, and so on. Of course, we would ordinarily say things like "Everest is Chomolungma", or "Kilimanjaro is Kaiser Wilhelm Spitze", and the like, using the "is" of identity, here.


Furthermore, and as noted earlier, The Isis (not to be confused with those violent insurgents in the Middle East!) is identical to the River Thames as it flows through Oxford (UK). Hence, whatever change The Thames undergoes in this stretch of the river, The Isis undergoes identically. [This again shows that identity is no enemy of change.]


Other examples come rapidly to mind: Vladimir Ilyich Ulyanov is Lenin, Lev Davidovich Bronstein is Trotsky, Yigael Gluckstein is Tony Cliff, Cicero is Tully, George Eliot is Mary Anne Evans, Carter Dickson is John Dickson Carr, Currer Bell is Charlotte Brontë, Edgar Box is Gore Vidal, Mumbai is Bombay, Calcutta is Kolkata, Peking is Beijing, Siam is Thailand, Burma is Myanmar, Gold Coast is Ghana, Ceylon is Sri Lanka, Persia is Iran, Upper Volta is Burkina Faso, Ayers Rock is Uluru, The First Battle of Manassas is The First Battle of Bull Run, The Battle of Sharpsburg is the Battle of Antietam --, and, in August 2015, President Barak Obama told us that Mt McKinley is now Denali, and so on.


[Dozens of examples of pen names and pseudonyms are listed here. My current favourite re-naming is the Kuiper Belt object, Ultima Thule, which is now 486958 Arrokoth. It was renamed because "Ultima Thule" was the name given by the Nazis to a mythical homeland for the 'Aryan people'.]


In addition, the ordinary examples given earlier about absolute identity apply in this case too: so, whatever is true of Mt Everest is absolutely identical to whatever is said about Chomolungma. Even if this mountain changes (which it plainly does), those changes will apply (absolutely) equally to Everest and Chomolungma. Indeed, anything true of anyone climbing Everest is identically true of that person climbing Chomolungma.


So, concerning any two individuals, NN and MM, climbing Chomolungma, the following will be true:


Let NN and NM begin to climb the same mountain at the same time and also let them remain on it for two days. While still on that mountain, whatever changes it experiences (and this includes changes to all its properties and relations over these two days), this will always be the case: NN and NM are on identically the same mountain as each other.


More precisely (adapting an argument from earlier):


A1a: During temporal interval, T -- lasting m hours --, a set of predicates, P, is true of, or can be used to form true propositions about, (i) Mountain, M, (ii) All who climb it, and (iii) M, all on that mountain during T, and their relations with everything in the universe -- where P is comprised of the following elements: {P1, P2, P3,..., Pn}, and where n is indefinitely large.


A2a: NN and NM are both climbing M during any sub-interval, tk, of T, of arbitrary length.


A3a: During tk, a subset of P, namely {Pi..., Pk}, can be used to form true propositions about (i) M, (ii) All on M, and (iii) M, all on that mountain during T, and their relations with everything in the universe, including any changes that occur to one or both during tk (m ε ).


A4a: So, during tk, NN and NM are on absolutely identically the same mountain as one another, since, while they are on M, the valid applicability of every element of {Pi,..., Pk} remains the case throughout.


Now, every element of {Pi,..., Pk} -- i.e., these predicates, or what they 'reflect' -- can change 'dialectically' all the time, but that won't alter the result, since any changed element of {Pi,..., Pk} will also be an element of that sub-set, by the above definitions.


So, here we have an example of absolute identity that would remain such even in a Heraclitean universe.


[The same comments are also true of the other items listed above. The reader is left to work out the details for herself.]  


Instead of consulting Trotsky on identity -- or even worse, Hegel --, DM-fans would do well to begin their study of identity by reading Frege. [I.e., Frege (1892). However, it has to be said, even Frege tended to confuse identity with equality!]


Excellent introductions to Frege's thought can be found in Kenny (1995), Noonan (2000) and Weiner (1990, 1999); for a useful guide to the philosophical issues involved, see Linsky (1977). [For a much more detailed bibliography, see here.]


We now read the following in a relatively recent issue of the New Scientist:


"In 2003, at team...[in] Moscow discovered two distant elliptical galaxies just a whisker apart. Detailed analysis of the twins known as CSL-1, suggested that they were images of the same galaxy. The team suggested that the duplicate images were being created by a 'cosmic string'.... If one of these cosmic strings were to pass between Earth and a giant galaxy, the warping of space-time by the string would create a gravitational lens and form two identical images of the galaxy -- exactly like CSL-1.... Unfortunately for the proponents of cosmic strings, observations with Hubble on 12 January have revealed that CSL-1 is actually two different galaxies...." [New Scientist 189, 2537, 04/02/2006, p.21. Quotation marks altered to conform with the conventions adopted at this site. Paragraphs merged.]


Now, this is an excellent example of genuine science at work. Instead of 'solving' this problem in an a priori manner (à la Trotsky, or à la Hegel), declaring that things are never equal to themselves, astronomers were able to show that these images weren't of the same object, but of two different galaxies.


[However, one wonders what dialecticians would have said if these two images had been shown to be of the same galaxy.]


In the article posted at the above link (to the New Scientist) several examples were given of multiple images of identically the same object. In relation to which, consider the following line-of-argument:


G1: Image I1 is identical to image I2.


G2: Image I1 and image I2 are both of the same object.


G3: But, image I1 is an image of object O1.


G4: Image I2 is an image of object O2.


G5: Therefore, O1 is identical to O2.


[I hasten to add that this isn't the line-of-argument employed by the scientists mentioned in the above article!]


It isn't easy to see how dialecticians would be able to tell G1 and G2 apart (if these scientists had in fact found they were of the same object), nor account for the conclusion recorded in G5.


And, it would be no use pointing to the alleged limitations of the LOI here, since, no matter how much, or to what extent, objects O1 and O2 changed, they would still be identical, since they would change at an identical rate as 'one another' (being one object, not two)! Certainly, the images of these objects may or may not be identical, but these 'two' objects can't fail to be, since there is only one of them!


[As should now be obvious, this is just another example of the sort given earlier, which also underlines the fact that identity is no enemy of change.]


Of course, if it were now maintained that, on the basis of what Trotsky or Hegel said, objects O1 and O2 were nonetheless not identical (or they were both identical and not identical(!)), then that would fatally undermine this part of Astrophysics since it would at the very least throw into doubt the use of gravitational lensing in the above manner. If these objects were always non-identical, then images I1 and I2 would plainly be of two different objects, and the above inference would fail.


As noted earlier, it isn't easy to see how dialecticians can hold on to their criticism of the LOI without compromising, at a minimum, this part of Physics.


Furthermore, as we will see later, a desperate appeal to "approximate identity" would be to no avail here, either.


Here is another recent example of the ordinary use of words for identity in one of the sciences:


"More than 600,000 plant species have been deleted from the dictionary of life after the most comprehensive assessment carried out by scientists.


"For centuries, botanists from different parts of the world have been collecting and naming 'new' plants without realising that many were in fact the same. The humble tomato boasts 790 different names, for example, while there are 600 different monikers for the oak tree and its varieties. The result was a list of more than 1 million flowering plant species. Although experts have long known that it included many duplicates, no one was sure how many. Later this year, the study team, led by UK and US scientists, will announce that the real number of flowering plant species around the world is closer to 400,000.


"The project -- which has taken nearly three years -- was the number one request made by the 193 government members of the Convention on Biological Diversity at their meeting in 2002. There were concerns that without this work, it would be impossible to work out how many plants were under threat and how successful conservationists were in saving them. The information will also be vital for any organisation or researcher looking at 'economically important' plants, such as those for food and nutrition or medicine, said Alan Paton, assistant keeper of the herbarium at the Royal Botanic Gardens at Kew, west London, one of the four leading partners in the project.


"'On average, one plant might have between two and three names, which doesn't sound a great deal, but if you're trying to find information on a plant, you might not find all [of it] because you're only looking at one name,' Paton said. 'That's even more critical for economically useful plants: because they are more used, they tend to have more names.'


"In one example, researchers calculated that for the six most-used species of Plectranthus, a relative of the basil plant, a researcher would miss 80% of information available if they looked under only the most commonly used name. On another database, they found only 150 of 500 nutritionally important plant species using the names cited in current literature. 'By going for one name, we missed the majority of information mankind knows about that plant, which isn't too clever,' said Paton. 'What's really a breakthrough is we have a place which allows people to search through all the names used.'


"Kew Gardens joined up nearly three years ago with Missouri Botanical Garden in the US, and experts on two of the biggest and most valuable plant families: legumes, or peas and beans, and Compositae, which include asters, daisies and sunflowers. They have since attempted to search existing plant lists and work out an 'accepted' name for each species, and then list all known variations. One of the databases was originally set up using £250 left in the will of Charles Darwin. The full results will not be published until the end of the year, but so far the researchers have found 301,000 accepted species, 480,000 alternative names, and have 240,000 left to assess.


"Although work will continue to assess smaller plant groups in more detail and check for missed duplications, Paton said they now believe that the true number of plant species will turn out to be '400,000 or just over. You can't give an absolute number of names, but we have narrowed the possibility,' he said. Previous estimates, without the help of a full assessment, put the figure at between 250,000-400,000.


"Most of the work of the study group was sifting and sorting different names allocated to one species, often because scientists were simply not aware of the work of rivals and colleagues who had previously 'described' the plant in a scientific journal, or because of confusion caused by superficial differences such as different sized leaves in different climates. In some cases, plants thought to be the same have also been judged to be different species because of differences which have been revealed by later scientific discoveries, such as DNA.


"As well as the likely 400,000-odd flowering plants, there are thought to be 15,000 species of ferns and their allies, 1,000 gymnosperms such as conifers, and 23,000 mosses and allies making up the plant kingdom. For comparison there are more than 1 million species of insects listed by science, 28,000 living species of fish, 10,000 birds and 5,400 mammals.


"A meeting of the Convention on Biological Diversity in October in Japan is likely to declare that targets to halt biodiversity loss by this year failed and set tougher new aims to halt the problem." [The Guardian, 20/09/10, p.1. Quotation marks altered to conform with the conventions adopted at this site. Bold emphases added. Several paragraphs merged.]


It isn't too clear how DM can accommodate several plant names all naming the same species. Once more, an appeal to 'abstract' and/or 'approximate' identity won't help, as we will soon see.


10. Some of these observations, of course, depend on the said weighing scales changing at the same rate as the sugar being weighed, which, while unlikely, isn't beyond the bounds of possibility. The point is, of course, that this is an empirical matter that can't be settled a priori, as Trotsky and Hegel both imagined it could.


10a. Anyone tempted to think that the present writer is an "empiricist" -- or even a "positivist" -- on the basis of the constant demand for confirmation should read this, and then perhaps think again.


10b. It is also important to point out that the 'truth' if this 'law' is equally problematical as its 'falsehood' (no irony intended), for both of these depend on treating the LOI as a sort of Super-Empirical Truth.


[There is more on this in Essay Twelve Part One, and Note 17, below. Why this is so is connected with Wittgenstein's comments about 'rule-following'; again, on this see the references given in Essay Twelve Part One, and in Essays on the nature of language to be published at a later date -- for example, here.]


11. The Material that used to be here has been moved to the main body of the Essay.


12. The Material that used to be here has been moved to the main body of the Essay.


13. The reader will no doubt have noticed that in order to interpret Trotsky we have had to use the dread LOI. In that case, a decision must be taken as to whether or not S8 is equivalent to (i.e., identical with) S11 or S12.


S8: A pound of sugar is equal to itself.


S9(b): [All bodies] are never equal to themselves.


S10: Let A1 be a pound of sugar at time, T1.


S11: Let A2 be a pound of sugar at time, T2. [T2 > T1.]


S12: S8 means A1 is equal to A1.


S13: S8 means A1 is equal to A2.


Of course, it is reasonably clear that Trotsky believed that a complete understanding of change (even if humanity never actually attained to it) would require the employment of concepts more adequate to the task -– i.e., those found in DL. Many of these 'more adequate concepts' have been examined throughout this site where they have been shown to be no less vague, confused, or just plain incoherent.


[IDM = In Defense of Marxism (i.e., Trotsky (1971)); DL = Dialectical  Logic.]


However, in his Notebooks, Trotsky added a number of important qualifications to his comments on the LOI found in IDM. Among which are the following:


"a = a is only a particular case of the law of a ¹ a…. Formal Logic involves stationary and unchanging quantities: a = a. Dialectics retorts: a ¹ a. Both are correct. A = a at every given moment. A ¹ a at two different moments. Everything flows, everything is changing." [Trotsky (1986), pp.86-87.]


This suggests that Trotsky might have accepted a version of S13 or S21(a):


S13: S8 implies A1 is equal to A2.


S21(a): There is an A and a time, t1 such that A at t1 is not equal to A at t2.


S13 was in turn dependent on S8, S10 and S11:


S8: A pound of sugar is equal to itself.


S10: Let A1 be a pound of sugar at time, T1.


S11: Let A2 be a pound of sugar at time, T2.


[UO = Unity of Opposites/Unities of Opposites, depending on the context.]


The above would appear to mean that Trotsky was only committed to the idea that an object isn't identical with itself at some later time, as opposed to adhering to the stricter principle that objects aren't self-identical at any given moment -– the latter of which options is in turn based on the doctrine that all objects are contradictory UOs. If so, the above passage would seem to suggest that Trotsky was rejecting a core DM-thesis: that is, that there is at least one UO in every object and process, which drives change because UOs somehow cause, constitute or create 'internal contradictions'. Clearly, it is highly unlikely that Trotsky denied this core DM-idea. For example, his emphasis on the contradictory nature of the former USSR (elsewhere in IDM) strongly suggests he accepted this doctrine.


On the other hand, since Trotsky nowhere (to my knowledge) refers to the idea that change is the result of the struggle between UOs, it is possible that he did reject, or didn't fully accept, this part of the theory.


[For a contrary inference from the same Notebooks, see below. Having said that, there are several things Trotsky does say later on in these same Notebooks, that suggest he did view everything as a UO; e.g., p.103 -- on that, see here.]


Nevertheless, since the above quotation is taken from notebooks which weren't intended for publication it would be unwise to rely too heavily on what they contain as an accurate picture of Trotsky's intentions. This is especially so since it (rather fittingly) appears to contradict what was said in IDM:


A1: "In reality 'A' is not equal to 'A'…. [O]bserve these two letters under a lens -- they are quite different from each other." [Trotsky (1971), pp.63-64.]


S9(a): All bodies change uninterruptedly. (b) They are never equal to themselves.


Compare this with a passage from the Notebooks (quoted above) where Trotsky now seems to say the opposite:


A2: "A = a at every given moment." [Trotsky (1986), p.87.]


If A1 and S9(a)(b) were correct, A2 couldn't be. At best, A2 would only tell half the story.


For example, in A2, the two letter "A"s are easy to distinguish without the aid of a lens: the second letter is in the lower, while the first is in the upper case. But here, in A2, Trotsky now argues that these "A"s are equal at "every given moment" -- even though they look different to the naked eye!


Conversely, in A1 Trotsky claims that the opposite of this is true with respect to two letter "A"s, which not only look identical but also are in the same upper case! He claims that the two capital letters in A1 above look different if examined under a lens, while a lower case "a" and a capital "A" in A2 are equal at every moment!


If A2 were correct, then Trotsky's reference in A1 to the physical appearance of these two letter "A"s when viewed under an eyeglass would be completely pointless. The only reasonable conclusion here seems to be that since A1 was intended for publication it must contain Trotsky's more considered thoughts.


Furthermore, as noted above, A2 seems to be inconsistent with the claim that change is the result of internal contradictions: that is, with the idea that at any given moment an object both is and is not self-identical, constituting a UO -- in the present case, presumably a unity of "A and not A" (i.e., "pound bag of sugar and not pound bag of sugar" -- or, maybe, "pound bag of sugar and pound bag of non-sugar", or even "pound bag of sugar and non-pound non-bag of non-sugar" (it is far from clear which of these is implied by what Trotsky said), if, as we are told, everything, including weights, bags and sugar change all the time and are "never equal to themselves"), or that a pound bag of sugar is both identical and not identical with its 'other', as Hegel might have put it. [On this, see Essay Eight Part Three, and Essay Twelve (summary here). On the confusions Hegel's ideas will always introduce in this area, see here.]


A2: "A = a at every given moment." [Trotsky (1986), p.87.]


[Precisely what one of these 'Hegelian others' is of a pound bag of sugar is somewhat unclear. A pound bag of tea? A half pound bag of tea? A pound bag of Quinine? However, if a pound bag of sugar has no 'other' (and no dialectical-logical 'other', either) then, according to the DM-classicists, it can't change. To be sure, sugar and what can happen to it are highly complex; there are any number of things it can and does change into, so it must have countless 'others' (which fact rather makes a mockery of Hegel's 'analysis' of change, one would have thought). More on that here.]


In addition, A2 is itself rather badly worded. When Trotsky wrote:


A3: "A = a at every given moment" (italic emphasis added),


he must have meant:


A4: "A = a at any given moment."


This is because the wording of A3 implies that "A" never changes; i.e., that at all times "A = a" -- something Trotsky certainly didn't believe. [Of course, he might have been alluding to the IED-thesis, that every object is both identical and different from itself. However, I have batted that idea out of the park, here.]


On the other hand, A3 might contain an indirect allusion to Trotsky's point about 'abstract moments' in time:


"A sophist will respond that a pound of sugar is equal to itself at 'any given moment'…. 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.]


But, according to this, if Trotsky were referring to 'abstract moments', it would mean that the objects and processes to which he was alluding couldn't exist. If so, it would be unclear how A3 itself could ever be true -– that is, always assuming Trotsky meant it to apply to these non-existent 'abstract moments'.


A1: "In reality 'A' is not equal to 'A'…. [O]bserve these two letters under a lens -- they are quite different from each other."


A2: "A = a at every given moment."


A3: "A = a at every given moment" (italic emphasis added).


A4: "A = a at any given moment".


Of course, if A2 and A3 were merely about letter variables (not their supposed denotations -- i.e., what they supposedly refer to) it might be possible to re-interpret them in a more viable form.


One such re-configuration would see A2 and A3 recording the fact that while objects in the world change, letters depicting them don't. But that would make Trotsky's other assertions about the "A"s in A1 decidedly odd, for the aim there had been to argue that these letters weren't in fact "equal" irrespective of what they referred to, and that was because, upon close examination, we would always find there were minor differences between them. That was the whole point of Trotsky's appeal to ocular inspection. And, since letters are physical objects in their own right, his claim surely was that they are just as susceptible to change and diversity as the things to which they supposedly refer. So, this option (that while objects change, letters referring to them don't) doesn't look like it is a reliable interpretation of Trotsky's intentions.


On the other hand, if Trotsky had wanted to argue for something more complex in this regard it would prove impossible to comprehend his point. For example, if he had meant something like the following:


T1: Variable letters and what they refer to both change, and that they do so as follows:


(a) Each letter "A" no longer refers to whatever it used to refer to moments earlier, and,


(b) The object that each old letter "A" once denoted is no longer the same as it was when first identified, and,


(c) Earlier and concurrent manifestations of any and all letter "A"s are never the same as the 'same' new letter "A" now on the page/screen (which page/screen also changes), and,


(d) Any two or more concurrent letter "A"s on the 'same' line (which also changes) are not only different from each other, they change at different rates, and,


(e) Each letter individually denotes in a different and changing manner objects in reality, which objects are also different and are all changing at different rates themselves.


If something like this had been Trotsky's intention then his entire point would become too obscure to make much sense of, for we wouldn't have a clue what the hell he was banging on about. And yet, if everything changes uninterruptedly in every respect (as Trotsky himself claimed), he must have 'meant' this!


It could be objected that Trotsky only needs to appeal to the relative stability of medium-sized objects in reality to neutralise criticisms like this. Hence, if both language and most medium-sized objects are relatively stable, points (T(a)) to (T(e)) above don't apply.


But, how could anyone committed to this theory know whether or not language is 'relatively stable' -- especially if they also believe that everything is in the grip of the Heraclitean Flux? In fact, as soon as language itself is implicated in this 'Flux', everything that might seem semantically solid must melt into thin air. In that case, it would be no good appealing to evidence (drawn from dictionaries, textbooks, personal memory, common usage, etc.) in support of the claim that language is 'relatively stable', for if everything is changing then so is the language in which this evidence is itself expressed, so are the notebooks and/or primary data sheets from which it has been retrieved, and so are the memories upon which all of these depend. Given this way of looking at things, for all anyone knows, every single word could change its meaning every fraction of a second (along with any and all memories of, or about, the objects that seem familiar to us, like dictionaries, journals and textbooks).


This extreme view of the mutability of language was in fact endorsed by Voloshinov:


"[T]heme must be unitary, otherwise we would have no basis for talking about any one utterance. The theme of an utterance is individual and unreproducible, just as the utterance itself is individual and unreproducible. The theme is the expression of the concrete, historical situation that engendered the utterance. The utterance 'What time is it?' has a different meaning each time it is used, and hence, in accordance with our terminology, has a different theme, depending on the concrete historical situation ('historical' here in microscopic dimensions) during which it is enunciated and of which, in essence, it is a part." [Voloshinov (1973), p.99. Bold emphases added. Quotation marks altered to conform with the conventions adopted at this site.]


[And, as we will see in Essay Thirteen Part Three, several other comrades seem to be of the same opinion (no pun or irony intended).]


Of course, it could be objected in response that if the above were the case, the world would be far too crazy and confusing for anyone to make any sense of. But, the fact that we can makes sense of the world tells us that this interpretation (i.e., that Trotsky's theory is inconsistent with relative stability) can't be correct.


In fact, Trotsky and other DM-theorists said the following:


"[A]ll bodies change uninterruptedly in size, weight, colour etc. They are never equal to themselves.... [E]verything exists in time; and existence itself is an uninterrupted process of transformation….


"For concepts there also exists 'tolerance' which is established not by formal logic…, but by the dialectical logic issuing from the axiom that everything is always changing….


"Dialectical thinking analyses all things and phenomena in their continuous change….


"Dialectics…teaches us to combine syllogisms in such a way as to bring our understanding closer to the eternally changing reality." [Trotsky (1971), pp.64-66. Italic emphases added.]


"Dialectics…prevails throughout nature…. [T]he motion through opposites which asserts itself everywhere in nature, and which by the continual conflict of the opposites…determines the life of nature." [Engels (1954), p.211. Italic emphases added.]


"[Among the elements of dialectics are the following:]…internally contradictory tendencies…in this [totality]…and unity of opposites…. [E]ach thing…is connected with every other…[this involves] not only the unity of opposites, but the transitions of every determination, quality, feature, side, property into every other….


"In brief, dialectics can be defined as the doctrine of the unity of opposites. This embodies the essence of dialectics….


"The splitting of the whole and the cognition of its contradictory parts…is the essence (one of the 'essentials', one of the principal, if not the principal, characteristic features) of dialectics….


"The identity of opposites…is the recognition…of the contradictory, mutually exclusive, opposite tendencies in all phenomena and processes of nature…. Development is the 'struggle' of opposites.


"…The unity…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.221-22, 357-58. Emphases in the original.]


"According to Hegel, dialectics is the principle of all life…. [M]an has two qualities: first being alive, and secondly of also being mortal. But on closer examination it turns out that life itself bears in itself the germ of death, and that in general any phenomenon is contradictory, in the sense that it develops out of itself the elements which, sooner or later, will put an end to its existence and will transform it into its opposite. Everything flows, everything changes; and there is no force capable of holding back this constant flux, or arresting its eternal movement. There is no force capable of resisting the dialectics of phenomena….


"At a particular moment a moving body is at a particular spot, but at the same time it is outside it as well because, if it were only in that spot, it would, at least for that moment, become motionless. Every motion is a dialectical process, a living contradiction, and as there is not a single phenomenon of nature in explaining which we do not have in the long run to appeal to motion, we have to agree with Hegel, who said that dialectics is the soul of any scientific cognition. And this applies not only to cognition of nature….


"And so every phenomenon, by the action of those same forces which condition its existence, sooner or later, but inevitably, is transformed into its own opposite….


"When you apply the dialectical method to the study of phenomena, you need to remember that forms change eternally in consequence of the 'higher development of their content'….


"In the words of Engels, Hegel's merit consists in the fact that he was the first to regard all phenomena from the point of view of their development, from the point of view of their origin and destruction…." [Plekhanov (1956), pp.74-77, 88, 163. Bold emphases alone added.]


"'All is flux, nothing is stationary,' said the ancient thinker from Ephesus. The combinations we call objects are in a state of constant and more or less rapid change….


"…[M]otion does not only make objects…, it is constantly changing them. It is for this reason that the logic of motion (the 'logic of contradiction') never relinquishes its rights over the objects created by motion….


"With Hegel, thinking progresses in consequence of the uncovering and resolution of the contradictions inclosed (sic) in concepts. According to our doctrine…the contradictions embodied in concepts are merely reflections, translations into the language of thought, of those contradictions that are embodied in phenomena owing to the contradictory nature of their common basis, i.e., motion….


"…[T]he overwhelming majority of phenomena that come within the compass of the natural and the social sciences are among 'objects' of this kind…[:ones in which there is a coincidence of opposites]. Diametrically opposite phenomena are united in the simplest globule of protoplasm, and the life of the most undeveloped society…." [Plekhanov (1908), pp.93-96. Bold emphases alone added.]


"Everything in motion is continually bringing forth this contradiction of being in two different places at the same time, and also overcoming this contradiction by proceeding from one place to the next….


"A moving thing is both here and there simultaneously. Otherwise it is not in motion but at rest….


"Nothing is permanent. Reality is never resting, ever changeable, always in flux. This unquestionable universal process forms the foundation of the theory [of dialectical materialism]….


"According to the theory of Marxism, everything comes into being as a result of material causes, develops through successive phases, and finally perishes….


"Dialectics is the logic of movement, of evolution, of change. Reality is too full of contradictions, too elusive, too manifold, too mutable to be snared in any single form or formula. Each particular phase of reality has its own laws…. These laws…have to be discovered by direct investigation of the concrete whole, they cannot be excogitated by the mind alone before material reality is analysed. Moreover, all reality is constantly changing, disclosing ever new aspects….


"If reality is ever changing, concrete, full of novelty, fluent as a river, torn by oppositional forces, then dialectics…must share the same characteristics….


"Nature cannot be unreasonable or reason contrary to nature. Everything that exists must have a necessary and sufficient reason for existence….


"The material base of this law lies in the actual interdependence of all things in their reciprocal interactions…. If everything that exists has a necessary and sufficient reason for existence, that means it had to come into being. It was pushed into existence and forced its way into existence by natural necessity…. Reality, rationality and necessity are intimately associated at all times….


"If everything actual is necessarily rational, this means that every item of the real world has a sufficient reason for existing and must find a rational explanation….


"But this is not the whole and final truth about things…. The real truth about things is that they not only exist, persist, but they also develop and pass away. This passing away of things…is expressed in logical terminology by the term 'negation'. The whole truth about things can be expressed only if we take into account this opposite and negative aspect….


"All things are limited and changing…. In logical terms, they not only affirm themselves. They likewise negate themselves and are negated by other things…. Such a movement of things and of thought is called dialectical movement….


"From this dialectical essence of reality Hegel drew the conclusion that constitutes an indispensable part of his famous aphorism: All that is rational is real….


"[M]ovement…from unreality into reality and then back again into unreality, constitutes the essence, the inner movement behind all appearance….


"Everything generates within itself that force which leads to its negation, its passing away into some other and higher form of being….


"This dialectical activity is universal. There is no escaping from its unremitting and relentless embrace. 'Dialectics gives expression to a law which is felt in all grades of consciousness and in general experience. Everything that surrounds us may be viewed as an instance of dialectic. We are aware that everything finite, instead of being inflexible and ultimate, is rather changeable and transient; and this is exactly what we mean by the dialectic of the finite, by which the finite, as implicitly other than it is, is forced to surrender its own immediate or natural being, and to turn suddenly into its opposite.' (Encyclopedia, p.120)." [Novack (1971), pp.41, 43, 51, 70-71, 78-80, 84-87, 94-95; quoting Hegel (1975), p.118, although in a different translation from the one used here. Bold emphases added. Quotation marks altered to conform with the conventions adopted at this site.]


[Many more passages like the above have been reproduced in Essay Two.]


It is hard to seen how anything could remain the same, even if only temporarily, given what the above have to say -- after all, Trotsky tells us that "All bodies change uninterruptedly in size, weight, colour etc. They are never equal to themselves...." and that it is an "axiom" that "everything is always changing." The other DM-theorists say more-or-less the same (no irony intended, once more).


Indeed, Engels characterised as "metaphysical" the belief that things are identical with themselves:


"To the metaphysician, things and their mental reflexes, ideas, are isolated, are to be considered one after the other and apart from each other, are objects of investigation fixed, rigid, given once for all. He thinks in absolutely irreconcilable antitheses. 'His communication is "yea, yea; nay, nay"; for whatsoever is more than these cometh of evil.' For him a thing either exists or does not exist; a thing cannot at the same time be itself and something else. Positive and negative absolutely exclude one another, cause and effect stand in a rigid antithesis one to the other." [Engels (1976), p.26. Quotations marks altered to conform with the conventions adopted at this site, Bold emphasis added.]


The above is a direct consequence of the doctrine that everything is a UO, a combination of what it is and what it is not; what it is and what it is changing into. Any who deny this are branded "metaphysicians" by Engels.


[I have said much more about this topic in Essay Eleven Part One; readers are directed there for more details.]


In view of the above, and because we can make (even limited) sense of the world, that must mean the opinions of these DM-worthies are about as wide of the mark as any could be.


Indeed, as Plato himself recognised, the Heraclitean Flux is no respecter of theories; in fact it completely mangles them.


[On that, see Note 10 above, and Note 15 below -- and, in this case, Essay Thirteen Part Three, where Voloshinov's ideas are subjected to sustained and destructive criticism. On Plato, see Note 49 of Essay Eleven Part One.]


Furthermore, the reasoning in the Notebooks appears to be somewhat confused:


"a = a is only a particular case of the law a ¹ a." [Trotsky (1986), p.86.]


It is difficult to see how "a = a" could be a particular case of the 'law' "a ¹ a", any more than "a + b = b + a", for instance, could be a particular case of the rule "a + b ¹ b + a"". If "a ¹ a" is a 'law', then "a = a" refutes it; it doesn't instantiate it. Of course, this would be the case unless the word "refute" has in the meantime changed its meaning. [Perhaps because of the local effects of the pesky Heraclitean Flux?] Naturally, in such a madcap Heraclitean world it isn't easy to see exactly what would or could either stay the same or change -- or, indeed, for how long a decision about even that possibility would itself remain stable!


And yet, contrary to what was intimated earlier, the above quote could be an indirect reference on Trotsky's part to a UO operating, namely this: the fact that "a = a" and "a ¹ a". If so, Trotsky's criticism of the LOI would collapse into irredeemable confusion. If "a" doesn't equal "a", that is, if "a" is in fact also "not a", then the following must be true:


C1: "a is not a" -- or "a = not a".


In which case, we can substitute "not a" every time we see an "a", giving this series of dialectical oddities (but using capital letter "A"s to make the point a 'little clearer'):


C2: "A = not A".


C3: "Not A = not (not A)".


C4: "Not (not A) = not (not (not A))".


C5: "Not (not (not A)) = not (not (not (not A)))".


C6: "Not (not (not (not A))) = not (not (not (not (not A))))".


C7: "Not (not (not (not (not A)))) = not (not (not (not (not (not A)))))".


And so on, substituting "not A" for each "A" in each C(n), to yield C(n+1). 


But, worse: whatever is true of "A" must also be true of "=" and "not" (and even "is"!), yielding the following ever-expanding bowl of dialectical spaghetti (as each "not" and each equal sign is replaced by its 'dialectical equivalent', "not (not)" and "= and ¹", in each case -- and then, of course, by "not not (not (not))", and "not (not = not and not ¹)", respectively):


C8: "A = not A".


C9: "Not A (= and ¹) not not (not A)".


C10: "Not Not (Not A) not (not = not and not ¹) not not not not (not not (not A))".


C11: "Not Not Not Not (Not Not (Not A) not not (not not not = not not not and not not not ¹) not not not not not not not not (not not not not (not not (not A)))."


And so on.


[The above also ignores the fact that each bracket, quotation mark, full stop, C(n), and space between each letter and word, should also be 'negated', à la Trotsky/Hegel -- but that task can be left to the reader to complete. (If letters are "never equal to themselves", the spaces between them aren't, either, and neither are brackets and other punctuation marks!)]


If "A" is never equal to itself (and nothing else is, either), then the above must follow -- which dire dialectical dénouement can only be avoided by those who reject Trotsky and Hegel's crazy ideas about identity and change.


But, we have been here already.


[Again, the 'relative stability' defence has been neutralised, here, here and here.]


Moreover, several other things Trotsky said in IDM indicate that the above passage from his Notebooks isn't a reliable guide to his thinking:


"A sophist will respond that a pound of sugar is equal to itself at 'any given moment'…. 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.]


This at least confirms the accuracy of the interpretation put on Trotsky's analysis of the LOI in this Essay -– that is, in so far as any sense can be made of it.


Finally, what TAR itself says about Trotsky's argument seems to agree with the interpretation given here, as do other commentators. [Cf., Rees (1998), p.273. See also: Novack (1971) and Woods and Grant (1995/2007).]


13a. Quantifiers in language are words like "all", "every", "any", "some", "most", "nothing", and "none", etc. Tensed quantifiers are terms like "always", "never" and "sometimes". On this, see here, here and here.


14. It could be objected to this that "moments in time" aren't objects; time is, it seems, one of the "modes of existence of matter" -- that is an inference of mine based on what Engels said:


"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…." [Engels (1976), p.74. Italic emphasis in the original.]


It is difficult to see that time isn't a "mode of the existence of matter" if it is so intimately connected with motion.


Be this as it may, if time can be measured (rather like bags of sugar can be weighed) then Trotsky's criticisms must apply to time, too. Moreover, if we accept what Lenin said about matter (i.e., that it is whatever exists "objectively outside the mind" (a doctrine examined in detail in Essay Thirteen Part One)), time must be material, too -- unless, of course, time 'exists' only in the mind. If that were so, time couldn't be 'objective':


"Recognising the existence of objective reality, i.e.., matter in motion, independently of our mind, materialism must also inevitably recognise the objective reality of time and space, in contrast above all to Kantianism, which in this question sides with idealism and regards time and space not as objective realities but as forms of human understanding....


"'Space and time,' says Feuerbach, 'are not mere forms of phenomena but essential conditions...of being' (Werke, II, S. 332). Regarding the sensible world we know through sensations as objective reality, Feuerbach naturally also rejects the phenomenalist (as Mach would call his own conception) or the agnostic (as Engels calls it) conception of space and time. Just as things or bodies are not mere phenomena, not complexes of sensations, but objective realities acting on our senses, so space and time are not mere forms of phenomena, but objectively real forms of being. There is nothing in the world but matter in motion, and matter in motion cannot move otherwise than in space and time. Human conceptions of space and time are relative, but these relative conceptions go to compound absolute truth. These relative conceptions, in their development, move towards absolute truth and approach nearer and nearer to it. The mutability of human conceptions of space and time no more refutes the objective reality of space and time than the mutability of scientific knowledge of the structure and forms of matter in motion refutes the objective reality of the external world." [Lenin (1972), p.202-03. Bold emphasis alone added. Quotation marks altered to conform with the conventions adopted at this site. Link added.]


Hence, it seems that not only is matter 'objective', so are time and space, according to Lenin; and if they are 'objective', they can be measured. In that case, even if time isn't an 'object', as we have seen, it can be measured. [On this, see Note 14a. More on this presently, too.]


14a. On the various systems for measuring time, see here.


It could be argued that since moments in time follow on from each other, it isn't possible to measure one of them in order to compare it with any other -- since the earlier one will no longer exist to make the comparison! This isn't the case with respect to objects that have to be weighed; they clearly exist side-by-side during the entire process, facilitating comparison.


Or, so this objection might proceed...


But, can't moments differ even if we are unaware of it? And, can't two objects be weighed simultaneously, and local to each other, with the duration of each weighing instance timed simultaneously, too, so that these durations can be compared just like their weights?


Be this as it may, Trotsky's argument still relies on some reference to the 'same moment', and that must involve a use of the LOI applied to time.


Concerning differing time intervals (or, at least, their measurement), this is what a recent New Scientist article had to say:


"Clocks that gain or lose no more than a fraction of a second over the lifetime of the universe could be on the way, thanks to a technique for cutting through the 'heat haze' that compromises the accuracy of today's instruments. The most accurate atomic clock we have now is regulated by the electrons of a single aluminium ion as they move between two different orbits with sharply defined energy levels. When an electron goes from the higher energy level to the lower it emits radiation of a precise frequency. That frequency is used to mark out time to an accuracy of better than 1 part in 1017, or 1 second in 3 billion years.


"That's pretty good, but it could be better. Infrared photons emanating from the background cause the two energy levels to shift by slightly different amounts, says Marianna Safronova at the University of Delaware. That affects the frequency of the emitted radiation to an unknown extent, adding a small uncertainty to the clock's tick. Safronova reported this month at a conference in Baltimore, Maryland, that by combining two different mathematical approaches, she and her colleagues have now managed to calculate how much the energy gap between the two levels changes.


"Using this information to correct an atomic clock could in principle increase its precision to around 4 parts in 1019, or about 1 second per 80 billion years. Such a clock could test whether the fundamental constants of nature are changing, Safronova suggests." [New Scientist 210, 2813, 21/05/2011, p.15. Bold emphases added. Quotation marks altered to conform with the conventions adopted at this site. Several paragraphs merged.]


Even with this new method, time intervals (or at least the processes by means of which we measure them) clearly change over billions of years, just as weights do (over shorter periods).


15. Again, it could be argued that all Trotsky requires is the relative stability of the words he used, which won't have changed significantly during the short intervals involved.


Unfortunately, Trotsky holed that response well below the water line, declaring that:


"[A]ll bodies change uninterruptedly in size, weight, colour etc. They are never equal to themselves… But everything exists in time; and existence itself is an uninterrupted process of transformation…. 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 added.]


In that case, since words are also material objects they must "change uninterruptedly" (as must their meanings) and hence they are "never equal to themselves". So, if, according to Trotsky, every letter "A" is subject to the Heraclitean Flux from moment-to-moment and each is "never equal" itself, words and entire sentences stand no chance. So, not only does this theory (DM) imply that there is no way of knowing whether or not words (and/or their meanings) have changed dramatically --, even while they are being uttered (including any of the words that might be used to argue for or against either possibility) --, it also implies that, whether we know it or not, they have changed in the above way.


[We saw earlier that Voloshinov also appeared to hold this view.]


Consequently, the word "identity" (and its meaning) must itself fail to be self-identical at one and the same moment (if Trotsky were to be believed), since everything (including every meaning, one supposes) is a unity of itself and its 'opposite' (its "other", according to Hegel and Lenin). Plainly, that implies the word "identical" must also mean and not mean "not identical", at the same time!


If that weren't the case, then dialecticians would have no way of accounting for the change in meaning of the word "identity" itself, which, according to their own theory, has to change, and it can only do that because of one or more of the following factors: (i) Its own 'internal contradictions', (ii) The 'internal contradictions' of (or in) the meanings we attribute to it, or (iii) A response to 'contradictions' in society-at-large (which we are told are reflected in language). So, given the truth of DM, unless "identical" now means "not identical", its meaning couldn't change.


The same argument in turn applies to anyone who uses this word. So, by "identical" they, too, must mean "identical" and "not identical". Moreover, assuming DM is true, it isn't easy to see how the understanding of these changed and changing meanings, allied with altered and altering intentions, could possibly be coordinated across an entire population of dialecticians, let alone the wider community. Naturally, that would completely undermine inter-personal communication, which in turn would prevent DM-theorists from communicating their ideas to the rest of humanity, or even to one another, since they would all mean something different by their use of this particular term, or, indeed, any word.


The same argument applies to anyone who uses this word. So, by "identical" they, too, must mean "identical" and "not identical". Moreover, if, for the purposes of argument, we assume DM is true, it isn't easy to see how the comprehension of these changed (and continually changing) meanings, allied with altered (and continually altering) intentions, could possibly be coordinated across an entire population of dialecticians, let alone the wider community. Everyone would mean something increasingly different by every word they used, and that would also be different from their own previous use of the 'same' words. Clearly, this would scupper inter-personal communication, which in turn would prevent DM-theorists themselves from communicating their ideas to the rest of humanity, never mind one another. They too would continually mean something different by their use of "identity", or, indeed, any word.


This would further imply that no one would or could possibly "understand" dialectics -- not Hegel, not Marx, not Engels, not Plekhanov, not Lenin, not Trotsky..., since the meaning of every single term used would be subject to unspecified changes, and hence consequent indeterminacies. Even worse, given the truth of DM, there is nothing that could be done to rectify the situation. Any attempt to do so would also be subject to very same the tender mercies of the dread Heraclitean Flux. [Irony intended.]


Furthermore, if we now apply DM consistently across the board, any such a 'rectification' (should one be attempted) would be both a 'rectification' and 'not a rectification' at the same time!


[On that, also see Essays Three Part Two, Eleven Part One and Thirteen Part Three.]


The only way to avoid 'ridiculous' conclusions like these is to abandon the doctrine that all things change all the time (as a result of their 'internal contradictions') --, or admit that some things remain identical (namely, at least the word "identical" and its meaning), indefinitely. Either way, DM would suffer yet another body blow.


Hence, in order to avoid the unremitting confusion that the Heraclitean doctrine of universal change would introduce into DM itself, Trotsky needed the LOI to apply to his own words and their meanings (as a rule of language or of practice) while he was using them. In addition, that would have to have been true for many years (possibly even for several centuries), too, so that his supporters/epigones would be able to understand him correctly (or, indeed, at all!). That would be the case especially when he hoped to employ certain words to question the application of identically same law to those letter "A"s and bags of sugar! [Irony intended, once more.] Otherwise, for all he knew, his words and their meanings could be non-self-identical from moment-to-moment. [His theory implies that anyway.]


In addition, anyone consulting Trotsky's words today must be able to read them with their original meanings intact or they wouldn't be able to agree with the originally intended content/message, and hence with what Trotsky had attempted to argue. If so, contemporary dialecticians who read Trotsky's words (or, indeed, Hegel's) must in effect take any argument against the application of a strict version of the LOI with a pinch of salt or risk failing to grasp the exact message Trotsky (or Hegel) had intended. Alternatively, they would have to admit that what those two had to say about identity and change can't be grasped by anyone if what they claimed about the LOI were the case. So, if Trotsky and Hegel's words about identity and change (etc.) were correct, the message they intended to convey wouldn't now be accessible, having changed in untold ways over the years -- possibly (probably!) into its opposite! Indeed, definitely into its opposite, if we were to believe what the DM-classics have to say about change.


It could be objected that our words do in fact remain relatively stable, so the above comments are entirely misguided. However, if Hegel and Trotsky are to be believed, there would be no way that either of them (or anyone else, for that matter) could possibly determine whether or not words remain 'relatively stable'. Indeed, if their theory were true, even the words in the previous sentence, along with their meanings, will have changed!


As should now seem obvious, if DM were a valid theory, there would be nothing to which anyone could appeal in order to base a single secure thought. Hence, if Trotsky and the other DM-theorists quoted earlier were speaking the truth, this couldn't be true:


"[A]ll bodies change uninterruptedly in size, weight, colour etc. They are never equal to themselves… But everything exists in time; and existence itself is an uninterrupted process of transformation…. 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." [Ibid. Bold emphasis added.]


If nothing in the entire universe -- including thoughts, words and their meanings -- is ever "equal to" itself, then there can be no secure foundation for a single DM-proposition, let alone anything else.


On the other hand, if there were something upon which DM-theorists could ground their thoughts, then Trotsky, Hegel and Heraclitus must have been mistaken, since, in that case, at least something would remain unchanged long enough for it to be of any use -- namely whatever it is that we could ground even that thought upon. So, given the validity of Trotsky's argument -- that nothing stays the same, that all things change uninterruptedly and are never equal to themselves, they are never self-identical -- there can't be any such grounding. An appeal to the memory we might have of a given word (of its use or its meaning), for example, would be to no avail, either. If everything is changing, then memory itself can hardly remain unscathed. Not even Cartesian 'clear and distinct ideas' would be available to anchor a single DM-cognition on solid epistemological bedrock. The words, concepts, and ideas retrieved in order to formulate these comforting 'Cartesian certainties' would themselves be non-self-identical from moment-to-moment, subject to unremitting and continual change, in like manner.


A moment's thought (no pun intended!) will confirm that even the phrase "relatively stable" must itself be subject to change -- along with its meaning -- if we were to believe what DM-theorists try to tell us. How could dialecticians to rule that out? In fact, it is implied by their own theory! This must be so if everything is subject to relentless change in the way that Heraclitus, Hegel, Lenin, Trotsky and all the rest imagined. In which case, it is the DM-doctrine of constant, universal change that must be rejected to save this theory from its own absurd implications and easy self-refutation. Of course, the only way to do that would involve an invocation of the LOI, interpreted now as a rule of language or of practice, not as a metaphysical, or any other sort of, truth.


Once again, DM-fans would have to appeal to FL and/or ordinary language to rescue their theory from itself.


Alternatively, if it is indeed a fact that language is stable, then the DM-theory of change must be wrong (and for reasons rehearsed above; see also here and here), since, at a minimum, A will equal A (even if only for few moments), thus refuting Trotsky, Hegel and all the rest.


As I point out in Essay Eleven Part One:


Of course, as is the case with other linguistically competent human beings, DM-apologists understand perfectly well how to use words for identity –- such as, "similar", "equal", "equivalent", "same", and "identical" -- along with their appropriate qualifiers (e.g., "exactly", "precisely", "accurately", "very", "nearly", "approximately" and "almost"). Plainly, a grasp of terms like these arises out of their employment in everyday life by each language user, not from a supposed 'law'. Nor does this facility follow from the 'negation', nor yet the double 'negation', of the LOI. [There is more on that here.] In fact, this everyday facility with words for identity (etc.) is what enables DM-theorists themselves to engage in a pretence that the LOI is either false or only 'approximately true' when it is applied to objects and processes in concrete reality. As competent language-users they understand the LOI perfectly well, and yet it is their misinterpretation of the socially-conditioned rules we have for the use of terms like the above as if they represented empirical truths about any object and itself that ultimately misleads them.


In short: dialecticians mistake the misinterpreted content of a social norm for reality itself, and then make a fetish of the result.


[I hasten to remind the reader that the above does not mean that I think there is no change anywhere in the universe! As I have noted already, this is an empirical question to be settled by science, not the confused musings of mystics.]


16. Again it could be argued that identity criteria for temporal instants could be specified by mapping them onto the Real Numbers; since the latter are distinguishable, the former must be, too. Given this scenario, such instants would be isomorphic to the Reals.


In response to this, several points are worth making:


(1) This view assumes that 'time itself' (as opposed to the measurement of time) is composed of discrete units, and that they can therefore be counted (or, at least, mapped onto the Positive Integers/The Positive Reals). But, that sits rather awkwardly with the idea that temporal instants can be measured, which appears to suggest that time must be both discrete and continuous.


[Dialecticians might be happy with that implication, but just watch them then (i) Squirm when asked to explain (in physical terms) how that is even possible, or (ii) Reach for yet another Nixon card.]


(2) The only 'evidence' for the validity of such a manoeuvre derives from the proposed isomorphism itself. In that case, any criteria of identity for instants in time that result from this mapping would clearly be a reflection of the imported properties of Real Numbers, which is precisely the point at issue. If 'instants' in time have no identity -- that is, if they aren't discrete (or, rather, if their ordering isn't the result of the application of an inductive law to a discrete variable; or, indeed, to any variable at all) -- an isomorphism like this would simply amount to their conventionalised re-description.


Hence, the proposed isomorphism could end up misrepresenting the very thing being mapped -- especially if this is considered to be the only way to view time -- since it makes something that appears to be continuous look as if it were discrete, imposing on time a structure it might not possess. [To be sure, mathematicians since Dedekind have regarded the Reals as both dense and continuous. But even then, there is no suggestion that Real Numbers merge into one another, that they have no discrete identities or that they can't be distinguished. Cf., Sanford (2005).] In that case once more, dividing time into temporal instants in this way would impose on it something it might not have.


It seems, therefore, that time can only be broken up into metaphysical instants if it is mapped onto something that is already fragmented (in the above sense), like the Reals. On the other hand, if time is only continuous (and isn't composed of discrete 'instants'), then, without distortion, it can't be mapped onto the Real Numbers, which are discrete but continuous (again, in the above sense -- that is, there are no 'gaps' between them, with "gap" defined in a specific way in Real Analysis (this links to a PDF)). Of course, any supposition to the contrary would suggest that it isn't in fact time which has been mapped onto Real Numbers, but Real Numbers that have been mapped onto themselves, and then misleadingly re-labelled "instants".


[That comment disposes of the claim that scientists are actually speaking about time when they talk this way. What they are in fact doing is talking about Real Numbers in drag. (On this in general, see Read (2007), pp.79-115, on which many of my own ideas have been based.)]


(3) A successful isomorphism would itself depend on an application of the LOI (interpreted as a rule, not as a 'philosophical truth'), making this attempt to patch up the argument of little use to DM-theorists --, or, at least to those who are still concerned to observe even a modicum of consistency.


[LOI = Law of Identity.]


Of course, all this is independent of the fact that isomorphisms are creatures of convention; they don't actually populate the universe. Any attempt to use them to shore up DM would be unwise, therefore, since it would imply that whatever is concluded about the LOI would likewise be a product of convention, and hence not at all 'objective'.


[There is an interesting, if somewhat metaphysical, discussion of this topic in Adamson (2002), pp.5-58. Nevertheless, Adamson's 'solution' (based on the radically confused writings of Henri Bergson) seems to be far worse than the problem it was meant to solve. (It is also worth adding that Adamson's characterisation of Analytic Philosophy is highly misleading. However, I don't propose to defend that accusation here.)]


[I have commented on this use of nested intervals of Real Numbers in Essay Five.]


17. Again, the reader mustn't assume that I accept that the LOI expresses an 'absolute truth' -- or indeed any sort of truth. The questions at this point in the main body of this Essay are directed at those who regard the LOI either as a profound metaphysical thesis (even if it is never instantiated in material reality -- or, at least, 'only within certain limits'), or as a highly confirmed empirical generalisation. If, as is maintained here, the LOI is in fact a misleading way of expressing several distinct grammatical rules (we have for sentences using words for identity and sameness, depending on when and where the LOI is applied), then the 'problems' associated with the traditional view of this 'law' simply disappear. There is no such 'law' to be true of anything, just a series of rules we have been socialised into using -- and, it is worth recalling that rules can't be true or false, just practical or impractical, useful or useless, applied or misapplied.


In this way a whole cloud of dialectics has been condensed from a drop of misconstrued grammar (to paraphrase Wittgenstein).


18. This doesn't contradict the earlier claim that Trotsky didn't comprehend the LOI. What is maintained here is that Trotsky -- just like any other competent language-user -- understood perfectly well how to employ ordinary words for identity, sameness, equality and difference in everyday life. It is only when he allowed himself to be led astray by the obscure doctrines he found in Hegel's Logic that his solidly-grounded material grasp of change was seriously compromised.


19. This is how the contrary argument will be put in Essay Twelve (some of this has already been posted in Essay Four, but it is presented here in a highly edited form):


John Rees put things this way:


"Ordinary language assumes that things and ideas are stable, that they are either 'this' or 'that'. And, within strict limits, these are perfectly reasonable assumptions. Yet the fundamental discovery of Hegel's dialectic was that things and ideas do change…. And they change because they embody conflicts which make them unstable…. It is to this end that Hegel deliberately chooses words that can embody dynamic processes." [Rees (1998), p.45.]


The problem with this passage is that it gets things completely the wrong way round. It is in fact our use of ordinary language that enables us to speak about change, movement and development. Complex philosophical jargon (especially terminology invented by Hegel) is completely useless in this regard, since it is wooden, static and of indeterminate meaning, despite what Rees asserts.


[Any who think differently are invited to reveal precisely which set of Hegelian terms is able do what the words listed below (or their equivalent in German) already achieve for us, only better.]


As is well-known (at least by Marxists), human beings managed to progress because of their interaction with nature, later constrained by the class war and the development of the forces of and relations of production. In which case, ordinary language -- the result of collective labour -- couldn't fail to have invented a range of words with the logical and semantic multiplicity that allowed its users to speak about changes of almost limitless complexity, speed and duration.


This is no mere dogma; it is easily confirmed. Here is a greatly shortened list of ordinary words (restricted to modern English, but omitting simple and complex tensed participles and auxiliary verbs) that allow speakers to talk about changes of almost unbounded complexity, rapidity or scope:


Vary, alter, adjust, adapt, amend, make, produce, revise, rework, advise, administer, allocate, improve, enhance, deteriorate, depreciate, edit, bend, straighten, weave, merge, dig, plough, cultivate, sow, reap, twist, curl, turn, tighten, fasten, loosen, relax, ease, tense up, slacken, fine tune, bind, wrap, pluck, carve, rip, tear, mend, perforate, repair, puncture, renovate, restore, damage, impair, scratch, diagnose, mutate, metamorphose, transmute, sharpen, hone, modify, modulate, develop, upgrade, appear, disappear, expand, contract, constrict, constrain, shrivel, widen, lock, unlock, swell, flow, glide, ring, differentiate, integrate, multiply, divide, add, subtract, simplify, complicate, partition, unite, amalgamate, fuse, mingle, disseminate, connect, entwine, unravel, link, brake, decelerate, accelerate, fast, slow, swift, rapid, hasty, protracted, lingering, brief, heat up, melt, freeze, harden, cool down, flash, shine, glow, drip, bounce, cascade, drop, pick up, fade, darken, wind, unwind, meander, peel, scrape, graze, file, scour, dislodge, is, was, will be, will have been, had, will have had, went, go, going, gone, return, lost, age, flood, swamp, overflow, precipitate, percolate, seep, tumble, plunge, dive, float, sink, plummet, mix, separate, cut, chop, crush, grind, shred, slice, dice, saw, sew, knit, spread, coalesce, congeal, fall, climb, rise, ascend, descend, slide, slip, roll, spin, revolve, circulate, bounce, oscillate, undulate, rotate, wave, splash, conjure, quick, quickly, slowly, instantaneously, suddenly, gradually, rapidly, briskly, hurriedly, absolutely, lively, hastily, inadvertently, accidentally,  carelessly, really, energetically, lethargically, snap, drink, quaff, eat, bite, devour, consume, swallow, gulp, gobble, chew, gnaw, digest, ingest, excrete, absorb, join, resign, part, sell, buy, acquire, prevent, block, avert, avoid, forestall, encourage, invite, appropriate, lose, find, search, pursue, hunt, track, explore, follow, cover, uncover, reveal, stretch, distend, depress, compress, lift, put down, fetch, take, bring, carry, win, ripen, germinate, conceive, gestate, abort, die, rot, perish, grow, decay, fold, empty, evacuate, drain, pour, fill, abduct, abandon, leave, abscond, many, more, less, fewer, steady, steadily, jerkily, intermittently, smoothly, awkwardly, expertly, very, extremely, exceedingly, intermittent, discontinuous, continuous, continual, emit, push, pull, drag, slide, jump, sit, stand, run, sprint, chase, amble, walk, hop, skip, slither, crawl, limp, swim, fly, hover, drown, submerge, immerse, break, abrogate, dismiss, collapse, shatter, split, interrupt, charge, retreat, assault, squash, adulterate, contaminate, purify, filter, clean, raze, crumble, erode, corrode, rust, flake, demolish, dismantle, pulverise, atomise, disintegrate, dismember, destroy, annihilate, extirpate, flatten, lengthen, shorten, elongate, crimple, inflate, deflate, terminate, finish, initiate, instigate, augment, replace, undo, redo, analyze, synthesise, articulate, disarticulate, reverse, repeal, abolish, enact, quash, throw, catch, hour, minute, second, instant, moment, momentary, invent, devise, teach, learn, innovate, forget, rescind, boil, freeze, thaw, cook, liquefy, solidify, congeal, neutralise, evaporate, condense, dissolve, process, mollify, pacify, calm down, excite, enrage, inflame, protest, object, challenge, confirm, deny, repudiate, reject, refute, expel, eject, repel, attract, remove, overthrow, expropriate, scatter, distribute, equalise, surround, gather, admit, acknowledge, hijack, assemble, attack, counter-attack, charge, repulse, defeat, strike, occupy, picket, barricade, revolt, riot, rally, march, demonstrate, mutiny, rebel, defy, resist, lead, campaign, educate, agitate, organise...


[In each case, where there is a noun form of a word its verb form has been listed (for instance, "object" as in "to object"). Moreover, where I have listed the word "ring", for example, I also intend cognates of the verb "to ring" -- like "ringing" and "rang". I have also omitted many nouns that imply change or development, such as "river", "runner", "wind", "lightning", "tide", "cloud", and "fire". Anyone who didn't know such words implied changing processes in the world -- that rivers flow, fires burn, runners run, tides ebb and flow and winds blow -- would have thereby advertised a lack of comprehension of English (or whatever language theirs happened to be), compounded by a dangerously defective knowledge of the world. So, not knowing that fires burn or rivers flow, for example, could endanger life. In addition, several of the above also have verb forms, such as "fired" or "winding". Other nouns also imply growth and development, such as "tree", "flower", "mouse", "day",  "human being". Anyone who thought "human being", for example, reflected a 'fixed and changeless' view of the world would perhaps be regarded as suffering from some form of learning disability; either that, or they were in the grip of an off-the-wall philosophical theory of some description.]


Naturally, it wouldn't be difficult to extend this list until it contained literally thousands of entries -- on that, see here and here --, all capable of depicting countless changes in limitless detail (especially if it is augmented with words drawn from mathematics, science and HM). It is only a myth put about by Hegel and DM-theorists (unwisely echoed by Rees, and others -- such as Woods and Grant) that ordinary language can't depict change adequately, since it is supposedly dominated by 'the abstract understanding', a brain module helpfully identified for us by Hegel without a scrap of supporting evidence, a brain scan or even the use of a consulting couch. By way of contrast, ordinary language performs this task far better than the incomprehensible and impenetrably obscure jargon Hegel invented in order to fix something that wasn't broken.


Dialecticians like Rees would have us believe that because of the alleged shortcomings of the vernacular only the most recondite and abstruse terminology -- concocted by Hegel, the meaning of much of which is still unclear, even to Hegel scholars! -- is capable of telling us what we already know, and have known for tens of thousands of years, that things change!


[There is much more on this here.]


The idea that the search for knowledge can be modelled asymptotically was a metaphor introduced by Engels, in the following passage:


"The identity of thinking and being, to use Hegelian language, everywhere coincides with your example of the circle and the polygon. Or the two of them, the concept of a thing and its reality, run side by side like two asymptotes, always approaching each other but never meeting. This difference between the two is the very difference which prevents the concept from being directly and immediately reality and reality from being immediately its own concept. Because a concept has the essential nature of the concept and does not therefore prima facie directly coincide with reality, from which it had to be abstracted in the first place, it is nevertheless more than a fiction, unless you declare that all the results of thought are fictions because reality corresponds to them only very circuitously, and even then approaching it only asymptotically." [Engels to Schmidt (12/03/1895), in Marx and Engels (1975), p.457, and Marx and Engels (2004), pp.463-64.]


[Lenin and other DM-theorists said more-or-less the same (no irony intended).]


There are many things wrong with the above, mention of which will have to be left for another time; but this way of depicting things is entirely misleading. The idea of an asymptotic approach in mathematics is connected with the concept of a limit -- if the limit in question can be shown to exist.


Unfortunately, if a series has no limit, a set of its partial sums can't in fact "approach" anything at all. Such a series is therefore said to be divergent -- not convergent. However, Engels's argument depends on the aptness of a metaphor that pictures knowledge as convergent on a limit, which limit he neglected to show exists. But worse: the limit in this case is quite unlike any found in mathematics. In order to show that this particular limit exists, Engels would have to have access to absolute knowledge. That would have to be so for him to know (not suspect, surmise or conjecture) that this alleged limit was indeed a limit of this particular sort (i.e., according to his own structures on "concepts" expressed in the above passage), and that his knowledge of it was thus absolutely reliable before he had access to it. He would thus have to know before he knew!


[Yes, I know that mathematicians have 'shown' that certain divergent series, those that are Cesàro summable, do 'have a sum', but this area of mathematics is controversial and it is far from clear that it will be of any help to Engels. That is because these series produce notoriously paradoxical and ridiculously implausible results -- such as the following: 1 + 2 + 3 + 4 + 5 +... = -(1/12) -- no misprint! If any DM-fans want to go down that route, I can only wish them good luck. But even if they do, they have yet to show that this is the case with respect to knowledge.]


In that case, another annoying dialectical inversion now confronts materialist dialecticians (I am ignoring for the present whether or not it confronts idealist dialecticians): given this metaphor, knowledge can't be asymptotically converging on an absolute limit, but diverging from it. If that is so, since the gap between a large finite and an infinite body of knowledge is itself infinite, we are forced to conclude that human beings must remain forever trapped in a bottomless pit of infinite ignorance -- even supposing we could assert this much with any confidence given the infinite distance between that supposition itself and the absolute truth about it (that is, if there is a limit, and if we were to believe what Engels claimed about "concepts").


Of course, it could be objected that iterative functions in mathematics might yield infinite sequences, and yet that doesn't mean that the distance between any intermediate value given by a partial sum of that function and the point toward which it is converging is itself infinite. For example, the sequence: 1 + 1/2 + 1/4 + 1/8 +...+ 1/2n-1 converges on 2 (as n +∞), but none of the rational numbers (formed from partial sums of this series) is infinitely far from 2.


This isn't strictly so (and that is because we are told there is an infinite number of Rational and Reals between any two numbers, let alone between a partial sum and the limit toward which it is converging), but even if it were the case, the above would only have been an effective response had Engels bothered to prove there is indeed a limit here (surely implied by his asymptote metaphor); but since he didn't, it isn't.


This means that if Engels's metaphor is apt, the sum total of our 'knowledge' of the specific characteristics of any and all parts of the Whole (or even of the Whole itself) will always be overwhelmingly outweighed by the black hole of infinite ignorance around which we must forever orbit, whose grip we can never shake off.


Kant's Noumenon by any other name…?


[There is more on that here.]


19a. It is it important to stress once again that I don't prefer this way of speaking about identity (that is, that it can be 'identical' (or not) with 'identity'; my reasons for saying that can be found here and here). This manner of speaking has only been adopted to expose the hidden non-sense lurking at the heart of traditional, metaphysical talk like this.


Moreover, both DM-, and Hegel-fans, who themselves confuse the relational form of identity with its nominal form -- that is, they conflate its use in sentences of the form, 'A is identical with B' with plain and simple 'Identity' (when they speak about "Identity" and "Difference", for example -- on that, see the above link) --, have little room to complain about my temporary use of such terms. They lost that particular right when they began to use the word "identity" as the name of the abstract idea or concept: Identity. [On that, see here.]


Now, Trotsky's 'analysis' of the LOI grew out of this confused tradition (albeit as part of what is in fact the runt of the litter of this wing of Traditional Thought, i.e., DM itself), treating it as a 'profound truth', which he was concerned to deny always (or ever(?)) applies to objects or processes in 'concrete' reality. So, for him (and for Hegel, so far as can be ascertained), the LOI is merely an 'approximate or relative truth' when used 'concretely'. Of course, this Essay is aimed at exposing the irredeemable confusion that arises from the shared assumption that the LOI is either (a) absolutely true, (b) approximately true, or (c) is any sort of truth to begin with. [On this, see Note 10, Note 13, Note 15 and Note 20.]


If, however, the LOI is simply a misleading expression of the many rules we have in the vernacular that enable us to speak of sameness and difference in everyday life, then treating it as any sort of truth would be to fetishise it. That is, it would amount to treating what had once been the product of the social relations between human beings (i.e., a family of practice-based rules about sameness and difference) as if it were the real relation between things (or, in this case, as a relation allegedly between an object and itself, even if it is only ever 'approximately true'), or even as those things themselves -- if we insist on treating Identity as an object or a concept named by the word "identity". For Hegel, this fetishisation appears in his fairy tale about the incestuous relationship between 'Being', 'Nothing' and 'Becoming'.


[In Essay Three Part One, we saw that that fantasy itself arose out of a crass misinterpretation of the "is" of predication as an "is" of identity. In Essay Eight Part Three, we also saw this egregious error was compounded by further syntactic confusions of a rather more radical nature.]


The ideological background to the fetishisation of language like this (initiated in the 'West' by Ancient Greek Philosophers, which was later turned into an art-form by Hegel) -- and how its reappearance in DM has helped cripple Dialectical Marxism -- is analysed in Essays Nine Part Two, Ten Part One, and Fourteen Part Two.


The idea that concepts are capable of changing (in the way that Hegel and/or DM-fans imagine they can) is discussed in Essay Four, where it was shown that they may only do so if they have been turned into abstract objects, which unfortunately destroys their role as concepts. Depicted in this way, DM-concepts don't in fact change, they vanish.


20. The material that used to be here has been moved to the main body of the Essay.


20a. The material that used to be here has been moved to the main body of this Essay.


21. This isn't to suggest that the words we have for identity and difference (in ordinary language) severally or collectively have only one meaning; the wide and expansive semantic diversity available to us (by the use of such words) has been highlighted by the numerous examples given in this Essay (for instance, here). In fact, countless alternative connotations have been omitted.


22. The term "non-sense" is being used here in a specific way, not in its more usual, disparaging sense to mean "ridiculous", "patently wrong" or "mere babble". Nor does it refer to syntactically-challenged symbols, such as: "the is but of", "XXX YYY ZZZ", or even "**##&&$£**%^&&*^%".


At this site, the epithet "non-sense" has in general been reserved for metaphysical 'propositions'. As we will see in detail in Essay Twelve Part One, the latter appear to communicate profound, 'Super-Empirical Truths' about 'Reality Itself'. However, upon examination each of them turns out to be (i) A disguised or misconstrued rule of language, or (b) Based on a distortion of ordinary language -- as we saw in Essay Three Part One, for example, where the "is" of predication was transformed into the "is" of identity (by Hegel and subsequent DM-theorists), and in this Essay in connection with ordinary words for identity, etc.


[There is as yet no definitive account of Wittgenstein's analysis of the words we have for identity (etc.), however Roger White's long-awaited book should cast considerable light on this topic. Cf., White (2006 and forthcoming). There is an excellent, albeit brief summary in Glock (1996), pp.256-64, with a more considered account in Diamond (1991), alongside several articles published in Crary and Read (2000) -- for example, Conant (2000). See also, Goldfarb (1997) and McGinn (1999).]


23. See Note 1, above.


24. 19th century mathematicians found that they couldn't construct a rigorous solution to problems they faced in Real and Complex Analysis until they had much clearer ideas about the nature of numbers, infinite series, limits, convergence and continuity. As the more astute of them began to realise, this would involve a more precise understanding of quantifiers, multiple generality, scope ambiguity and the logic of relational expressions. This meant that many of the advances subsequently made in Analysis and foundational work over the last hundred years or so wouldn't have happened without the new Mathematical Logic introduced by logicians and mathematicians (such as Boole, Frege, Peano, Pierce, Whitehead and Russell). [On this, see Giaquinto (2004), and Grattan-Guinness (1970, 1997, 2000a, 2000b). See also the excellent summary in Kitcher (1984), pp.227-71.]


In addition, as was pointed out in Essay Four, modern FL has played a key role in contemporary technological innovation, in the development of computers, for example, whereas DL has yet to motivate a single successful practical application --, save, of course, that of radically confusing Dialectical Marxists, or, indeed, screwing around with Russian agriculture.


In that case, this isn't a dry, academic, impractical issue.




Several of Marx and Engels's works listed below have been linked to the Marxist Internet Archive, but since Lawrence & Wishart threatened legal action over copyright infringement many no longer work.


However, all of their work can now be accessed here.


Adamson, G. (2002), Philosophy In The Age Of Science And Capital (Continuum).


Aristotle, (1984a), The Complete Works Of Aristotle, Two Volumes, edited by Jonathan Barnes (Princeton University Press).


--------, (1984b), Metaphysics, in Aristotle (1984a), Volume Two, pp.1552-1728.


Baghavan, R. (1987), An Introduction To The Philosophy Of Marxism (Socialist Platform).


Barnes, J. (2009), Truth, Etc. Six Lectures On Ancient Logic (Oxford University Press).


Beaney, M. (1997), The Frege Reader (Blackwell).


Brading, K., and Castellani, E. (2003) (eds.), Symmetries In Physics. Philosophical Reflections (Cambridge University Press).


Bukharin, N. (1925), Historical Materialism (George Allen & Unwin).


Butterfield, J. (1993), 'Interpretation And Identity In Quantum Theory', Studies in History and Philosophy of Science, 24, pp.443–76.


Castellani, E. (1998) (ed.), Interpreting Bodies. Classical And Quantum Objects In Modern Physics (Princeton University Press).


Caulton, A., and Butterfield, J. (2012), 'On Kinds Of Indiscernibility In Logic And Metaphysics', British Journal for the Philosophy of Science 63, 1, pp.27-84.


Conant, J. (2000), 'Elucidation And Nonsense In Frege And Early Wittgenstein', in Crary and Read (2000), pp.174-217.


Conner, C. (1992), Dialectical Materialism. The Philosophy Of Marxism (Walnut Publishing).


Crary, A., and Read, R. (2000) (eds.), The New Wittgenstein (Routledge).


Deutsch, H. (2018), 'Relative Identity', Stanford Encyclopedia of Philosophy, edited by Edward N. Zalta (Fall 2018 Edition).


Deutscher, I. (1970), The Prophet Outcast. Trotsky 1929-1940 (Oxford University Press).


Diamond, C. (1991), The Realistic Spirit (MIT Press).


Dieks, D., and Versteegh, M. (2007/2008), 'Identical Quantum Particles and Weak Discernibility', paper deposited at the University of Pittsburgh Philosophy of Science Archive. [This has now been published in Foundations of Physics 38, October 2008, pp.923-34.]


Dietzgen, J. (1906), The Positive Outcome Of Philosophy (Charles Kerr).


Dirac, P. (1967), The Principles Of Quantum Mechanics (Oxford University Press, 4th ed.).


Engels, F. (1888), Ludwig Feuerbach And The End Of Classical German Philosophy, reprinted in Marx and Engels (1968), pp.584-622.


--------, (1954), Dialectics Of Nature (Progress Publishers).


--------, (1976), Anti-Dühring (Foreign Languages Press).


Feferman, A. (1993), From Tarski To Gödel. The Life Of Jean Van Heijenoort (A K Peters Ltd.).


Fine, G. (1999) (ed.), Plato 1: Metaphysics And Epistemology (Oxford University Press).


--------, (2000), 'Inquiry In The Meno', in Kraut (2000), pp.200-26, reprinted with minor modifications as Fine (2003b).


--------, (2003a), Plato On Knowledge And Forms. Selected Essays (Oxford University Press).


--------, (2003b), 'Inquiry In The Meno', in Fine (2003a), pp.44-65.


--------, (2007), 'Enquiry And Discovery: A Discussion Of Dominic Scott, Plato's Meno', in Sedley (2007), pp.331-67.


--------, (2018), The Possibility Of Inquiry: Meno's Paradox From Socrates To Sextus (Oxford University Press).


Frege, G. (1892), 'On Sense And Meaning', in Frege (1980), pp.56-78; reprinted in Beaney (1997), pp.151-71, as 'On Sinn And Bedeutung', and in Frege (1980), pp.56-78.


--------, (1980), Translations From The Philosophical Writings Of Gottlob Frege, translated by Peter Geach and Max Black (Blackwell, 3rd ed.).


French, S. (2019), 'Identity And Individuality In Quantum Theory', Stanford Encyclopedia of Philosophy, edited by Edward N. Zalta (Winter 2019 Edition).


French, S., and Krause, D. (2006), Identity In Physics: A Historical, Philosophical And Formal Analysis (Oxford University Press).


French, S., and Redhead, M. (1988), 'Quantum Physics And The Identity Of Indiscernibles', British Journal for the Philosophy of Science, 39, pp.233–46.


Gallois, A. (2003), Occasions Of Identity. The Metaphysics Of Persistence, Change and Sameness (Oxford University Press).


Geach, P. (1967), 'Identity', Review of Metaphysics 21, pp.3-12; reprinted in Geach (1972), pp.238-47.


--------, (1968), Reference And Generality (Cornell University Press, 2nd ed.). [This links to the 3rd (1980) edition.]


--------, (1972), Logic Matters (Blackwell).


--------, (1973), 'Ontological Relativity And Relative Identity', in Munitz (1973), pp.287-302.


--------, (1975), 'Names And Identity', in Guttenplan (1975), pp.139-58.


--------, (1990), 'Identity Theory: A Reply To Muller, Williams And Dummett', in Lewis (1990), pp.276-99.


Giaquinto, M. (2004), The Search For Certainty. A Philosophical Account Of Foundations Of Mathematics (Oxford University Press).


Glock, H-J. (1996), A Wittgenstein Dictionary (Blackwell).


Goldfarb, W. (1997), 'Metaphysics And Nonsense', Journal of Philosophical Research 22, pp.57-73.


Grattan-Guinness, I. (1970), The Development Of The Foundations Of Mathematical Analysis From Euler To Riemann (MIT Press).


--------, (1997), The Fontana History Of The Mathematical Sciences (Fontana).


--------, (2000a), The Search For Mathematical Roots 1870-1940. Logics, Set Theories And The Foundations Of Mathematics From Cantor Through Russell To Gödel (Princeton University Press).


--------, (2000b), From Calculus To Set Theory 1630-1910. An Introductory History (Princeton University Press).


Griffin, N. (1977), Relative Identity (Oxford University Press).


Guttenplan, S. (1975) (ed.), Mind And Language (Oxford University Press).


Hahn, S. (2007), Contradiction In Motion. Hegel's Organic Concept Of Life And Value (Cornell University Press).


Hale, R., and Wright, C. (1997) (eds.), A Companion To The Philosophy Of Language (Blackwell).


Hanna, R. (1986), 'From An Ontological Point Of View. Hegel's Critique of Common Logic', Review of Metaphysics 40, pp.305-38; reprinted in Stewart (1996), pp.253-81.


Hegel, G. (1975), Logic, translated by William Wallace (Oxford University Press, 3rd ed.).


--------, (1999), Science Of Logic (Humanity Books).


Kenny, A. (1995), Frege (Penguin Books).


Kitcher, P. (1984), The Nature Of Mathematical Knowledge (Oxford University Press).


Hilborn, R., and Yuca, C. (2002), 'Identical Particles In Quantum Mechanics Revisited', British Journal for the Philosophy of Science 53, 3, pp.355-89.


Houlgate, S. (2006), The Opening Of Hegel's Logic (Purdue University Press).


Kraut, R. (2000) (ed.), The Cambridge Companion To Plato (Cambridge University Press).


Ladyman, J., and Bigaj, T. (2010), 'The Principle Of The Identity Of Indiscernibles And Quantum Mechanics', Philosophy of Science 77, 1, pp.117-36.


Lear, J. (1980), Aristotle And Logical Theory (Cambridge University Press).


Lefebvre, H. (1968), Dialectical Materialism (Jonathan Cape).


Lenin, V. (1961), Collected Works, Volume 38 (Progress Publishers).


--------, (1972), Materialism And Empirio-Criticism (Foreign Languages Press).


Lewis, H. (1990) (ed.), Peter Geach: Philosophical Encounters (Kluwer Academic Press).


Linsky, L. (1977), Names And Descriptions (University of Chicago Press).


Lowe, E. (1989), Kinds Of Being: A Study Of Individuation, Identity And The Logic Of Sortal Terms (Blackwell).


--------, (2015), More Kinds Of Being. A Further Study Of Individuation, Identity And The Logic Of Sortal Terms (Wiley-Blackwell).


Magee, G. (2008), Hegel And The Hermetic Tradition (Cornell University Press). [The Introduction to this book is available here.]


Mandel, E. (1979), Introduction To Marxism (Pluto Press, 2nd ed.).


Marion, M. (1998), Wittgenstein, Finitism, And The Foundations Of Mathematics (Oxford University Press).


Marx, K., (1976), Capital Volume One (Penguin Books).


Marx, K., and Engels, F. (1968), Selected Works In One Volume (Lawrence & Wishart).


--------, (1970), The German Ideology, Students Edition, edited by Chris Arthur (Lawrence & Wishart).


--------, (1975), Selected Correspondence (Progress Publishers, 3rd ed.).


--------, (2004), MECW, Volume 50 (Lawrence & Wishart).


McGinn, M. (1999), 'Between Metaphysics And Nonsense', Philosophical Quarterly 49, pp.491-513.


Modrak, D. (2001), Aristotle's Theory Of Language And Meaning (Cambridge University Press).


Molyneux, J. (2012), The Point Is To Change It. An Introduction To Marxist Philosophy (Bookmarks).


Moravcsik, J. (1971), 'Learning As Recollection', in Vlastos (1971), pp.53-69.


Muller, F., and Saunders, S. (2008), 'Discerning Fermions', British Journal for the Philosophy of Science 59, 3, pp.499-548.


Muller, F., and Seevinick, M. (2009), 'Discerning Elementary Particles', Philosophy of Science 76, 2, pp.179-200.


Munitz, M. (1973) (ed.), Logic And Ontology (New York University Press).


Noonan, H. (1980), Objects And Identity (Martinus Nijhoff).


--------, (1997), 'Relative Identity', in Hale and Wright (1997), pp.634-52.


--------, (2001), Frege. A Critical Introduction (Polity Press).


--------, (2022), 'Identity', Stanford Encyclopedia of Philosophy, edited by Edward N. Zalta (Fall 2022 Edition).


Novack, G. (1965), The Origins Of Materialism (Pathfinder Press).


--------, (1971), An Introduction To The Logic Of Marxism (Pathfinder Press, 5th ed.).


Ollman, B. (2003), Dance Of The Dialectic: Steps In Marx's Method (University of Illinois Press).


Perkins, D. (2000), Introduction To High Energy Physics (Cambridge University Press).


Pippin, R. (1978), 'Hegel's Metaphysics And The Problem Of Contradiction', Journal of the History of Philosophy 16, pp.301-12; reprinted in Stewart (1996), pp.239-52.


Plato, (1997a), Complete Works, edited by John Cooper (Hackett Publishing).


--------, (1997b), Meno, in Plato (1997a), pp.870-97.


Plekhanov, G. (1908), Fundamental Problems Of Marxism (Lawrence & Wishart). [The Appendix to this work -- which in fact formed part of Plekhanov's Introduction to Engels (1888) -- can be found here under the title 'Dialectic and Logic'. It can also be found in Plekhanov (1976), pp.73-82.]


--------, (1956), The Development Of The Monist View Of History (Progress Publishers). This is reprinted in Plekhanov (1974), pp.480-737.


--------, (1974), Georgi Plekhanov. Selected Philosophical Works, Volume One (Progress Publishers, 2nd ed.).


--------, (1976), Georgi Plekhanov. Selected Philosophical Works, Volume Three (Progress Publishers).


Priest, G. (2000), Logic. A Very Short Introduction (Oxford University Press).


Read, R. (2007), Applying Wittgenstein, edited by Laura Cook (Continuum Books).


Rees, J. (1998), The Algebra Of Revolution (Routledge). [This links to a PDF.]


Sanford, D. (2005), 'Distinctness And Non-Identity', Analysis 65, 4, pp.269-74.


Saunders, S. (2006), 'Are Quantum Particles Objects?', Analysis 66, 1, pp.52-62. [This links to a PDF.]


Scott, D. (1999), 'Platonic Recollection', in Fine (1999), pp.93-124.


--------, (2007), Plato's Meno (Cambridge University Press).


Sedley, D. (2007) (ed.), Oxford Studies In Ancient Philosophy, Volume XXXII (Oxford University Press).


Somerville, J. (1946), Soviet Philosophy (Philosophical Library).


--------, (1967), The Philosophy Of Marxism (Random House).


Stewart, J. (1996) (ed.), The Hegel Myths And Legends (Northwestern University Press).


Thalheimer, A. (1936), Introduction To Dialectical Materialism. The Marxist World-View (Covici, Friede Publishers).


Tomassi, P. (1999), Logic (Routledge).


Trotsky, L. (1971), In Defense Of Marxism (New Park Publications).


--------, (1973), Problems Of Everyday Life (Monad Press).


--------, (1986), Notebooks, 1933-35 (Columbia University Press). [Parts of this book can be accessed here; this links to a PDF.]


Van Heijenoort, J. (1978), With Trotsky In Exile. From Prinkipo To Coyoacán (Harvard University Press).


Vlastos, G. (1971) (ed.), Plato. A Collection Of Critical Essays. 1: Metaphysics And Epistemology (Doubleday Anchor).


Voloshinov, V. (1973), Marxism And The Philosophy Of Language (Harvard University Press). [Chapters One and Two can be found here.]


Weiner, J. (1990), Frege In Perspective (Cornell University Press).


--------, (1999), Frege (Oxford University Press).


White, R. (1978), 'Wittgenstein On Identity', Proceedings of the Aristotelian Society 78, pp.157-74.


--------, (2006), Wittgenstein's Tractatus Logico-Philosophicus. A Reader's Guide (Continuum).


--------, (forthcoming), The General Form Of A Proposition.


Williams, C. (1979), 'Is Identity A Relation?', Proceedings of the Aristotelian Society 80, pp.81-100.


--------, (1989), What Is Identity? (Oxford University Press).


--------, (1992), Being, Identity, And Truth (Oxford University Press).


Wittgenstein, L. (1958), The Philosophical Investigations, translated by Elizabeth Anscombe (Blackwell, 2nd ed.).


--------, (1972), Tractatus Logico-Philosophicus, translated by David Pears and Brian McGuinness (Routledge, 2nd ed.).


--------, (1993), Philosophical Occasions, 1912-1951, edited by James Klagge and Alfred Nordmann (Hackett Publishing Company).


Woods, A., and Grant, T. (1995/2007), Reason In Revolt. Marxism And Modern Science (Wellred Publications 1st/2nd ed.). [The online version still appears to be the First Edition.]


Internet Articles


[1] French, S. (2019), 'Identity And Individuality In Quantum Theory', Stanford Encyclopedia of Philosophy, edited by Edward N. Zalta (Winter 2019 Edition).


[2] Lowe, E. (No date), Philosophical Logic. [Accessed 16/09/11.]


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