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

Visitors are encouraged to read this Essay in conjunction with Essays Four and Five.

Technical Preliminaries

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Preface

[DM = Dialectical Materialism/Materialist, depending on the context; HM = Historical Materialism/Materialist, again depending on the context.]

As is the case with all my work, nothing here should be read as an attack either on HM -- a scientific 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-eight years ago. The difference between DM and HM, as I see it, is explained here.

Some readers might wonder why I have quoted extensively from a wide variety of DM-sources in the Essays published at this site. In fact a good 10-20% of the material in many of them is comprised of just such quotations. Apologies are therefore owed the reader in advance for the length and extremely repetitive nature of most of these passages. The reason for their inclusion is as follows: Long experience has taught me that Dialectical Marxists simply refuse to accept that their own classicists -- e.g., Engels, Plekhanov, Lenin, Trotsky and Mao --, alongside countless 'lesser' DM-theorists actually said the things I have attributed to them. That is especially the case after they are confronted with the absurd consequences that follow from their words. That remains the case unless they are shown chapter and verse and in extensive detail. Partial quotes, paraphrases and summaries in my own words they reject as "distortions", "mis-interpretations", "mis-representations", and even "manipulations".  Furthermore, in debate when I quote only one or two passages in support of what I allege they are simply brushed off as "outliers" or "atypical". Indeed, in the absence of dozens of proof texts drawn from a wide range of sources (drawn from all areas of Dialectical Marxism), and reproduced in full, they tend to regard anything that a particular theorist has to say -- regardless of who they are, even if they turn out to be one of the aforementioned classicists -- as either "far too crude", "unrepresentative" or even(!) unreliable. Failing that they often complain that any such quotes have been "taken out of context". Many in fact object since -- surprising and sad though this is to say --, they are largely ignorant of the fine detail of their own theory, or, perhaps even worse, they simply haven't read the DM-classics! The only way to counter such attempts to deflect, reject and deny is to quote DM-material frequently and at length.

But even that doesn't always work!

Furthermore, because of the highly sectarian and partisan nature of Dialectical Marxism I have also found it necessary to quote a wide range of sources from across the entire 'dialectical spectrum'. Trotskyists object if I quote Stalin or Mao; Maoists and Stalinists complain if I reference Trotsky -- or even if I cite "Brezhnev era revisionists". Non-Leninist Marxists bemoan the fact that I haven't confined my remarks solely to what Marx or Hegel had to say, advising me to ignore the confused, even "simplistic", ideas expressed by Engels, Plekhanov, Lenin, Stalin, Mao and Trotsky! This often means I have to quote the lot!

That itself has had the (indirect) benefit of revealing how much and to what extent they (the classicists and subsequent epigones across all areas of Dialectical Marxism) largely agree with each other (despite sectarian rhetoric to the contrary), at least with respect to DM (but not so much over how to apply it)!

Some critics have complained that my linking to Wikipedia completely undermines the credibility of these Essays. When I launched this project on the Internet in 2005, there was very little material easily available on-line related to the vast majority of topics I could link to other than Wikipedia. In the intervening years alternative sites have become available (for example, the excellent Stanford Encyclopedia of Philosophy and the Internet Encyclopedia of Philosophy), so I have been progressively replacing most of the old Wikipedia links with links to these other sources. Having said that, I haven't done so for some of the Wikipedia links -- for instance, those that are connected with geographical, historical, scientific, biographical (etc.) topics where the latter aren't deemed controversial, at least among fellow Marxists. In every case, I have endeavoured to avoid linking to Wikipedia in relation to key areas of my arguments against DM so that at no point does my criticisms depend exclusively on such links. [However, it will take several years to complete this change-over!]

In addition to the above (as readers can easily check if they consult the Bibliography and End Notes attached to each Essay), I have provided copious references to other published academic and non-academic books and articles (posted on-line or printed in hard copy) that further develop, amplify or substantiate anything I argue, claim, allege or propose.

Several others have complained about the sheer number of links there are attached to these Essays, because they say it makes them very difficult to read. Of course, DM-supporters can hardly grumble about that since they believe everything is inter-connected, which must surely apply to Essays that attempt to debunk that very idea. However, to those who find this does make these Essays difficult to read I say this: ignore the links(!) -- unless, of course, you want to access further supporting evidence and argument related to a particular point, or a specific topic fires your interest.

Still others wonder why I have linked to familiar subjects and topics that are part of common knowledge -- such as the names of recent US Presidents, UK Prime Ministers, the names of rivers and mountains, the titles of popular films or the definition of certain words in common usage. I have done so for the following reason: My Essays are read all over the world by people from all 'walks of life', so I can't assume that something which is part of common knowledge in 'the west' is 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 aren't native English speakers, either, so any help I can give them I will continue to provide.

Finally on this specific issue, several of the aforementioned links connect to web-pages that regularly change their URLs or which even 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. I would greatly appreciate it, therefore, if readers informed me of any dead or incorrect 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.

I have also linked to Woods and Grant's book, Reason in Revolt, in this Essay, but the link I used currently only takes the reader to parts of the Second Edition instead of the entire book, which used to be the case. However, anyone who wants to access a complete version of that specific version can now do so here. I haven't changed the scores of links to the old site because they used to take the reader to specific chapters of that book, but that faculty is no longer available.

Formal Logic [FL] is a highly technical subject, but since the vast majority of revolutionaries know little or no FL -- that isn't to put them down; it is possible to know zero FL and still be an excellent revolutionary! --, in what follows I have had to keep such complexities to a minimum. What little FL I have used has deliberately been kept rather simple, basic and straightforward. Any technical terms I have used I have endeavoured to explain or provide links to where they are explained. If there is anything I have left obscure, unclear or unexplained, please let me know and I will do my best to rectify the problem.

It is also worth adding 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, readers are directed here, here and here for further 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 important to point out that up to a half 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.]

Two final points:

First of all, throughout this site I have used the term "Dialectical Marxism"; I have done so in order to distinguish Marxism from the 'dialectical aberration' that has completely dominated (and I claim has helped cripple) revolutionary socialism for over 150 years. So, the terms 'Dialectical Marxism' and 'Dialectical Marxist' refer to the mainstream 'Marxist' tradition that has accepted, in one form or another, the validity of DM and as a result has either applied it to all of 'reality' or has restricted it to social and historical development (i.e., where it has largely or even exclusively been inserted into HM). However, I distance Marx himself from this toxic amalgam -- to which conclusion many Dialectical Marxists have taken great exception (perhaps more so than anything else at this site!). Why that is so has been explained here and here. Furthermore, why this Hegelianised Hybrid (but not Marxism itself) has been an abject failure for well over a century is explained here and here. [That argument has been summarised here and here.]

Second: Anyone puzzled by the unremittingly hostile tone I have adopted toward DM and Dialectical Marxism might find it instructive to read this first.

~~~~~~oOo~~~~~~

As of October 2025, this Essay is just under 79,000 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: 06/10/25.]

Quick Links

Anyone using these links must remember that they will be skipping past supporting argument and evidence set out in earlier sections.

If your Firewall/Browser has a pop-up blocker, you will need to press the "Ctrl" key at the same time or these and the other links here won't work!

I have adjusted the font size used at this site to ensure that even those with impaired vision can read what I have to say. However, if the text is still either too big or too small for you, please adjust your browser settings!

(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 Had This Base Covered -- Or Did He?

(5) Bags Of Sugar Actually Refute Trotsky

(a) Trotsky's Answer

(b) Was This 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) Is Trotsky's Analysis 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) Criticisms Of Ordinary Language And The LOI Are Unequal To The Task

(i)  Materially-Induced Dialectical Misery

(8) The Knock-Out Blow

(9) Traditional Logic Versus Modern Formal Logic

(a) Is Traditional Logic Defective?

(b) Is Dialectical Logic Superior To Formal Logic?

(c) Does Formal Logic Represent A Fragmented, 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

Introduction

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.

As this Essay unfolds it should become increasingly obvious to attentive readers that Trotsky not only confused equality with identity, he conflated two different 'problems' connected with identity itself: (i) The diachronic (i.e., the temporal) identity of an object or process with itself as it changes through time, and (ii) The synchronic identity of two or more objects/processes (at the same moment in time). This confusion alone makes much of what he wrote about identity completely worthless (and I say that as a Trotskyist!). It also raises serious questions about the intellectual credibility and honesty of his epigones and followers, who have uncritically swallowed and unthinkingly propagated his garbled claims about the LOI (each of which will be examined in painstaking detail in what follows).

So, by the end of this Essay it should be clear to all but the most partisan and sycophantic that:

(a) Identity is no enemy of change; and that,

(b) Dialectical Marxists would be well advised to ignore everything Trotsky had to say about this entire topic.

[The same will be the case with respect to Hegel's claims about the LOI, but that task has been devolved to Essay Twelve Part Five.]

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 borrowed (directly or indirectly) from Hegel, which 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 100 years leading up to his intervention on this topic.² Clearly, he didn't think his failure to check the 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!

That 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. That is hardly surprising given the allegations advanced above.

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 this way:

"[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, ran as follows:

"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 the following way, I will highlight them in bold to distinguish them from the ordinary use of the same (no irony intended).]

So, there are at least a dozen substantive points Trotsky was trying to make 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.

There are three initial points that are worth making (which will be expanded on as this Essay proceeds):

First, as noted in the Introduction, Trotsky is here conflating (i) The diachronic (i.e., the temporal) identity of an object or process with itself as it changes, and (ii) The synchronic (i.e., the same time) identity of two or more objects/processes. Whether or not two physical examples of A are identical has no bearing on whether or nor A is identical with itself over time. But as with most things, that itself depends on what the key term used to make these points themselves mean. This Essay will seek to untangle the many linguistic and logical confusions that have bedevilled this topic for centuries (particularly as they apply to Trotsky's words, and, derivatively, to Hegel's).

Second: also as noted in the Introduction, instead of focussing on identity, Trotsky immediately deflected his attention onto equality. The two are not the same (no irony intended), as will be demonstrated in what follows.

Finally: Trotsky nowhere supports any of his claims with actual evidence -- or, rather, none that isn't itself based on further thought experiments. That (otherwise) fatal defect rarely seems to bother his epigones. In earlier Essays, we saw why they airily brush aside the need to substantiate their theories with anything that even remotely resembles proof: in their eyes it seems that if the universe is governed at every level by DL, a simple 'thought experiment' is all the 'evidence' a Dialectical Marxist really needs.

[DL = Dialectical Logic.]

Naturally, only consistent materialists will object at this point.

As George Novack pointed out:

"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 adding 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 also 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"

An Interpretative Error 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 no doubt be appreciated, that reply is itself problematic -- not the least because it reveals that the latter individuals have an equally insecure grasp of the issues involved. [Irony intended.]

Well:

(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, 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, that 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 also have an insecure grasp of the nature of abstraction, and are largely content to be told what to think (on this score) by notorious Mystics and 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 anyone's serious attention.

Ordinary Language Once Again Thwarts Dialectical Casuistry

Our comprehension of words for identity, sameness, equality and difference clearly depends on the use of certain expressions in ordinary life, whatever technical modifications we might subsequently introduce, and for whatever reason that might be done. 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 only approximate 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 of, 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 else 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 rapid 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 by Marx right across the planet, at the Marxist Internet Archive. Despite the fact that they might consult 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. 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.]

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It could be argued that these examples of identity aren't in fact examples of strict identity since all of 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 (here, here, 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, 10^2000000 -- 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, Two and Three. 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 theory 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 and relations: B₁, B₂, B₃,..., B_n.

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 -- other than being informed that they change into their 'opposites' -- i.e., into not (B₁, B₂, B₃,..., B_n), or perhaps (B₁*, B₂*, B₃*,..., B_n*), where each term with an attached asterisk is the 'opposite' of the 'same' term without one).

[However, even that possibility is closed off in Essay Seven 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, B_i, changes, it becomes, say, B_i*, in the first instance, and then B_i** 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*,..., B_n*}. 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 Essay Seven Part One and Essay Eight Parts One, Two and Three (links above).]

[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, B_k -- 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; they have been set out in detail in Essay Eleven Part One.

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

A nice unity of opposites here, one feels!

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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.]

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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 Tipp-ex or the delete key.

And deletion clearly 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 any permitted by the obscure and lifeless language Hegel inflicted on his readers. That shouldn't surprise anyone; ordinary, technical and semi-technical languages have been developed over countless centuries by working people and/or scientists in their interaction with the world and with one another. These systems of communication reflect the complex inter-relationships 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.

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Clearly, Trotsky and Hegel created serious problems for themselves when they erected an insecure 'logical' edifice on such insubstantial linguistic foundations. This predicament was further compounded by their choice of an extremely narrow range of examples 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 easily confirmed fact. This 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 x 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 only be flanked 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 (i.e., from 'identity' to 'equality') allowed Trotsky to deflect onto irrelevant claims about things like bags of sugar, which is another point raised in the Introduction -- i.e., that Trotsky deliberately confused diachronic identity (whether A is equal to itself as it changes) with synchronic identity (whether one A is equal to another A at the same time). Equality introduces issues that involve (i) the measurement of objects as opposed to their being counted, and (ii) the interpretation of each A in S1 as quantities of sugar. Both factors heavily biased Trotsky's criticism of the LOI, allowing him to focus on aspects of equality that aren't necessarily, or even typically, 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 (diachronic identity) even while it was unequal in weight to a second seemingly identical bag (synchronic identity). [How that is possible will become clear as the argument unfolds.] And, it is entirely possible for two different bags of sugar to be equal in weight (even if only momentarily) as far as our most sensitive instruments are concerned. Not only that, two separate bags could both have their weights changing in exactly the way and at the same rate that Trotsky described (no irony intended). Indeed, 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 distinct 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 pile 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 clear distinctions like these; and it is worth adding that such distinctions are easily drawn in ordinary language (indeed, as they have been, here) -- differences that 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. 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 that number, even if other aspects of the said objects 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 this sense alone most do.

Of course, it could be objected that not only do some things divide as they change, others merge; 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 an 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 divided or merged. 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 specific 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 the above objection (i.e., 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 greatly increased 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 2025 (namely, three (or six)).

L2: The number of volumes of Das Kapital on any one day in 2025 is identical to the number of volumes of Das Kapital on the same day in 2025 (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 2025" (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 2025", which noun 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" (in L3 or S1) fails to capture the much wider use of words we have for sameness, equality and 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 criticisms of the LOI.

It is also worth recalling that the volumes of Das Kapital are just as material as bags of sugar. So, this isn't an 'abstract' counter-argument.

In response, 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 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 2025, 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 language-user 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 and difference -- those derived from that notorious Idealist, Hegel -– as use that plainly failed to  match their employment 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.]

As pointed out several times, Trotsky is here diverting attention away from the diachronic identity of an object with itself (which is what the LOI is actually about!), onto the identity of two (separate) letters, both labelled "A" (which it isn't!).

Independently of that major mis-step, 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 way, 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 this 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 Had This Base Covered -- Or Did He?

Some might object and claim that Trotsky had anticipated the above points when he said the following:

"[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 faced by anyone operating in this (changing) world. In that case, since no two symbols could be absolutely identical, Trotsky's point remains valid.

Or so this pro-DM-response 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 that term -- is itself parasitic on ordinary identity, or on a surreptitious material application of the LOI (as a rule, not as a truth), as we are about to 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 might have 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 the relatively minor quibbles aired in earlier sub-sections, 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.]

Was This Mere Guesswork On Trotsky's Part?

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

Indeed, there are several alternatives he failed to consider. For example, the weighing scales themselves could alter slightly; that could be the real cause of the inferred/alleged change in the weight of the sugar. Indeed, the weighing scales could alter to such an extent that they actually compensate for the change in the weight of the sugar, so that in the end no overall difference was detected. How could Trotsky possibly rule either of these out? Plainly, he couldn't do so if constant change -- including that undergone by instruments -- is a central postulate of DM. 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 their being genuine phenomena representing real 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 a dogmatic stipulation to that effect. And, as seems plain, a stipulation is different from an imposition on the facts in name alone.¹⁰

Of course, that doesn't mean Trotsky was wrong. 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 these 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 such factors --, or, indeed, whether or not they were attributable other proximate causes he failed to consider.

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 theory 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 any such claim can't actually be verified. And that 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 extrapolated from a few (theoretical) observations about local conditions connected with weighing that bag 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 his 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 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 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 could ever be the same, anyway. In that case, there would be no accurate, exactly repeatable way to test his or anyone else's predictions!

[Once more, an appeal to "approximately identical instructions" has been dealt with, below.]

On the other hand, if Trotsky had been confronted by 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 contrary-experimenter must have been mistaken.

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

(3) Alleging 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 Options (1) and (2), above), Trotsky would only be able to criticise this apparent 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' the LOI!

In which case, Trotsky's predictions about bags of sugar 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/diluted version of it!

Hence, an appeal to 'approximate identity' here 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 -- and not just because there is no evidence that he performed these experiments himself. Moreover, anyone who did, or who might perform them would be caught in the above practical refutation of his criticisms of this 'law' each time they performed one of these experiments exactly as he instructed.

[It is worth adding that the development of science has at least suggested a much more cautious approach (than one adopted by Trotsky) 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 law -- that is, into a 'fundamental truth' supposedly valid across all regions 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 claims 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 the LOI (or highlight its assumed 'limitations') 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 try to refute, earlier results, or even check measurements and hence confirm the accuracy of Trotsky's predictions.

Nor would they be able to learn to employ 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' (again, applied as a rule of language, or which helped guide experimentation), 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 DM-claims about 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 that 'law', only now operating as a rule governing the use of words to make this 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 expresses a rule (and a rather badly stated rule about our use of words for identity, sameness or difference, too). In that case, just like any rule, it can't be true or false, only practical or impractical, useful or useless, obeyed or broken.

In fact, as we have seen, it is impractical -- or, rather, it is the height of folly -- even to attempt to reject, or even criticise, 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 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 other words we have for identity, sameness and difference. Without some comprehension concerning 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 might fall short of a standard that presupposed the applicability of words we already 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 is one that the "abstract understanding" employs seemingly uncritically, or which is 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 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 these 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 so 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 applied in any 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 to gather any evidence aimed at questioning it, and so are the arguments used to undermine it. None of the latter could remain the same from moment-to-moment if DM-theorists (and Hegel) are correct. Even worse, none of their words would remain the same from moment-to-moment, either. And if that is so, none of their conclusions (which must alter, too, in like manner) could be trusted.

Which is indeed what we have seen.

[There is more on this in Note 15, below. See also Note 13.]

Naturally, a grudging acceptance of the above (linguistic and socially-sanctioned) conventions -- along 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 bags 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, exactly and to the letter, would, in that very act, refute such ill-considered 'thought experiments' - again as we 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 obscure 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 those remarks 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, or imply, W1.

The above counter-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 concerning any two objects, not only can workers see that they aren't the same, they also know they can't make two distinct objects that turn out to be identical. He didn't say, or even imply, 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 that.

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 that direction 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 specific topic at greater length 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." [Trotsky (1971), 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 objects) 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 certain 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 them 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 exactly 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, thereby showing the latter aren't identical!

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

[Earlier we saw there was a similar predicament facing experimenters.]

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/union officials could be promoted to the same rank/post as one another (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 these unfortunate 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 were no longer restricted to considering only two items, 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 or both of the former.

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

It could be objected that these supposed counter-examples themselves beg the question, since, if Trotsky is right about the defects of the LOI, none of them would be genuine identity statements (which is simply assumed to be the case, above).

["Beg the question" is here being used with its older meaning -- i.e., "To assume what was to be proved" -- not with is contemporary meaning, "To prompt or elicit a question".]

However, as argued earlier, our ordinary lexicon of words we have for identity (for instance, "the same as", "exact", "similar", "identical", "not different", "precisely", etc.) is itself highly complex. It is far more sophisticated than Trotsky or Hegel imagined in their 'theoretical' deliberations -- although in their everyday speech they couldn't have been unaware of this. They would have used sentences that employed terms like those above countless times throughout their lives.

The vernacular -- which, it is worth reminding ourselves (yet again) is derived from, and even based on, everyday, material practice -- allows for the expression of a wide range of complex and varied identities; those given above amounting to a tiny fraction of those possible. [There are many more of the same itemised 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 plainly be incapable of recognising, using or comprehending the same words in the same way as the rest of us -- or even as they themselves). In which case, they would hardly be in any position to criticise the 'law' [i.e., the LOI] that supposedly operates behind, or which informs, such words -- or even that expresses a relation between certain objects and processes.

Indeed, the employment of words/phrases like those above (in the sort of contexts indicated) tells us more about their meaning than could be gleaned by reading the same passages in Hegel's work over and over (irony intended). His extremely narrow, metaphysical employment of an unrepresentative sample of words we have for identity, difference and change shares nothing with their ordinary use.

Some might object that many of the above examples fall foul of a 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 highly peculiar and oft repeated sentiment. Had Heraclitus said that it wasn't possible to step into the same body of flowing water twice, he might have had a point. But, despite what he said,, it is easy to step into the same river, and many times over, too! Indeed, without that capacity, not even Heraclitus could test his own 'theory' (or even imagine such a test being performed in his 'mind's eye'), since he wouldn't be able to characterise -- or even recognise -- anything as "the same river" to test his thought on, even in theory!

As now seem clear, Heraclitus got into a terrible mess over the criteria of identity we use in relation to mass nouns and the criteria of identity we use for count nouns. [I have explained the relevant use of these terms elsewhere in this Essay.]

Consider the following conversation:

R1: NN -- "I swam in the Thames this morning and then 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. I rather think he didn't mean that."

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 actually be made of it!) 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, 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 that 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 no less 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 to 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, that isn't to deny we sometimes use words in different ways (or that the meaning of words doesn't change over time), but that can't be typical (or apply unrestrictively across the board, and at all times) otherwise communication would be impossible. Moreover, anyone who rejected this simple point would, plainly, fail to get their thoughts across -- again, having used words differently! Alternatively, if they managed to get their point across, they will have used their words in the same way as the rest of us, undermining the very point they thought they wanted to make!

On the other hand, if 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 and its associated terms. [The reasons for saying this are explained in more detail in Note 19.]

[To those who think we never really communicate, anyway, I say this: "I'm sorry, are you trying to communicate something?" (I have said much more about that hackneyed objection in Essay Thirteen Part Three, here.)]

As this Essay will show, it is in fact impossible to decide what (if anything) Trotsky actually meant by his criticism of the LOI. That in itself 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 epigones. 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 require.

It is certainly all that workers require.

Identical A Priori Tactics

In Essay Five we saw how Engels had (irresponsibly and illegitimately) 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 just "equal"! From this he also sought 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 (real) 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 apparent) actually undermines his own hyper-bold conclusions !

And this is supposed to be cutting-edge philosophy/science?

Is someone pulling our legs?

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. That can be seen if consideration is given 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 have ever made of bodies in recorded history, that would still fail to substantiate the hyper-bold claim recorded in S9, and that is because even 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, and that in turn is because that data is equivocal, at best. For example, since the 'quantum revolution', it is now thought that several sub-atomic particles are (exactly) 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 don't 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 he claimed, it is all too easy to make two (or more) identical objects -- indeed, every single one of us does so when we throw a light switch. Here is some material that was 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 has been neatly illustrated here). Here is how Philosopher of Science, Steven French, put 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."

However, the above was published in 2011, but French has now (i.e., in 2019) re-written this as follows (over at the Stanford Encyclopedia of Philosophy):

"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." [French (2019). Bold emphases and links added. Referencing conventions altered in line with those adopted at this site.]

The above passage has been slightly modified, again; this is the new version [i.e., French and Bigaj (2024)]:

"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, it may be argued that 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), Dieks and Versteegh (2008).] 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." [French and Bigaj (2024). Bold emphases added. Referencing conventions altered in line with those adopted at this site.]

Admittedly, 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, as pointed out here, one might well wonder how anyone could possibly know whether or not two particles had been interchanged if they are all 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 distinguishable.

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 did.

Naturally, contentious anti-DM 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) body of data that would be required 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 the facts.

It could be objected that photons don't occupy the same spatio-temporal co-ordinates as other photons, 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 weren't identical (nor did Hegel). Trotsky appealed to their assumed physical differences to make his point. But, photons are physically indistinguishable, according to the above physicists.

Independently of that, photons falsify Trotsky's claim that we can't make two identical objects (which must be located in different spatio-temporal locations), for here we can.

Despite this, hardcore DM-fans might still maintain that the above pro-DM spatio-temporal objection means that photons aren't identical, but that objection is based on a certain definition of identity forced on nature in defiance of the assurance 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), as well as French and Krause (2006). See also Hilborn and Yuca (2002), Ladyman and Bigaj (2010), and the Wikipedia entry here.]

It might 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; 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 Dialectical Marxist 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. too.

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. Paragraphs merged.]

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 modified 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:

"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 $\pi$ 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; paragraphs merged.]

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, that 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 Classics 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", 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 of a Jackson Pollock than it is a 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 and Bigaj  (2024), 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-of-thought operating here; one that holds 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 another that claims 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, they nevertheless 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 possess.

[That 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 a key tenet 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 that supposedly changes it, or them. [So-called "Quantum Entanglement" is discussed here.]

And:

(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 (such changes are the result of what are often called "internal relations" by Idealists and DM-fans alike) then, once more, that re-definition would be no different from imposing on nature something that it might not have. If dialecticians have any evidence that there are any such "internal relations" in nature and society, they need to produce it or resist the temptation to keep foisting these Idealist-inspired concepts on the facts.

Of course, there are DM-theorists (these are mainly drawn from the HCD-wing of Dialectical Marxism) who re-configured a handful of 'arguments' derived from the aforementioned Idealists aimed at showing that such relations do indeed exist (between objects and between 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 'reasoning' wouldn't be needed if DM-fans had any scientific evidence to back up these mystical theories.

It is, of course, a Hermetic doctrine that everything is inter-connected and a union of opposites, as Glenn Magee pointed out:

"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. Links added; paragraphs merged. Quotation marks altered to conform with the conventions adopted at this site. There is more on this here and here.]

Bertell Ollman's book on this topic is just the latest DM-addition to this mystical genre -- Ollman (2003). Significant parts Ollman's work has been analysed in detail in Essay Three Part Two.

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

As of September 2025, the above is still the case: Proton decay has yet to be observed.

Moreover, we have already seen that ordinary language itself contains countless (simple and complex) expressions that enable 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 over the two hours), the following 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 L, and about (iii) L and the relation its contents have with everything in the universe -- where P is comprised of the following elements: {P₁, P₂, P₃,..., P_n}, and where n is indefinitely large (m ε ℝ).

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

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

A4: So, during t_k, NN and NM are on absolutely identically the same train as one another, since, while they are on L, every element of {P_i,..., P_k} remains the case throughout.

Now, every element of {P_i,..., P_k} -- i.e., these predicates, or what they 'reflect' -- can change 'dialectically' constantly and as much as they like, but that won't alter the result, since any changed element of {P_i,..., P_k} 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 utilised in order to assist in its long-overdue demise.]

Finally, what is true of passengers on trains applies in equal measure (irony intended) to those in or on ships, boats, aeroplanes, spacecraft, buses, taxis, rickshaws, bikes, hovercraft, cars, skateboards, rollerblades and scooters 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 this planet, at the same time. And if we go for broke, it applies to everything in the entire universe, at the same time. [Except the latter might have to be relativised in accord with Special Relativity!]

[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 grains of sand in The Sahara.

Of course, it could be objected that this sort of identity isn't what interests dialecticians, so the above remarks are irrelevant. But, as we will see as this Essay unfolds, we don't (yet) know what DM-theorists actually mean by the odd things they say about identity, so not even they may consistently advance that objection. At least not this side of saying for the very first time with any clarity what the dickens 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 Trotsky had in mind when he asserted S9(b) (repeated below). Consider the following interpretations of S8 as S9(b)'s possible targets:

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. (This means T₂ is later than 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/usual 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 (expressed by 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 or target that Trotsky had in mind. He must have read S8 as equivalent to S12 (i.e., identical with S12 -- but, note the irony here!), 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 (in such circumstances). That is, he had to assume the LOI was reliable in order to try to show it was unreliable. Clearly, if the above interpretation is correct, it means his 'demonstration' is radically defective since this 'analysis' only succeeded in undermining itself.

In order 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 perhaps expresses the content (or intent) 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. That 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 now 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 focuses on 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 connected with the identity of temporal instants. This is highlighted by his use of the phrase "that 'moment'" (which clearly means "the same moment", no matter how short it happens to be), 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 "the same 'moment'".]

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

That being so, Trotsky's argument clearly requires something to remain the same (in this case, 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" is for the "A"s in S1. That can be seen if the following sentence stencils 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 simply a less specific version of S17, but it is nevertheless still a legitimate (interpreted) 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 shows he needs the LOI to apply unrestrictedly to some 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 manifestly wasn't about "temporal instants" themselves, or their supposed identity -- which are abstractions, anyway. Indeed, he was merely pointing out that no matter how brief the time frame, change still takes place. That is abundantly clear from these words of his:

"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 the very same moment during which the object in question changes; if "that moment" weren't the same, then his argument would fall apart, for the assumed change in question would have taken place during 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 -- i.e., if Trotsky's other claims about measuring things are taken as read. Temporal intervals are just as measurable as are the weights of bags of sugar. It isn't easy to see how Trotsky could consistently criticize the LOI on the basis of hypothetically differing weights of such bags without it 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 measurable intervals staying the same while measurable weights 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 (in weight) occur.

A Turn To The Concrete?

However, even if the above serious problems were put to one side (for whatever reason), 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 possible substitution instance of S14, expressed in S21(a), and compare it with S21(b):

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 (as they are in S21(b)), 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 employ the same material objects -- i.e., 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:

S22(a): Variable t₁ is equal to (or identical with) variable t₁.

S22(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 considered.¹⁵

Furthermore, Trotsky's analysis can't be seen as an 'immanent critique' -- i.e., as one that undermines the LOI from within -- as it were. That is because his argument depends on that '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 must 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 it applies either to temporal instants or to tensed variable letters -- or even to ordinary words that give expression to one or both. In the latter two instances (i.e., concerning variable letters and ordinary words), this will involve a use of very real material objects (written in ink or even represented as 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 they 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, criteria governing the concrete use 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 to ordinary words that have been identically understood and 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 instance without also appealing to its unrestricted application somewhere else.

Is 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 rendered far less grandiose than they might otherwise seem, or they would have collapsed into incomprehensibility.

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

S23(a): No object is identical with itself at a later time.

S23(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 S23(a) and S23(b) express the banal truth expressed in S13 and S21(a), with which few would want to take issue.

Clearly, therefore, S23(a) and S23(b) are almost certainly not what Trotsky had in mind. He clearly wanted to argue that no object is self-identical at the same instant/moment -- presumably because every object is subject to its own internal struggle, 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, the second option would apply whenever, for example, 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 DM-IED ploy). That would then have Trotsky meaning one or other of the following:

Where the word "same", as it appears in S24(a) and S24(b), has been replaced in italics by its (assumed) 'dialectical' equivalent/meaning -- "same and non-same", or "same and not the same" -- in S24(c) and S24(d) respectively, in order to make explicit the radical confusion this 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, stand any chance of objecting successfully, at this point.]

Any who feel confident in their ability to explain what S24(c) could possibly mean shouldn't be given the benefit of that doubt until they manage to do likewise with S24(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"?

When the same word "same" is given similar treatment in S24(d), it rapidly collapses into the following linguistic monstrosity:

S24(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 S24(d) is replaced with its 'dialectical-equivalent/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' patience, eyesight and sanity prevents me from extending it any further.]

Of course, some might complain that the above is a ridiculous explication/expansion 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 no help now 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 attempt to apply the IED gambit explodes it in the face of those reckless enough to try.

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 creates even greater confusion!

In that case, it is clear that it is impossible to challenge or even refute the LOI in the same crude way attempted by Trotsky (irony intended). As the above shows, it can't be done without using the LOI in any attempt to do so. 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 prosecute 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 semaphore, telepathy or even an Aldis Lamp?

Did Trotsky Understand Identity?

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

As seems reasonably clear, 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 weren't 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 and by 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/inaccurate 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 adequately to depict change. 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 of reflecting fundamental aspects of change or capturing the fluid nature of reality. This has nothing to do with understanding or even with failing to understand anything.

Or so an objection might proceed...

However, this apparent 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 that is so will be examined in Essay Twelve Part One. Those who might be tempted to refer to Hegel's criticisms of the LEM at this point should check this out, 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. In such circumstances 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 concocted), 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 has 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'. Given that argument, no material copy of S1 or S1a would ever be identical to itself, or to any subsequent copy of it. Furthermore, 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 (perhaps for decades!), which is something S9a, S9b and S1a rule out.

In that case, we have (as yet) no idea what 'absolute', or even 'abstract', identity could possibly be, since no material embodiment of it (in a sentence) would fully express these 'concepts', given this theory.

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 these Hegelianesque 'propositions', 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 sentential letter, "P", (i.e., if "P" now stands for "A is both equal to non-A and to non-non-A"), we are instantly faced with serious problems -- as S1a is applied to its own material embodiment -- the very material sentence, P:

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

[So, when fully spelled out, S1b is in effect this (as each occurrence of P is replaced by A is both equal to non-A and to non-non-A):

S1c: A is both equal to non-A and to non-non-A is equal to non-A is both equal to non-A and to non-non-A and to non-non-A is both equal to non-A and to non-non-A.]

And then, if we repeat this twice and replace each occurrence of P with its 'dialectical' equivalent -- non-P -- we end up with something like these:

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

S1e: 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.

[We could go all the way here and replace each occurrence of P in S1e with A is both equal to non-A and to non-non-A, just as we did with S1c, but that dialectical task will be left to the reader.]

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 S1e. Hence, only the rejection of S1a will prevent this logical monstrosity from propagating itself. If A itself fails to be identical with itself (according to Hegel and Trotsky), so must S1a -- and the rest follows.]

Furthermore, such material inscriptions can't tell us what 'approximate identity' is even an approximation to. So, if identity statements are never absolutely true (in this changing world), how could anyone learn what the implications of accepting or rejecting sentences like S1 amount to? Again, what exactly would they be accepting or rejecting?

[Note the significance of the highlighted 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! There is no way around this short of never saying anything.]

Even to ask the question requires a use of the dreaded 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 is to be believed), 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 an (approximate) empirical truth, we would still be owed an explanation of what it approximates to. And how could that be specified if we never encounter the absolute limit of identity, if we don't know what it is (even in theory) and can't express it in any known language without having to use absolute identity in order to do so? We would have to be able to specify exactly what it is we are trying to approximate in the very act of denying we were doing just that! If we couldn't do that, we wouldn't be talking about absolute identity but something entirely different (irony intended!).

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 correct. In which case, if DL is valid, we couldn't even theorise about a correct/exact application of that 'law' -- or conceptualise 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 as soon as they were formed) of anyone who raised doubts about the material application of this 'law' (again, as 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 that 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 content -- since nobody would have the slightest idea what was being ruled out.

Hence, the earlier question about what Trotsky did or didn't understand turns out to be 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.

Or so it might be countered...

However, as we will soon see, in order for anyone to be able to say that such a statement is never true (or never absolutely 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 easy to see there are several applications of the LOI (as a rule of language) over many years, too.

Once more, any rejection of the LOI must self-destruct (and that includes any attempt to declare this 'law' 'true only within certain limits'). If it isn't possible for at least one identity statement to remain exactly the same over a given length of time, then that applies to any sentence that a 'dialectical objector' might try to use, too. In that case, even what Hegel had to say about this 'law' must also have changed, which implies 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 is discussed below.]

Again, it could be objected that identity statements are in fact both true and false (that would seem to an implication of any attempt to repudiate the universal, unqualified application of the LEM; it also seems to be what lies behind S1a). However, it is entirely unclear what that could mean in relation to the LOI. If true, it would either imply that material identity only approximates to absolute identity (since, on this view, no one would then have a clue what absolute identity amounted to); if it is false then material identity would still only approximate to absolute identity, because, again on this view, identity propositions are (so we are told) merely 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.]

Or so it might be argued...

But, both of the above amount to the same criticism of the LOI (i.e., that it only captures approximate identity), and since we still have no idea what ordinary identity approximates to, this particular objection is itself devoid of content. We still remain in the dark over exactly what is being ruled in or out (that is, we still have no idea what 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 Dialectical Marxists, what exactly would the any material example of the LOI be both true and false of? But, no answer to that question can be given short of a clear grasp of absolute identity, which, alas, no one is supposed to have. Again, exactly what is being criticised here? As soon as we are told, the LOI will have been used (as a rule of language). On the other hand, if we aren't told, then this criticism of the LOI also falls flat.

Hence, this is independent of whether or not the LEM is valid.

Again, it could be objected 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. The following is what is associated 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 material reality.

Or so a counter-argument 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-after-year, generation-after-generation, and if that is so, there will be 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 from moment-to-moment, we wouldn't have a 'clear idea' of absolute identity even to criticise!

Once again, it could be objected that propositions about identity could still be both true and false so that one option (i.e., the truth of an identity statement) 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 decides to do with, or to, it. In that case, if no one has a clue what the subject of a denial or an 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, or about, 'it'.

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

Those of my readers who can't rise to that challenge will now have got the point.

[Spoiler: "Meskonator" is a totally meaningless word; a complete invention of mine!]

Plainly, if no one knows what a Meskonator is, then no one could possibly know if anything approximated to 'it' or what was or wasn't absolutely identical to 'it'.

Of course, in relation to 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. That term would then be implicated in its own lack of meaning (since no one would know what meaning was or wasn't identical to it!), as opposed to being implicated in the meaning of different word, such as "Meskonator".

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 was the case in Essays Three Part One, Four Part One and Eight Part Three), dialecticians are the last ones who can legitimately raise this 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 -- on paper, on a screen, or in some form in a central nervous system somewhere) must change and thereby become different -- since we are told that all objects and processes "are never equal to themselves" (bold added). But in that case, how could anyone tell whether that stored idea (supposedly) of identity had actually changed without being able to compare it 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, (i) Coincided with absolute identity itself, (b) Fell slightly short of absolute identity itself, or (c) Was totally different from absolute identity.

And if instead we were to concentrate on words or concepts that are somehow stored 'in the mind' -- contemplated/processed or not by 'speculative reason', à la Hegel --, even that would be to no avail. 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, 'true within certain limits', DM-'approximation' to it), not anything. But, if that is the case, then not even the last sentence could be securely grasped, for it, too, contains the very phrase we have yet to comprehend: "absolute identity", and we can't even do that since it would change just as soon as it was cognised. Either that, or it will have to be exempted from the Heraclitean Flux [HF], which would mean it was (supposedly) the only changeless 'thing' in the entire universe that had been given such a pass.^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 undermined. If the letter "A" is never the same, what chance has the phrase "absolute identity"?

An appeal to the 'relative' stability of words and concepts at this late stage would prove 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) -- and we have already claimed that not even the letter "A" remains the same as itself from moment-to-moment -- then we surely have no way of knowing that any words or phrases -- such as "absolutely identical", or even "approximately identical" -- remain 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 given what Trotsky (and Hegel) have to say we have no way of knowing by how far, or how little, each word in the verb phrase "minor changes in words over short periods of time can be ignored" has or hasn't altered. That is especially so since we still haven't a clue what absolute identity is so that we might compare any such altered words with exemplars of them that haven't changed.

Once again if the letter "A" is never the same from moment-to moment, "minor changes in words over short periods of time can be ignored" stands no chance.

Some might try to argue that it is reasonable to assume our words remain the same from moment-to-moment; but, given DM we don't yet understand the word "same" (let alone "assume"). If so, no one would know whether or not others were employing such words in the 'same' way as anyone else -- or even in the 'same' way as they themselves had used them only a few seconds earlier!

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

It is worth repeating this, if not even the letter "A" is the same as any other letter "A" on the page in front of you, if Trotsky is to be believed, then no two (supposed) copies of the word "same" (or even "assume") will be the same in two different heads, and even those two different (supposed) copies will themselves change from moment-to-moment!

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

Once identity is questioned (at any level -- and in howsoever a nuanced way this is attempted), all such comparisons and contrasts fall apart -- and to such an extent that it now becomes impossible to say anything at all determinate.

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

Even if we could say what Hegel meant without it having changed!

'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 -- i.e., 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 about Hegel (no irony intended).

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. That certainly didn't commit him to using the LOI in order to criticise it, contrary to what was claimed earlier. Trotsky was obviously arguing dialectically, using everyday notions to show how they become contradictory when applied beyond the usual boundaries of commonsense or the confines of ordinary life.

Or, so a counter-argument might go...

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

However, and once again, if sense is to be made of approximate identity, some grasp must surely be had of absolute identity so that (even) a vague idea might be formed of what this watered-down version of it actually approximates to. If that 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 (hidden) world. In the absence of a clear account of this abstract notion of identity, we would still have no idea what ordinary identity was 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 by 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, but 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 a day or a week later?

Consider further the following 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 target was (i.e., identity itself)? For all anybody knew, an intentional target like this could, or would be entirely different in the minds of two different abstractors, or even in the mind of one of them at the same or different times. It isn't as if either of them could check inside each other's skulls, or access their own present or past thoughts, in order 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 happened to be -- and, indeed, whether there is actually 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" --, even the day before (never mind several years ago) --, how might any of us ascertain if we ever recall this word (and its meaning) 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 brand of deodorant or a new species. Nor can we appeal to ordinary language for assistance. If this theory (i.e., DM) undermines the vernacular -- which we have seen it does -- then it can't possibly help fix the very thing that caused the problem.

Of course, 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 words and phrases 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.

The serious problems associated with abstractionism were set out in detail 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 that 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 some attempt might be made to assemble even a vague idea of an 'Abstract Cat'. With such accommodating felines, abstractionists at least have something on which to set their metaphysical sights.

And, it isn't as if absolute identity is "what all forms of material identity have in common", since, according to Dialectical Marxists, they don't. Nothing in reality instantiates absolute identity, so nothing could possibly share it. Compare that with the 'Abstract Cat', which is supposed to be what all cats have in common. Absolute identity can't underlie all forms of approximate identity; absolute identity can't be what all forms of approximate identity instantiate 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 (since everything possesses some form of identity, relative, concrete or whatever). There thus seems to be no grist for the 'dialectical mill' to grind away at, here.

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.

Admittedly, we already use words for identity, and these are put to use in practices in connection with which rules for the use of such terms 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 Hegel conned his publishers into inflicting on the German people. [On this, see also Note 20a.]

In that case, howsoever it is conceived, the process of abstraction is clearly surplus to requirements; in order to abstract absolute identity successfully into existence abstractors must already have access to that very concept in order to be able to decide if they have arrived at the right target. They will need to know whether any of the concepts they have lined-up in their thoughts are ready to assume the title as the rightful heir or whether they are all pretenders to the throne. And, how are they going to do that without already having the 'concept' they were looking for?

So much was clear at least to Plato.

It is worth pointing out here that that doesn't mean I agree with Plato (I don't!), only that the traditional- (and the DM-) view of abstract ideas (or the Proper Names thereof) pictures them in a way that makes them resemble physical objects (or the Proper Names thereof) that do exist in the natural or social world -- even if 'abstractions' themselves don't exist there (and no one seems to know where they do hang out!). In that case, it makes sense to pose this 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."

"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.]

"Meno: How will you look for it, Socrates, when you do not know 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.]

In Plato's case, knowing the Forms was a type of recognition, somewhat like recalling the faces of old friends. Hence, knowledge was simply recollection, a recalling to mind of the 'faces of long lost concepts', as it were, but reified now as quasi objects in Platonic Heaven that every single one of us had 'met' before birth. In that pre-existing state we all 'knew' the Forms because, as denizens of this Ideal World, we could see them face-to-face, since they, too, were Ideal Objects (or were 'Exemplars' somehow masquerading as Ideal Objects; Plato commentators are divided on this question!). This was a case of like recognising like -- a pre-existent soul recognising a fellow 'Ideal Object'. This is the moment that Representationalism took its first breath and rationalism was given a shot in the arm. [On that, see Essay Three Part Two, link above. Apologies for those mixed metaphors!]]

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.]

[But, there can be no generality associated with such 'concepts' (even if they were invented to explain generality!): they are 'abstract objects' (and are therefore singular in form), reified in our 'pre-existent presence'. The importance of that observation was underlined in Essay Three Part One. As intimated above, these comments will be connected with sustained criticism of "Representationalism", another dominant ruling-class form-of-thought based on just this confusion (i.e., of concepts with objects), in Essays Twelve Part One and Thirteen Part Three.]

Knowing a Concept or a 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 Plato this took place in our pre-existing state. Unfortunately, this confused propositional knowledge (savoir) -- "NN knows that p" -- with personal knowledge (connaitre) -- "NN knows MM" --, conflating both now as different varieties of acquaintance. [This confusion cropped 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 all based on the conflation of objects and concepts.] This mix up 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 he 'intellectualised' it).

Knowledge is now to be viewed as a relation of some sort -- it is supposed to be a relation each Knower is said to have with the ideas, thoughts, 'representations', or 'qualia' they supposedly have in their 'consciousness'/'minds'. Alternatively, this is a relation each Knower has with facts, propositions, statements, sentences, concepts, categories or 'representations' in their heads, in 'heaven' (in the 'Mind of God'), or even in this world (somewhere), all of which are 'reflected' in each head. This overarching idea 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.]

Hence, what had been for Hegel a set of 'self-developing concepts in the mind' now became 'objects in the world reflected in our minds', just like ordinary objects supposedly are. This is the same confusion, only now in reverse. All forms of 'knowledge by reflection' depend on this ancient conflation -- in forward gear, according to Plato and Hegel, or in reverse gear according to Marx, Engels, Lenin and Trotsky).

A confusion by any other name is still a confusion.

"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.]

Traditionally, this meant that knowledge was in fact a relationship between the 'Knower' and the 'Known' -- which 'objects' now faced each other like two acquaintances. This had the damaging effect of deflecting attention away from the special nature of propositional (and hence scientific) knowledge. This in turn meant that both propositional and scientific knowledge were conflated and hence interpreted along similar lines, as a relation between the Knower and a peculiar sort of object, a proposition. This was now objectified and/or reified (as an object -- either a sort of shadowy entity in an 'extra-mental reality' or as something stored '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 will be explored in Essay Twelve Part Seven (summary here).]

This antiquated, shop-worn 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, given DM-Epistemology, humanity is collectively struggling toward these 'Forms' (albeit, re-christened as 'abstractions'/'concepts'), this prolonged 'dialectical ramble' subsequently(?) re-birthing humanity in a future Ideal State of 'Objective Knowledge', à la Hegel (upside down or the 'right way up'). 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!

And here it is in all its glory, DM-Reverse Platonism:

"'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. Bold emphasis alone added; paragraphs merged.]

"The reproaches you make against the law of value apply to all concepts, regarded from the standpoint of reality. The identity of thought and being, to express myself in Hegelian fashion, 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 yet 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. But although a concept has the essential nature of a concept and cannot therefore prima facie directly coincide with reality, from which it must first be abstracted, it is still something more than a fiction, unless you are going to declare all the results of thought fictions because reality has to go a long way round before it corresponds to them, and even then only corresponds to them with asymptotic approximation." [Engels to Conrad Schmidt (12/03/1895), in Marx and Engels (2004), pp.463-64. Bold emphases added. I have used the on-line version here, which differs slightly from the published text.]

"Systematics impossible after Hegel. The world clearly constitutes a single system, i.e., a coherent whole, but the knowledge of this system presupposes a knowledge of all of nature and history, which man will never attain. Hence he who makes systems must fill in the countless gaps with figments of his own imagination, i.e., engage in irrational fancies, ideologise." [Marx and Engels (1987), p.597. Bold emphasis alone added.]

"But there are more than these two properties and qualities or facets to [any material object]; there are an infinite number of them, an infinite number of 'mediacies' and inter-relationships with the rest of the world…. [I]f we are to have true knowledge of an object we must look at and examine all its facets, its connections and 'mediacies'. That is something we cannot ever hope to achieve completely…. [D]ialectical logic requires that an object should be taken in development, in change, in 'self-movement' (as Hegel sometimes puts it). This is not immediately obvious in respect of such an object as a tumbler, but it, too, is in flux, and this holds especially true for its purpose, use and connection with the surrounding world." [Lenin (1921), pp.92-93. Bold emphases alone added. Paragraphs merged.]

"Dialectics as living, many-sided knowledge (with the number of sides eternally increasing), with an infinite number of shades of every approach and approximation to reality (with a philosophical system growing into a whole out of each shade) -- here we have an immeasurably rich content as compared with 'metaphysical' materialism, the fundamental misfortune of which is its inability to apply dialectics to the Bildertheorie [theory of reflection -- RL], to the process and development of knowledge....

"Human knowledge is not (or does not follow) a straight line, but a curve, which endlessly approximates a series of circles, a spiral. Any fragment, segment, section of this curve can be transformed (transformed one-sidedly) into an independent, complete, straight line, which then (if one does not see the wood for the trees) leads into the quagmire, into clerical obscurantism (where it is anchored by the class interests of the ruling classes). Rectilinearity and one-sidedness, woodenness and petrification, subjectivism and subjective blindness -- voilà the epistemological roots of idealism. And clerical obscrutantism (= philosophical idealism), of course, has epistemological roots, it is not groundless; it is a sterile flower undoubtedly, but a sterile flower that grows on the living tree of living, fertile, genuine, powerful, omnipotent, objective, absolute human knowledge." [Lenin (1961), pp.360-61. Bold emphases alone added.]

"Cognition is the eternal, endless approximation of thought to the object." [Ibid., p.195. Bold emphasis 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.... 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.]

"Dialectical materialism insists on the approximate, relative character of every scientific theory of the structure of matter and its properties; it insists on the absence of absolute boundaries in nature, on the transformation of moving matter from one state into another." [Lenin (1972), p.312. Bold emphasis added.]

"'Here once again we find the same contradiction as we found above, between the character of human thought, necessarily conceived as absolute, and its reality in individual human beings with their extremely limited thought. This is a contradiction which can only be solved in the infinite progression, or what is for us, at least from a practical standpoint, the endless succession, of generations of mankind. In this sense human thought is just as much sovereign as not sovereign, and its capacity for knowledge just as much un-limited as limited. It is sovereign and unlimited in its disposition..., its vocation, its possibilities and its historical ultimate goal; it is not sovereign and it is limited in its individual expression and in its realisation at each particular moment....'

"'Truth and error, like all thought-concepts which move in polar opposites, have absolute validity only in an extremely limited field, as we have just seen, and as even Herr Dühring would realise if he had any acquaintance with the first elements of dialectics, which deal precisely with the inadequacy of all polar opposites. As soon as we apply the antithesis between truth and error outside of that narrow field which has been referred to above it becomes relative and therefore unserviceable for exact scientific modes of expression; and if we attempt to apply it as absolutely valid outside that field we really find ourselves altogether beaten: both poles of the antithesis become transformed into their opposites, truth becomes error and error truth'.... Here follows the example of Boyle's law (the volume of a gas is inversely proportional to its pressure). The 'grain of truth' contained in this law is only absolute truth within certain limits. The law, it appears, is a truth 'only approximately'.

"Human thought then by its nature is capable of giving, and does give, absolute truth, which is compounded of a sum-total of relative truths. Each step in the development of science adds new grains to the sum of absolute truth, but the limits of the truth of each scientific proposition are relative, now expanding, now shrinking with the growth of knowledge." [Ibid., pp.150-51, quoting Engels (1976), pp.108-09, 114. The on-line and published translations are slightly different. Quotation marks altered to conform with the conventions adopted at this site. Bold emphases and link added.]

"Idealism and mechanical materialism, opportunism and adventurism, are all characterized by the breach between the subjective and the objective, by the separation of knowledge from practice. The Marxist-Leninist theory of knowledge, characterized as it is by scientific social practice, cannot but resolutely oppose these wrong ideologies. Marxists recognize that in the absolute and general process of development of the universe, the development of each particular process is relative, and that hence, in the endless flow of absolute truth, man's knowledge of a particular process at any given stage of development is only relative truth. The sum total of innumerable relative truths constitutes absolute truth. The development of an objective process is full of contradictions and struggles, and so is the development of the movement of human knowledge. All the dialectical movements of the objective world can sooner or later be reflected in human knowledge. In social practice, the process of coming into being, developing and passing away is infinite, and so is the process of coming into being, developing and passing away in human knowledge. As man's practice which changes objective reality in accordance with given ideas, theories, plans or programmes, advances further and further, his knowledge of objective reality likewise becomes deeper and deeper. The movement of change in the world of objective reality is never-ending and so is man's cognition of truth through practice. Marxism-Leninism has in no way exhausted truth but ceaselessly opens up roads to the knowledge of truth in the course of practice. Our conclusion is the concrete, historical unity of the subjective and the objective, of theory and practice, of knowing ant doing, and we are opposed to all erroneous ideologies, whether 'Left' or Right, which depart from concrete history." [Mao (1961b), pp.307-08. Bold emphasis added; quotation marks altered to conform with the conventions adopted at this site.]

The above view isn't limited to the DM-classicists, either. Here, for example, is Abram Deborin:

"The idea does not coincide immediately with reality, but it is derived from reality: reality corresponds to the results of thinking, the idea corresponds to reality, only asymptotically approaching it, to use Engels's expression. As we have seen, Lenin, too, develops the same point of view." [Quoted from here; accessed 08/06/2023. Bold emphasis added.]

This represents Maurice Cornforth's two-cent's worth:

"What then is truth? It is correspondence between ideas and objective reality. Such correspondence between our ideas and reality is only gradually established, and then the correspondence is often no more than partial or incomplete.... In such cases, we should not say that our idea was false, but yet it would not be absolutely -- completely and in all respects -- true. Truth, therefore, is not a property which an idea, or a proposition, either possesses or does not possess; it may belong to an idea to a certain degree, within certain limits, in certain respects....

"This characteristic of truth...is very well known to science. The laws which science establishes certainly reflect objective processes; they correspond to the real motion and interconnection of things in the external world. Yet science has established few laws which can claim to be absolute truths.... [M]any erroneous views in science and philosophy, which have had to be, not modified, but rejected as errors, concealed a certain truth which received in them an erroneous distorted expression....

"We should recognise, then, that certain erroneous views, including idealist views, could represent, in their time, a contribution to truth -- since they were, perhaps, the only ways in which certain truths could first begin to come to expression.... Complete, full, absolute truth -- the whole truth and nothing but the truth about everything -- is something we can never attain. But it is something toward which we are always approximating.... The correspondence is never complete, exact, absolute. But it continually approaches yet is always infinitely distant from that absolute limit as truth and knowledge continually advance...." [Cornforth (1963), pp.135-45. Bold emphases added; several paragraphs merged.]

Here, too, is faithful echo-chamber denizen, Henri Wald:

"A 'concrete' truth is a logical system of abstractions multilaterally reflecting the real concrete. One truth is more concrete than another to the extent to which it reflects more essential traits of the investigated object. Concrete truth like absolute truth, can only be reached asymptotically ad infinitum." [Wald (1975), p.35. Quotation marks altered to conform with the conventions adopted at this site. Bold emphasis alone added.]

Reverse Platonism thus appears to be about as definitive a DM-Dogma as 'change through internal contradiction'.

Hence, it seems the DM-Elect, gathered together at some far distant terminus atop 'Epistemological Mount Olympus' at 'the end of time' -- that is, should any humans 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, a 'Great Teacher' perhaps, will lift the scales from their eyes allowing 'dialectical enlightenment' to flood in. In that case, DM is also a form of Teleological Platonism; humanity will regain what it lost by their 'epistemological fall from grace' -- or, to use DM- buzzwords, it will wave 'goodbye' to 'partial' and 'relative' knowledge, and ascend to these sunny uplands bathed by the warm, reassuring rays 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". On this, see the next sub-section.]

The other form of knowledge -- i.e., savoir, or propositional knowledge -- can't be squeezed into this bourgeois individualist straight-jacket, but it nevertheless harmonises seamlessly with what we now know about the social nature of science. [That was one of the main topics of Essays Three Parts One and Two. It will be covered in more detail in Essay Thirteen Part 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 that section of Magee's book, published at a right-wing site (note especially the précis of Chapter Three of the book).) See also, here.]

This 'allows' Hegelians (and Dialectical Marxists) to talk about concepts 'developing' as knowledge grows, since for them propositional knowledge (savoir) has now been fused (à la Plato) with the personal knowledge (connaitre) of concepts, which also seem to have developed somehow 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 now joined together in a grandiose version of the infamous 'Chymical Wedding of Christian Rosenkreutz' (which off-the-wall 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.]

As should by now seem obvious from all that has been said at this site, I reject this entire way of speaking about knowledge. I am merely utilising these ideas in order to help undermine the conservative thought-form ['dialectics'] that DM-supporters have imported into our movement from Hegel and Traditional Philosophy. It is certainly not the case that I think we need to know our concepts before we know them! But this is a very real paradox implied by DM-Epistemology.

"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. Bold emphasis added.; paragraphs merged.]

"[Philosophical problems] are, of course, not empirical  problems; they are solved, rather, by looking into the workings of our language, and that in such a way as to make us recognize those workings: in despite of an urge to misunderstand them. The problems are solved, not by giving new information, but by arranging what we have always known. Philosophy is a battle against the bewitchment of our intelligence by means of language." [Wittgenstein (1967), p.47e, §109. Bold emphasis added.]

Given what he said about Philosophy and Ordinary Language, it is hard to believe Marx would have disagreed with the above.

Dialectical Dilemma

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

Either:

(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, or via 'cognition', 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, Option (1) barely conceals its Platonic/mystical provenance, which, as we also saw, appears to lie behind Engels's 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. Bold emphasis alone added; paragraphs merged.]

"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 both the object of knowledge and knowledge itself as "concept"-based:

"[T]he 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 appears to mean that for him such "concepts" are independent of our knowledge of them -- in that they are somehow equated with "reality". How would it be possible to "approach" these "concepts" asymptotically if that weren't so? Hence, they must predate human existence in some form. Which is why the DM-theory of knowledge was labelled a form of Teleological Platonism. [That is because, as was the case with Plato, these concepts, or Forms, pre-exist humanity, but according to DM humanity didn't encounter them in a pre-existing state; instead we are "approaching" them "asymptotically".]

Of course, it would be unwise (and unfair) to place too much weight on these words of Engels -- since they appear in an irredeemably unclear passage taken from an unpublished letter. 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", which is Lenin's example) in fact expresses an identity relation between the predicate expression and the subject term, or, indeed, what they supposedly 'reflect in extra-linguistic reality'. So, "John is a man " became "John is identical with Manhood". In that case, 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: a 'Platonic Form', a 'Universal', a 'concept', an 'Idea', or an 'abstraction'. Exactly where these were supposed to exist depended on how consistent or 'Absolute' an Idealist a given philosopher happened to be.

This slide helped motivate the above confusion of linguistic expressions with whatever they supposedly refer to/reflect extra-linguistically. Hegel swallowed that confusion whole, without blinking:

"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 descent from there to the doctrine that predicates are extra-linguistic, extra-mental entities in their own right was further accelerated by the adoption of the 'reflection theory of knowledge' by Dialectical Marxists. Predicate expressions in some as-yet-to-be-explained way now supposedly 'reflect' real material objects and processes in the 'extra-mental 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' 'outside the mind'. [I have entered into this in more detail, here.]

For the above DM-theorists predicate expressions also referred to 'abstractions'/'concepts'/'representations' apprehended by, or cobbled-together in, 'the mind' (depending on which DM-theorist was telling the tale), all this accomplished by the mysterious 'process of abstraction'. Unfortunately, this now presented DM-theorists with a knotty problem: the above dilemma, but with an added twist -- which was the problem of explaining where exactly these 'abstractions'/'concepts'/'representations' are supposed to exist. If they only exist 'in the mind', what 'objectivity' do they possess? Alternatively, if 'abstractions'/'concepts' 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 (by Lenin) to be abstract, just as human concepts are also supposed to be 'objective'! Of course, this aspect of DM doesn't sit at all well with what Lenin also had to say about 'objectivity':

"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.]

From the above 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. But if that is so, some -- perhaps many -- must have pre-existed the evolution of the human species.

I can see no way of avoiding that conclusion.

[If anyone can, 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 over-worked 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". Hence, the term, "a man", is now both a linguistic expression and an extra-linguistic entity in its own right. The two have now been conflated, which is the only reason the predicate itself 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' -- as "abstract identity". I have given other further examples of the 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 the above dilemma DM-fans finally grasp (if they ever actually make up their minds -- or are ever much clearer than Engels 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 that 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 above suggests they should. Admittedly, this is a controversial claim but it may only be neutralised when it becomes clear what that metaphor implies -- more specifically when it becomes clear whether or not it means that as far as individual dialectical truth-seekers are concerned, there is a goal (identical for all of them) they have correctly targeted toward which they are all slowly homing in on. Naturally, a positive answer to that puzzle would sink the dialectical theory of identity, since it would: (i) Make plain that this metaphor implies there is something unchanging called "absolute identity" that all dialectical truth-seekers are gravitating toward, about which they are all already --, or about which they will one day be --, in equal agreement. On the other hand, a negative response would undermine DL equally quickly. That is because, 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 Dialectical Marxists were aiming at the very same intentional target. Indeed, they would all 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, to put it bluntly, their search would be aimless; it is headed nowhere, since there is nothing for it to head toward. On the other hand, if there is a way of delving into the mental depths of any two randomly selected dialectical abstractionists and ask what it is that enables any of them decide whether or not they are all in pursuit of the very same goal, the first of the above points would apply (i.e., (a(i))), since it would then be obvious that these truth-seekers had employed the dread 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 if they were to reject -- one or other of Alternatives (a) and (b), 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, indeed, 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, it 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 were talking among themselves about the same thing (or even the same target as Hegel and/or Trotsky) --, any reference to 'it' by critic and believer alike would be entirely empty.

Otherwise, they would be well-advised to acknowledge their irresolvable differences and then 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 mutant. Without that, of course, they would lose the right to say that absolute identity has, or might have, changed. Without an accurate idea of absolute identity it would surely be impossible to decide whether or not it had changed!

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