The 'Law' Of Identity
Readers need to make note of the fact that this Essay does not represent my final view on any of the issues raised. It is merely 'work in progress'.
If you are viewing this using Mozilla Firefox, some of the codes that Microsoft has put into FrontPage (the editor I have used) have made some of the font colours in the second half of this Essay change erratically. In addition, you might not be able to read all the symbols I have employed.
This Essay is over 46,000 words long; a summary of its main ideas can be found here.
This Essay is principally about Trotsky's objections to this 'law', but much of its content also applies to Hegel's 'critique'. [His 'theory' will be considered, though, in detail in Essay Twelve Part Five, when it is published.]
Quick Links
Anyone using these links must remember that they will be skipping past supporting argument and evidence set out in earlier sections:
(2) Identical Is Not Identical With Equal
(a) A Mistake Most Dialecticians Make
(b) Ordinary Language Thwarts Dialectics
(c) Trotsky Changes The Subject
(3) Trotsky's Argument Dissected
(a) Precisely What Is Trotsky Denying?
(b) Trotsky Has This Base Covered -- Or Has He?
(c) Bags Of Sugar Refute Trotsky
(i) Mere Guesswork On Trotsky's Part?
(ii) Yet Another Misidentification
(iii) Wrong Anyway
(iv) Identical A Priori Tactics
(v) Superscience From Mere Words
(vi) Incomprehensible Or Just Trivial?
(4) Trotsky Uses Identity To Criticise Identity
(a) Same Moment
(5) Did Trotsky Understand Identity?
(a) Can Anyone Learn Identity If None Exists?
(c) 'Approximate' Identity And 'Abstract' Identity
(e) Trotsky's Exact Words Self-Destruct
(f) Trotsky's Attack Unequal To The Task
(g) Materially-Induced Dialectical Misery
(7) Dialectical Logic Superior?
(8) Physicists Discover Identical Particles
(9) Notes
(10) References
Abbreviations Used At This Site
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', many have in fact attacked the wrong target.
[FL = Formal Logic; LOI = Law of Identity.]
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, but the objections I raise here against the highly repetitious and misguided comments dialecticians make about it will also apply indirectly to his work.
Since these Essays have been written from within the Trotskyist movement, and because Trotsky's comments on this 'law' are far more influential on active revolutionaries than are those of Hegel, it makes sense therefore to begin with his widely quoted remarks.
In his debate with Burnham, Trotsky rehearsed an argument that was aimed at exposing what he took to be serious limitations of the LOI, one he had lifted directly or indirectly from Hegel and one that has resurfaced almost verbatim in the writings of other DM-theorists who claim to be Trotskyists.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 written in the last 200 years.2 Clearly, he did not think that this disqualified him from passing opinion on the subject. By the same token therefore we may suppose him an expert in High Energy Physics and brain surgery.
This damning criticism applies equally to most 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.
[AFL = Aristotelian Formal Logic; MFL = Modern Formal Logic.]
Most of the criticisms DM-theorists make of FL were examined in Essay Four and were shown to be based on a serious misunderstanding even of AFL, let alone MFL. This is hardly surprising given the allegations made in the previous paragraph. Nevertheless, in this Essay I plan to concentrate on Trotsky's criticisms of the LOI, which DM-theorists -- at least those in that wing of the revolutionary tradition -- generally regard as definitive. John Rees, for example, outlined one key issue in the following 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.]
The main part of Trotsky's own argument, however, went 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.]
One puzzling fact about this passage -- which it shares in common 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 quote.3
Nevertheless, there appear to be at least a dozen substantive claims that Trotsky is making here:
T1: AFL begins with "A is equal to A".
T2: This "postulate" applies quite well in most practical situations.
T3: Close inspection under a lens will show that any chosen letter "A" is not exactly the same as any other letter "A".
T4: Similar observations apply if these letters stand for material objects like pound bags of sugar.
T5: Any two weighings of seemingly equal bags of sugar will always reveal minor differences.
T6: All bodies undergo constant change; 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 is an interval, then any object will undergo inevitable change in that interval.
T9: If that moment is not 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 do not change.
T12: Objects that do not change, do not exist.
Trotsky nowhere backs any of these up with evidence (or, any worthy of note) -- but that serious scientific defect rarely bothers dialecticians. In earlier Essays, we saw why DM-theorists airily brush aside the need to substantiate their theses with anything that remotely resembles proof: this is because, if the universe is governed by DL, a simple 'thought experiment' is all the 'evidence' that needed.
Naturally, only consistent materialists will object at this point.
Lest anyone object to the above, the 'evidence' that Trotsky and/or his followers advanced in support of their contentious claims is examined below.
However, Trotsky's quasi-Hegelian observations were based on a serious 'misunderstanding' even of AFL -- a defect compounded by an ironically appropriate mis-identification of the LOI, further complicated 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 consequential on the above.]
Oddly enough, "Identical Is Not Identical With "Equal"
A Mistake That Applies Equally To Most Dialecticians
Trotsky's initial characterisation of the LOI is itself rather strange. His paraphrase of it was as follows:
S1: A is equal to A.4
But, as an accurate depiction of identity, S1 is not 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, DM-apologists tend to say, "So what? What is the difference between the two?" As will be appreciated, that response is itself problematic (not the least because it reveals that they too have an insecure grasp of the issues involved):
(1) If there is no difference between the two, then they are identical, which means that at least here we would now have a genuine example of the LOI on which all could agree.
(2) If they are different, then Trotsky attacked the wrong target.
Now, when challenged with this dilemma, dialecticians tend either to ignore it, or they retreat into the "It's just abstract" defence (accompanied or not by the "This law only applies to objects and processes in nature" ploy).
As we will soon see, this retreat is itself a step back too far, for there is also a clear difference between abstract equality and abstract identity, which dialecticians have likewise failed to notice. So, abstract or concrete, the two notions are not the same.
Furthermore, as we discovered in Essay Three Parts One and Two, dialecticians have a somewhat insecure grasp of the nature of abstraction, and are largely content to be told what to think on this score by ruling-class thinkers.
As we will also find out, our grasp of words that attempt to depict or criticise the nature of 'abstractions' depends on the employment of material correlates in this world. For example, the above objections have to be committed to paper, or propagated in 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' cannot apply to any material embodiment of his ideas, for in that case we would once again have a use of this 'law' in the material world which undermined all he had to say about it, for here we would have sentence that were identical in content. On the other hand, if they do not, then once more: Trotsky attacked the wrong target.
Finally, the fact that dialecticians -- who are supposed to be developing 'cutting edge' science -- failed to notice this serious mistake, and who still try to ignore it no matter how many times they are told about it, that fact seriously undermines their credibility. Indeed, these major interpretive blunders fatally compromise the claim that DM is a science, let alone a philosophical theory that merits serious attention.
Ordinary Language Once Again Thwarts Dialectical Casuistry
Our comprehension of words for identity, sameness, equality and difference clearly revolves around our use of such words in ordinary life, whatever technical modifications we want to add on later, and for whatever reason.
But, the ordinary use of terms like "equal", "identical", "same" and "different" is highly complex. And yet, this is not the impression one gets from reading Trotsky's comments (or those of his epigones); nor is it the impression one receives from reading what Hegel had to say.
[More on this later. The importance of ordinary language will be highlighted in Essay Twelve (summary here and here).]
Whatever one thinks of the limitations or otherwise of the vernacular, unless we begin with an accurate or representative view of the use of such terms in ordinary language we stand in real danger of making fundamental mistakes. As we will see this is exactly what has hobbled the criticisms DM-theorists make of this 'law'.
It would be a mistake to think "equal", "identical", "same" (and related terms) all meant the same (no irony intended). But, because of his cavalier attitude to the vernacular Trotsky either ignored, or was oblivious to, the conceptual space ordinary language opens up to its users, a flexibility that allows them to make complex and intricate allusions to identity, equality, similarity difference, and much more besides, with ease.
Consider several 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 not be identical.
Again, two separate 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 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 the Stop the War Coalition (STWC)). In that case we could say that "The Unison rep is identical with the STWC Treasurer", and since NN is both at once, change would not affect this identity statement (unless of course she resigns from one or both).5
Furthermore, two or more things can be the same even if they are not at all alike: for example, two copies of identically the same book (e.g., of Das Kapital) in radically different languages (say, English and Chinese) are easily recognised as the same book even if they are totally dissimilar. Minor differences between the two are irrelevant. So, while these books may not be identically the same physical object, they are identically the same work by Marx.
This indicates that our application of identity criteria in different areas of discourse change depending on the substantival terms involved.5a This shows that there is in the vernacular no such thing as the meaning of any of our terms for identity, sameness and 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 distinct athletes who cross the winning line together would both be equal joint-winners of the Gold medal, say; or, two women at the front of two different queues in the same or different Post Offices would both be equally first in line; or, 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 punters could share (equally) 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 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
Of course, only those who take their philosophical cue from ordinary (material) language will be impressed with the above examples. On the other hand, since ordinary language is the means of communication invented and maintained by ordinary workers (as they interface with one another and with the material world), only those with a preference for non-materialist language -- for instance, those with an inexplicable fondness for the jargon invented by Philosophers, or even worse, by Hegel -- over the materially-grounded vernacular, would have reason to cavil. Annoyingly, each of the latter would be doing so for identically the same ideologically-compromised reasons. [On this see, Essay 12 (summary here).]
But, such is the cunning of ordinary discourse.
Clearly, Trotsky and Hegel created serious problems for themselves by erecting an insecure 'logical' edifice on such an insubstantial linguistic base. And this predicament was further compounded by their choice of an extremely narrow range of examples, compared to the countless available to them (and to ordinary speakers) --, which permit talk of equality, sameness, identity and difference with ease.
Equally annoyingly, traditional Philosophers have done exactly the same.6
As is no doubt apparent to any competent user of language, "equal" and "identical" are not synonymous. Several examples given above illustrate this fact; the distinction can also be seen if "equal" is substituted for "identical" in either of the following two 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 that taken (perhaps presented to her as a cheque).
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: cos(3θ) + sin(θ) º 4sin(θ)cos2(θ).
[In S6 "º" is the sign for identity or equivalence.]
Now, nobody confuses "=" with "º" in mathematics. And in S5, just because x = 7 or x = -6, it does not mean that x is identical with either -- otherwise it could never become another number, and would not be called a variable. In that case, equality does not prevent change, nor does it even imply that things cannot change, at least, not in mathematics.7
So, the question remains: 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 his glaring omission -- coupled with his subsequent and irrelevant digression over bags of sugar and eye-glasses, and his failure to consider the wider use of identity words in the material language of everyday life -- tends to suggest that Trotsky did not 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 that was not identical with it (irony intended).
Perhaps one answer to this 'puzzle' lies in the fact the change of subject recorded in S1 allowed Trotsky to go on to make what turn out to be largely irrelevant claims about things like bags of sugar. Because the latter involve items that can be measured (as opposed to their being counted), the interpretation of the "A"s in S1 as quantities of sugar heavily biased Trotsky's criticism; it allowed him to focus his attention on one particular aspect of equality that is not necessarily connected with identity.
S1: A is equal to A.
For example, one and the same bag of sugar could be 'self-identical' and equal to itself in weight even while it was unequal in weight to a second seemingly identical bag. [How this is possible will become clear as the argument unfolds.] And two different bags of sugar could be equal in weight (even if only momentarily), as far as the most sensitive instruments could tell. Not only that, two separate bags could both have their weights changing in exactly the way Trotsky described (no irony intended); the first bag could have its weight falling, and in the second it could be rising. At some point, therefore, their two weights could momentarily be identical. How could this possibility be ruled out?
Furthermore, in two separate piles, bag A in pile one, and bag B in pile two, could be the heaviest in their respective heaps. In that case, each would be equally the heaviest in their respective groupings while still being non-identical in weight. No doubt the reader can imagine other cases Trotsky failed to consider.
Clearly, Trotsky's analysis blurred these clear distinctions -– ones, incidentally, that are easily made in ordinary, material language (as they have been here), and which are readily understood even by working-class children.
More importantly, Trotsky clearly failed to notice that even though objects might vary in weight, they could still be identical in number. Indeed, as is patently obvious, any object is identical to itself in number -- so much so that close inspection over an extended period of time will fail to reveal any relevant difference here, even if other aspects of the said object change markedly. Trotsky overlooked this obvious counter-example to his claim that things cannot remain the same while they change: in at least this sense most do.
Of course, it could be objected here that not only do some things divide as they change, others merge together; in such cases, their number would not be identical from moment to moment. This is undeniable. However, descriptions of divisions and mergers depend on the said objects being identifiable first, which process clearly depends on the application of the LOI. If we cannot count objects before or after they divide/merge, we are surely in no position to judge that they have changed in this respect. Since counting depends on identification under a given general term (so that we can say we have, say, 2 bags of sugar -- or one amoeba, then two), this aspect of the rejoinder itself depends on an application of the LOI as a rule of language.
Even if the above response were rejected for some reason, there are uses of numerical identity that are not susceptible to this simple rebuttal. 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). Even though the number of copies of Das Kapital has increased markedly over the years, and each copy might have changed in the meantime, the number of volumes of Das Kapital remains steadfastly fixed on three. 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 2006 (namely, three).
L2: The number of volumes of Das Kapital on any one day in 2006 is identical to the number of volumes of Das Kapital on the same day in 2006 (namely, three).
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 2005" (when interpreted as they are in L2), it is clear that it is not possible to map the same "A"s consistently onto anything analogous in L1. This 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 2006", which phrases are quite clearly not typographically identical, even though they are part of an expression here of simple rules we have for identity.
This demonstrates that Trotsky's narrow interpretation of the variable letter "A"s (in L3 or S1) does not capture the wider uses of words we have for identity in ordinary language -- some of which were considered above (and more will be below). Even so, both L1 and L2 surely count as further counter-examples to Trotsky's charges against the LOI. And it is worth recalling that the volumes of Das Kapital are just as material as bags of sugar are.
Again, it could be objected that number is an abstract property of objects, making the above points irrelevant. But, according to Lenin anything that enjoys objective existence external to the mind is material:
"[T]he sole 'property' of matter with whose recognition philosophical; materialism is bound up is the property of being an objective reality, of existing outside our mind." [Lenin (1972), p.311.]
Well, the three volumes of Das Kapital surely exist just as objectively "outside the mind" as do pound bags of sugar. Moreover, if Trotsky is allowed to refer to some of the measurable properties of bags of sugar -- such as weight; is this not equally 'abstract'? --, critics of the above cannot 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 connected 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 What Is To Be Done?
L6: The Morning Star is identical to the Evening Star.
L7: The population of the United Kingdom on any day, at mid-day, in January 2007, is identical to a whole number somewhere between 50 and 60 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 will be an act of the working class.
L10: The editor of International Socialism in 2007 is identical to the author of A People's History Of The World.
L11: Mount Godwin-Austen is identical to K2.9
Naturally, once again, sentences like these can be multiplied indefinitely. As a highly competent user of language, Trotsky cannot have been unaware of this. So why did he feign ignorance? Was his analysis of this 'concept' biased by his extremely narrow focus on a particular philosophical use of words for identity -- one derived from a notorious Idealist (Hegel) -– and one that did not match their application in ordinary material language?
As we will see, these suspicions are not all that easy to dismiss.
Precisely What Is Trotsky Denying?
However, returning to Trotsky's actual argument:
"In reality 'A' is not equal to 'A'. This is easy to prove if we observe these two letters under a lens -– they are quite different to each other…." [Trotsky (1971), p.63.]
But, not even a lens-wielding Trotsky would consider making the same point in relation to the following legitimate example of the use of the "=" sign:
S7: y = 2x + 1.
[S1: A is equal to A.
S1a: A is equal to itself.]
But, if not why not? In S7, the two sides of the equation do not 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. This use of "=" is not therefore susceptible to Trotsky's 'microscope argument'. This suggests that this particular point about the "A"s in S1 was equally misguided.
It could be objected that S7 is an 'abstract' example, which exempts it from such criticism. But, Trotsky queried whether or not these two "A"s in S1 were equal before he specified what they referred to. Indeed, for Trotsky at first, the "A"s in S1 were merely letters. And yet, if the symbols in S7 were to be interpreted in the same light, his lens-inspired criticism would make no sense. Who in their left mind would use a magnifying glass to check whether "y" was exactly equal to "2x + 1" 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 contain. Hence, despite the fact that the symbols appearing in S7 look totally different to the naked eye no one would question their use to express a rule for a one-one function.
In that case, why did Trotsky use such a crass argument against the expression of a linguistic rule in S1? If mathematicians were to scrutinise each other's work in the same crude way, they could dispense with proof and simply resort to inspecting manuscripts with magnifying glasses. Mathematical advancement would then depend on proof-reading and not on proof!
Trotsky Has This Base Covered -- Or Has He?
Some might claim that Trotsky anticipated this point when he wrote:
"[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 these are constrained by limitations imposed on anyone operating in the material world. In that case, no two symbols would be absolutely identical. Hence, Trotsky's point remains valid -- or so the argument might go.
However, in the vast majority of cases in mathematics symbols like "=" and "º" occur between symbols that do not even look remotely the same. Several examples were given above. Anyone who doubts this should consult a mathematics text (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. Trotsky's analysis thus fails completely to account for this use of symbols. In fact, not only are mathematicians not really interested in "approximate equality", the notion of "abstract identity" -- if any sense can be made of it -- is itself parasitic on ordinary identity, or on a (surreptitious) material application of the LOI (as a rule, not as a truth), as we shall soon see.
Bags Of Sugar refute Dialectics
Again, some readers might think that Trotsky had anticipated these minor quibbles, since he went on to consider the objection that the two "A"s in S1 might really be "symbols for equal quantities, for instance, a pound of sugar". In response to this, he pointed out that in the real world a pound of sugar is never equal to a pound of sugar, since any apparent equality would vanish upon closer examination:
"But, one can object, the question is not the size or the form of the letters, since they are only symbols for equal quantities, for instance, a pound of sugar. The objection is beside the point; in reality a pound of sugar is never equal to a pound of sugar -– a more delicate scale always discloses a difference. Again one can object: but a pound of sugar is equal to itself. Neither is true (sic) -– all bodies change uninterruptedly in size, weight, colour etc. They are never equal to themselves. A sophist will respond that a pound of sugar is equal to itself at 'any given moment'…. How should we really conceive the word 'moment'? If it is an infinitesimal interval of time, then a pound of sugar is subjected during the course of that 'moment' to inevitable changes." [Trotsky (1971), p.64.]
Mere Guesswork?
The problem with this is that Trotsky was clearly guessing; he had no way of knowing for sure that greater accuracy in weighing would always reveal detectable differences.
Indeed, there are several possibilities he failed to consider. For example, the weighing scales used could alter slightly, thus compensating for the inferred change in the weight of the sugar so that in the end no overall difference was observed. How could he rule this out? Plainly, he could not do so if constant change -- including that which instruments undergo -- is a central postulate of dialectics. How then could he be so sure that these hypothetical differences were not artefacts of the machines themselves -- or of other ambient causes -- as opposed to their being genuine phenomena representing actual changes in the weight of the sugar? For all he knew the sugar itself could remain the same for a few seconds (or minutes), with any apparent change being the result of other incipient factors. In fact, as seems clear, Trotsky could only be 100% confident that subsequently detectable differences were always and only the result of changes to the sugar itself because of an a priori stipulation to that effect. And, as seems plain, a stipulation is different from an imposition on nature in name only.10
Of course, that does not mean Trotsky was wrong in this case. No doubt if a series of identical experiments -- note the use of the italicised 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 an ensemble of some or all of these --, or, indeed, they were attributable to other proximate causes.
Some might think this irrelevant; if things change, who cares what causes it? But, Trotsky is here appealing to the results of an experiment -- one that he clearly did not carry out -- to substantiate a claim about all objects everywhere in the universe and for all of time. It now turns out that because of that thesis itself, it might not be possible to verify some of his claims. If so, we are still owed an explanation as to why Trotsky thought it correct to say everything changes all the time, when this cannot be confirmed. And this is not 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 do not have access to most regions of space and time!
And, as we shall soon see, any attempt to plug the gaps in Trotsky's argument merely drills larger holes in it.
This is quite independent of the fact that Trotsky seems to have confused the LOI with something 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 he should have used the following schema:
W1: A is equal to B.
But not:
W2: A is equal to A.
In the quoted passage above, Trotsky referred to the manufacture of "cone bearings" as part of his argument against the unrestricted application of his own simplified version of the LOI. In this, he was clearly interpreting the two "A"s of W2 as standing for different (even if 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 is not an instance of his own garbled version of the LOI!
Some might regard this 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 W2.
This objection has some force -- but not much. This is because Trotsky began with the following assertion:
W3: Every worker knows that it is impossible to make two completely equal objects.
The idea seems to be that workers often (invariably?) realise that the LOI is of limited (or zero) applicability when they make things. However, even if this 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', and that they were 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 does not refer to two separate objects being the same; in its classical form (and sometimes in Trotsky's version, too) it is manifestly about an object's 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 claiming would have been clear to all; no worker (or anyone else for that matter) would form such a crazy idea.
However, in W1, Trotsky's point is completely different; there he was arguing that different objects are not identical, and that workers know this. in this case, he was not saying that any one specific object is not self-identical, but that of any two objects, not only can workers see that they are not the same, they also know they cannot make two that are identical. He did not say that workers are aware that they cannot 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 -- that it is impossible to make an item that is self-identical.
In any case, Trotsky's point (in W3) is not even derivable from his own criticism of the LOI. W3 is not even a DM-thesis! And this is quite independent of whether or not workers conclude all he said they should. As seems clear, it is not relevant to claim that workers are automatic dialecticians because they assent to a banal truth that is not actually part of DM. It is not a DM-thesis that two objects are different -- only that one object is not the same as itself. What was wanted here was an example taken from DM that workers could assent to before they were talked into it by a fast-talking Dialectical Missionary. What we actually have here is a truism that any card-carrying member of the ruling-class could accept: even George W Bush knows that two apples are not one apple!
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 this is what Trotsky meant then it is certainly unexceptionable, but it's not what he said. And even if he had have said it, it would not have distinguished a DM-description of reality from one available to anyone using ordinary language or anyone cognizant of 'bourgeois' science. Indeed, we can go further: no sane Capitalist believes that all commodities are identical or that things do not change.
Moreover, Trotsky failed to notice that the alleged limitation he thought he noticed in the making of two identical items does not appear to affect whoever it is that is responsible for applying these "norms of tolerance". According to Trotsky's own description, such workers are at least able to determine what constitutes the same application of these norms to different cone bearings. But, that surely means that such workers would have to use a norm encapsulating the dread LOI in order to apply that norm equally between cases. That is, they would have to know (in practice) what constituted an identical application of that norm over time, since an approximate application to two very similar cones might very well pass them off as identical!
Hence, in order for a worker to do what Trotsky says, he or she would have to know precisely what constitutes the correct application of the same norm to at least two different cone bearings. Even if these workers rejected the LOI (which is doubtful), they would still have to use a norm expressing it in order to be able to agree with Trotsky that this 'Law' fails to apply to cone bearings! In fact, they could only concur with Trotsky after completing a practical refutation of what he declared they all implicitly knew!
Despite this, what Trotsky actually said is patently incorrect. His comments clearly ruled out the possibility that two different objects could become the same, that a worker could make two distinct objects into one and the same thing, and that workers know this. In fact, ordinary language and common experience allows for both eventualities (many of which workers will be well aware of already).
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/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 (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 have been made to the pig, 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 comprise the same outlet.
(6) Two miscounted Widgets can create the same excuse for a strike.
(7) Two sentences can form the same paragraph of the same or different strike leaflets.
(8) Two (or more) of the above can form the same excuse for dialecticians to ignore them.
Of course, if we are no longer restricted to considering only two items then it is possible to multiply the above examples indefinitely. For instance, one hundred thousand workers could form the same revolutionary column, or two million people could form the same march against the war in Iraq. Or even: two thousand police officers could constitute the same panic-stricken retreat from either of the former.
It could be objected here that these 'counterexamples' beg the question since, if Trotsky is right about the defects of the LOI, none of the above would be genuine identity statements.
However, as was argued earlier, our ordinary use of words for identity (i.e., "the same as", "exact", "similar", "identical", "not different", "precisely", etc.) is highly complex. It is far more involved than Trotsky imagined in his 'theoretical' deliberations -- although in his everyday speech he could not have been unaware of this fact, and he would have used sentences employing terms like the above countless times throughout his life.
The vernacular --, which, it is worth reminding ourselves yet again, is derived from everyday material practice -- allows for the expression of all manner of complex identities; the lists given above outline only a few of these (there are more given here). Anyone who could not recognise these as examples of sameness and identity (etc.) would be deemed not to understand their own language (since they would be incapable of recognising and using and comprehending the same words from that language in the same way as anyone else); indeed, they could in some circumstances become a danger to themselves. In which case, they would hardly be in a position to criticise the 'law' that supposedly operates behind such words.
Indeed, the employment of these words in contexts like the above tells us more about their meaning than could be learnt from reading the same comments in Hegel an indefinite number of times (irony intended). His narrow metaphysical use of a few of our words for identity and change shares nothing with their ordinary employment; as such his use is devoid of meaning. [Why this is so will be explained in Essay Twelve Part One.]
If, on the other hand, these examples do not tell us what our words for identity (etc.) mean, if they are defective in some way, then even those who criticise the use of such terms must fail to grasp what they themselves are criticising (i.e., the ordinary use of a word they have just failed to grasp), since they will not be able to put into words what constitutes the same use of either that word or its associated terms. [The reasons for saying this are outlined in more detail in Note 19.]
As this Essay shows, it is in fact impossible to decide what (if anything) Trotsky actually meant by his attack on the LOI. All this suggests that the above examples represent a far more legitimate use of words for identity than the severely limited range found in Hegel, Trotsky or his latter-day clones. Hence, as far as ordinary language is concerned, it is quite easy to speak about making two or more things exactly the same -- which is all that us non-Idealists need.
It is certainly all that workers need.
In Essay Five we saw how Engels had extrapolated wildly from a sketchy thought experiment about moving bodies -– complemented by an idiosyncratic understanding of ordinary words like "motion", "place" and "contradiction" -- to universal theses that were supposed to be true of all bodies everywhere and for all of time. Here, we see Trotsky doing something similar based on his own idiosyncratic interpretation of a severely restricted set of ordinary-looking words for identity. From these he extracted several substantive theses about every object and process also valid for all of space and time, thus attempting to derive Superscientific truths from a superficial and demonstrably misguided conceptual analysis of what he assumed were the meanings of words like "identical", "change", "equality", "time", "moment" and "measure". And he too based his cosmically-ambitious conclusions on an alarmingly narrow set of words for identity, none of which turned out to be about identity to begin with -- supported by a 'thought experiment' about bags of sugar that (as it turns out below) undermines its own rationale!
And this is supposed to be the cutting-edge science?
yet More A Priori SuperScience
Even if these serious difficulties are put to one side, Trotsky's analysis is deeply flawed for other reasons. This can be seen if consideration is give to the rejoinder Trotsky himself advanced (in S9) to a hypothetical objection (recorded in S8):
S8: A pound of sugar is equal to itself.
S9(a): All bodies change uninterruptedly. (b) They are never equal to themselves.
However, 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. He can't have based this on all the observations humans beings have made of bodies in recorded history, since these only amount to a vanishingly small fraction of all the bodies there are, have ever been, or will ever be. Nor could it have been based on scientific evidence itself, since that is equivocal, at best. For example, it is now thought that certain sub-atomic particles are equal to themselves for unimaginably long periods of time. Protons, for instance, have an estimated life span in excess of 1032 years, which is approximately 1018 times longer that the age of the Universe (if we accept the BBT). During that time they do not change (as far as we know), and as such they are surely equal to themselves.12 And this are not the only example. [On this, see Note 11]
[BBT = Big Bang Theory.]
Moreover, we have already seen that the material language of ordinary human beings has programmed into it complex expressions permitting talk about objects and processes that can and do remain identical. Hence, neither human experience nor scientific theory agrees with Trotsky's analysis.
Using The LOI To Criticize The LOI
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.
For instance, it is far from certain which target he had in mind when he asserted S9(b). Consider the following interpretations of S8 as possible alternatives:
S8: A pound of sugar is equal to itself.
S9(b): [All bodies] are never equal to themselves.
S10: Let A1 be a pound of sugar at time T1.
S11: Let A2 be a pound of sugar at time T2.
S12: S8 means A1 is equal to A1.
S13: S8 means A1 is equal to A2.
[S1: A is equal to A.]
At first sight, it seems that Trotsky might have had S13 in his sights when he wrote S9(b), since it compares a pound bag of sugar with itself as it changes over time, which is perhaps the normal way of regarding change. But, S13 does not even look like a classical formulation of the LOI; nor does it look like Trotsky's simplistic version (recorded in S1), either. It more closely resembles a quasi-empirical claim about the temporal continuity of material substances. Clearly, if Trotsky had wanted to use S9(b) to refute S13, then S12 (surely, a more likely target) would have been left unscathed. This suggests that S13 was not the interpretation that Trotsky had in mind. He must have read S8 as equivalent to (i.e., identical with) S12, which he plainly thought was refuted by S9(b):13
S8: A pound of sugar is equal to itself.
S12: S8 means A1 is equal to A1.
S13: S8 means A1 is equal to A2.
S9(b): [All bodies] are never equal to themselves.
If this is so then it is possible to show that Trotsky had to assume the truth of the LOI in order to declare it false; he had to assume the LOI was reliable in order to try to show it was unreliable. Clearly, if this is so, it would mean that such a 'demonstration', based on this law, would be defective, since Trotsky's 'analysis' would have undermined itself.
To see this more clearly, it is worth trying to make 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.
S14 expresses the content of S9(b) a little more clearly; indeed, Trotsky himself employed a tensed ordinary language quantifier expression in S9(b) (viz., "never").13a
Unfortunately, this change of emphasis introduced a serious problem Trotsky failed to notice. This can be seen if we refer back to S1, S9 and S14:
S1: A is equal to A.
S9(a): All bodies change uninterruptedly. (b) They are never equal to themselves.
S14: For any object A, at any time t, A at t is not equal to A at t.
Clearly, S9(b) -- when interpreted along lines suggested by S14 -- implies that S1 must be rejected because:
S15: It is never true that A is equal to A.
However, S15 appears to imply the following:
S16: For any time t, and any A, A at t is not equal to A at t.
But, this now transfers the emphasis onto the temporal aspects of identity, which underlines the points Trotsky himself tried to make about time and change:
"Again one can object: but a pound of sugar is equal to itself. Neither is true (sic) -– all bodies change uninterruptedly in size, weight, colour etc. They are never equal to themselves. A sophist will respond that a pound of sugar is equal to itself at '