Essay Six: The 'Law Of Identity'
Does Not Exclude Change -- Trotsky On Identity
Readers should take 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'.
It's also worth pointing out that a good 50% of my case
against Dialectical Materialism [DM] has been relegated to the
End Notes. This has been done 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 raised objections (some obvious, many not -- and some that will
have occurred to the reader) to my own arguments -- which I have then answered.
[I explain why I have done this in
Essay One.]
If readers skip this material, then my answers to any
objections they might have to my arguments will be missed. [Since I have been
debating this theory with comrades for over 25 years, I have heard all the
objections there are! Many of the more recent debates are listed
here.]
If you are viewing this
using Mozilla Firefox, some of the codes that Microsoft have put into FrontPage
(the editor I have used) will make several 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. As far as I know, there are no problems with Internet
Explorer.
This Essay is over 52,000 words long; a summary of its
main ideas can be found
here.
This Essay principally concerns Trotsky's objections to
this so-called 'Law', but much of it applies to Hegel's own 'critique', too. It
should be read in conjunction with Essays
Four and
Five.
Quick Links
Anyone using these links must remember that
they will be skipping past supporting argument and evidence set out in earlier
sections. [If you have a pop-up blocker in your Firewall, you will need to press
the "Ctrl" key at the same time or these and the other links here won't work!]
(1)
Trotsky On Identity
(2)
Identical Is
Not Identical With Equal
(a) A Mistake Most Dialecticians
Make
(b) Ordinary Language
Thwarts Dialectics
(c)
Mathematical Equality Vs Mathematical Identity
(d) Trotsky Changes The Subject
(3)
Trotsky's Argument Dissected
(a) Precisely What Is Trotsky
Denying?
(b) Trotsky Has This Base Covered --
Or Has He?
(4)
Bags Of Sugar Refute
Trotsky
(a)
Mere Guesswork On Trotsky's Part?
(b)
Yet Another Misidentification
(c)
Wrong Anyway
(d)
Identical A Priori Tactics
(e)
Superscience From Mere Words
(f) Incomprehensible Or Just Trivial?
(5)
Trotsky Uses
Identity To Criticise Identity
(a) Same Moment
(b) Turn To The Concrete
(6) Did Trotsky
Understand Identity?
(a) Can Anyone Learn What Identity
Is If
None Exists?
(b) The Sting In The Tail
(c) 'Approximate' And
'Abstract' Identity
(d) Identity Schmidentity
(e) Trotsky's Exact Words Self-Destruct
(f) Trotsky's Attack Unequal
To The Task
(g) Materially-Induced
Dialectical Misery
(7) The Knock-Out
Blow
(8) Dialectical Logic
Superior?
(9)
Shock! Physicists Discover Identical Particles
(10)
Plato, Hegel And
'Abstract' Ideas
(11) Notes
(12) References
Abbreviations Used At This
Site
Few other areas of FL cause dialecticians more
problems than the LOI. For many, it's 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
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; even so, the objections I have raised here against the highly repetitious and
misguided comments that dialecticians range against 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 those of Hegel (few of whom have
read or studied his work), it makes
sense to begin with his widely quoted remarks.
Trotsky On Identity
In his debate with
Burnham, Trotsky rehearsed an
argument aimed at exposing what he took to be serious limitations of
the LOI, which he had lifted directly or indirectly from Hegel and which has
resurfaced almost verbatim in the writings of 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 an opinion on the
subject. By the same token therefore we may suppose him an expert in
High Energy Physics
and perhaps even brain surgery!
This 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 were examined in Essay Four,
where they were shown to be based on
a serious misunderstanding even of AFL, let alone MFL. This is hardly surprising
given the allegations made in the previous paragraph.
Be this as it may, in
this Essay I plan to focus on Trotsky's criticisms of the LOI, which
DM-theorists -- at least those in the Trotskyist wing of Marxism -- generally
regard as definitive. John Rees, for example, outlined one key issue 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.]
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.]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 quote.3
Nevertheless, there appear to be at least a dozen substantive
claims Trotsky was making here:
T1: AFL begins with "A is equal
to A".
T2: This "postulate" applies quite well in most
practical situations. But, in reality "A is not equal to A".
T3: Close inspection under a lens. for example, will show that any
chosen letter
"A" is not exactly the same as any other letter "A".
T4: A similar observation applies if these
letters stand for material objects like pound bags of sugar.
T5: Any two weighings of seemingly equal bags of
sugar will always reveal minor differences.
T6: It's not use saying 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 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 not
based on
thought experiments) -- but that fatal 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 -- which is: if the universe is governed by DL, a simple 'thought
experiment' is all the 'evidence' the trained dialectician needs.
Naturally, only consistent materialists will object
at this point.
As George Novack notes:
"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, what little 'evidence'
Trotsky and/or his epigones have advanced in support of their contentious
claims will be
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 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 went 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, that means Trotsky's criticisms
of the LOI were misconceived from the start.
However, when confronted with the above,
DM-apologists tend to respond, "So what? What's the difference between the two?" As will
be appreciated, that reply is itself problematic (not the least because
it reveals that such individuals also 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 we would now have a genuine
example of the LOI on which all could agree, namely here.
(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 appeal to the "It's just abstract" defence
(accompanied or not with the "This law only applies to objects and processes in
nature" ploy, sometimes compounded with a clichéd accusation of "pedantry").
As we will soon see, this retreat is itself a
backward step too far, since
there is also a clear difference between abstract equality and
abstract identity, which dialecticians have also failed to notice. So,
abstract or concrete, the two notions are not the same. [Anyway,
'Abstract/Absolute Identity' will be dealt with
later.]
Furthermore, as we discovered in Essay Three Parts
One and
Two,
dialecticians have an insecure grasp of the nature of abstraction, and are
largely content to be told what to think on this score by Idealist, ruling-class
thinkers (like Aristotle,
Spinoza and Hegel).
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 situated in this world. For example, the above objections have to be
committed to paper (or computer screens), 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' can't apply to any material
embodiment of his ideas --, since, if they did, 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 very material sentences that are identical in content.
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 developing 'cutting edge' science --
failed to notice this serious error,
and who continue to ignore it no matter how many times it is brought to their
attention, seriously undermines their credibility. Indeed, these major interpretive
blunders fatally compromise the claim that DM is a science to begin
with, let alone
a philosophical theory that merits serious attention.
Ordinary Language Once Again Thwarts Dialectical
Casuistry
Our comprehension of words for identity,
sameness, equality and difference clearly revolves around our use of such
expressions
in ordinary life, whatever technical modifications we might subsequently want to
introduce, 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 Hegel, either.
[More on this later. The importance of
ordinary language was highlighted in Essay
Four, and Essay Twelve
Part One (as well as the rest of Essay
Twelve, summary 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 in more complex areas. As we will see this is exactly what hobbles the
criticisms DM-theorists make of this 'law'.
It would be a mistake to
think "equal", "identical", "same" (and related terms) all mean 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 in this regard, 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. [If they were identical
they could not be opposite.]
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 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 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 of
these at once,
change would not 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 are not 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. Only a fool would deny that a copy of Das Kapital was in fact a copy
of that book because of a different coloured cover, for example. So, while these books may not be identically the same physical
object, they are identically the same work by Marx. Indeed, countless different
readers can now access the same works of Marx's right across the planet at the
Marxist Internet Archive.
Despite the fact that they might access his work using different browsers,
screen resolutions or text magnification, few would claim that these facts prevent
readers accessing identically the same work.
This indicates that
our
application of identity criteria
in different areas of
discourse change depending on the
substantival terms involved.5a
This shows that there is in the vernacular no such thing as
the meaning of any of our terms for identity, sameness or difference --, which
further implies that Hegel and other dialecticians focussed their attention on an
entirely spurious target.
To continue, two
totally different things can be equal. For example, two non-identical athletes who
cross the winning line together would both be equal joint-winners of the Gold
medal, say. Two women at the front of two different queues in the same or
different Post Office 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 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
Of
course, only those (like
Marx) who take their philosophical cue from ordinary
(material) language will be impressed by 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 the
material world), only those with a preference for non-materialist
language -- for instance, those with an incurable fondness for the jargon
invented by Philosophers, or even worse, by Hegel -- over the
materially-grounded
vernacular, would have reason to cavil. Annoyingly, any individuals who opt for
the latter the latter will be
doing so for identically the same ideologically-compromised reasons. [On
this see, Essay Twelve (summary
here).]
But, such is the
cunning of ordinary discourse.
Clearly, Trotsky and Hegel created serious
problems for themselves when they erected an insecure 'logical' edifice on such an
insubstantial
linguistic
base. This predicament was further compounded by their choice of an extremely
narrow range of examples, 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 will no doubt be 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 in cheque).
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 does not
mean x is identical
with
either -- otherwise it could never become another number, and would not
therefore 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's also true that 0 + 0 = 0, and 0 x 0 =
0 -- even though it's 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 stay the same. For example, it's easy to transform 1/√n into √n/n
thus: 1/√n x
√n/√n = √n/n. But, 1/√n does not 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 do not prevent change, nor do they even imply that things cannot 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 (this is flanked only by
Proper names (or other singular expressions, such as Definite Descriptions)), whereas "º"
and is a
truth-functional operator (and can be flanked only by propositions).
[Of course, these distinctions are not the same as those
that feature in
ordinary language (no irony intended), nor yet those found in traditional Philosophy
-- more on this below.
"Truth-functional"
is technical term that relates to the logical relation between propositions the
alteration of which changes the conditions under which they are true or false.]
[MFL = Modern Formal
Logic.]
It could be objected once again that these example are all
abstract, in which case, the reader is re-directed to my
earlier response.
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 irrelevant digression over bags of
sugar and eye-glasses, and his failure to consider the wider use of identity
words in the language of everyday life -- tends to suggest that
he 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 (equality) that was not identical with it
(irony intended).
Trotsky Changes The Subject
One answer to this
'puzzle' could lie 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 this involves 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 our
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, the second 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 bag 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 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 (and it is identical in number with any other
object, too). Moreover, close inspection over an
extended period of time will fail to reveal any relevant
difference here (with respect to their number), even if other aspects of the said object(s) 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),
that aspect of this objection
itself depends on an application of the LOI as a rule of language.
Anyway, the above comments are still applicable to objects that
do not divide or merge. Plainly, there
are uses of numerical identity that are not susceptible to this objection
(concerning objects that divide or merge). For example, if we consider, say, the
number of volumes of
Das Kapital, it's clear that there are just as many volumes today
as there were 100 years ago (viz., three -- that is, if we do not count
Theories of Surplus Value; six if we do). Even though the number of copies
of Das Kapital has increased markedly over the years, and most
copies will have changed markedly in the meantime, the number of volumes of Das
Kapital remains steadfastly fixed on three (or six). Hence, the following statements are true:
L1: The number of volumes of Das
Kapital in the year 1900 is identical to the number of volumes of Das
Kapital in 2011 (namely, three (or six)).
L2: The number of volumes of Das
Kapital on any one day in 2011 is identical to the number of volumes of
Das Kapital on the same day in 2011 (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 2011" (when
interpreted as they are in L2), it's clear that it's 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 2011", 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
examined below). Even so, both L1 and L2 surely count as further counter-examples to
Trotsky's charges against the LOI. And it's worth recalling that the volumes
of Das Kapital are just as material as bags of sugar.
Again, it could be objected that number
is an abstract property of objects, making the above points irrelevant.
But, according to Lenin, anything that enjoys objective existence external to the
mind is material:
"[T]he sole
'property' of matter with
whose recognition philosophical; materialism is bound up is the property of
being an objective reality, of existing outside our mind." [Lenin (1972),
p.311.]
Well, the three volumes
of Das Kapital
surely exist just as objectively "outside the mind" as do pound bags of sugar.
Moreover, if Trotsky is allowed to refer to some of the measurable
properties of bags of sugar -- such as their 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 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 on any day, at noon, in January 2011, is identical to
a whole number somewhere between 50 and 70 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 must be the act of the workers themselves.
L10: The editor of
International Socialism
in 2009 is identical to the author of
A People's History Of The World.
L11: Mount
Godwin-Austen is
identical to K2.9
Once again, sentences like these
can be multiplied indefinitely. As a highly competent user of language, Trotsky
cannot 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 particular
philosophical use of words for identity -- one derived from a notorious
Idealist (Hegel) -– and one that did not match their application in ordinary
language?
As we will see, these suspicions are not easy to dismiss.
Trotsky's Argument
Precisely What
Is Trotsky Denying?
However, returning to Trotsky's actual argument:
"In
reality 'A' is not equal to 'A'. This is easy to prove if we observe these two
letters under a lens -– they are quite different to each other…." [Trotsky
(1971), p.63.]
But, not even a lens-wielding Trotsky would
consider making the same point (no irony intended) in relation to the following legitimate
example of the use of the "=" sign:
S7: 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. In which case, this use of
"=" is not susceptible to Trotsky's 'microscope argument'. That suggests this
particular point about the "A"s in S1 was equally misguided (irony not
intended).
It could be objected that S7 is an 'abstract'
example, which exempts it from such criticism. But, Trotsky queried whether or
not the "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 employ.
Hence, despite the fact that the symbols appearing in S7 look totally different to
the naked eye no one would question their use in expressing 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, not on proof, but on proof-reading!
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 this world. In that case,
since no two symbols would be absolutely identical, 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 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. 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
Trotsky
Again, some readers might still think that Trotsky
had anticipated these relatively minor quibbles, since he went on to consider the
reply that
the two "A"s in S1 might really be "symbols for equal quantities, for
instance, a pound of sugar". 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 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 is clearly guessing here. He had no way of knowing
for sure that greater accuracy in weighing would always reveal detectable differences.
Indeed, there are
several possibilities that he
failed to consider. For example, the weighing scales 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 observable. How could he rule this out? Plainly, he
couldn't do so if constant change -- including that experienced by instruments
-- is a central postulate of dialectics. How could he be so sure that the
hypothetical differences he says must exist between these bags (or between a bag
and itself) weren't artefacts of the machines themselves -- or of
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 any subsequently detectable differences were always
and only the result of changes to the sugar itself because of an a priori
stipulation to that effect. And, as seems plain, a stipulation is different from an
imposition on nature in name alone.10
Of course, that doesn't
mean Trotsky was wrong in this case. No doubt if a series of identical
experiments -- note the use of that 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 all or some of these
--, or, indeed, they
were attributable to other proximate causes.
Some might think
the above considerations
are irrelevant;
if
things change, who cares what causes it? But, Trotsky is here appealing to the
results of an experiment
-- one that he
clearly did not carry out -- to substantiate a claim about all objects everywhere
in the entire universe, for all of time. It now turns out that because of that thesis itself, it might
not be possible to verify what he said. If so, we are still owed an
explanation as to why Trotsky thought it correct to say
everything changes all the time, when this cannot be confirmed. And
this
is not just because
many of the above complications could cancel each other out or even mask a temporary lack of change
in other things, it's because we only have access to a tiny fraction of space
and time!10a
And,
as we shall soon see, any attempt to plug the gaps in
Trotsky's argument will only succeed in punching larger holes in it.
Yet Another Misidentification
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 (two "cone bearings", in the above example) 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" 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 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 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 W1.
This objection has some force -- but, fortunately, not much. That is because
Trotsky began with the following assertion:
W3: Every worker knows that it is impossible to
make two completely equal objects.
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 are aware of its
limitations. In order to do this, he advanced the claim that workers in general know that
it's 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
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 claiming would have been clear to
all; no worker (or anyone else for that matter) would entertain 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 particular 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. 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 -- i.e., that it's impossible to make
an item that is self-identical. He manifestly failed to do this.
In any case, Trotsky's point (in W3) can't even be derived from
his own criticism of the LOI. W3 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's 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 no object
is self-identical. What's wanted here is 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!
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 this is what Trotsky meant then it's 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 (or even two of them) 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 the "norms of tolerance" he mentioned. 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 it 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
reject 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 short, they could only concur with Trotsky after completing a practical
refutation of what he declared they all implicitly knew!
Wrong Anyway
Despite this, what Trotsky actually said is patently incorrect.
His comments clearly ruled out the possibility that 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 (of which workers will be well aware 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
had been made to the pigs, of course.
Examples of two items being made into one include the following:
(1)
Two rivets can be made into the same seal between two plates
of metal.
(2)
Two buckets of paint can be mixed to form the same colour
(i.e., green and red making brown).
(3)
Two wooden posts can form the same support in a
mine.
(4)
Two ropes can form the same towline.
(5)
Two plastic pipes can 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 would be possible to multiply the above examples indefinitely. For instance,
one hundred thousand workers could form the same revolutionary column, or two
million people could form the same march against the war in Iraq. Or even: two
thousand police officers could constitute the same panic-stricken retreat from
either of the former.

Figure One: London February 15th 2003 -- Two
Million 'Unconscious' Anti-Dialecticians?
It could be objected here that these
'counterexamples' beg the question since, if Trotsky were right about the defects of the LOI, none of the
above would be genuine identity statements.
However, as 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's far more involved
than Trotsky and Hegel imagined in their 'theoretical' deliberations -- although in
their
everyday speech they could not have been unaware of this fact, and they would have
used sentences employing terms like the above countless times throughout their
lives.
The vernacular --,
which, it is worth reminding ourselves yet again, is derived from everyday
material practice -- allows for the expression of all manner of complex
identities; the lists given above outline only a few of these (there are
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 sense. [Why this is so will be
explained in Essay Twelve Part One.]
It could be objected that none of us use our words in the same
way, so the above observations are misguided. However, that objection itself is
(at least) based on that objector using the words I have employed in the
'offending' paragraphs above in the same way. If it isn't, then that
objection itself is misguided since it's aimed at the wrong target. And the same
can be said of any other general attempt to advance the same objection --
since it will not be the same objection if the words it contains
are employed in a different way. Of course, this is not to deny that we
sometimes use words in different ways, but this can't be typical otherwise
communication would cease. And anyone seeking to deny this would, plainly, fail
to get their point across!
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 will show, 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 selection 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.
Identical A Priori Tactics
In Essay
Five we saw how Engels had wildly
extrapolated 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 are supposedly 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 (in fact, equality!). From these he too
attempted to extract several
substantive theses about every object and process, valid for all of space and
time -- thus deriving Super-scientific 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". Like Engels, too, he based his cosmically ambitious conclusions on an alarmingly narrow set of words for identity,
none of which turn out to be about identity
to begin with -- supported by a 'thought experiment' about bags of sugar, which
(as will soon become apparent) undermines his entire argument!
And this is supposed to be 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.
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. He can't have based this on all the observations humans beings have
made of bodies in recorded history, since they only amount to a
vanishingly small fraction of all the bodies there are, have ever been, or will
ever be -- nor had he access to them, anyway. And neither could it
have been based on scientific evidence itself, since that is equivocal, at best. For
example, it's 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 than the currently accepted age of the Universe. During that time they do not change (as far as we know), and as such
they are surely equal to themselves.12
And this isn't the only example. [On this, see
Note
11]
Moreover, we have
already seen that the
material language of ordinary human beings has programmed into it
simple and complex
expressions that enable talk about objects and processes, which can and do remain
identical. Hence, neither human experience nor scientific theory agrees with
Trotsky's analysis.
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's
far from certain what he had in mind when he asserted S9(b). Consider the
following interpretations of S8 as 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.]
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 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 own 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 wasn't 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.
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; he had
to assume the LOI was reliable in order to try to show it was unreliable. Clearly,
that means this 'demonstration' -- based on this 'law' -- is defective,
since Trotsky's 'analysis' has undermined itself.
To see this more clearly, it's 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's a stylistic monstrosity. Not much hangs on the difference.
Given the above
caveat, S14 expresses the content of S9(b) a little
more clearly; indeed, Trotsky himself employed a tensed ordinary language
quantifier expression in S9(b) (viz., "never").13a
Unfortunately, this change of
emphasis introduced a serious problem Trotsky failed to notice. This
can be seen if we refer back to S1, S9 and S14:
S1: A is equal to
A.
S9(a): All bodies change
uninterruptedly. (b) They are never equal to themselves.
S14: For any object A,
at any time t, A at t is not equal to A at t.
[S9(b): [All bodies] are never equal to
themselves.]
Clearly, S9(b) -- when interpreted along lines
suggested by S14 -- implies that S1 must be rejected because:
S15: It is never true that
A is equal to A.
However, S15 appears to imply the following:
S16: For any time t,
and any A, A at t is not equal to A at t.
But, this now transfers the emphasis onto the
temporal aspects of identity, which underlines the points Trotsky himself
tried to make about time and change:
"Again one can object: but a pound of
sugar is equal to itself. Neither is true (sic) -– all bodies change
uninterruptedly in size, weight, colour etc. They are never equal to
themselves. A sophist will respond that a pound of sugar is equal to itself
at 'any given moment'…. How should we really conceive the word 'moment'? If it
is an infinitesimal interval of time, then a pound of sugar is subjected during
the course of that 'moment' to inevitable changes. Or is the 'moment' a
purely mathematical abstraction, that is, a zero of time? But everything exists
in time; and existence itself is an uninterrupted process of transformation;
time is consequently a fundamental element of existence. Thus the axiom 'A' is
equal to 'A' signifies that a thing is equal to itself if it does not change,
that is if it does not exist." [Trotsky
(1971), p.64. Bold emphases added.]14
It's here that Trotsky unwittingly
introduced the serious difficulty alluded to above, one that is now connected with the
identity
of temporal instants. This is highlighted by his use of the phrase "that
'moment'" (which clearly means "that same moment", no
matter how short it is), during which an
object supposedly changes. But, in referring to time this way, the phrase "that moment"
suggests that we can make sense of the following schematic sentence:
S17: For some instant t,
t is identical to t.
[S17 is just a formal paraphrase of
"that [same] 'moment'".]
If Trotsky did in fact rely on S17, it
would be fatal to his argument since it claims that the instant in question is self-identical
and thus subject to the LOI!
That being so, Trotsky's argument clearly requires
something to remain the same (i.e., an instant in time during which an
object changes) so that his
objection to the LOI can gain some purchase. The serious problems Trotsky's
analysis now faces become a little clearer just as soon it is realized that the
following sentences follow from S16:
S16: For any time t, and any A,
A at t is not equal to A at t.
S18: There is a time t1
and an A such that, A at t1 is not equal to A
at t1.
S19: At one and the same
instant, A is not equal to A.
Now, the phrase "instant in time" --
represented in S18 by the use of "t1" --
is just as legitimate a substitution instance for the "A"s in S1 as was the phrase
"pound bag of
sugar". This can be seen if the following are compared (and if we wave for the
time being the fact that "identical" is not identical to "equal"):
S20: t1 is
equal to t1.
S1: A is equal to
A.
S17: For some t, t
is identical to t.
[S20 of course is just a vaguer form of
S17, but it is nonetheless a legitimate version of S1.]
This means that Trotsky actually
requires S20 to be true so that he can reject the LOI as false! As noted
above, this implies he needs the LOI to apply in an
unrestricted sense to certain things (i.e., to temporal instants) so that he can
deny it of others (i.e., pound
bags of sugar, and slightly varying letter "A"s).
It could be objected here that Trotsky was
merely criticising the LOI as it applies to objects and processes that change
in time; his argument was certainly not about "temporal instants", which are
abstractions anyway. Indeed, he was simply pointing out that no matter how brief
the time frame, change takes place. This is clear from what he says:
"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.]
Nevertheless, if "that moment" is interpreted exactly as Trotsky
says, he
still has to be able to refer to that very same moment during which the object
in question changes; if "that moment" were not the same, then his argument would
plainly fail, for the assumed change in the object in question would have occurred
in a different moment. Clearly, no one would object to that.
Moreover the counter-claim that moments in time are abstractions will not work
either; if bags of sugar can be weighed and
found to vary, moments in time can be measured and shown to differ/vary. They
are just as measurable as bags of sugar are. It is not easy to see how
Trotsky could consistently criticize the LOI on the basis of the hypothetically
differing weight of material objects without this spilling over
into a general criticism of anything at all that is capable of being measured,
like time. If time is to be exempted from his critique of measurement, why not weight?14a
It seems therefore that
Trotsky's argument relies on these measurable intervals staying the same while those
measurable bags of sugar don't.
In fact, no matter how short the interval
within which a given change is supposed to occur, Trotsky has to be able to
refer to identically the same one to make his case. Without that, his
whole analysis collapses. So whether or not these "moments" are extensionless
temporal points, or extended intervals in time, Trotsky still has to use
the LOI so
that he can assert that at least one of these is the very same moment during in which
the assumed changes occur.
A Turn To The
Concrete
However, even if this latest difficulty is
put to one side, and it's conceded that reference to identical temporal
instants is an unfair criticism of Trotsky, and S14 is examined without the
quantifier switch
(this links to a PDF)
recorded in S16, the same conclusions follow (no irony intended).
Consider this putative
substitution instance of S14 expressed in S21(a):
S14: For any object A, at any time
t, A at t is not equal to A at t.
S21(a): There is an A and a
time t1
such that A at t1 is not
equal to A at t2.
S21(b): There is an A
and a time t1 such that, A at t1 is
not equal to A at t1.
[S16: For any time t, and any A,
A at t is not equal to A at t.]
As the reader will no doubt have noticed,
S21(a) is in fact an illegitimate substitution instance of S14 because
the variable letters "t1"
and "t2" are
not identical, which violates certain conventions governing the
interpretation of
bound variables in quantified contexts in MFL.
[MFL = Modern Formal Logic.]
Now, Trotsky has to
make use of something like 21(b) -- or its colloquial equivalent -- if his criticisms of the LOI
are to work. This means that even if the objections noted
above (about abstract "temporal instants" etc.) are ignored/rejected, Trotsky's argument still
has to rely on the identity of quantified tensed variables -- or their
equivalent phrases in ordinary language (such as "same moment") --, if it is to
succeed. In short he has to use the same material objects -- words or
symbols -- to make his point.
In that case, his argument depends on one or both of the
following:
S20(a): Variable t1
is equal to (or identical with) variable t1.
S20(b): In referring to the same moment
during which an object changes, critics of the LOI have to use the identical
phrase "same moment", and they all have to mean exactly the same by it each
time.
But, an appeal to the identity
of symbols like these -- the linguistic expression of which must surely
count as concrete and material -- would be just as good an instance of the application
of the LOI as any Trotsky himself considered.15
Furthermore, Trotsky's analysis cannot be seen as
an 'immanent critique' -- i.e., as one undermining the LOI from
within
-- as it were. This is because his argument depends on this 'law' being
absolutely correct (for the above reasons) 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
uses are not identical in meaning as he uses them, then his conclusions fail
too.
[Identity criteria
for words (as opposed to letters) are considered in more detail
below.
Objections to the above, based on the 'relative stability' defence, are
neutralised in Note 15, and
here.]
In that case, it's not easy to
see how Trotsky's attack on the LOI could either proceed or succeed -- expressed in any
language (technical, scientific or ordinary) -- without an implicit or explicit use of the very
'law' under scrutiny. In order to reject the LOI as it relates to objects
it now looks like Trotsky has to admit that it applies either to temporal
instants or to tensed variable letters -- or, to ordinary words that give
expression to either (or neither). In the latter two cases, that would involve a
definite use of material objects (written in ink, perhaps), just like those
"A"s, to which Trotsky took exception.
Either way, Trotsky's analysis is now
involved in intractable problems; identity criteria for temporal instants (even if
these interpreted as discrete) are notoriously difficult to define -– even if you
accept the LOI. They are far more problematic than identity criteria for
material objects.16
And, of course, those governing the
concrete employment of tensed variables are governed by convention. In which case, it looks like Trotsky has to
appeal to the identity of tensed variables -- or to identical marks on the
page, or to ordinary words identically applied -- if his argument is to work
against the very same 'law' he used in his criticism of it!
Hence, in order to make
his case, Trotsky had to ignore in practice what he had earlier concluded in
theory, undermining what he said about the LOI by disregarding his own
strictures against it.
So, if truth is confirmed in practice,
Trotsky effectively scuppered his entire criticism of the LOI by having to
apply, in practice, that very same 'law'.
It seems, therefore, that it is not
possible to attack this 'law' without also appealing to its unrestricted application somewhere
else.
Trotsky's Analysis --
Incomprehensible, Or Just Trivial?
On the other hand, if Trotsky had been aware
of these problems and had still rejected the LOI (as it supposedly applied to
temporal instants, tensed variables or even ordinary words for identity), his
criticisms would have either (1) become far less grandiose than they seem, or they
would have (2) collapsed into incomprehensibility.
As far as (1) is concerned, if reference to
absolutely the same instant (or the same anything) is to be regarded as
illegitimate (since it would clearly require yet another application of the
LOI, as noted above), then Trotsky could only have meant one or more of the following:
S22(a): No
object is identical with itself at a later time.
S22(b): No object is
identical with itself at a different time.
[S13: S8 means A1
is equal to A2.
S21(a): There is an
A and a time t1
such that A at t1
is not equal to A at t2.]
But, both S22(a) and S22(b) express the banal
truth found in S13 and S21(a), with which few would want to quarrel.
Clearly,
therefore, S22(a) and S22(b) are almost certainly not what Trotsky had in mind;
he surely wanted to argue that no object is self-identical at the same
instant -- presumably because all objects are subject to their own internal
struggles,
each generated by a UO.
Unfortunately, as we have just seen, without recourse to the LOI applied to temporal instants, tensed variables or their
ordinary language equivalents, he cannot assert this.
[UO = Unity of Opposites; IED = Identity-in-Difference
i.e., 'Improvised Explanatory Device').]
On the other hand, option (2) would be
applicable whenever, say, each and every reference to
the LOI was dialectically 'made and un-made', as it were, at the same
time. That is, it would apply where "same" and "different" are said to
"interpenetrate" one another: as in "same and non-same" or in "the same and not
the same" (using the IED ploy). This would then have Trotsky meaning one or more
of the following:
S23(a): No object is self-identical at
the same non-self-identical instant in time.
S23(b): No object is self-identical at
the same and not the same self-identical instant in time.
But, as we have
already seen with other dialectical principles, this threatens to explode in
the following manner:
S23(c): No object is self-identical at
the same and non-same non-self-identical instant in time.
S23(d): No object is self-identical at
the same and not the same and not the same and not the same
self-identical instant in time.
Here, the word "same", as it appears in 23(a)
and 23(b), has been replaced in italics by its (assumed) 'dialectical' meaning
-- "same and non-same", or "same and not the same" -- in 23(c) and 23(d)
respectively, in order to make explicit the radical confusion this option would
create.
But, what could any of these
possibly mean? What precisely is a "same and non-same non-self-identical instant
in time"? Either it surreptitiously employs the LOI again by the use of
the word "same". or it's meaningless. [Of course, only those who
reject the IED ploy, and thus who also reject classical DM) will object
(successfully) at this point.]
Any who feel
confident in their ability to explain what 23(c) could possibly mean should not
be given the benefit of that
considerable doubt until they have done likewise with 23(d). What on earth could this
mean: "the same and not the same and not the same and not the same
self-identical instant in time"?
If now the same word "same" is given similar
treatment in 23(d), it rapidly collapses into the following linguistic mess:
S23(e): No object is
self-identical at the same and not the same and not the same and not the same
and not the same and not the same and not the same and not the same
self-identical instant in time.
As each "same" in 23(d) is
replaced with its 'dialectical meaning': "same and not the same".
[As should seem clear, this process of
'dialectical' explication can be continued indefinitely. However, an "excessive
tenderness" for my readers prevents me from extending it any further.
Some might feel that the above is a ridiculous
explication of "same" as that word is used in
DL. Maybe so, but until
dialecticians tell us what they mean by their sloppy use of words, it will
have to do.]
Moreover, it will not do 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 cannot be the "same, yet different", they can only be "different,
yet different". Of course, if they are indeed the "same, yet different"
then it cannot also be true that they are never the same. Either way, the use of this IED explodes in the face of whoever
attempts to use it.
It could be objected that Trotsky really meant that objects could
not simply be the same, but that they are always the same yet
different. However, we have seen In Essay Four that this results in even
greater confusion!
In that case, it is impossible to refute the
LOI in the same crude way attempted by Trotsky (irony intended). As the
above shows, this can't be done without using the LOI in the 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: having to reproduce an
identical copy of that argument; second: using the LOI applied to tensed
variable letters (or their equivalent in ordinary language) to establish their case.
The question now is: How might anyone who
agrees with Trotsky accomplish the 'very same' task without falling into this
double trap?
By means of
semaphore,
telepathy or
Aldis lamp?
Did Trotsky Understand Identity?
Learning Identity In A World With None
As seems reasonably plain, in
order to draw conclusions from a putative identity statement like S1, Trotsky
must have been able to understand the words it contained -- even if he
subsequently claimed that such propositions were not strictly or always true. Clearly,
therefore, Trotsky had to be able to comprehend the LOI before he
could criticise it (and the same goes for Hegel).17
S1: A is equal to
A.
But, this raises another serious question:
From where did Trotsky's understanding of the LOI originate? It's not easy
to see how he could have grasped the concepts involved if he only ever
encountered them in false sentences, or if their use was never
absolutely legitimate, as was claimed in S9:
S9(a): All bodies change uninterruptedly.
(b) They are never equal to themselves.
On the surface, it would seem impossible for
anyone to learn what something meant if all they ever experienced were not
examples of the very thing they were being taught, or if they were all incorrect
instances of it.18
The Sting In The Tail
Of course, it could be objected here that
Trotsky's ability to understand everyday words for identity is irrelevant since
the concepts they express are only valid "within certain limits"; even
dialectical concepts only approach the truth "asymptotically".19
Hence, although Trotsky clearly knew how to
use ordinary language, it could be argued that
DL reveals that the
vernacular is in fact unable to depict change adequately. Indeed, the whole point of philosophical criticism is
to demonstrate that everyday notions (which are perfectly legitimate in their
own sphere of application) are incapable of reflecting fundamental
aspects of change or the fluid nature of reality. This has nothing
to do with understanding or with failing to understand anything. Or so the objection
might go.
However, this problem does not
just affect the vernacular; the same considerations apply to technical and
scientific language. They would be unusable unless it was
possible to specify the conditions under which their empirical propositions were
true or the conditions under which they were false (should there be any). [Why this is so will be examined in
Essay Twelve Part One. Those who
might be tempted to refer to Hegel's criticisms of the LEM should read
this, and then think
again.]
But, if identity statements cannot ever
be true -- not just as a matter of fact, but of logic --, it
would then 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 this were the case. That is because it would be impossible for anyone to
comprehend in what way such ordinary identity statements could fall short of
something (i.e., "absolute identity") that was never anywhere
instantiated in reality; hence, no one would have any idea what "approximate
identity" actually approximated to.
This is not easy to see, so some elaboration is necessary.
Trotsky described things 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 word "axiom"; indeed, Trotsky is merely
paraphrasing a bogus 'logical' argument of
Hegel's), how would anyone
be able to declare with any confidence
that approximate identity failed to match this ideal by so much or so little?
Worse still, if no one had any grasp at all of perfect identity, how
could
dialecticians be so sure that it is never instantiated in nature or society?
In such circumstances, exactly what is being ruled out?
It is no good, either, appealing to sentences
like S1a to answer queries like this since all such sentences are material objects
in their own right, and because of that are susceptible
to Trotsky 'lens argument'. On that view, no material copy of S1 or S1a would
identical to itself, or to any further copy of it, and no copy of S1 or S1a would stay
the same over time. In that case, S1a-type sentences cannot inform us exactly
what Trotsky and/or Hegel were ruling out. Clearly, in order to do so, their content
would have to remain the same, which is something S9(a/b) and S1a rule out.
So, we have as yet no idea what 'absolute', or even 'abstract' identity could
possibly be, since no material embodiment of it could fully express this
'concept', given this view.
S1: A is equal to
A.
S1a: A is both equal
to non-A and to non-non-A.
S9(a): All bodies change uninterruptedly.
(b) They are never equal to themselves.
[Here, in this
Hegelianesque
'proposition', the "non-non-A" refers to the negation of the negation of
A, as new content 'emerges'. The 'relative stability of words and meanings'
response will be considered presently. Also, see
Note 15.]
Again, that is because, if we now represent S1a
by the propositional letter "P", we are instantly faced with serious problems,
as S1a is applied to its own material embodiment -- the very material
sentence expressed by P:
S1b: P is equal to
non-P and to non-non-P.
And then (replacing each "P" with what it is
'dialectically' identical with) we would have:
S1c: Non-P is equal to
non-non-P and to non-non-non-non-P.
S1d: Non-non-P is equal to
non-non-non-P and to non-non-non-non-non-non-non-non-P.
And so on, ad infinitem.
And there is little point irritated DM-fans arguing that this is
ridiculous, since the Hegelian rule (expressed in S1a) implies the bowl of
logical spaghetti contained in S1d, etc. Hence, only the rejection of
S1a will stop this logical bindweed from propagating itself.
Worse still, such material embodiments cannot tell us what
'approximate identity' is even an approximation to.
And, if identity statements are never
absolutely true (in this material world), how could anyone learn what the
implications of accepting or rejecting sentences like S1 amounted to? Again, what exactly would they be
accepting or rejecting?
[Note the significance of the italicised word,
here. No irony is intended; just a reminder that we cannot evade 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'd have to use/accept the LOI in that very act of repudiation!]
Even to ask the question requires a use of
the dread LOI. If no one has a clue what absolute identity is (and on this
theory, just as soon as anyone claimed they did have a clue, that very idea
itself would change, if DL
were correct),
the question naturally arises: What precisely is being rejected in the
repudiation of sentences like S1?
If, on the other hand, the LOI encapsulates merely an empirical
truth -- even if only an approximate one -- we would still be owed an
explanation of what this empirical truth approximates to. And how could that be settled in
this world if we never encounter the absolute limit of identity -- and
worse: if we do not know what it is?
Questions like these apply equally well to
any rejection of propositions expressing identity (as false) -- or even
those that question its material application (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 this is only to point out how limited or "one-sided" it is.
But, if there is
no way of saying what would make an identity statement true, its denial must
lack a sense, for, once more, no one would have the slightest idea what was
being ruled out.
Hence, the earlier question about what Trotsky did or did not
understand was relevant after all.
It could be argued that it is possible to say what would
make an identity statement true; it's just that they are never absolutely true.
However, as we will soon see, in order for anyone to be able to say that they
are never true, the content of at least one identity statement would have to
remain exactly the same over the same length of time, otherwise, anyone who
criticised the absolute application of the LOI would not be using/appealing to
the same principle from day to day, year to year, century to century. In this,
it is quite easy to see several applications of the LOI -- and over many years, too.
Once more, a rejection of the LOI self-destructs, for if it is not possible for
at least one identity statement to remain exactly the same over a given length
of time, then that will apply to any sentence an objector uses, too. In that
case, what Hegel had to say about this 'law' must have changed (and hence we
cannot now access what he meant to say). If it hasn't, then what Hegel had to
say was false. [Abstract identity will be
discussed below.]
Again, it could be objected that identity
statements are in fact both true and false (this would seem to be the
implication of repudiating the LEM as it applies to material processes without
qualification -- and this seems to be what lies behind S1a); but it's entirely unclear what that
could mean in relation to the LOI. It
would either imply that material identity only
approximates to absolute identity (since no one would then have a clue what
absolute identity is, on
this view), or if it is false then material identity would still only approximate to
absolute identity, because, again on this view, identity propositions are mere approximations.
Either way, no identity statement would be absolutely true.
S1a: A is both equal
to non-A and to non-non-A.
[LEM = Law of Excluded Middle.]
But both of
these amount to the same criticism of the LOI (i.e., that it merely captures
approximate identity in this material world), and since we still have no idea
what ordinary identity approximates to, this objection is completely
empty. We still remain in the dark over exactly what is being ruled in or out (that is, we
still have no idea what it is that approximate identity approximates to) -- with or without an
appeal to the truth or falsity of the LEM.
So: even if the LOI were both true and
false, and the LEM was as unsafe to use (unrestrictedly) as we are told, what exactly would the ascriptions
contained in any material example of the LOI be both true and false of?
But, no answer to that can be given short of a clear grasp of absolute identity,
which, alas, no one is supposed to have.
In that case, this new set of DM-difficulties is independent of
whether the LEM is safe, unsafe, or both.
Again, it could be pointed out that we do
have a clear idea what absolute identity amounts to; it's just that it doesn't
apply to anything in reality. And here is what we associate with it:
S1: A is equal to
A.
We can surely have a clear idea of something even if it is never
true, if it is only partially true, or if it is never instantiated in reality. But, in that case, as pointed
out above, even this clear idea will have to remain the same from year to year,
generation to generation, and if that is true, then there is something in
reality that instantiates an absolute application of the LOI, namely this
clear idea. On the other hand, if this clear idea does not remain the
same over many centuries, then we do not have a clear idea of absolute
identity!
Once again, it could be objected that propositions about identity
could still be true and false, so that one option (i.e., the truth of
identity statements) need not necessarily mean that the other must be set in
opposition to it. Maybe so, but if we still have no idea what absolute identity
is then neither option is viable, whatever anyone does with it. In that case, if no
one has a clue what the subject of both this denial and this affirmation amounted
to,
if no one knew what the 'concept' of absolute identity is, then nothing true or
false (or both, or neither) could be said of it.
In order to see this, the reader should try to say something true or false
about meskonators (over and above a handful of trivialities,
perhaps relating to the spelling of
that word, say).
["Meskonator" is a meaningless word; it's a pure
invention.]
Plainly, if no one knows what a meskonator is, then no one could
possibly know if anything approximated to one, or what was absolutely identical to,
or totally different from, one.
Of course, with identity itself, the situation
is even worse, for the last sentence could not even be formed coherently if
"meskonator" is replaced by "identity", and no one knew what absolute identity
was. The latter term would then be implicated in its own lack of meaning, as
opposed to being implicated in that of another word.
To be sure, identity is not an object -- but who
says 'a' meskonator is? If we have no clear idea of the nature of either of
these, or
no idea at all, then we can say nothing coherent about them. Worse still,
the way that relational terms (such as "identical") are depicted in DM (i.e., as abstract objects
-- we saw this in Essay Three
Part One,
Essay Four and Essay Eight
Part Three), then
dialecticians are the last ones who can legitimately raise this particular objection.
However, the situation is even worse than this might suggest;
as hinted above, 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, and/or in the central nervous system) must alter, and thus become different. But in that case,
how could anyone to tell whether that idea had changed without being
able to compare that altered idea of absolute identity with another idea of
absolute identity that had not
changed? Without that comparison, no one would be able to say whether their idea
of absolute identity coincided
with absolute identity itself, fell short of it, or was totally different from it.
And, concentrating on words/concepts 'in the
mind'--, contemplated or not by 'speculative reason', a là Hegel --, would be to
no avail, either. If DM is correct then nothing remains the same: not words, not
thoughts (whether these are or are not about absolute identity, or even a wishy-washy
DM-approximation to it), not anything. Now, if that is true, then
not even the last sentence could be securely grasped, for it too contains a phrase we
have yet to comprehend: "absolute identity" -- since it would change just as soon as
it was either conceptualised or vocalised.19a
Furthermore, if words and/or concepts actually
stay the same (even if only temporarily), then Trotsky's (and Hegel's)
criticisms of the LOI would be seriously compromised. Hence, an appeal to the
'relative' stability of words and/or concepts would be of little assistance to
beleaguered DM-fans. If we have no idea what the original phrases "absolute
identity" and "approximate identity" relate to (i.e., if we have as yet no clue
how far short of, or how close to, the intended target they lie), then we would surely have
no way of knowing if and when such words remained stable --, i.e., "absolutely
identical", or even "approximately identical" with themselves over time. Hence,
if this view of the LOI is correct, 'relative stability' is itself parasitic on a concept we do not yet comprehend:
identity (in any of its forms).
And it's little use saying that minor changes in
words over short periods of time can be ignored, for on this view we would have
no way of knowing by how far or how little each word altered if we still haven't
a clue what absolute identity is so that we might compare such altered words
with an exemplar that did not change.
Some might want to argue that it's
reasonable to assume our words remain the same from moment to moment,
but,
on this view, since we do not yet understand the word "same", no one would know if
anyone else was employing it in the 'same' way as anyone else, or even in the
'same' way as we ourselves had used it a few seconds earlier! [Unless, of
course, we grant our brains or our memories with an exemption certificate. More
on this in the next section.]
Once identity is questioned, all such
comparisons/contrasts fall apart.
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).
'Abstract'
Versus 'Approximate' Identity And Relative Material Stability
In response, it could be argued 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
material reality. He only needed to appeal to
approximate identity in the first instance -- the sort of identity we meet
in everyday life (from which the LOI has been abstracted) -- to underline
the limitations of the Ideal/abstract version of LOI when it is confronted with
concrete reality and change. The same perhaps could be said of Hegel (no
irony intended).
[To save on needless repetition, whenever I refer to "absolute
identity" I also mean "abstract identity", and vice versa.]
To that end, therefore, the relative
apparent stability of material objects allowed Trotsky to refer coherently to such
things as the "same" instant, or the "same" object changing over time, and so
on (perhaps employing the IED
argument, once more). This certainly did not commit him to using the LOI in order to
criticise it. Trotsky was obviously arguing dialectically, accessing
everyday notions to show how they become contradictory when they are applied
beyond the usual boundaries of commonsense.
Or so the argument might go.
However, if sense is to be made of
approximate identity, some grasp must surely be had of absolute identity so
that a vague idea might be formed of what this watered-down version of the LOI
actually approximates to. If this is to be achieved by a retreat
into the abstract then we are no further forward.
Indeed, if there is a problem about material
identity in this world, there must surely be an even more intractable one
about abstract identity in an Ideal world. In the
absence of a clear account of this abstract notion of identity, we still
have no idea what ordinary identity is approximating to. But, how might
that be determined without another surreptitious appeal to the LOI?
Hence, we
would surely need to know by how much or how little approximate identity
was or was not absolutely identical to absolute identity 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
approximates to.
Consider this sentence:
A1: Approximate identity is approximately
identical with absolute identity.
Until we understand absolute identity, any approximation to it
will remain an
empty notion.
Moreover, and more concretely: Just how is the sense of
this abstract notion of identity fixed so that two or more references to it at
different times 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 today of absolute identity was or was not absolutely/approximately
identical with the one he had accessed only yesterday?
Consider further these sentences:
A2: Comrade NN means by "abstract identity"
exactly the same as comrade NM.
A3: Comrade MM means by "abstract identity"
approximately the same as comrade MN.
A4: Comrade NP means by "abstract identity" exactly the same as
she did yesterday.
A5: Comrade PN means by "abstract identity" approximately the
same as he did yesterday.
A6: Comrade PQ's memory of "abstract identity" is exactly the same as it was
yesterday.
A7: Comrade NQ's memory of "abstract identity" approximately the same as it
was yesterday.
How could anyone determine what the word "same"
meant in these contexts before they knew what the intended goal was
(i.e., identity itself)? For
all anybody knew, an
intentional target like this could be entirely different in
the minds of two different abstractors, or in the mind of even one. It's not as if either of them could
check inside each other's brain, or 'read' his or her thoughts to monitor their
precision. But, how do dialecticians themselves lock on to
identical ideas of abstract identity, those they supposedly share exactly with one
another (or with Hegel or with Trotsky or with Lenin...) -- even before they have determined what that
target is (and whether there is in fact one target, or many -- or if there is
indeed one at all)? Is it just luck? Or do they know something the rest of us
don't?
Moreover, if our memories are to access what we meant by "identity" only the day
before (never mind several years ago), how might we tell if we recalled this
word
the same? 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 did not 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".
And we cannot just assume this is possible; if no one knows what absolute
identity is, then assuming it means this or it means that would be about as useful as
assuming that a meskonator is a new brand of deodorant. Nor can
we appeal to
ordinary language for assistance. If this theory undermines the vernacular --
which we have seen it does -- then it can't possibly help rescue the very thing that caused the
problem.
To be sure, as language users we already know what terms for
identity and difference mean. 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 given earlier in this Essay of the use of these terms
in everyday contexts. However, in these new and rarefied 'dialectical' contexts, where change
and obscurity rule the day, ordinary terms fare rather badly. Even worse, jargonised
expressions survive not at all. In
fact, they commit
hara-kiri, as we have just seen.
As we also saw in Essay Three Parts
One and
Two, theorists who appeal to the existence of "abstract concepts"
to justify the "objectivity" of human knowledge find themselves in a serious
dilemma at this point. Either (1) they admit that such concepts already
exist, toward which knowledge advances or
approximates, or (2) they concede that it is we who construct such
notions out of experience by a mysterious process of 'abstraction' (that
still awaits explication).20
Option (2), it seems, underlies the proffered 'dialectical' response rehearsed above, while
(1) barely conceals its Platonic/mystical provenance. But, whichever horn of
this dilemma dialecticians grasp, neither is conducive to the dialectical analysis of
the LOI (Trotsky's version or Hegel's alternative).
There are at least two reasons for saying this:
(1) 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 hunted down. In that case, the LOI
must apply to it (since it does not change). But, no self-respecting DM-theorist
can possibly agree to such a Platonic view of
abstract concepts, even though their asymptotic approach metaphor suggests that
they should. Admittedly, this is a controversial allegation, but it
may only be neutralised when it becomes clear what the asymptotic approach metaphor itself implies --
more pointedly, when it becomes clear whether or not it means that as far as
individual dialectical truth-seekers are concerned, there is indeed a goal (identical
in the case of each
dialectical pilgrim), which they have targeted aright -- collectively or
severally -- toward which they are all slowly gravitating. Naturally, a positive
answer here would sink the dialectical analysis of the LOI,
since it would make plain that the asymptotic metaphor implies that there is
something unchanging called "absolute identity" that all dialectical
truth-seekers are zooming in on, and upon which all are in equal agreement.* A negative response,
on the other hand, would undermine DL equally
quickly: if there is no such goal then approximate identity must approximate to
nothing at all.**
(2) Alternatively, if human knowledge is
dialectically conditioned, and there are no abstract concepts that exist
independently of our knowledge of them, then there would be no objective way to
decide whether or not any two randomly selected dialecticians were aiming at the very same
target. Indeed, we would be hard-pressed to say what could count as
the same intentional goal in such circumstances (without using the LOI to
decide). But, if such
dialectical detectives haven't locked on to the very same target, then the
second of the above conclusions (i.e., point (1)** above) must surely apply. In
that case, their search is, to put it bluntly, aimless.
On the other hand, if there is a way of delving into the minds of any two
randomly selected dialectical abstractionists, which enables one and all to
decide whether they are in pursuit of an identical goal, the first of
the above points would plainly follow (i.e., (1)*) -- for it would then be
obvious
that such truth-seekers had used the LOI to identify exactly the same target,
and had done it with equal accuracy.
In fact, if anyone were to advocate or
reject either of the above options they would still have to appeal to the very
same 'law' in question in order to maintain that approximate identity was
more or less identical with abstract identity (whether or not "abstract identity"
was understood Platonically, or as a quasi-Hegelian/dialectical construct). 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, and
in real time, too.
If that wasn't so, this 'target' must surely be
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 can specify under what
conditions their notion of absolute or abstract identity does not change
over time -- but remains absolutely self-identical in the minds of
supporters and critics alike, over many centuries, for them to be able to
say with confidence that they are talking about the same thing --, any
reference to 'it' by critic and believer alike would be entirely empty.
Otherwise they should acknowledge their irresolvable differences, and cease
their pointless blather.
And even if the concept of 'abstract identity' were
to change (as it is apprehended by one or all), then in order to express this fact, some way must
be found to
say 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 it as just such a mutant.
Without that, of course, we would lose the right to say that absolute identity
has, or might
have, changed. Indeed, we would need it not to change in order to say that it had!
[Once more, this just underlines the intimate connection there is
in language between change and identity, contrary to what dialecticians tell
us.]
Moreover, an implicit
reference to the LOI would have to be made within each claim that any randomly
chosen dialectical bloodhound had a concept of 'abstract identity' which was identical with that of
any other, so that it could be said that they were referring to the same 'abstract concept' in
making the 'same' point even about "approximate identity" --,
even if
they were disagreeing!
But, if these assumed ideas of "abstract identity"
(or even of "approximate identity") were not exactly the same, then
agreement/disagreement over what they were talking about would be illusory, too. On
the other hand, of course, if their ideas were 'absolutely identical' then
their criticisms of the LOI would plainly self-destruct. [Hence, the reference to
hara-kiri, above.]
Furthermore, if the
supposed subject of enquiry were only 'roughly identical' (in the minds of the
many DM-fans sat round the dialectical table), not only would that fact be
untestable and
unverifiable, it would mean that the topic of discussion would be
indeterminate, too -- and for the same reason. Again, this would mean that any
and all criticisms levelled against the LOI would have been misdirected, for not only would no
one know exactly what "abstract identity" was so that it could be criticised
equally the same -- and by the use of identical arguments -- by those
who do not believe in the absolute validity of the LOI, no two critics of the
LOI would be able to say that they had the very same thing in mind when they
were even so much as pointing out its possible or actual limitations. Indeed, they could not even use the word
"same" with any clear meaning in this or any other context --, and, annoyingly,
this would be so for the same reason.
On the other hand, if it were now conceded
that any two notions of strict (or even approximate) identity were exactly the
same in the minds of any two intrepid dialectical abstractors, so that it
could be said of one or both that they were talking
about the very same thing, there would be no point in criticising the
LOI, for it would be correct -- and admitted to be correct -- at least here,
by its severest critics.
Worse still, if were denied
that anyone had an exact notion of strict identity (based on the claim that
everyone holds only an approximate version of it), we should still
want to know exactly what was being ruled out. In that case we
(they) would have to have an idea of strict identity to be able to deny they
(we) had any such idea!
Identity Schmidentity!
In order to underline this point, consider an analogy: let us
suppose that someone introduced a word into the language -- say "schmidentity"
-- but could not give an example of anything in reality that exhibited
"schmidentity". If we were then told that certain things were "approximately
schmidentical" (or even "schmidentical only within certain limits") we would
still have no clear idea of what this new word meant; if we do not know what
"schmidentity" is, we certainly do not know what "approximate schmidentity" is.
And calling this new 'concept' "abstract schmidentity",
"absolute schmidentity", or even "relative schmidentity" would be equally
futile.20a
In that case, when dialecticians presume to tell us that a word (or set of
words) in ordinary material language connected with sameness and identity, which
we all know how to use, does not mean what we usually take it to mean, then the
onus is on them to tell us what they do mean by their new word (or set of words). Until they do,
they might as well be talking about schmidentity.
21
And it's little point referring to Hegel's criticisms of the LOI; as I have
demonstrated here,
he badly misconstrued this 'law', compounding his folly with a series of crass
errors concerning the nature of propositions.
Indeed, for all DM-fans know, they could very well be talking about schmidentity -- or,
alternatively, about
nothing whatsoever.
For example, how do they know that their notion of
identity is not absolutely identical with schmidentity? Or, indeed with nothing?
The fact that I have not defined "schmidentity" is no objection. They have yet to
tell us what they mean by their use of words for identity. In fact, they
mis-identify this word right from the start, and they copied this exact
misidentification from Hegel! [Irony intended.]
In which case,
they probably are talking about nothing.
Nevertheless, there are other intractable
problems faced by the objection outlined at the beginning of
the last sub-section. What these are can be seen
if we consider the exact words Trotsky himself used to
criticise the LOI over 60 years ago (irony intended, once more).
Trotsky's Exact Words Now Dialectically Implode
The claim that our concepts are only
approximately true -- if true itself -- would undermine DM more
effectively than anything that has been written here.
In order to see this, let us introduce
the term "adequate" to describe the language belonging to any theory, but
understood in the following manner:
S24: A language is adequate to a theory
if, when expressed in that language, the empirical propositions of that theory
can be deemed true (by appropriate means).
However, if it is impossible to
develop a language adequate to a theory no matter what we do, then it
would surely be impossible to grasp that theory's content, or even determine
what the theory was about. In the case of identity and DM, this problem is
particularly acute.
With respect to the matter in hand, this
fatal defect can be highlighted by a consideration of Trotsky's own words:
"The
Aristotelian logic of the simple syllogism starts from the proposition that 'A'
is equal to 'A'…. In reality 'A' is not equal to 'A'. This is easy to prove if
we observe these two letters under a lens -– they are quite different to each
other. But one can object, the question is not the size or the form of the
letters, since they are only symbols for equal quantities, for instance, a pound
of sugar. The objection is beside the point; in reality a pound of sugar is
never equal to a pound of sugar -– a more delicate scale always discloses a
difference. Again one can object: but a pound of sugar is equal to itself.
Neither is true (sic) -– all bodies change uninterruptedly in size, weight,
colour etc. They are never equal to themselves." [Trotsky
(1971), pp.63-64.]
The question is: are the above words
identical to the ones Trotsky actually wrote? Now, even though I have
carefully copied the above quotation of Trotsky's words from my own copy of IDM, and
checked them against the versions found in TAR and RIRE (and now on-line), as no doubt others have done with their
copies --,
which copies themselves were copies of successive generations of further copies
of the originals --, despite all this, in one sense they aren't the very same words that
Trotsky committed to paper over 60 years ago: they are just copies! So, in that
sense they aren't exactly his words. Hence, if we count words as ink marks on
the page, they are manifestly not physically identical with the
originals. How could they be?
[IDM = In Defense of Marxism; i.e.,
Trotsky (1971); TAR = The Algebra of Revolution (i.e., Rees (1998); RIRE
= Reason in Revolt (i.e., Woods and Grant (1995); UO = Unity of
Opposites.]
On the other hand, in another
perfectly ordinary sense of the term, the words quoted above are
identical with the originals; they are Trotsky's words, and that is why his
followers constantly quote them. We have other criteria of identity for words
(and also for ink marks) that Trotsky had to rely on in order to present his case
against the LOI to
those interested in reading it, and upon which his followers also depend.
And I am now about to use his words against
him/them.
The fatal consequences mentioned above derive from the import of the last few sentences
above and from the
paradoxical conclusions that arise from rejecting what they say -- as it seems
DM-critics of the
LOI must do to remain consistent with their own precepts.
For example, any acknowledgement that the
above quoted words are identical with Trotsky's own would mean that
anyone reading his words now, and accepting his case against the LOI as
correct, must have (implicitly) employed that very 'law' in order to criticise
it! They would have to say (in effect) that in their copy, these words are
exactly the same as the ones Trotsky typed or penned all those years ago,
which words now undermine (an absolute application) of the 'law' they had just used to arrive at that conclusion!
Clearly, Trotsky and his epigones
failed to take into account this perfectly ordinary (hence material)
sense of identity: anyone who reads Trotsky's words today has before them
material objects (i.e., ink marks on the page/pixels on a screen) that are clearly identical with
countless other such objects separated from one another in space and time (i.e.,
other ink marks on paper/pixels on a screen that represent still other copies of the very same
article he wrote), by means of which an indefinite number of readers may access
the very same ideas that Trotsky intended they should, but which marks
are not numerically or materially identical to the ones he originally penned.
This is as clear a case of ordinary 'identity-in-difference' as one should wish
to find, but one that does not commit us to a belief in those terminally obscure DM-UOs
(in fact, here there are none). Unfortunately, this is an example of
'identity-in-difference' that completely undermines
Trotsky's criticism of the LOI.
In this case, countless manifestly (and
optically) different material objects -- separated in space and time --
are nonetheless uncontroversially identical. That fact could only be denied by those
who possess a defective copy of Trotsky's writings! Even better: the
admission of this fact does nothing to undermine anyone's belief in
change.
[Annoyingly, that belief itself
could remain the same even while agreement with Trotsky on the LOI changed as a
result of the above argument!]
Of course, anyone who disagreed with
the above line-of-thought would then perhaps be committed to the view that the
words in their copy of Trotsky's writings weren't identical with the
ones Trotsky authored all those years ago. Indeed, that idea might itself have
been prompted
by Trotsky's writings and the message they conveyed, which these very same
words seem to express: that nothing is identical with itself, or with anything else --
including those very words!
Someone could object***: the above considerations actually support Trotsky's case, for
here we have several objects that we ordinarily call identical, which are manifestly not
the same. So, our ordinary grasp of identity is not as secure as we might
think.
In response, it's worth pointing out that we count certain words,
phrases and symbols as identical even though they use different letters, or none
at all. For example, few of us would say most or all of the following were not
the same word/symbol, or didn't mean the same:
B1: Cat and CAT; Trotsky and TROTSKY; colour and color; maths and
math; In Defence of Marxism and In Defense of Marxism; Das Kapital
and Capital; vixen and female fox; one, eins, un, uno, een, en,
and egy; 10 minus 9, 2 divided by 2, and 1; 3 multiplied by 2 and 6; February 12th and 12th of February; Beta and β; dog and
chien; red and
red;
red and
red;
red and rot; bold and bold;
italic and italic; and and and; an empty space and
; arrow and →; 6 and upside down 9; half
full and half empty; non and uou (written upside down); and so on, and so on...
This is because our criteria of identity for words, symbols and
letters are not all the same (no irony intended). With respect to words and symbols,
a whole host of criteria apply; physical form is clearly not the only one (as
the above objection*** seems to assume). Hence, even though the physical form of
the words and symbols used could be different, we nonetheless recognise all or
most of them as identical, despite the fact that they look radically
different. B1 above contains a shortened list of these.
And it can't be the meaning of the above words in B1 (etc.) that makes them
all identical, although
this is clearly true of some. Hence, we would regard
"schmidentity" and "Schmidentity" as identical words even though they have no
meaning. That in turn cannot be because they both lack any meaning,
otherwise on that basis "schmidentity" and "meskonator" would have to be counted as the
same word!
Now, when meaning is introduced, the situation
becomes even more problematic for dialecticians.
In that case, the meaning of Trotsky's words today must be identical to
their meaning 60 years ago, even if their
physical form is not, otherwise it
would suggest that what Trotsky intended to convey (e.g., that nothing is
identical over time) itself meant that that very message was not the same as the
one he propagated 60 years ago. That is because his message implies that not even messages
are identical over time! If, therefore, a reader had
in fact accessed the identical message put out by Trotsky urging them to come to
the conclusion that even Trotsky's message must change over time, then they cannot have understood that message
at all well if they now
agree with a corrupted copy of it -- which original for all they know
might support an absolute, unqualified belief in the LOI!
In either case, such a critic would not be
able to ascertain exactly what Trotsky had written or meant, even
while they feigned assent to the exact import of what he said to arrive
at that sceptical conclusion, which
had just prompted those very doubts about messages corrupting over time.
Alarmingly, that must mean that such latter-day critics of the LOI
must have
access to Trotsky's precise thoughts by other means, over and above
the physical text they now concede cannot be exactly the same as that which
Trotsky had originally authored. This alternative route to Trotsky's thoughts must
therefore go beyond the confines of the physical document itself --,
since the latter (on this view) exists now in this corruptible world, and in a
corrupted form.
This alternative means
of communication could only be 'ethereal' or 'telepathic'. So, such Telepathic Trotskyists
and Ethereal Epigones seem to be able to intuit Trotsky's
exact meaning -- which now unfortunately prompts them to question any such
exactitude!
But, even that
would imply that while this 'ethereal' message (issuing somehow from Trotsky)
was
identical to the original transmitted through the ether all those years ago
by the master -- enabling these very doubts about identity to be accessed
exactly and with no loss of meaning over the years in this occult manner by contemporary recipients
-- the physical message wasn't!
Naturally, that would
still commit such individuals to the accuracy of LOI -- only now applied to
'occult' messages and ethereal identities.
An Attack On The LOI That Is Unequal To The Task
A moment's thought would confirm the fact that the
idea that our concepts are somehow inadequate to the tasks we set them cannot
be correct, even if the LOI were defective. That is because, if
our words are inadequate in some way then so were Trotsky's when he
criticized the LOI, and so are those of anyone who echoes such doubts. But,
that would mean that all such attacks on language would be
defective in virtue of that very assault. Indeed, any words that
expressed
even an abstract disquiet about the adequacy of ordinary words would be inadequate to that task, which would mean
that such inadequacies could never adequately be expressed.
Short of
saying nothing at all, this impasse will always block the thoughts of anyone who
thinks their words are permanently inadequate to any task we/they set them. Of course, that would
mean that scepticism about the adequacy of language (howsoever mildly expressed)
would either be disingenuous or self-refuting. [Those who doubt this
should read this,
and then think again.]
[And this includes anyone who tries to impugn
ordinary language, however so slightly -- as we will see in Essay Twelve. Some thoughts on this are
expressed in Note
19, below, others can be found in
Essay Three Part Two, and
at the link in the previous paragraph.]
Furthermore, if what Trotsky had
intended to say about the limitations of the LOI was in principle
impossible to express in any language, if even its physical embodiment was not
identical with his thoughts as they were taking physical form (when he
committed them to paper) -- because
of his claim that all objects (including words 'in the mind', or those on
paper) are "never equal to themselves", let alone anything else --, then Trotsky
himself could not have intended to mean anything by such words! That is because
there would have been nothing for him to have intended to have meant in such a
scenario; a faded simulacrum of his own thoughts on the LOI would have been
of no use even to Trotsky. Hence, even one nanosecond after being thought,
Trotsky's own words would be
non-self-identical. On paper, or in the mind, they would not be the same as those he had thought
seconds earlier, and he would thus have no access to his own precise intentions, for
the same reason -- unless, that is, we are supposed to exempt memory, claiming
that it is the only thing in the entire universe that
does not 'dialectically' change. Hence, his thoughts would be forever lost, as if they had
never existed.
Hence, if what Trotsky said were correct then
not even he could
have affirmed or confirmed whether or not he was criticising the same 'law' from
moment to moment -- without surreptitiously appealing to that 'law', or to an
identical 'mental' copy of it, while he was attempting to do so.
More generally: if it's
impossible to specify what it is that is being attacked (on the basis that
whatever is thought about 'it' is in principle not identical to what had
just been thought about 'it', which 'it' also has changed!), no intention to criticise 'it' can crystallise,
for there would be no such 'it' to denigrate.
Once again, an appeal to 'approximate identity' might
suggest itself here, but that would be to no avail. As was argued above,
the latter notion is parasitic on concepts of identity not
semantically-challenged in this Idealist manner. Hence, we would need some idea of what was
being approximated to if the notion of 'approximate identity' is capable of
doing any real work. But, ex hypothesi, this cannot be achieved without an
appeal being made to the validity of the LOI as a linguistic rule (and
not a metaphysical truth) -- and, moreover, to the same rule applied
repeatedly. [Irony intended.]
Trotsky (or Hegel -- or indeed anyone who
agrees exactly with either author), would be forced therefore to use such
linguistic criteria to formulate their self-refuting Idealist theories about a
'law' they seriously misidentified
to begin with!
Yet More Materially-Induced
Dialectical Misery
Unfortunately, these fatal
defects do not end there: anyone who consults Trotsky's words today, and who
agrees with his case against the LOI, owes us an explanation as to why on the
one hand the "A"s in S1:
S1: A is equal to
A
are
subject to the following objection:
"But in reality 'A' is
not equal to 'A'. This is easy to prove if we observe these two letters under a
lens -– they are quite different from each other." [Trotsky
(1971), p.63.]
when Trotsky's own words (written many years
ago) are not. Such a person would in effect be saying:
"I agree that the
(abstract version of the) LOI is defective because of what Trotsky's words say.
That is because the marks on paper that I am now
reading in IDM express exactly what Trotsky was thinking all those years
ago; they convey the very same message he intended. And yet, in at least
this respect the LOI must be correct for me to understand Trotsky, agree with
him and arrive at exactly the same conclusion he did, and which he intended
his readers should. That allows me now to disagree with the LOI that I have just
used in arriving at that identical result!
"On
the other hand, if my conclusions are only approximately the same as his, I must have some
grasp of his exact intentions so that I may truthfully say with what my own
opinions are
in fact approximating. And this I must know, for I assent to
the idea that all identity statements are approximations because I agree with
exactly what Trotsky says. Without his words I would still be under the
illusion that identity was absolute -- er..., which it must be if I have just used
it to get to this point...".
But, if the LOI is correct at least once,
and the same person arrives at exactly the same conclusion as Trotsky -- only
decades later -- then what Trotsky says in S9, for example, cannot be true:
S9(a): All bodies change uninterruptedly. (b) They
are never equal to themselves.
[And, of course, words are also "bodies".]
In that case, such a person could only
agree with Trotsky's criticism of the LOI by appealing (implicitly) to the
validity of the very same defective 'law' every time they accessed his words,
drawing identical conclusions from it, many times over, throughout their
no doubt very confused dialectical
lives. Naturally, this would mean that anyone agreeing with Trotsky, and who
derived the same conclusion, would only be able to do so on condition
that they then promptly disagreed with Trotsky in practice, implying that the exact opposite result was correct,
a result now not
identical with the one Trotsky had obtained!
This further implies that
Trotsky's claims are right only if they are not, and that if what he had
intended to say were true, it would then become impossible for anyone (including
Trotsky) to determine what it was he wanted to say, whether it was right or
wrong, or whether it had been committed to paper accurately, transposed
correctly between copies -- or even whether these questions were themselves
defective or not --, and for the same reasons.
Once more:
dialectical confusion has been exposed in actual practice.
This new (and ironic) dialectical inversion
(whereby a rejection of the LOI depends on its successful application in
practice by anyone trying to ascertain exactly what Trotsky or Hegel
thought they wanted to say)
just confirms how complex the conventions of material language really are
(to paraphrase Wittgenstein), and how it's not possible to criticise those
conventions without that attack itself falling apart for want of words with
which it might accomplished.
And we can now see why one of the maxims of this site is
the following: In this neck of the woods, the Owl of
Minerva flies only at its peril.
[On the Owl of Minerva, see
here.]
Clearly, this paradoxical result is a
consequence of the cavalier attitude to ordinary language adopted by dialecticians like Trotsky,
seriously compounded by a direct assault on the LOI. It
is not possible to criticise that 'law' in this manner -- that is, by treating it
as a putative truth.
Admittedly, traditional thinkers have always
regarded the LOI a deep metaphysical 'thesis' about everything in existence --
that is, as a 'necessary truth'. However, as will be argued in Essay Twelve
Part One, all
such metaphysical theses are non-sensical. If so, their denial is equally non-sensical.22 Little wonder, then, that Trotsky's
'analysis' collapsed into incoherence.
The weaknesses of the
LOI in fact lie elsewhere.23
The Anti-Dialectical Knock-Out
Punch
Finally, it's worth noting that the fact
that objects in the world undergo constant change cannot in general be used to
refute any of the above points since no matter how fast anything changes
whatever it is identical with will change equally quickly. In that case, the LOI is no enemy
of change.
With that observation, much
of 'materialist dialectics' falls apart.
[The details are spelt out in Note
5b, and in Essay Eight
Parts One and
Two.]
Traditional Versus Modern FL
There is a serious point at stake here. Traditional
AFL, criticised by Trotsky, not only ignored complex inferences
inexpressible in
syllogisms, it dramatically failed to cope with
relational
expressions,
quantifiers expressing
multiple generality, internal and external
negation and
scope ambiguity.
[This links to a PDF.] That was partly because of the way that quantifier
expressions themselves had been interpreted by earlier logicians --, who, with their slavish
adherence to the traditional grammar of
subject and predicate, helped
cripple logic for over two thousand years.
[On the origin of some of these confusions, see Barnes (2009).]
It's no exaggeration to say that
much of traditional Philosophy (i.e., Metaphysics) depends on antiquated logic
like this. In which case, two millennia of philosophical confusion --
including that found in Hegel -- largely derives from what is in effect
Stone Age Logic.
Now, many of the 'difficulties' outlined in the
last three Essays
(i.e., Essay Four,
Essay Five and this one) are a direct consequence of the crude way that quantifiers, relational
expressions and tense operators had been interpreted (or ignored) both by traditional
and dialectical
logicians. In fact, progress toward unravelling these pseudo-problems could only
begin after
Frege had completely re-laid the foundations of FL about 120
years ago. As noted earlier, this salient fact has still to
register with most dialecticians -- no matter how many times they are told.
At first sight, considerations like
these might appear to be dry, impractical and academic, but of no interest to
revolutionaries. However, if Marxists plan to use a radically flawed system (DL)
in their endeavour to help change the world then this is plainly not
irrelevant.24
Indeed, our astounding lack of success so far suggests that
DL has in fact been a millstone around
our necks.
History has so far delivered an
unambiguous verdict: DL has been tested in practice and
found wanting.
Is DL Superior To FL?
Essay Four began by asking which of these two
rival logics (DL or
FL) is the more adequate for use in science,
and which one could
most easily accommodate change, identity and motion. It's now quite clear (from
Essays Four to Seven) that DL is vastly inferior to FL in every single department; it
is incapable of handling even the simplest examples of change -- or even of describing them!
This is partly because it relies on a garbled version of AFL (compounded by a confused
metaphysic, which has itself been crippled by the impenetrable jargon invented by
Hegel), and partly because it unwisely attempts to undermine ordinary language.
In fact, DL is so limited it
cannot even cope with a simple bag of sugar
-- let alone "long drawn out processes".
Small wonder then that it has
so far hindered the scientific
development of Marxism.
FL
And A Fragmented View Of Reality
It might be felt at this stage that it is not surprising that the
views expressed here reach the conclusions they do since they depend on
analysis -– that is, they are based on a fragmentary view of reality, one that
splits the world and its contents into separate and un-mediated parts. Naturally, when divorced from the
whole, such things are going to appear paradoxical. Only against a wider
background is it possible to comprehend the world correctly. In broader
contexts, the contradictory nature of objects and processes is easier to see, as
indeed are the inadequacies of FL and the LOI.
This objection introduces the centrally
important DM-concept: the Totality. This terminally nebulous notion will be discussed
in detail in Essay Eleven Part One.
Notes
1.
As we will see, Hegel did at least mention identity in his
critique of the LOI, even though it's clear from what he said that not only had
he given this topic insufficient thought, he advanced superficial and erroneous
claims about it. [A fuller consideration of Hegel's 'analysis' of identity can be found in Essay Twelve
(summaries
here,
here
and
here), and in Essay Eight Part
Three.]
Hegel's alleged denial of the LOI and the
LOC are examined in
Pippin (1978) and Hanna (1986). Even though these two authors struggle
heroically to make Hegel comprehensible on this and other issues, it's difficult
to tell whether they have succeeded or not, or both.
It's also worth pointing out that the LOI was unknown to
Aristotle.
Unfortunately, since this Essay was originally published, the
article at the above link has been changed; the one I referenced above can be found
here.
On this, see also
here.
After another edit: The Wikipedia article
now
attributes this 'law' to Aristotle's Metaphysics, but it's quite
clear that this 'law' is not
stated there in its hackneyed form (i.e., A = A). Indeed, if anything, Aristotle
actually derides this 'law':
"Now 'why a thing is itself' is a meaningless
inquiry (for -- to give meaning to the question 'why' -- the fact or the
existence of the thing must already be evident -- e.g., that the moon is
eclipsed -- but the fact that a thing is itself is the single reason and the
single cause to be given in answer to all such questions as why the man is man,
or the musician musical, unless one were to answer 'because each thing is
inseparable from itself, and its being one just meant this' this, however, is
common to all things and is a short and easy way with the question)."
[Metaphysics
Book VII, Part 17.
This can be found in Aristotle (1984b), p.1643.]
Quoting this as an example of the use of, or
as an allusion to, this 'law' would be to distort what Aristotle says; so I have
added the following
comment
(reproduced here in blue and red)
to the Wikipedia article:
And the quotation takes this 'law' out of
context, for not only does Aristotle not mention 'identity', he specifically
talks about predication (and since identity is a relation, he cannot be talking
about identity here):
"Let us state what, 'i.e. what kind of thing, substance should be said to
be', taking once more another starting-point; for perhaps from this we shall get
a clear view also of that substance which exists apart from sensible substances.
Since, then, substance is a principle and a cause, let us pursue it from this
starting-point. The 'why' is always sought in this form -- 'why does one thing
attach to some other?' For to inquire why the musical man is a
musical man, is either to inquire -- as we have said why the man is musical, or
it is something else. Now 'why a thing is itself' is a meaningless inquiry (for
-- to give meaning to the question 'why') the fact or the existence of the thing
must already be evident -- e.g. that the moon is eclipsed -- but the fact that a
thing is itself is the single reason and the single cause to be given in answer
to all such questions as why the man is man, or the musician musical, unless
one were to answer 'because each thing is inseparable from itself, and its being
one just meant this'; this, however, is common to all things and is a short and
easy way with the question). But we can inquire why man is an animal of such and
such a nature. This, then, is plain, that we are not inquiring why he who is a
man is a man. We are inquiring, then, 'why something is predicable of
something' (that it is predicable must be clear; for if not, the
inquiry is an inquiry into nothing). E.g. why does it thunder? This is
the same as 'why is sound produced in the clouds?' Thus the inquiry is about the
predication of one thing of another. And why are these things, i.e. bricks
and stones, a house? Plainly we are seeking the cause. And this is the essence
(to speak abstractly), which in some cases is the end, e.g. perhaps in the case
of a house or a bed, and in some cases is the first mover; for this also is a
cause. But while the efficient cause is sought in the case of genesis and
destruction, the final cause is sought in the case of being also." [Ibid., bold
emphasis added.]
So, I think the article needs amending.
Finally, since this 'law' is foreign to Aristotle, how can the author of this
article say:
"The law of identity has deep impact on
Aristotle's ethics as well. In order for a person to be morally praiseworthy
or blameworthy for an action, he or she must be the same person before the act
as during the act and after the act. Without the law of identity, Aristotle
notes, there can be no responsibility for vice."
Personal identity is not the same as the 'law of identity'.
However, I could not find in the Nicomachean Ethics
anything like this reference to personal identity; so perhaps the author of this
article will provide an exact quotation?
In fact the Wikipedia article is highly
misleading. For example, it attributes this phrase to Aristotle, which can't be
found in Metaphysics VII Part 17: "a fixed constant
nature of sensible things", and as such misrepresents what he was saying.
August 2011: I have just read Deborah
Modrak's book on Aristotle (i.e., Modrak (2001)); she devotes an entire section
to Aristotle's views on identity -- pp.194-98. However, Modrak concentrates on
Aristotle's views on sameness; identity is conspicuous by its absence.
Certainly, there is no mention of the LOI.
Nevertheless, the
defects of the LOI lie elsewhere; these
are outlined in Wittgenstein (1972), pp.97, 105-07, and Wittgenstein (1958), pp.84-85,
91, 111. [Cf., Glock (1996), pp.164-69.]
The best analysis of
Wittgenstein's
criticisms of identity can be found in White (1978). See also Marion (1998),
pp.48-72 for an extended discussion. On identity in general, see Geach (1967,
1970, 1973, 1975, 1990), Griffin (1977), Noonan (1980, 1997,
2009) and Williams
(1979, 1989, 1992). See also
Deutsch (2007).
Here are a few examples of the extremely
repetitive nature of this part of dialectics:
"[T]he first of [the
universal Laws of Thought], the maxim of Identity, reads: Everything is
identical with itself, A = A…." [Hegel (1975),
p.167, §115.]
"In this remark, I will
consider in more detail identity as the law of identity which is usually adduced
as the first law of thought.
"This proposition in its positive expression
A = A is, in the first instance, nothing more than the expression of an empty
tautology." [Hegel (1999),
p.413,
§875.]
"Abstract Identity (a = a…) is
likewise inapplicable in organic nature. The plant, the animal, every cell is at
every moment of its life identical with itself and yet becoming distinct from
itself….The law of identity in the old metaphysical sense is the
fundamental law of the old outlook: a = a." [Engels (1954),
pp.214-15.]
"The 'fundamental laws of
thinking' are considered to be three in number: 1) The Law of Identity… [which]
states that 'A is A' or A = A…." [Plekhanov (1908),
p.89.]
"…Hegel elucidates the
one-sidedness, the incorrectness of the 'law of identity' (A = A)…." [Lenin
(1961),
p.134.]
"Formal Logic starts from the
proposition that A is always equal to A. We know that this law of identity
contains some measure of truth…. Now…when we go to reality and look for evidence
of the truth of the proposition: A equals A…we find that the opposite of this
axiom is far closer to the truth." [Novack (1971), pp.32-33.]
"Formal Logic asserts:
'A is A'.
Dialectical Logic is not saying 'A is not-A'…. It says: A is indeed A, but A is
also not-A precisely so far as the proposition 'A is A' is not a tautology but
has real content." [Lefebvre (1968), p.41.]
"The Law of identity is
usually expressed in the form, A is A. That is, each thing is identical with
itself." [Somerville (1946), p.183.]
"The Aristotelian conception
of the laws basic to correct thinking may be stated as follows: 1. Law of
Identity: Each existence is identical with itself. A is A…." [Somerville (1967),
pp.44-45.]
"Classical, Aristotelian
logic takes as its fundamental premise the Law of Identity, the statement that a
thing is identical with itself. Expressed in a formula: A is A…. In Aristotle's
formal logic A is A, and never non-A. In Hegel's dialectics A is A as well as
non-A." [Baghavan (1987), pp.75-76.]
"The biggest contradiction of
all lies in the fundamental premises of formal logic itself…. The basic
laws…are:
1) The law of Identity ('A' =
'A')…." [Woods
and Grant (1995),
pp.90-91.]
"Dialectics, or the logic of motion, is distinct from formal or static logic.
Formal logic is based on three fundamental laws:
"(a) The
law of identity: A is equal to A; a thing is always equal to itself." [Mandel
(1979), p.160.]
"The laws
of logic are based on two main propositions. The first is that of identity or of
self-conformity. The proposition very simply states: 'A is A,' that is, every
concept is equal to itself. A man is a man; a hen is a hen; a potato is a
potato. This proposition forms one basis of logic. The second main proposition
is the law of contradiction, or as it is also called, the law of the excluded
middle. This proposition states: 'A is either A or not A.' It cannot be
both at the same time. For example: Whatever is black is black; it cannot at the
same time be black and white. A thing -- to put it in general terms -- cannot at
the same time be itself and its opposite. In practice it therefore follows that
if I draw certain conclusions from a given starting point and contradictions
arise, then there are errors in thinking or my starting point was wrong. If from
some correct premises I come to the conclusion that 4 is the same as 5, then I
deduce from the law of contradiction that my conclusion is false.
"So far
all appears to be clear and certain. What can be a clearer law than that man is
man, a rooster a rooster, that a thing is always the same thing? It even appears
to be absolutely certain that a thing is either large or small; either black or
white, that it cannot be both at the same time, that contradictions cannot exist
in one and the same thing.
"Let us
now consider the matter from the standpoint of a higher doctrine of thought,
from the standpoint of dialectics. Let us take the first law which we have
developed as the foundation of logic: A is A. A thing is always the same thing.
Without testing this law, let us consider another one which we have already
mentioned, the law of Heraclitus which says 'Everything is in flux,' or 'One
cannot ascend the same river twice.' Can we say that the river is always the
same? No, the law of Heraclitus says the opposite. The river is at no moment the
same. It is always changing. Thus one cannot twice nor, more exactly, even once
ascend the same river. In short: the law 'A is A' in the last analysis is valid
only if I assume that the thing does not change. As soon as I consider the thing
in its change, then A is always A and something else; A is at the same time
not-A. And this in the last analysis holds for all things and events."
[Thalheimer (1936),
p.88-89.]
"The central principle on
which formal logic is built can be expressed in a simple formula that at first
glance appears to be a self-evident truth 'A equals A'." [Conner (1992), p.22.]
Examples like these can be multiplied almost
indefinitely. Even though there are several minor differences in emphasis
between them, the basic point of the above comments is reasonably clear:
DM-theorists have fixated on a superficial
form of the LOI, one they copy from each other generation after generation. Seldom do they bother to check that what they are criticising
even remotely resembles
anything taken from a logic text written in the last 120 years, or even one
written by Aristotle!
In at least this respect, DM-authors are
(ironically) identical. And, as we will see, in response
to Hegel (here),
identity statements are not tautologies. And, as we will see in this
Essay, the LOI does not preclude change.
Essay
Four showed in
detail that this tactic is part of a long and sorry tradition among 'Materialist
Dialecticians': Define the basics of
logic in a completely fanciful way, ridicule them, and then advertise the
superiority of DL over this sub-straw man.
Here again is Novack's attempt:
"There are three fundamental
laws of formal logic. First and most important is the law of identity. This law
can be stated in various ways such as: A thing is always equal to or identical
with itself. In algebraic terms: A is equal to A.
"…If a thing is always and
under all conditions equal or identical with itself, it can never be unequal or
different from itself. This conclusion follows logically and inevitably from the
law of identity. If A always equals A, it can never equal non-A." [Novack
(1971), p.20.]
Clearly Novack failed to consider these
counter-examples to his "logical" conclusion:
N1: The number of volumes of Das
Kapital is equal to the number of goals in a hat-trick.
N2: There were equal numbers of Union and
non-Union members at the meeting last night.
N3: Although NN and MM have different
disabilities they came equal first in the 100 metres final at the Para-Olympics,
sharing the Gold Medal.
N4: Those two comrades sold equal
numbers of papers on two different demonstrations last week.
N5: The author of Novack
(1971) is identical to the comrade who penned the words in the last quotation
above.
None of these suggests that the items they
allude to can never change -- but, when they do change, anything identical to them
will change equally quickly.
Or, of course, it will cease to be identical with whatever it
used to be!
Apart from Woods and Grant (1995), none of
the above theorists refers his readers to a single logic text (save those
written by Hegel and other 'dialectical logicians'); worse, not one of them
bothers to quote Aristotle even though many attribute this idea to him!
However,
several of the above dialecticians at least mention the word "identity", but
they then confuse it
with equality. Hence most of the criticisms levelled against Trotsky
in this Essay
apply equally
to them (no pun intended) --, except that, in this area,
some of the former appear to be
slightly less benighted than Trotsky.
But, Woods and Grant go further:
"Firstly, let us note that the appearance of a
necessary chain of reasoning, in which one step follows from another, is
entirely illusory. The law of contradiction merely restates the law of identity
in a negative form. The same is true of the law of the excluded middle. All we
have is a repetition of the first line in different ways. The whole thing stands
or falls on the basis of the law of identity ('A'='A'). At first sight this is
incontrovertible, and, indeed, the source of all rational thought. It is the
Holy of Holies of Logic, and not to be called into question. Yet called into
question it was, and by one of the greatest minds of all time (sic).
"There is a story by
Hans-Christian Andersen
called The Emperor's New Suit of Clothes, in which a rather foolish emperor is
sold a new suit by a swindler, which is supposed to be very beautiful, but
invisible. The gullible emperor goes about in his fine new suit, which everyone
agrees is exquisite, until one day a little boy points out that the emperor is,
in fact, stark naked. Hegel performed a comparable service to philosophy in his
critique of formal logic. Its defenders have never forgiven him for it." [Woods
and Grant (1995), p.91. Quotation marks altered to conform to the
conventions adopted at this site.]
This is typical hyperbole from Woods and Grant, who seem to think
that modern logicians will be in any way bothered by the confusions that litter Hegel's
badly
misnamed book (on 'logic'). Indeed, the vast majority of them pay no more attention to
Hegel, or Woods and Grant, than
Woods and Grant themselves pay to the work of, say,
Frédéric
Bastiat.
Furthermore, based on Woods and Grant's execrable book, they are not likely to change
(irony intended) -- even if they could be bothered to read it.
Moreover, as we will see
(here), the LOC
and the LOI are not connected in the way that Woods and Grant say they are; these
two have simply copied this error from Hegel and Engels, without bothering to check whether
the one implies the other.
And, as we saw in Essay
Four,
neither AFL nor MFL is based on the LOI (although there are systems of
logic that
are).
In their haste to blame FL for
everything but the
Black Death, Woods and Grant failed to
acknowledge this, always assuming they were aware of it.
[LOC = Law of Non-Contradiction; AFL =
Aristotelian Formal Logic; MFL = Modern Formal Logic.]
2.
There is very strong evidence that Trotsky read and studied Hegel's Logic
(see the Introduction to Trotsky (1986), and the reference to the Logic
in Trotsky (1986) itself -- e.g., pp.98),
so this observation is not entirely correct. Nevertheless, even if questions about
the accuracy of the title of Hegel's book on 'logic' are put to one side for now, it's
reasonably clear that apart from the Hegel's Logic Trotsky seems not
to have consulted a single logic text before he began issuing
ex cathedra pronouncements about it.
[Having said that,
Isaac
Deutscher tells us that in the early 1930s, in preparation for a book he
intended to write on Lenin, Trotsky "went back to classics of logic and
dialectics, Aristotle and Descartes, but especially to Hegel" (Deutscher (1970),
p.267). If so, Trotsky might have studied Aristotle's logical texts, but if he
did it is plain that view of Aristotle was heavily skewed by Hegel's
misrepresentations.
It is, however, possible that
Jean
van Heijenoort, a member of Trotsky's entourage, and later an expert
logician, gave him some advice -- but if he did, there is precious little
evidence that any of it sank in. On this, see Van Heijenoort (1978), and Feferman
(1993).]
In so doing, Trotsky was not alone; DM-theorists in general are only too happy to
regale us with their home-spun ideas about FL -- fables whose pristine
simplicity has not been complicated by the arduous task of opening a
single book on MFL, and then attempting to master some of what it contains.
Finally, those who think that AFL
is based on the LOC (etc.) should consult Lear (1980), pp.98-114, where they
will find a more balanced and scholarly account.
2a. Although I have
quoted part of this above, there is a passage in Engels's
DN which closely
resembles what Trotsky was saying:
"Abstract
identity (a = a; and negatively, a cannot be
simultaneously equal and unequal to a) is likewise inapplicable in
organic nature. The plant, the animal, every cell is at every moment of its
life identical with itself and yet becoming distinct from itself, by
absorption and excretion of substances, by respiration, by cell formation and
death of cells, by the process of circulation taking place, in short, by a sum
of incessant molecular changes which make up life and the sum-total of whose
results is evident to our eyes in the phases of life -- embryonic life, youth,
sexual maturity, process of reproduction, old age, death. The further physiology
develops, the more important for it become these incessant, infinitely small
changes, and hence the more important for it also the consideration of
difference within identity, and the old abstract standpoint of formal
identity, that an organic being is to be treated as something simply identical
with itself, as something constant, becomes out of date.
[In the margin of the manuscript occurs the remark: 'Apart, moreover, from the
evolution of species.'] Nevertheless, the mode of thought based thereon,
together with its categories, persists. But even in inorganic nature identity as
such is in reality non-existent. Every body is continually exposed to
mechanical, physical, and chemical influences, which are always changing it and
modifying its identity. Abstract identity, with its opposition to
difference, is in place only in mathematics -- an abstract science which is
concerned with creations of thought, even though they are reflections of reality
-- and even there it is continually being sublated. Hegel, Enzyklopädie,
I, p. 235. [This is a reference to Hegel (1975),
pp.169-70, §117;
see below -- RL.] The fact that identity contains difference within itself is
expressed in every sentence, where the predicate is necessarily
different from the subject; the lily is a plant, the rose, is red,
where, either in the subject or in the predicate, there is something that
is not covered by the predicate or the subject. Hegel, p. 231. [This is a
reference to Hegel (1975),
pp.166-68, §115;
see below -- RL.] That from the outset identity with itself requires
difference from everything else as its complement, is self-evident.
"Continual
change, i.e., sublation of abstract identity with itself, is also found in
so-called inorganic nature. Geology is its history. On the surface, mechanical
changes (denudation, frost), chemical changes (weathering); internally,
mechanical changes (pressure), heat (volcanic), chemical (water, acids, binding
substances); on a large scale – upheavals, earthquakes, etc. The slate of today
is fundamentally different from the ooze from which it is formed, the chalk from
the loose microscopic shells that compose it, even more so limestone, which
indeed according to some is of purely organic origin, and sandstone from the
loose sea sand, which again is derived from disintegrated granite, etc., not to
speak of coal.
"The
law of identity in the old metaphysical sense is the fundamental law of the
old outlook: a = a. Each thing is equal to itself. Everything was
permanent, the solar system, stars, organisms. This law has been refuted by
natural science bit by bit in each separate case, but theoretically it still
prevails and is still put forward by the supporters of the old in opposition to
the new: a thing cannot simultaneously be itself and something else. And
yet the fact that true, concrete identity includes difference, change, has
recently been shown in detail by natural science (see above).
"Abstract
identity, like all metaphysical categories, suffices for everyday use,
where small dimensions or brief periods of time are in question; the limits
within which it is usable differ in almost every case and are determined by the
nature of the object; for a planetary system, where in ordinary astronomical
calculation the ellipse can be taken as the basic form for practical purposes
without error, they are much wider than for an insect that completes its
metamorphosis in a few weeks. (Give other examples, e.g., alteration of species,
which is reckoned in periods of thousands of years.) For natural science in its
comprehensive role, however, even in each single branch, abstract identity is
totally inadequate, and although on the whole it has now been abolished in
practice, theoretically it still dominates people’s minds, and most natural
scientists imagine that identity and difference are irreconcilable opposites,
instead of one-sided poles which represent the truth only in their reciprocal
action, in the inclusion of difference within identity." [Engels (1954),
pp.214-16. Italic emphases in the original; bold emphases added.
Quotation marks altered to conform to the conventions adopted here.]
Which merely confirms the allegation made above: that
DM-theorists uncritically copy ideas off one another.
Here are the relevant sections of Hegel's Shorter Logic:
"Difference is first of all (1) immediate difference, i.e.
Diversity
or Variety. In Diversity the different things are each individually what they
are, and unaffected by the relation in which they stand to each other. This
relation is therefore external to them. In consequence of the various things
being thus indifferent to the difference between them, it falls outside them
into a third thing, the agent of Comparison. This external difference, as an
identity of the objects related, is Likeness; as a non-identity of them, is
Unlikeness.
"The gap
which understanding allows to divide these characteristics is so great that
although comparison has one and the same substratum for likeness and unlikeness,
which are explained to be different aspects and points of view in it, still
likeness by itself is the first of the elements alone, viz., identity, and
unlikeness by itself is difference.
"Diversity
has, like Identity, been transformed into a maxim: 'Everything is various or
different': or 'There are no two things completely like each other'. Here
Everything is put under a predicate, which is the reverse of the identity
attributed to it in the first maxim: and therefore under a law contradicting the
first. However, there is an explanation. As the diversity is supposed due only
to external circumstances, anything taken per se is expected and
understood always to be identical with itself, so that the second law need not
interfere with the first. But, in that case, variety does not belong to the
something or everything in question: it constitutes no intrinsic characteristic
of the subject: and the second maxim on this showing does not admit of being
stated at all. If, on the other hand, the something itself is, as the maxim
says, diverse, it must be in virtue of its own proper character: but in this
case the specific difference, and not variety as such, is what is intended. And
this is the meaning of the maxim of
Leibnitz (sic).
"When understanding sets itself to study Identity, it has
already passed beyond it, and is looking at Difference in the shape of bare
Variety. If we follow the so-called law of Identity, and say, The sea is the
sea, The air is the air, The moon is the moon, these objects pass for having no
bearing on one another. What we have before us therefore is not Identity, but
Difference. We do not stop at this point, however, or regard things merely as
different. We compare them one with another, and then discover the features of
likeness and unlikeness. The work of the finite sciences lies to a great extent
in the application of these categories, and the phrase 'scientific treatment'
generally means no more than the method which has for its aim comparison of the
objects under examination. This method has undoubtedly led to some important
results; we may particularly mention the great advance of modern times in the
provinces of comparative anatomy and comparative linguistics. But it is going
too far to suppose that the comparative method can be employed with equal
success in all branches of knowledge. Not -- and this must be emphasised -- can
mere comparison ever ultimately satisfy the requirements of science. Its results
are indeed indispensable, but they are still labours only preliminary to truly
intelligent cognition.
"If it be the office of comparison to reduce existing
differences to Identity, the science which most perfectly fulfils that end is
mathematics. The reason of that is that quantitative difference is only the
difference which is quite external. Thus, in geometry, a triangle and a
quadrangle, figures qualitatively different, have this qualitative difference
discounted by abstraction, and are equalised to one another in magnitude. It
follows from what has been said formerly about mere Identity of understanding
that, as has also been pointed out (s.99),
neither philosophy nor the empirical sciences need envy this superiority of
Mathematics.
"The story is told that when Leibnitz (sic) propounded the
maxim of Variety, the cavaliers and ladies of the court, as they walked round
the garden, made efforts to discover two leaves indistinguishable from each
other, in order to confute the law stated by the philosopher. Their device was
unquestionably a convenient method of dealing with metaphysics -- one which has
not ceased to be fashionable. All the same, as regards the principle of Leibnitz
(sic), difference must be understood to mean not an external and indifferent
diversity merely, but difference essential. Hence the very nature of things
implies that they must be different." [Hegel (1975),
pp.169-70, §117.]
"The
Essence lights up in itself or is mere reflection: and therefore is only
self-relation, not as immediate but as reflected. And that reflex relation is
self-identity.
"This
identity becomes an Identity, in form only, or of the understanding, if it be
held hard and fast, quite aloof from difference. Or, rather, abstraction is the
imposition of this Identity of form, the transformation of something inherently
concrete into this form of elementary simplicity. And this may be done in two
ways. Either we may neglect a part of the multiple features which are found in
the concrete thing (by what is called analysis) and select only one of them; or,
neglecting their variety, we may concentrate the multiple character into one.
"If we
associate Identity with the Absolute, making the Absolute the subject of a
proposition, we get: The Absolute is what is identical with itself. However,
true this proposition may be, it is doubtful whether it be meant in its truth:
and therefore it is at least imperfect in the expression. For it is left
undecided, whether it means the abstract Identity of understanding- abstract.
that is, because contrasted with the other characteristics of Essence -- or the
Identity which is inherently concrete. In the latter case, as will be seen, true
identity is first discoverable in the Ground, and, with a higher truth, in the
Notion. Even the word Absolute is often used to mean more than 'abstract'.
Absolute space and absolute time, for example, is another way of saying abstract
space and abstract time.
"When the
principles of Essence are taken as essential principles of thought they become
predicates of a presupposed subject, which, because they are essential, is
'everything'. The propositions thus arising have been stated as universal Laws
of Thought. Thus the first of them, the maxim of Identity, reads: Everything
is identical with itself, A = A: and negatively, A cannot at the same time be A
and Not-A. This maxim, instead of being a true law of thought, is nothing
but the law of abstract understanding. The propositional form itself contradicts
it: for a proposition always promises a distinction between subject and
predicate; while the present one does not fulfil what its form requires. But the
Law is particularly set aside by the following so-called Laws of Thought, which
make laws out of its opposite. It is asserted that the maxim of Identity, though
it cannot be proved, regulates the procedure of every consciousness, and that
experience shows it to be accepted as soon as its terms are apprehended. To this
alleged experience of the logic books may be opposed the universal experience
that no mind thinks or forms conceptions or speaks in accordance with this law,
and that no existence of any kind whatever conforms to it.
"Utterances after the fashion of this pretended law (A planet is a planet;
Magnetism is magnetism; Mind is Mind) are, as they deserve to be, reputed silly.
That is certainly a matter of general experience. The logic which seriously
propounds such laws and the scholastic world in which alone they are valid have
long been discredited with practical common sense as well as with the philosophy
of reason.
"Identity is, in the first place, the repetition of what
we had earlier as Being, but as become, through supersession of its
character of immediateness. It is therefore Being as Ideality. It is important
to come to a proper understanding on the true meaning of Identity; and, for that
purpose, we must especially guard against taking it as abstract identity, to the
exclusion of all Difference. That is the touchstone for distinguishing all bad
philosophy from what alone deserves the name of philosophy. Identity in its
truth, as an Ideality of what immediately is, is a high category for our
religious modes of mind as well as all other forms of thought and mental
activity. The true knowledge of God, it may be said, begins when we know him as
identity -- as absolute identity. To know so much is to see all the power and
glory of the world sinks into nothing in God's presence, and subsists only as
the reflection of his power and his glory. In the same way, Identity, as
self-consciousness, is what distinguishes man from nature, particularly from the
brutes which never reach the point of comprehending themselves as 'I'; that is,
pure self-contained unity. So again, in connection with thought, the main thing
is not to confuse the true Identity, which contains Being and its
characteristics ideally transfigured in it, with an abstract Identity, identity
of bare form. All the charges of narrowness, hardness, meaninglessness, which
are so often directed against thought from the quarter of feeling and immediate
perception rest on the perverse assumption that thought acts only as a faculty
of abstract Identification.
"The Formal Logic itself confirms this assumption by
laying down the supreme law of thought (so-called) which has been discussed
above. If thinking were no more than an abstract Identity, we could not but own
it to be a most futile and tedious business. No doubt the notion, and the idea
too, are identical with themselves: but identical only in so far as they at the
same time involve distinction." [Hegel (1975),
pp.166-68, §115.
Bold emphasis added.]
Hegel's rather weak attempt to 'analyse' the LOI is examined in
detail in Essay Eight Part Three,
and Essay Twelve Parts Five and Six.
3.
That is, if they are aware of
them! on this, see Note
8, below.
4.
In fact, Trotsky's version was:
S1(a): 'A' is equal to 'A'.
However, since not much seems to hang on
Trotsky's use of single quotation marks (over and above his rather odd reference
to the microscopic examination of the letters in question), I have ignored them
in what follows.
5.
Change in, or to, NN is irrelevant here; that is because, howsoever much her two
roles altered, since NN occupies both at the same time it will always be true that "The Unison rep is identical with the
STWC Treasurer". The same
point applies
mutatis
mutandis
to the other examples listed in the main body of
this Essay. See also Note 5b,
below.
5a. A substantival term is a common
noun that in general (but not always) admits of number, e.g., three books, two
people, five comrades. Where they do admit of number, they are often
called 'count nouns'. The distinguishing mark of such terms centres on
the
criteria of identity we use in each case.
This is what
Professor Lowe had to say:
"...[N]ot
all general terms are common names -- for instance, adjectival or characterizing
general terms such as 'red' and 'circular' are not, nor are abstract nouns such
as 'redness' and 'circularity' (if indeed the latter are deemed to be general
terms, for an alternative view is that they are singular terms referring to
abstract individuals). The distinguishing feature of common names -- sometimes
also called substantival or sortal general terms -- is that they
have associated with them, as a component of their meaning, a criterion of
identity for the individuals to which they apply (see Lowe 1989, Ch. 2). A
criterion of identity for individuals of a kind K is a principle which
determines, for any individuals x and y of kind K, whether
or not x and y are one and the same K. Thus, the criterion
of identity for cities tells us that Paris and London are different
cities, since they occupy different locations; and the criterion of identity for
rivers tells us that
the Isis and the Thames are the same river, since
they flow from the same source to the same mouth. Different kinds of
individuals, denoted by different sortal terms, very often have different
criteria of identity governing them -- and in some cases there is philosophical
debate as to precisely what these criteria are (for example, in the case of
persons). Credit is once more due to
Frege for recognizing the important
role that criteria of identity have to play in the semantics of sortal terms."
[Lowe, internet resource [2].]
Count nouns are to be distinguished from
other common nouns which do not admit of number, e.g.
mass nouns (such as chalk,
cabbage, meat, etc. -- it makes no sense to refer to "two meat", or "three
chalk"; of course, these can be converted into count nouns, as in "three chalks"
or "two meats" (which are in general short for "three pieces/sticks of chalk"
and "two examples/portions/slices/cuts of meat). However, some mass nouns are also substantivals, e.g., gold,
lead, and plastic.
Many of the mistakes dialecticians make over
identity originate from their failure to notice the different logic that applies
to these two sorts of nouns. [The same lack of attention to detail vitiates, for
example, Heraclitus's
comments about stepping into a river. I will say more about this in Essay
Twelve.]
On this in general, see
here and
here.
However, as with most things in philosophical
logic, things are never quite so simple. On this, see Geach (1970), pp.39-41.
5b. It could be argued that these examples
of identity are not in fact examples of strict identity, since all the
items listed will change in small ways, as will their relation to countless other
local (and distant) 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, above, but will be completely laid to rest below (here, here,
and here), and in the closing part
of this Essay.
However, it's worth pointing out that
the examples given in the main body of this Essay are merely being used to show
that abstract and material identity are not the same as abstract and material
equality, and that ordinary material language (but not the obscure jargon
philosophers employ) is our best guide to what we mean by
identity, sameness, equality and difference.
[The primacy of ordinary language is
taken for granted here; 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 Part One
and Part Two,
no more will be said about them here. However, on this, see
here.
Nevertheless, a few things need to be said
about the doctrine of universal change.
Naturally, it would be perverse to deny that
things change; not only is this given in ordinary language and common
understanding (a highly truncated list of ordinary words for change can be found
here), it's a familiar feature of everyday life
and highly
confirmed by science.
However, even if the evidence we now have were to be
multiplied by several million orders of magnitude (i.e., by a factor of, say, 102000000
--
or more), that would still not 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.]
"Dialectics…prevails throughout nature….
[T]he motion through opposites which asserts itself everywhere in nature,
and which by the continual conflict of the opposites…determines the life of
nature." [Engels (1954),
p.211. Italic emphases added.]
"[Among the elements of dialectics are the
following:]…internally contradictory tendencies…in this [totality]…and
unity of opposites…. [E]ach thing…is connected with every other…[this
involves] not only the unity of opposites, but the transitions of
every determination, quality, feature, side, property into
every other….
"In brief, dialectics can be defined as the
doctrine of the unity of opposites. This embodies the essence of dialectics….
"The splitting of the whole and the cognition of
its contradictory parts…is the essence (one of the 'essentials', one of
the principal, if not the principal, characteristic features) of dialectics….
"The identity of opposites…is the recognition…of
the contradictory, mutually exclusive, opposite tendencies in all
phenomena and processes of nature…. Development is the 'struggle' of opposites.
"…The unity…of opposites is conditional,
temporary, transitory, relative. The struggle of mutually exclusive opposites is
absolute, just as development and motion are absolute." [Lenin (1961),
pp.221-22,
357-58. Emphases in the original.]
"According to Hegel, dialectics is the
principle of all life…. [M]an has two qualities: first being alive, and
secondly of also being mortal. But on closer examination it turns out that
life itself bears in itself the germ of death, and that in general
any phenomenon is contradictory, in the sense that it develops out of
itself the elements which, sooner or later, will put an end to its existence and
will transform it into its opposite. Everything flows,
everything changes; and there is no force capable of holding back this
constant flux, or arresting its eternal movement. There is no force
capable of resisting the dialectics of phenomena….
"At a particular moment a moving body is at a
particular spot, but at the same time it is outside it as well because, if it
were only in that spot, it would, at least for that moment, become motionless.
Every motion is a dialectical process, a living contradiction, and
as there is not a single phenomenon of nature in explaining which we do
not have in the long run to appeal to motion, we have to agree with Hegel, who
said that dialectics is the soul of any scientific cognition. And this
applies not only to cognition of nature….
"And so every phenomenon, by the
action of those same forces which condition its existence, sooner or later, but
inevitably, is transformed into its own opposite….
"When you apply the dialectical method to the
study of phenomena, you need to remember that forms change eternally
in consequence of the 'higher development of their content'….
"In the words of Engels, Hegel's merit consists
in the fact that he was the first to regard all phenomena from the point
of view of their development, from the point of view of their origin and
destruction…." [Plekhanov
(1956), pp.74-77,
88,
163. Bold emphases alone added.]
"'All is flux, nothing is
stationary,' said the ancient thinker from Ephesus. The combinations we call
objects are in a state of constant and more or less rapid change….
"…[M]otion does not only make objects…, it is
constantly changing them. It is for this reason that the logic of
motion (the 'logic of contradiction') never relinquishes its rights
over the objects created by motion….
"With Hegel, thinking progresses in consequence
of the uncovering and resolution of the contradictions inclosed (sic) in
concepts. According to our doctrine…the contradictions embodied in
concepts are merely reflections, translations into the language of thought,
of those contradictions that are embodied in phenomena owing to the
contradictory nature of their common basis, i.e., motion….
"…[T]he overwhelming majority of phenomena
that come within the compass of the natural and the social sciences are among
'objects' of this kind…[:ones in which there is a coincidence of opposites].
Diametrically opposite phenomena are united in the simplest globule of
protoplasm, and the life of the most undeveloped society…." [Plekhanov (1908),
pp.93-96. Bold emphases alone added.]
"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.]
[Several more quotations from the dialectical classics (that echo
the above sentiments) can be found
here.]
Now, the evidence supporting these recklessly 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 the above comrades.
However, the uncontroversial admission that most things change does not amount to
any sort of concession to DM, since dialectical change is supposed to be the result of
'internal contradictions'. Now, that doctrine is demolished in Essay Eight
Parts One and
Two. In
Essay Eleven Part One, Heraclitean change will be destructively criticised, too.
So, non-dialecticians can agree with dialecticians on the reality of change; where they differ is over
the cause of change: 'internal contradictions'.
It's worth underlining here that the denial of universal change does
not imply that everything is changeless; just that some things might be
(and probably are) changeless, some things not; on this see
Note 11 and
Note 12,
below. Plainly, this is an empirical matter which cannot be settled by appealing
to the
'authority' of a handful Idealist Philosophers -- nor simply by dogmatic assertion.
Nevertheless, let us suppose that an object
B has the following properties, qualities or relations: B1,
B2,
B3,...,
Bn.
According to several of the above
dialectical worthies, all of these properties, qualities and relations must change
all the time (into what they do not say, but presumably it's into
their 'opposites',
not (B1,
B2,
B3,...,
Bn)
-- or perhaps it's (B1*,
B2*,
B3*,...,
Bn*).
[However, that possibility is closed off in
Essays Seven and Eleven
Part One.]
Nevertheless, even as B changes it is still
identical with itself. In order to see this, let us suppose that when each
property, Bi,
changes, it becomes Bi*,
in the first instance, and then Bi**
in the next, and so on. But at any moment, B's identity will be given by its
set of properties, qualities or relations (if we must view identity
traditionally). So, in the first case, for example, B will have changed into {B1*,
B2*,
B3*,...,
Bn*}.
But, even though B has changed, it retains its changed identity. Hence, as
long as B exists it is identical to itself (albeit, its changed self). So,
when
viewed this way, identity is no enemy of change.
[Dialecticians often appeal to the existence
of UOs to defuse this sort of objection; this topic is examined in Essays
Seven
and Eight Parts One and
Two.]
[UO = Unity of Opposites.]
Of course, the above scenario (which is
called Maximal Heracliteanism (or MAH) in Essay Eleven -- link below) might not be the option
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's because, if just one of the properties, qualities or relations
B enjoys -- say, Bk
--
remains the same even for a few
nanoseconds
then the LOI must apply to it, and the dialectical game is up -- for here we would have something that remained the
same, and was identical to itself, even if only momentarily.
By way of contrast, the maximalist option (i.e.,
MAH -- this is explained at the following link) has even worse consequences for DM; these are spelt-out in detail in Essay Eleven
Part One.
Either way, Heraclitus is no friend of
dialectics --
or if he is, he's also its enemy.
A nice unity of opposites that, one feels!
6.
Hegel's egregious logical blunders were exposed
here and
here (and will be completely
demolished in Essay Twelve Parts Five and Six -- summaries
here and
here).
The use of ordinary
words for identity and difference is surprisingly varied,
and 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., first)
place in the English word, and the different letters "d" and "g" appear in the
same English word). Moreover, the
same word, in the same or different place in the same or different sentences, can mean the same or
different things (e.g., 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 sub-phrase: "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); 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.).
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, 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 pieces, and
the "9" in 1911 stood for 900 one cent pieces)). And the very same
number can change in other ways even while it stays the same -- for example, a
"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 I now add another zero
(to yield 500), the very same 5 will now mean five hundreds, and so on. [The
reader is to imagine me typing all this in one go, say into a calculator) so
that the very same figure 5 on the screen changes while it remains the same.] 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 of 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, year, decade, century, millennium, geological time period,
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, 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 its 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) goes for packing cases, bags, holes, garages, flats, sheds,
houses, cars, taxis, trains, aeroplanes, ships, buses...
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!
We needn't concentrate,
either, on examples that
some might still consider "abstract"; two physical ink marks on a page (two letter
"A"s, say) which are not 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 identically sandwiched between two other non-identical
letters (i.e., both in the middle -- and it's irrelevant whether or not either
letter is closer to either the "p", or the "t", the "P" or the "T"; if it's
anywhere between these letters, it's still in the middle of each word). Now the physical position of material ink marks on a page (or even
those electronically produced as pixels on your screen) is not abstract, it's
manifestly
material --, so much so that one or both can be obliterated by the non-dialectical
application either of
Tipp-ex or the delete key.
Such a
deletion would not be removing an abstraction.
As
noted elsewhere, ordinary and
technical/semi-technical languages have seemingly limitless capacities for allowing users to express complex and
subtle differences in meaning way beyond that permitted by the obscure and
lifeless language found in Hegel. This is not surprising; ordinary and
technical/semi-technical languages have been formed over countless centuries
by working people/materially-motivated scientists in their interaction
with the world and with one another. These systems of communication reflect our
species' complex inter-relationship
with one another and with changing reality -- and 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 consciousness (cobbled-together as part of a dubious, class-compromised
intellectual tradition that stretches back over two thousand years), and (in
Hegel's case) was invented by
a man who in his theoretical activity was far more concerned with his relation to the
world of ideas than he was with his interaction with objects and
processes in the material world. Small wonder then that his ideas
cannot cope with
changing reality.
7.
However,
as is the case with other philosophical 'problems', Trotsky is in good company here; Philosophers
and modern Logicians also manage to confuse equality with identity. In fact, this is a
highly neglected area in the Philosophy of Logic. There are signs, though, that this is beginning
to change;
on this see
Sanford (2005).
In fact, astute logicians have been aware of
such complexities for years, but have been opposed by traditionalists reluctant
to change (no irony intended). On this, see Geach (1967, 1973, 1975, 1990), Griffin (1977), and
Noonan (1980, 1997,
2009).
See also
Deutsch (2007).
8.
As
pointed out above, DM-theorists' comments on
FL are exceedingly repetitive
(Graham Priest's work being a notable exception). In
stark contrast to their
attempts to master other areas of knowledge -- for example,
classical and modern economics, science, history, politics and current affairs --,
when it comes to FL, dialecticians
display little or no comprehension even of
Elementary Logic.
Indeed, their writings almost invariably contain highly superficial and
inaccurate characterisations of what turn out to be obsolete forms of
AFL. Not surprisingly, such
'logical straw men' are quite easy to knock over.
[On this, see Essay Four,
here and
here.]
It's instructive to compare this dishonest
approach to FL with the justifiable condemnation that the very same DM-theorists level at analogously
distorted views of revolutionary socialism found in the writings
of the vast majority of Marx's bourgeois critics.
[For some reason, the words "sauce", "goose"
and "gander" come to mind here.]
In this regard, it's quite clear that John
Rees, for instance, cannot possibly have checked a single logic text
(other than those
mis-titled works written by Hegel, perhaps) before he wrote what he did about FL and the LOI in
TAR. As also seems to
be the case with most other Marxist critics of FL, Rees appears
to have confined his 'research' on this topic to reading only what
previous dialecticians had written about it, and simply copying what he found.
Indeed, a supporter of this site tried to point this out to him at a large
public meeting in London in 1990, but that was clearly a waste of breath.
In modern symbols, one form of
Leibniz's Law
is
as follows:
[1] (∀x)(∀y)((x
= y) º
(Fx
® Fy)).
[1] is otherwise known as the
"Indiscernibility of Identicals". Translated, it reads roughly: "Any two objects
are identical if and only if they share the same properties" -– or, "…whatever
is true of one is true of the other." This particular 'Law' will not be
defended here for reasons outlined in Note
1. Its translation into ordinary language is not happy on any reading. That
alone shows it's not equivalent to the ordinary use of such phrases as "equal
to", "the very same as", "identical with", or even "numerically identical with".
[For complications, see Gallois (2003).]
Nevertheless, it's important to note
that the use of the "=" sign in [1] has been strengthened by the presence of the
bi-conditional "º";
hence, it's not identical with Trotsky's use of the former sign, either (no irony
intended).
Contrast [1] with the following version of
the same 'Law':
[2] (∀x)(∀y)((∀F)(Fx
º
Fy)
®
(x = y)).
[2] is otherwise known as the "Identity of
Indiscernibles". Loosely translated it reads: "Any two objects that share every
property in common are identical." However, this version requires
quantification across properties, which is controversial.
One of Trotsky's mistakes was to suppose that
this 'Law' is empirically testable. He did this when, for
example, he supposed that the truth of S1 [i.e., "A is equal to A"]
could be checked directly by the use of an eyeglass, and when he referred to weighing
bags of sugar. He clearly regarded S1's failure to pass such tests as
sufficient grounds for rejecting this 'Law'. However, it did not seem to occur to
Trotsky that an empirical test of Leibniz's Law is wholly
inappropriate -– it would be just as misguided as an empirical test of, say,
"a + b = b + a" (i.e.,
commutativity over addition) in Mathematics. Anyone who
thought to check such a rule in this manner would be regarded as hopelessly
confused, and rightly so.
[On the other hand, if this 'Law' is regarded as an
expression of a rule, then the temptation to think it can be tested simply
vanishes. On this see Note 10,
below. Of course, it is possible to use this 'Law' to test whether or not
two or more bodies are identical, but that's a different matter (no irony
intended!).
Incidentally, on this issue it will not do to point
out that certain operators in mathematics do not
commute since no one in their
right or left mind would empirically test these, either. Of course, tests might be
performed to see whether or not certain systems in nature observed/obeyed
commutativity (for example, in
Matrix Mechanics), but no test would or could be run on
the principle itself. And despite what dialecticians say, the same goes
for the LEM, as it is allegedly used in QM. More on that later.]
[QM = Quantum Mechanics;
LEM = Law of Excluded Middle.]
Compare the two versions of Identity
outlined above with the following:
[3] j(y)
º
[(∃x)((x
= y)
&
j(x))].
[3] appears in Griffin (1977), p.1, which
also contains a strengthened version of Leibniz's Law:
[4] (∀x)(∀y)[(x
= y) º
(∀j)(j(x)
º
j(y))].
[4] roughly reads: "Any two objects are
identical if and only if for any property, one has it if and only if the other
has it, too." Griffin also gives other versions of [1] and [2] above, (ibid., p.2).
Incidentally, [3] roughly says "Anything true of some object is equivalently
true of any object identical to it." [In fact, I use this version in the
present Essay to show that
identity is no enemy of change. Because this rule does not require quantification
across properties, it is, in my view, preferable to [2] and [4].]
["∀"
is the universal
quantifier, equivalent to
"All", "Any" or "Every"; "∃"
is the existential quantifier, equivalent to "Some" or "At least one"; "º"
is the sign for logical equivalence, i.e., "If and only if"; "j"
and "F" are predicate variable letters; "®"
is the implication arrow, equivalent to "if...then"; "x" and "y" are
quantifier
variables. For more on this, see, for example, Priest (2000) and
Tomassi (1999). On quantifiers, see
Note 13a.
However, on variables, see
here.]
8a.
And the same could be said -- with slightly less justification --, about
Hegel; at least he used the right word, even if it is clear that he
failed to grasp the complexity of this logical 'law', just as he failed to do
justice to the ordinary words we have for identity, etc.
9. This from the
UK Guardian newspaper (Wednesday, 18/10/95):
"K2 appeared over the 40 million years or so
that India has been colliding with greater Asia. It was 'discovered' (i.e. by
the British) and designated K2 (Karakoram Peak 2) in 1856. The peak was granted
the name [Mount Godwin-Austen] in 1888, after its first surveyor, Col Henry
Haversham Godwin-Austen (1834-1923). The previous title is now preferred as
being less imperialistic. Ordinarily a mountain would revert to its local name,
but K2 is so remote that it appears never to have gained one." [Letter from
S McDiarmid, reprinted in Notes and Queries. See also
here.]
This is perhaps a more interesting
example of ordinary ascriptions of identity than those considered by Frege (for
instance, with
respect to the Evening Star and the Morning Star,
etc.), in Frege (1892).
And this is from the same newspaper:
"In Nepal they
call it Sagarmatha. To the people of Tibet, it is
Chomolungma,
though the ruling Chinese prefer the variant Qomolangma. When the British
first began mapping India, they knew it as Peak B, then as Peak XV.
But in 1865, to honour the surveyor-general of India who first mapped it,
Peak XV was given the name Mount Everest. And Everest the
mountain has remained throughout much of the rest of the world to this day.
"Now China is
launching a fresh effort to outlaw the name Everest. Accusing British
colonialists of 'raping the sacred mountain of Tibetans by giving it a false
name', Chinese newspapers are calling on the world to 'respect Tibetans' by
using the 50th anniversary of the first ascent next year to recognise the
mountain henceforward as Qomolangma.
"At first
sight, the proposal does not seem unreasonable. There are, after all, lofty
precedents for such renaming. The highest point in Africa, the summit of
Kilimanjaro, which was once known as
Kaiser Wilhelm Spitze, properly became Uhuru Peak. The world's
second-highest peak, once Mount Godwin-Austen to the British, has become
K2 (ironically this also dates from imperial survey days). But doubts
about the new proposal soon creep in. If Everest is unacceptable, why should the
world not prefer the Nepalese name to the Chinese or Tibetan one? Who are the
Chinese, of all people, to accuse others of raping Tibet? And how is the
'English language hegemonism' of which China complains worse than its Chinese
language equivalent?
"We hold no
great brief for the name Everest, though it has to be said that the word
has a fine ring to it. But the answer is to live and let live. If people prefer
Chomolungma, let them use that name. If others want to stick with
Everest, let them do so too. We have no problem with diversity, though the
Chinese may. In the end, the world's greatest mountain is surely more important
than any name that mere mortals give to it." [Guardian,
20/11/02. Italics added; quotation marks
altered to conform to the conventions adopted at this site.]
So, here we have several perfectly ordinary
identities: Everest is identical to
Chomolungma, Qomolangma, Peak B and
Peak XV; Kilimanjaro is identical to Kaiser Wilhelm Spitze and
Uhuru Peak, and so on.
Instead of reading Trotsky -- or
far worse, Hegel -- on
identity, comrades would do well to study Frege. [I.e., Frege (1892).] [It
has to be said, though, that even Frege confuses identity with equality!]
Excellent introductions
to Frege's thought can be found in Kenny (1995), Noonan (2000) and Weiner
(1990, 1999); for a useful guide to the philosophical issues involved, see Linsky (1977).
And we find this in a recent issue of the
New Scientist:
"In 2003, at team...[in]
Moscow discovered two distant elliptical galaxies just a whisker apart. Detailed
analysis of the twins known as CSL-1, suggested that they were images of the
same galaxy.
"The team suggested that the
duplicate images were being created by a 'cosmic string'.... If one of these
cosmic strings were to pass between Earth and a giant galaxy, the warping of
space-time by the string would create a
gravitational lens and form two
identical images of the galaxy -- exactly like CSL-1....
"Unfortunately for the
proponents of cosmic strings, observations with
Hubble on 12 January have
revealed that CSL-1 is actually two different galaxies...." [New Scientist
189, 2537, 04/02/06, p.21. Quotation marks
altered to conform to the conventions adopted at this site.]
Now, this is a clear example of science at
work. Instead of 'solving' this problem in an a priori manner (a là
Trotsky or a là Hegel), declaring that things are never equal to themselves, astronomers have
been able to show that these images were not of the same object, but of two
different galaxies.
[However, one wonders what dialecticians would
have said if these two images had been shown to be of the same galaxy.]
In the
article posted at the above link several examples are given of multiple images
of identically the same object.
In relation to which consider the following line of argument:
G1: Image I1
is identical to image I2.
G2:
Image I1
and image I2
are both of the same object.
G3:
But, image I1
is an image of object O1.
G4: Image I2
is an image of object O2.
G5: Therefore, object O1
is identical to object O2.
[This is not the line of argument used by the scientists
mentioned in the above passage!]
It's not easy to see how dialecticians would be able to tell G1 and G2 apart
(if these scientist had found they were of the same object),
nor account for the conclusion recorded in G5.
And it would be no use pointing to the alleged limitations of the LOI
here, since, no matter how much objects
O1
and O2
changed they would still be
identical, since they would change at an identical rate as 'one another' (being
one object, not two)! Certainly, the images of these objects may or may not be
identical, but these 'two' objects cannot fail to be, since there is only one!
Of course, if it were now claimed that,
on the basis of what Trotsky or Hegel said, objects O1
and O2
were nonetheless not identical
(or they were both identical and not identical(!)),
then that would fatally undermine this part of Astrophysics, since it would
nullify the application of gravitational lensing in the above manner. If the
objects here are always non-identical, then images
I1
and I2
would plainly be of two different objects;
the above inference would falter.
As noted above, it's not
easy to see how dialecticians can hold on to their criticism of the LOI without
undermining at least this part of Physics.
And, as we will see later, a desperate appeal to "approximate
identity" here would be to no avail.
Here's another recent example of the ordinary use
of words for identity in the sciences:
"More than 600,000 plant species have been
deleted from the dictionary of life after the most comprehensive
assessment carried out by scientists.
"For centuries, botanists from different
parts of the world have been collecting and naming 'new'
plants
without realising that many were in fact the same. The
humble tomato boasts 790 different names, for example, while
there are 600 different monikers for the oak tree and its
varieties.
"The result was a list of more than 1
million flowering plant species. Although experts have long
known that it included many duplicates, no one was sure
how many. Later this year, the study team, led by UK and US
scientists, will announce that the real number of flowering
plant species around the world is closer to 400,000.
"The project -- which has taken nearly
three years -- was the number one request made by the 193
government members of the
Convention on Biological Diversity
at their meeting in 2002. There were concerns that without this
work, it would be impossible to work out how many plants were
under threat and how successful conservationists were in saving
them.
"The information will also be vital for
any organisation or researcher looking at 'economically
important' plants, such as those for food and nutrition or
medicine, said Alan Paton, assistant keeper of the herbarium at
the
Royal Botanic Gardens at Kew,
west London, one of the four leading partners in the project.
"'On average, one plant might have
between two and three names, which doesn't sound a great
deal, but if you're trying to find information on a plant, you
might not find all [of it] because you're only looking at one
name,' Paton said. 'That's even more critical for economically
useful plants: because they are more used, they tend to have
more names.'
"In one example, researchers calculated
that for the six most-used species of
Plectranthus, a
relative of the basil plant, a researcher would miss 80% of
information available if they looked under only the most
commonly used name. On another database, they found only 150 of
500 nutritionally important plant species using the names cited
in current literature.
"'By going for one name, we missed the
majority of information mankind knows about that plant, which
isn't too clever,' said Paton. 'What's really a breakthrough is
we have a place which allows people to search through all the
names used.'
"Kew Gardens joined up nearly three years
ago with
Missouri Botanical Garden
in the US, and experts on two of the biggest and most valuable
plant families: legumes, or peas and beans, and
Compositae,
which include asters, daisies and sunflowers.
"They have since attempted to search
existing plant lists and work out an 'accepted' name for each
species, and then list all known variations. One of the
databases was originally set up using £250 left in the will of
Charles Darwin. The full results will not be published until the
end of the year, but so far the researchers have found 301,000
accepted species, 480,000 alternative names, and have 240,000
left to assess.
"Although work will continue to assess
smaller plant groups in more detail and check for missed
duplications, Paton said they now believe that the true number
of plant species will turn out to be '400,000 or just over'.
"'You can't give an absolute number of
names, but we have narrowed the possibility,' he said. Previous
estimates, without the help of a full assessment, put the figure
at between 250,000-400,000.
"Most of the work of the study group was
sifting and sorting different names allocated to one species,
often because scientists were simply not aware of the work of
rivals and colleagues who had previously 'described' the plant
in a scientific journal, or because of confusion caused by
superficial differences such as different sized leaves in
different climates. In some cases, plants thought to be the same
have also been judged to be different species because of
differences which have been revealed by later scientific
discoveries, such as DNA.
"As well as the likely 400,000-odd
flowering plants, there are thought to be 15,000 species of
ferns and their allies, 1,000
gymnosperms such as conifers, and
23,000 mosses and allies making up the plant kingdom. For
comparison there are more than 1 million species of insects
listed by science, 28,000 living species of fish, 10,000 birds
and 5,400 mammals.
"A
meeting of the Convention on Biological Diversity in October in
Japan is likely to declare that
targets to halt biodiversity loss by this year failed and set
tougher new aims to halt the problem." [The
Guardian, 20/09/10, p.1. Quotation marks altered to
conform to the conventions adopted at this site. Bold emphases added.]
It's
not too clear how DM can accommodate several plant names all
naming the same species. Once more, an appeal to 'abstract'
and/or
'approximate' identity here
will not help, either,
as we sill see.
10.
This
observation, of course, depends on the said weighing scales changing at the same
rate as the sugar being weighed, which, while unlikely, could conceivably happen.
The point is, of course, that this is an empirical matter that cannot be settled
a priori, as Trotsky attempted to do.
Nevertheless, it's worth
remembering that Trotsky's argument here revolves around the accuracy of measuring a
pound bag of sugar (etc.); he then extrapolated from a few (theoretical)
observations about local conditions to 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.]
Once more: how could Trotsky possibly have
known all this? His only 'evidence' appears to have been these
badly worded thought experiments.
Again, it could be objected that Trotsky was
merely drawing out the consequences of accumulated human experience of change,
deriving reasonable conclusions from thousands of years of our growing knowledge
of the world.
In response to this (and in addition to the
points made in this section) it might be worth trying
to reconstruct the reasoning behind Trotsky's claims. Presumably he argued
as follows: 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 his -- would embroil Trotsky in having to apply the LOI yet again, only
now applied to his own
instructions and to the measurements he would recommend
to anyone who wanted to test his ideas.
Clearly, these instructions would have to be carried
out identically and to the letter each time if they are to count as a verification of what Trotsky said -- but
now aimed at demonstrating that the LOI is
defective after all!
Of
course, given Trotsky's strictures on the LOI, no two attempts to carry out any
set of instructions would ever be the same. In that case there would be no
accurate way to test his or anyone else's predictions! [Once more, an appeal to
"approximately identical instructions" is batted out of the park,
below.]
On the
other hand, if Trotsky had been faced with someone who claimed that at least two
of
their results were
identical,
he could only have responded in one or more of the following ways:
(1) Insisting
that this experimenter must have been mistaken.
(2) Pointing
out that the machines used were not accurate enough.
(3) Maintaining that his instructions had not been carried out exactly to the letter.
(4) Arguing that identically the same experiments had not been performed each
time.
In other words, in the absence of a mistake (and if the same results were
recorded on more accurate scales) -- i.e., ruling out (1) and (2) above --, Trotsky would only be able to criticise the
above
reported experimental verification of the LOI by an appeal
to that very same 'law',
but now applied to his
own instructions! Hence, in order to counter
results that would disconfirm
his
forecast (about varying weights)
he would have to argue that only those who followed his instructions
identically and to the letter
could
disprove the LOI!
The irony is thus 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 weights of bags of sugar, confirming this 'law'!
In which case, Trotsky's predictions about
such objects would end up being refuted by anyone who adopted this
diluted version of the LOI!
Such experimenters would succeed in
confirming the absolute form of the 'LOI by employing a weaker version of it!
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 instructed, using the LOI applied to instructions in order to disconfirm it as
applied to bags of sugar!
Relying on evidence alone, therefore,
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 space and time -- not least
because he performed no experiments himself.
[Indeed, the
development of science has justified a much more cautious approach to theory. On that, see
Note 11 and
Note 12,
below.]
Hence,
Trotsky's claim
that
all objects everywhere
change
all the time, if extrapolated beyond the aforementioned conventions and
scientific facts, would
transform
the LOI into a metaphysical truth. But, there could be no body of evidence
large enough to support
an extrapolation as bold as this --
or, that is,
none that wasn't also based on those very
same
conventions relating to identically performed experiments and the use of
ordinary words for identity, etc.
Extrapolation beyond these -- by means of them -- to universal theses
that are
applicable everywhere and for all of time would convert them into universal
truths -- the very thing Trotsky affected to disavow. In that case, Trotsky
would have to appeal to the LOI as a universal truth to justify his
general conclusions about everything in existence behaving exactly as he
said it did, with every human being measuring objects identically
throughout all of human history, in order to show that no one could do
this!
So,
any evidence from either the past or the present used (by DM-theorists) to
undermine this 'law' would automatically call into question the methods by which
it had been collected, processed and checked. Without the LOI applied as a rule
of language, or as a rule guiding practice, no one would be able repeat the same
experiments to verify or refute earlier results, check measurements or confirm the
accuracy of his predictions. Nor would they be able to learn to use the very same
theories as others had, or appeal to the very same 'law', or
its alleged refutation (in the
way that, say, Trotsky and Hegel claimed to have done).
Without this 'law' applied as a rule of
language, or of practice, there would be no conceptual space within which
science or ordinary practice could develop --, and thus no reliable data, and no
settled theories --, for anyone even to begin to think about confirming the DM-hypothesis of
universal change.
In that
case, no viable science/philosophy could question the application of this 'law'
as a rule of language (or of practice) while remaining viable.
[The
problems in fact go much deeper than this, but their consideration would take us
too far into areas covered by other Essays posted at this site.]
This, of
course, explains the
origin of the difficulties
highlighted
above
(in connection with the postulated refutation of
Trotsky's predictions about the weights of bags of sugar) -- and it also
reveals why Hegel got into such a tangle in his attempt to half accept and
half reject this 'law',
and why this entire topic became such a puzzle to
him.
If this 'law' is treated as a metaphysical truth (which has generally been the
case with traditional theorists) -- i.e., as a 'necessary truth' --, then its
falsehood becomes impossible to state (as we have seen throughout this Essay)
--, as does its truth --, at
least in comprehensible language, or in language that does not implicitly rely
on this very 'law' to state what was intended of it.
[And it's
worth pointing out that the 'truth' if this 'law' is no less problematical than its 'falsehood', for both
of these depend on treating the LOI as a sort of Super-empirical thesis; more on this in
Essay Twelve Part One.
Why this is so is connected with Wittgenstein's
comments on 'rule-following'; again, on this see the references given in Essay Twelve
Part One, and in Essays on the
nature of language to be published here at a later date, for example,
here.]
On the
other hand, the 'truth' of this 'law' does not seem (to some) to be at all
trivial. If
it is viewed
(traditionally)
as a 'law' that is said to determine or depict
the 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 form of language that does not also rely on
this very 'law',
only
now
operating
as a rule governing the use of words to make that very point!
[As we have also seen.]
Alternatively, if this 'law' is viewed rightly as a rule of language, or of
practice (without which humanity could not have developed a single coherent
idea), then these pseudo-problems simply vanish. On this basis, the LOI is not
a universal truth because of what Hegel or anyone else has said about it;
it's not
a truth to begin with!
Because it's a rule,
it cannot be true or false, only practical or impractical, applicable or not.
[In fact, as we have seen it's impractical (or rather, it's the height of
folly) even to attempt to reject this rule.]
In that case, without a clear idea of how to use
words for identity (etc.), it would be impossible even for DM-theorists to
begin to wonder whether our words were approximately stable from moment to
moment, or if they altered in other alarming ways. If the LOI is rejected
(as a rule of language) --,
or if it is held to be an approximate
truth
-- then all of the above points go by the board.
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, just as we would also have to know the same
with respect to our other words for identity, sameness and difference. Without some notion as to what counts as
identically the same employment in all these cases, we could not even begin to
say what would constitute an approximation to anything whatsoever, or, indeed,
in what way anything fell short of a standard that itself presupposed the
applicability of such words for identity.
[Incidentally, I have use "know how to use this
word" in a pragmatic sense; there is no suggestion that we use such words
according to some theory we hold. In fact, I have borrowed this locution from
Ryle. On that, see here.]
Now,
this fundamentally important point sailed right over Hegel's head -- Lenin, Trotsky and other
DM-theorists also failed to notice it. They regarded the LOI as a truth (even if it was one that the
"abstract understanding" employed, or which was valid "only within
certain limits"). But it cannot be a truth (for the reasons outlined above); our
use of words for identity express rules
constitute our capacity to utter sentences that we can even so much as begin to regard
as true or false, or a bit of both (should we so choose).
[This
is not a 'transcendental argument', merely a reminder that we have to use words
to express our thoughts, and we may only do certain things with such words given
contingent facts about our history and our social nature.]
So,
this 'law', applied as a rule of language (or of practice), has to be employed even to make the
point that there is anything which is approximately true (since anyone doing
so has to at least use the word "truth" in a consistent way to make that point!); in which case that 'law' can
neither be true nor false itself.
And this is what makes the comments of dialecticians
in this area valueless, and it's 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 cannot 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, to Trotsky's own writings and to those of experimenters (etc.).
But, when
the LOI is
deployed in this manner in an attempt to expose its alleged
empirical limitations (a là Trotsky) -- or to reveal its supposed theoretical short-comings (a là
Hegel) --, then
that attempt itself will self-destruct. For
if that 'law' (applied as a rule) is unsafe then so are the methods used gather
any evidence aimed at questioning it, and so are the arguments used to
undermine it.
Which
is indeed what we have seen.
[More on this in
Note 15, below. See also
Note 13.]
Naturally, a grudging acceptance of the
above (linguistic and social) conventions (on the following lines "Ok, so it's a
rule, but that doesn't show that objects in the real world obey this rule; 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.) -- which would undermine
their own ideal
'thought experiments' aimed at revealing the alleged deficiencies of FL, and of
the LOI. Of course, it would be up to them whether they preferred their
ideal thought experiment to the identical, but real experiments carried
out in this world to test their ideas, which, once they had been performed to
the letter, would in the act refute that ill-considered thought experiment!
Now, DM-theorists might sincerely believe that all objects change constantly, 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 was not itself dependent on conventions of
measurement (counting and comparing) that were not themselves subject to
Trotsky's
(or Hegel's) strictures
-- should the latter continue to be accepted. And that is why both Hegel and
Trotsky had to use this 'law' 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'. [Irony intended.]
10a. Anyone who accuses
me of "empiricism" -- or even "positivism" -- ,
on
the basis of my constant demand for confirmation, should re-read this,
and then think again.
11.
Physics And Identical Objects
[This
forms part of Note 11.]
However,
contrary to what Trotsky says, it's very easy to
make two identical objects -- everyone does so when they throw a light switch.
Here is some material devoted to this topic copied from another
Essay posted at this site:
Physicists tell us that every photon, for example, is identical to every other
photon. This how Steven French puts things:
"It
should be emphasised, first of all, that quantal particles are indistinguishable
in a much stronger sense than classical particles. It is not just that two or
more electrons, say, possess all intrinsic properties in common but that -- on
the standard understanding -- no measurement whatsoever could in principle
determine which one is which. If the non-intrinsic, state-dependent
properties are identified with all the
monadic
or relational properties which can be expressed in terms of physical magnitudes
associated with
self-adjoint operators that can be defined for the particles, then it can
be shown that two
bosons or two
fermions in a joint symmetric or anti-symmetric state respectively have the
same monadic properties and the same relational properties one to another.
[French and Redhead (1988); see also Butterfield (1993).] This has immediate
implications for
Leibniz's Principle of the Identity of Indiscernibles which, expressed
crudely, insists that two things which are indiscernible, must be, in fact,
identical." [French (2011).
Bold emphases added. Referencing altered to conform to the conventions adopted
at this site.]
Of course, French offers his own solution to this difficulty, but
it isn't one that challenges the identity of quantal particles, just their lack
of individuality.
Paul Dirac
put the same 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.]
In that
case, every time a worker turns on a light, he or she makes/generates countless trillion
identical objects per second -- which must mean that such workers are
"unconscious" anti-dialecticians, if we apply the same sort of reasoning
here as Trotsky.
Naturally, contentious
claims like these can only be neutralised by an a priori stipulation to
the effect that every photon in existence (past, present and future) must
be non-identical -- despite what scientists tell us and in abeyance of the
impossibly large finite amount of data that would be needed to support such a cosmically ambitious
claim. At this point, perhaps, even hardnosed dialecticians might be able
to see in this a blatant attempt to impose DM on reality.
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), and particularly French and Krause (2006). See also
Hilborn and Yuca (2002),
Ladyman and Bigaj
(2010), and the Wikipedia entry
here.
It could
be objected that
Trotsky would surely have been unaware of developments in Physics after he died,
but, as the references given show, such facts were largely true of classical
particles, too; quantum particles merely present a more extreme form of strict
identity. And Lenin it was who reminded us that science is ever revisable; hence,
no dialectician (who agrees with Lenin) could consistently rule out the
possibility that scientists would one day discover identical particles -- as indeed they
have.
Even so, Trotsky was quite
happy to impose this view on nature before all (or most of) the evidence was in,
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.]
Also see Note 12, below.
12. Once
more, this is material that has been posted in another Essay at this site:
In fact, the half-life of a proton is
reckoned to be in excess of 1032 years (estimates
vary, but this is approximately 1020
times longer than the age of the known universe, if current theory is correct).
Experimental evidence suggests its half-life is probably longer than that. Apparently, electrons are even
less 'dialectical'. Predicted
proton decay
has so far not been observed. 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 and Trotsky seem to
have thought -- an
a priori truth based on the musings of an Idealist who died over
2500 years ago (i.e., Heraclitus)!
As far as protons are concerned, we are told the following:
"Along with neutrons, protons make up the
nucleus, held together by the
strong force. The proton is a
baryon and is
considered to be composed of two up
quarks and one down quark.
"It has long been considered to be a
stable particle, but recent developments of
grand unification models have
suggested that it might decay with a
half-life of about 1032 years.
Experiments are underway to see if such decays can be detected. Decay of the
proton would violate the conservation of
baryon number, and in doing so would be
the only known process in nature which does so." [Quoted
from here.]
Wikipedia
adds:
"In
particle physics,
proton decay is a hypothetical form of
radioactive decay in
which the
proton decays into
lighter
subatomic particles,
usually a neutral
pion and a
positron. Proton decay
has not been observed. There is currently no evidence that proton decay occurs."
Of course, it could be objected to this that
particles such as protons (i.e.,
hadrons) are composed of even more fundamental
particles, which do enjoy a contradictory life of their own 'inside' each
host 'particle'; their interactions would therefore mean that apparently
changeless protons are in fact changing 'internally' all the time. But, this
response simply pushes the problem further back, for these other, more
fundamental particles (i.e.,
quarks --, in the
case of protons, two "up"
and one "down"
quark), are themselves changeless, as far as is
known. Moreover, since protons are
baryons -- i.e., they are
composed of three
quarks --, it is not easy to view their inner lives as in any way
"contradictory" (with three terms?). 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. Unless acted upon
externally, their 'lifespan' is, so we are informed, infinite; hence, if they change,
it's not because of any "internal contradictions".
[An appeal to
antiquarks here, to save the dialectical day, would be to no avail, either. That is
because quarks do not turn into antiquarks, nor vice versa, which is
what the Dialectical Holy Books tell us should happen to all such 'opposites'.
On that, see
here
and here. (Anyway, since
there is very little
antimatter
in the entire universe, so we are told, this is an 'academic' question.)]
So, it
seems that the picture of reality painted by dialecticians is more a
Jackson
Pollock than it is a
Van Eyck.

Figure Two:
DM-style 'Realism'

Figure Three: Non-DM-style Realism
On protons, see
here,
here, and
here; on
electrons,
here;
Leptons
here; photons,
here. On this topic in general, cf., Perkins
(2000), and French and Krause (2008) -- but more specifically Saunders (2006).
See also Ladyman and Bigaj (2010).
In fact there appear to be two schools here, those who hold that all such
particles are identical and indiscernible (rather like the dollars/pounds in
your bank account, not the dollars/pounds in your wallet or pocket), and those
who claim they are identical and discernible. On this see the above references
and Muller and
Seevinick
(2009), and Muller and Saunders (2008).
Naturally, dialecticians might want to object
to the above on the lines that electrons, for example, are not really particles
--,
or that they are probability waves, 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.
Furthermore, if they change, they do not do so as a result of their
'internal
contradictions'.
[More on that
here.]
This comment puts paid to much of the
confused ruminations on sub-atomic 'particles' found in, for example, Woods and Grant
(1995). More details on this will
be posted in Essay Seven Part Two at a later date. On change though 'internal
contradiction', see Essay Eight Parts
One, Two and
Three.
Of course, the above considerations will only be of offence to those who,
for some odd reason, might want to foist dialectics on nature.
But, who on earth would
want to do that?
There is more on this topic,
here.
13. The reader will
no doubt have noticed that in order to interpret Trotsky, we have to use the
dread LOI. In that case, we have to decide whether S8 is equivalent to (i.e.,
identical with -- irony intended) S11 or S12.
S8: A pound of sugar is equal to
itself.
S9(b): [All bodies] are never equal to
themselves.
S10: Let A1
be a pound of sugar at time T1.
S11: Let A2
be a pound of sugar at time T2.
[T2
> T1.]
S12: S8 means A1
is equal to A1.
S13: S8 means A1
is equal to A2.
Of course, it's true that Trotsky believed that a complete understanding of change (even if
humanity never actually attained to it) would require the employment of concepts
more adequate to the task -– i.e., those found in DL. Many of these 'concepts'
are examined throughout this site, where they are shown to be no less confused.
[IDM = In Defense of
Marxism (i.e.,
Trotsky (1971)); DL = Dialectical Logic.]
However, in his Notebooks
Trotsky added a number of important qualifications to his comments on the LOI
found in
IDM. Among which are the following:
"a = a is only a particular case of the law
of a
¹
a…. Formal Logic involves stationary and unchanging quantities: a = a.
Dialectics retorts: a
¹ a. Both are
correct. A = a at every given moment. A
¹
a at two different moments. Everything flows, everything is changing." [Trotsky
(1986), pp.86-87.]
This suggests that Trotsky might have
accepted a version of S13 or S21(a):
S13: S8 implies A1
is equal to A2.
S21(a): There is an A and a
time t1
such that A at t1 is not
equal to A at t2.
S13 was in turn dependent on S8, S10 and S11:
S8: A pound of sugar is
equal to itself.
S10: Let A1
be a pound of sugar at time T1.
S11: Let A2
be a pound of sugar at time T2.
[UO = Unity of Opposites.]
This would appear to mean that Trotsky was
committed merely to the idea that an object is not self-identical at some
later time, as opposed to adhering to the stricter principle that objects
are not self-identical at any given moment -–, which (latter) belief is in turn
based on the doctrine that all objects are UOs. If so, the above passage would seem
to suggest that Trotsky was rejecting a core DM-idea: that UOs exist in
every object and process, which drive change because of such 'internal
contradictions'. Clearly, it's highly unlikely that Trotsky denied this
core DM-thesis. For example, his emphasis on the contradictory nature of the
former USSR
(elsewhere in IDM) strongly suggests he accepted this doctrine.
On the other hand, since Trotsky
nowhere (to my knowledge) refers to the idea that change is the result of the
struggle between UOs, it is possible that he did reject, or did not
fully accept this doctrine. [For a contrary inference from the same Notebooks,
see below.
Moreover, there are several things Trotsky says later on in these Notebooks
that suggest he did view everything as a UO; e.g., p.103 -- on that, see
here.]
Nevertheless, since this quotation is taken
from notebooks not intended for publication it would be unwise to rely too
heavily on what they say as an accurate indication of Trotsky's intentions. This
is especially so since it (rather fittingly) appears to contradict what was said in IDM:
A1: "In reality 'A' is not equal to 'A'…. [O]bserve these two letters under a lens
-- they are quite
different from each other." [Trotsky
(1971), pp.63-64.]
S9(a): All bodies change
uninterruptedly. (b) They are never equal to themselves.
Compare this with a passage from the
Notebooks (quoted above) where Trotsky now seems to say the opposite:
A2: "A = a
at every given moment." [Trotsky (1986), p.87.]
If A1 and S9 were correct, A2 couldn't be;
at best, it would represent only half the story.
For example, in A2, the two letter "A"s
are easy to distinguish without the aid of a lens: the second letter is in the
lower, while the first is in the upper, case. But here in A2, Trotsky now argues that these
"A"s
are equal at "every given moment" -- even though they look different to
the naked eye!
Conversely, in A1 Trotsky claims the opposite of this is
true with respect to two letter "A"s that not only look identical
but also are in the same upper case! He claims that the two capital
letters in A1 above look different if examined under a lens, while a lower case "a"
and a capital "A" in A2 are equal at every moment!
If A2 were correct,
then Trotsky's reference in A1 to the physical appearance of these two letter
"A"s
when viewed under an eyeglass would be entirely pointless. The only reasonable conclusion
here seems to be that since A1 was intended for publication it must
contain Trotsky's more considered thoughts.
Furthermore, as noted above, A2
seems to be inconsistent with the claim that change is the result of internal
contradictions: that is, with the idea that at any given moment an object
both is and is not self-identical, thus constituting a UO -- in this case
presumably a unity of "A and not A" (i.e., "pound bag of sugar and not pound bag
of sugar"), or that a pound bag of sugar is both identical and not identical
with its 'other', as Hegel might have put it. [On this, see Essay Eight
Part Three,
and Essay Twelve (summary
here). On the confusions Hegel's view must always introduce in this
area, see here.]
[Precisely what the 'other' is of a pound bag
of sugar is somewhat unclear. A pound bag of tea? A half pound bag of tea?
A pound bag of
Quinine? However,
if a pound bag of sugar
has no 'other' (and
no logical 'other', either)
then,
according to DM,
it can't change.
To be sure,
sugar is highly
complex; there are any number of things it can and does change into, so it must
have countless 'others' (which fact rather makes a mockery of Hegel's 'analysis'
of change). More on that
here.]
In addition, A2 is itself rather
badly worded. When Trotsky wrote:
A3: "A = a at every
given moment" (emphasis added)
he must have meant:
A4: "A = a at any
given moment."
This is because the wording of A3 implies
that "A" never changes; i.e., that at all times "A = a"
-- something Trotsky certainly did not believe.
On the other hand, A3
might contain an indirect allusion to Trotsky's point about abstract moments in
time:
"A sophist will respond that
a pound of sugar is equal to itself at 'any given moment'…. How should we really
conceive the word 'moment'? If it is an infinitesimal interval of time, then a
pound of sugar is subjected during the course of that 'moment' to inevitable
changes. Or is the 'moment' a purely mathematical abstraction, that is, a zero
of time? But everything exists in time; and existence itself is an uninterrupted
process of transformation; time is consequently a fundamental element of
existence. Thus the axiom 'A' is equal to 'A' signifies that a thing is equal to
itself if it does not change, that is if it does not exist." [Trotsky
(1971),
p.64.]
But,
according to this,
if Trotsky was referring to abstract moments in time in A3, it would mean that
the items he mentioned could not exist.
If so, it would be
unclear how
A3 could ever be true
-– that is, always assuming it was about such non-existent
things.
Of course, if A2 and A3 were
merely about letter variables (not their supposed referents, what they
supposedly refer to) it might
prove
possible to re-interpret them in a more viable form. One such re-configuration could see
A2 and A3 recording the fact that while objects in the world change, letters
depicting them do not. But that would make Trotsky's other assertions about the
"A"s in A1 decidedly odd, for the aim there had been to argue that these letters
were not in fact "equal" irrespective of what they refer
to. That was the whole point of Trotsky's appeal to ocular inspection.
And since variable letters are physical objects in their own right, his claim
surely was that they are just as susceptible to change and diversity as are the
things to which they supposedly refer. That option therefore does not look at all
promising.
On the other hand, if Trotsky had wanted to
argue for something more complex in this regard it would definitely be
impossible to comprehend his point. For example, if he had meant something like
the following:
(1) Variable
letters and what they refer to both change, and that they do so as
follows:
(2) Each letter "A" no longer refers
to whatever it used to refer to moments earlier, and,
(3) The object that each old letter "A"
once denoted is no longer the same as it was when first identified, and,
(4) Earlier and concurrent
manifestations of any and all letter "A"s are never the same as the
'same' new letter "A"
now on the page/screen (which page/screen also changes), and,
(5) Any two or more
concurrent letter "A"s
on the 'same' line (which also changes) are not only different from each other,
they change at different rates, and,
(6) Each letter individually denotes in a
different and changing manner objects in reality, which objects are also
different and are all changing at different rates themselves.
If something like this had been Trotsky's
intention then his entire point would become too obscure to assess for we
wouldn't have a clue what the hell he was banging on about. But, if all things
change uninterruptedly in every respect (as Trotsky himself claimed) then
he must have 'meant' this!
It could be
objected that Trotsky only needs to appeal to the relative stability of
medium-sized objects in reality to neutralise criticisms like this. Hence, if
language and most medium-sized objects are relatively stable, points (1)
to (6) above do not apply.
But, how could anyone
committed to this theory know whether or not language
is 'relatively stable'
-- especially if they also believe that everything is
in the grip of the
Heraclitean Flux?
In fact, as soon as language itself is implicated in this 'Flux', everything
semantically solid melts into thin air. Hence, it would be no good appealing to
evidence (drawn from dictionaries, textbooks, personal memory, common usage, etc.) in
support of the claim that language is 'relatively stable', for if everything is
changing then so is the language in which this evidence is itself expressed, so are the
notebooks and/or primary data sheets from which it has been retrieved, and so are the memories
upon which all
of these depend. Given this way of looking at things, for all anyone knows, every single word could change its
meaning every fraction of a second (along with any and all memories of, or about, the objects
that seem familiar to us, like dictionaries, journals and textbooks). This is almost certainly what
Voloshinov believed:
"[T]heme must be unitary, otherwise we would have
no basis for talking about any one utterance. The theme of an utterance is
individual and unreproducible, just as the utterance itself is
individual and unreproducible. The theme is the expression of the concrete,
historical situation that engendered the utterance. The utterance 'What time is
it?' has a different meaning each time it is used, and hence, in accordance
with our terminology, has a different theme, depending on the concrete
historical situation ('historical' here in microscopic dimensions) during
which it is enunciated and of which, in essence, it is a part." [Voloshinov
(1973), p.99. Bold emphases added. Quotation marks altered to conform to the
conventions adopted here.]
And, as we will see in Essay Thirteen
Part Three, several other
comrades say the same sort of thing.
However, as Plato himself was quick to recognise, the
Heraclitean Flux is no respecter of theories; in fact it comprehensively mangles them.
[On this, see
Note 10 above, and
Note 15 below
-- and, in this case,
Essay Thirteen
Part Three, where Voloshinov's ideas
are subjected to sustained and destructive
criticism.]
Furthermore, the reasoning
in the Notebooks appears to be somewhat confused:
"a = a is only a particular case of the law
a ¹
a." [Trotsky (1986), p.86.]
It's not
easy to see how "a = a" could be a particular case of the 'law' "a
¹ a", any more than "a + b = c", for instance, could
be a particular case of the rule "a + b
¹ c".
If "a ¹ a" is a
'law', then "a = a" refutes it; it does not
instantiate it. Of course, this is so unless the word "refute"
has changed
its meaning (perhaps because of the local effects of that pesky Heraclitean Flux). Naturally, in such a madcap Heraclitean world it's not easy to see exactly what would, or could either stay the same or change -- or,
indeed, for how long a decision about even that possibility would
remain stable, too!
And yet,
contrary to what was intimated earlier, this could be an indirect reference on
Trotsky's part to a UO operating here: the fact that
"a = a" and "a
¹
a". If so, Trotsky's criticism of the LOI would collapse into irredeemable
confusion. If "a" does not equal "a", that is, if "a" is "not a", then the
following must be true: "a is not a", or "a = not a". In which case we can
substitute "not a" every time we see "a", giving this series of dialectical
oddities (using capital letter "A"s):
C1: "A = not A".
C2: "Not A = not (not A)".
C3: "Not (not A) = not (not (not A))".
C4: "Not (not (not A)) = not (not (not (not
A)))".
C5: "Not (not (not (not A))) = not (not (not (not
(not A))))".
C6: "Not (not (not (not (not A)))) = not (not
(not (not (not (not A)))))".
And so on.
But worse, whatever is true of "A" must also be true of "=" and
"not" (and even of "is"), yielding this ever-expanding bowl of dialectical
spaghetti (as each "not" and each equal sign is replaced by its dialectical
equivalent, "not (not)" and "¹",
and then, of course, "not
¹", respectively):
C7: "A = not A".
C8: "Not A = and
¹
not (not) (not (not A))".
C9: "Not Not (Not A) = and
¹
and ¹
and not
¹
not not (not not) (not not (not not (not A)))".
C10: "Not Not Not Not (Not Not (Not A)) = and
¹
and ¹
and not
¹
and ¹
and not not
¹
and not not
¹
and not not
¹
not not not not (not not not not) (not not not not (not not not not (not not
(not A))))".
And so on.
Of course, the same will need to be done with each "and", too
(the brackets are optional, as are the line numbers , "C1", "C2", "C3", etc.),
but that can be left to the reader.
If "A" is never equal to itself (and nothing else is,
either), then the above must follow (which dire dialectical dénouement can only
be denied by those who disagree with Trotsky's diktat).
But, we have been here
already.
[The 'relative stability' defence has already been neutralised,
here,
here and
here.]
Moreover, other things that Trotsky said in
IDM indicate that the above passage from his Notebooks is not a reliable
guide to his thinking:
"A sophist will respond that
a pound of sugar is equal to itself at 'any given moment'…. How should we really
conceive the word 'moment'? If it is an infinitesimal interval of time, then a
pound of sugar is subjected during the course of that 'moment' to inevitable
changes. Or is the 'moment' a purely mathematical abstraction, that is, a zero
of time? But everything exists in time; and existence itself is an uninterrupted
process of transformation; time is consequently a fundamental element of
existence. Thus the axiom 'A' is equal to 'A' signifies that a thing is equal to
itself if it does not change, that is if it does not exist." [Trotsky
(1971),
p.64.]
This at least confirms the accuracy
of the interpretation put on Trotsky's analysis of the LOI in this
Essay
-–
that is, in so far as any sense can be made of
what he was trying to say.
Finally, what TAR itself says about Trotsky's
argument seems to agree with the interpretation given here, as do other
commentators. [Cf., Rees (1998),
p.273.]
13a. Quantifiers in language are words like
"all", "every", "any", "some", "nothing", and "none", etc. Tensed quantifiers are
terms like
"always", "never" and "sometimes". On this, see
here,
here
and
here.
14. It could be objected to this that
"moments in time" are not objects, and
that time is simply one of the "modes of existence of matter". But, if time can
be measured (just like bags can be weighed) then Trotsky's criticisms must apply to it. Moreover,
if we accept what Lenin said about matter (that it is whatever exists
"objectively outside the mind" (a doctrine examined in detail in Essay Thirteen
Part One)),
time must be material, too -- unless, of course, it 'exists' only in the mind.
But, even if time isn't an object, Trotsky had to use objects to
make his point, namely, words. More on that
presently.
14a.
On the various systems for measuring time, see
here.
It could be argued that since moments in time follow on from each
other, it's not possible to measure one of them in order to compare it with an
other -- since the earlier one will no longer exist to make the comparison! This is not the case with
respect to objects that have to be weighed; they clearly exist side by
side during the entire process, facilitating comparison. Or so the objection
might go.
But can't moments differ even if we are unaware of it? And can't
two objects be weighed simultaneously, and the duration of each weighing
instance timed
simultaneously, too, with the latter durations compared just like their weights?
Concerning differing time intervals (or, at least, their
measurement), this is what a recent New Scientist article had to say:
"Clocks that gain or lose no more than a fraction of a
second over the lifetime of the universe could be on the way, thanks to a
technique for cutting through the 'heat haze' that compromises the accuracy of
today's instruments.
"The most accurate atomic clock we have now is regulated
by the electrons of a single aluminium
ion as they move
between two different
orbits
with sharply defined energy levels. When an electron goes from the higher energy
level to the lower it emits radiation of a precise frequency. That frequency is
used to mark out time to an accuracy of better than 1 part in 1017,
or 1 second in 3 billion years.
"That's pretty good, but it could be better. Infrared
photons emanating from the background cause the two energy levels to shift by
slightly different amounts, says
Marianna Safronova at the University of Delaware. That
affects the frequency of the emitted radiation to an unknown extent, adding a
small uncertainty to the clock's tick.
"Safronova reported this month at a conference in
Baltimore, Maryland, that by combining two different mathematical approaches,
she and her colleagues have now managed to calculate how much the energy gap
between the two levels changes.
"Using this information to correct an atomic clock could
in principle increase its precision to around 4 parts in 1019,
or about 1 second per 80 billion years. Such a clock could test whether the
fundamental constants of nature are changing, Safronova suggests." [New
Scientist 210, 2813, 21/05/11, p.15. Bold emphases added.
Quotation marks altered to conform to the conventions adopted at this site.]
Even with this new method, time intervals clearly change (over
billions of years), just as weights do (over shorter periods).
15.
Again, it could be insisted that all Trotsky needs is
the relative stability of the words he used, which won't have changed
significantly during the short intervals involved.
Unfortunately, Trotsky holed
that response well below the water line by declaring that:
"[A]ll bodies change uninterruptedly in size,
weight, colour etc. They are never equal to themselves… But everything exists in
time; and existence itself is an uninterrupted process of transformation…. Thus
the axiom 'A' is equal to 'A' signifies that a thing is equal to itself if it
does not change, that is if it does not exist." [Trotsky
(1971), p.64.]