Re: [OPE-L] is algebra dialectical and vice versa?

From: glevy@PRATT.EDU
Date: Tue Sep 13 2005 - 17:42:36 EDT

---------------------------- Original Message --------------------------
Subject: is algebra dialectical and vice versa?
From:    "Jurriaan Bendien" <>
Date:    Tue, September 13, 2005 3:29 pm

One way to look at it is, that dialectical reasoning is *non-arbitrary*
reasoning, which complements formal logic, yet evades formal logic, because
it does not by logical necessity lead to any specific conclusion. The idea
here is that a way of thinking or action can be very reasonable, without
conforming to formal-logical principles.

It is a characteristic of most formal-logical systems and procedures that
the conclusions are true, if the premisses are true, yet, the logical
system itself provides no formal-logical method  for *inducing* those
premisses themselves, only instructions of the type that if X holds, Y
must apply or be true. This problem is frequently encountered in
modelling, where the realism of the model depends on how realistic the
assumptions are - yet the model itself cannot usually provide compelling
reasons for why certain assumptions should be imported, that is the
choice of the modeller. The model can only show the logical or empirical
implications of different assumptions. While the model may seem
"objective" and "neutral", it is embedded in a theory which may or may
not be made explicit.

Of course, you can teach a computer program to look for a set of premisses,
to check whether they apply, and if they apply, draw inferences from them.
But from what I understand, the program typically cannot be made to perform
logical operations such that it unpredictably "goes its own way" in the
data, according to principles it "invents for itself"; it is limited to the
logical possibilities of the program, in the implications or data that it
can identify.

Without wanting to specify exhaustively what dialectics is all about, I
tend to think that in general, Marx intended by dialectical logic
something like a *non-arbitrary* inferential or abstractive system, which
moves between different logical levels (for example, the part and the
whole, or the particular and the general), in order to introduce premisses
in a non-arbitrary way. It aims to capture facets of everyday reality
which are non-logical in the formal sense, yet also non-arbitrary and
reasonable, since they occur within definite limits, with definite
dynamics, which are unambiguously and practically specifiable.

In Aristotelian logic, it is often observed, you have the law of identity,
the law of non-contradiction and the law of the excluded middle term, yet
these principles, which correspond to the primary cognitive processes of
stimulus identification, stimulus discrimination and stimulus
generalisation, and which provide conditions for "making sense" in a stable
system of meanings, cannot do full justice to realities which, even
although they can be meaningfully understood, in fact do not conform to
the Aristotelian laws at all. As Heraclitus said, you cannot step in the
same river twice; things change. The reason why these realities don't
conform to Aristotelian logic is, as is also often noted, because they
involve motion, transition, development, antagonism and mediations; at the
most basic level, there are things which could not exist without each
other, yet also negate each other in some important sense. If ordinary
language is used, this must lead to paradoxes, yet these paradoxes exist,
only because of the semantic requirement of constancy in meaning (as a
condition for making sense).

Hegel then explores the idea of how meanings could mutate and evolve, but
in a non-arbitrary way, for which he claims to have discovered and
elucidated a series of principles, or modus operandi. Marx's initial
criticism of this dialectical procedure is that it is speculative, and
that it could, in truth, end up in all sorts of different ways. True,
meaning is not fixed once and for all, but by that very fact, meaning can
evolve in all sorts of ways. The real question is, why meaning evolves in
certain ways and not others, but that cannot be understood purely in terms
of the rules of logic or semantics internal to those meanings.

Marx provides many examples of this. Thus, Hegel and the Hegelians claimed
to have proved "dialectically" than a certain event must necessarily occur,
or existed by necessity, but in truth there is no such necessity at all, it
is more or less a linguistic trick, relying on word-usage. The way out of
that, says Marx, is to abandon speculation about reality, and base oneself
solidly on empirically and historically verifiable facts. Instead of an
object in thought, we focus on a real object, the substantive thought being,
that this object has its own specific dialectical properties, which have to
be discovered rather than assumed, and which can be known. This presumes
some kind of scientific realism. Hegel however seems to fall into the
illusion that Thought or the Logos evolves the universe out of itself, and
then pretends that he can have a true cognition of this Logos through
applying a "dialectical method". This, Marx suggests, ends up in mysticism,
precisely because it is thought totally divorced from practical experience.

The next step for Marx is then a "dialectics of inquiry" or a "logic of
discovery" in the Popperian sense. It is not formally logical, yet the
inferences and implications drawn from working over the empirical material
are non-arbitrary and reasonable - at each step of the way, it is possible
to specify, why or how the path of inquiry has moved here or there, and why
or how certain conclusions are arrived at. A good illustration of this, is
the available preparatory manuscripts for Das Kapital, such as the
Grundrisse. Marx aims to creatively and critically draw a whole lot of new
distinctions, but in a non-arbitrary way, moving between different logical
levels and different known facts. He is thinking simultaneously what does
this mean to me, what does it mean to others, and what should it mean or
what does it really mean.  In this process, what is particularly important,
is the vantage point from which the subjectmatter is approached, and for
Marx that vantage point is materialistic, empirical and practical.

This is followed by a "dialectics of presentation or communication" (a
"logic of justification" in the Popperian sense, perhaps, though without a
pejorative connotation), which aims to tell a rational story, which fully
reveals the dialectical properties of the subjectmatter. Marx thus says his
aim in Das Kapital was to "bring a science by criticism to the point where
it can be dialectically presented". He must now show, why some conditions
must necessarily follow from other conditions, in a way which is sound in
terms of formal logic. But he must simultaneously show the dialectical
properties of the subjectmatter in a non-arbitrary way, and introduce new
concepts and facts in a non-arbitrary way. Thus, he aims to unfold the
totality of capitalism from what he calls its "cell form", i..e. the
commodity, the object of commerce, showing how each practical contradiction
leads, through the way that it is resolved, to the next contradiction. The
same story could obviously have been told in innumerable different ways,
but in ordering his discoveries and conclusions, he felt that this had to
be the starting point, from which everything else could be shown to
follow, without arbitrarily adducing new premisses, or making wild

The dialectical process is then completed by a dialectic of theory and
action, or some kind of praxis; having reordered a complex reality in
thought, using non-arbitrary methods, with distinctions that reveal its
dialectical properties, the conclusions reached reflect back on the knowing
subject, who then aims to orient his activity with the knowledge gained, so
that action likewise occurs in a non-arbitrary, yet theoretically informed
way. In particular, this appears to mean for Marx both that meanings are no
longer confused or conflated with the reality they represent, so that one
is mentally trapped in a language or ideology used to talk about that
reality, and that the appropriate meanings are attached to the real object
of dialectical inquiry: a "way of perceiving" which unlocks stasis, and
permits the object to be changed, in accordance with a human purpose.

I think that just as Marx rejected a "philosophy of history", he likewise
rejected dialectics as a "lever for theoretical constructions and schemas".
It was not a question of spinning out ideas in thought, through evolving
subtle distinctions from word-meanings, but of developing one's thought
systematically, step by step, from a real, empirically verifiable object.
It was that real object which provided a non-arbitrary, practical basis
making conceptual distinctions in the first place. Reality could be carved
up in zillions of different ways, but only some ways could reveal the true
essence of the real object, and depict it in an objective way, which did
full justice to the contradictory realities it contained. The whole aim of
the method was to arrive at those distinctions which were truly adequate to
the object, i.e. not sucked out of one's thumb, but grounded in the
verifiable reality of the object.

There could therefore never be only "one dialectical method", anymore than
there can be "one scientific method"; this was a nonsense, because the
method applied had to be adequate to the specific real object of study, and
was in part developed from the cognition of that real object. Most
importantly, the dialectical properties had to be *discovered* in the real
object itself, rather than superimposed on the real object. In this sense,
science uses dialectical methods all the time, to the extent that insights
are mooted which are perfectly reasonable, but are not *necessary*
formal-logical inferences.

Dialectics is frequently accused of being metaphysical, but in that case,
we are all metaphysicians to some extent or other, in the sense that we
need some premisses to get our logical systems off the ground, yet the
logical systems themselves do not provide a foolproof method for making
the *choice* about what premisses to adopt. That is the problem. Certain
aspects of the materialist interpretation of history can without
qualification be regarded as metaphysical - e.g. statements that a
knowable, mind-independent world exists, or that historical development
can be understood rationally as a law-governed process, or that people
typically do things first, and only later fully comprehend why. There is
no way to avoid that, all we can say is that the findings of research
prove the validity of the method of inquiry used, or not; there is no
other way of validating any particular method.

In addition, we can say that, whereas we will adopt a certain metaphysic
anyway, we are aware that we do it, and that it could be wrong. We must all
believe something, and the only rational justification for that can be,
that certain beliefs are experientially more adequate or effective to our
practical purpose than others. In this sense, Engels mentions "the test of
practice" because there is ultimately no other test possible; and for that
reason, it is necessary to keep a truthful and honest record of our
experiments, and not present practical evidence other than it truly is.

But what about algebra? I would say, that algebra becomes dialectical, when
it is able to resolve an algebraic paradox without totally invalidating the
meaning of the algebraic terms that constitute the paradox. In this sense,
I do not rule out that dialectical properties could be discovered in the
application of the theory of algebra.


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