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

From: Ian Wright (iwright@GMAIL.COM)
Date: Wed Sep 14 2005 - 14:38:24 EDT

Hi Jerry

And another (more basic) question is: *what is dialectical?* I think you
> are using this term, for instance, in a very different way that those who
> refer to "systematic dialectics" do. 

I may well be. I was trying to use it in the Hegelian sense, although, as I 
mentioned, I'm no expert on Hegel, so I'm happy to be corrected. As far as I 
can understand Hegel, "dialectical" describes the nature of "being". Being 
is dialectical. I interpret this in a similar way to the claims of, say, 
"atomism", that being is ultimately composed of corpuscles, or such like, 
that interact. According to atomism, being is atomistic, according to Hegel 
being is dialectical. Hegel's categories, are, I think, a deduction of some 
necessary properties of being, but using the innovative methodology that 
valid inferences, the logical moves, must themselves be derived, organically 
so to speak, from the prior categories. Unlike formal logic, which takes a 
collection of rules of valid inference as independent variables, Hegel 
starts at an earlier point, and tries to self-referentially infer the valid 
rules of inference in an incremental fashion. In this sense, Hegel's 
categories are dependent variables, necessary consequences of his simple and 
abstract starting point. His aim, I suppose, to find the abstract necessary 
features of both nature and mind, and unify natural and logical necessity. 
So I think "dialectical" refers to both the method of derivation and the 
results of derivation.

Of course, this is very hard to understand, and the Science of Logic is a 
real puzzler. But as a computer scientist trying to make sense of this I'm 
repeatedly hit by the strong analogies to computation. We could perhaps 
mirror Hegel's method, but start, not from being, but from the Turing 
Machine, which is an incredibly simple (feedback) device. By examining how 
the TM operates we can very quickly deduce a complex ontology (or 
categories) of states and processes that it supports. As we know from 
practical and theoretical experience, the kinds of causal processes that a 
TM can support is in some sense "universal".

The level of abstraction that the theory of computation operates at is 
similar to the level of abstraction of Hegelian dialectics. The analogies do 
not end here: Paul's latest one, about the oscillator required to 
synchronize causal updates, is new to me, and very neat. A simple flip-flop 
mechanism is indeed the negation of the negation, and if you wish to 
implement a Turing machine in reality, you need something like this.

The Church-Turing thesis, or at least realist interpretations of it along 
the lines of Deutsch, conjecture that the universe is a computation, or that 
being is computational. It is called a thesis, not a theorem, because it 
cannot be proved.

Can Hegel really "prove" the necessity of his categories?

> Perhaps. And another question: is *calculus* dialectical and 
> vice versa? Recall the debates in recent years about the merit of 
> non-linear mathematical methods in terms of whether they are 
> superior (than linear algebra, for instance) in terms of explaining the 
> subject (capitalism). I.e. what is the math method that can best 
> express the character of capitalism?

Particular mathematical formalisms, at least if they are constructive in 
some sense, are particular Turing machines that may be implemented on the 
universal Turing machine. The theory of computation is separate from 
mathematics, more general than any particular formalism, whether linear or 
non-linear. The concept of computation causes controversy in the philosophy 
of mathematics: the mathematical reality of some mathematical objects is 
questionable from a computational point of view.

But to answer your question: I think it depends on what you're trying to do.

Within the context of *systematic dialectics* (of which Hegelian thought
> is an example) computation (and algebra and calculus) may have _a_
> role, but I am skeptical of a claim that a systematic reconstruction of
> the subject matter of capitalism in thought can be expressed through
> any of these math methods. However, computer languages and 
> computation are not my strong suit (a huge understatement!) so I
> will listen carefully to what others including yourself have to say on 
> this topic.

I am trying to distinguish computation from mathematics, in the sense it is 
more general than any particular mathematical formalism, and is more like a 
very general theory of causality, applying to all domains, natural and 
logical, very much like "dialectics". 


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