Date: Fri, 28 Jun 1996 00:21:10 +0100
Subject: Re: Waiting for Goedel


Zeynep wrote:

>Godel's famous theorem (as paraphrased into English by Douglas Hofstadter) is :
>"All consistent axiomatic formulations of number theory include undecidable
>propositions".
>
>Which means, that within each -closed- system, one may make a claim, that
>can not be proven to be true or false *within* that system.
>
>I think one may believably claim that human brain does not work in a way
>different than computers, but is a much more complicated network, resulting
>in associative thinking yet unknown to computers. (Something like an ever
>intertwined hypertext, to draw an anology from the internet).
>
>I don't know what the original quote was related to.
>
>To make it clearer, let me confess I'm lying. Am I? If so, I'm not. If I'm
>not lying then I am.


Well, our brain operates as a biological machine to help us in

1) orientation (which is really an aspect of

2) goal satisfaction

An interesting thing about it is that it's utterly insensitive, while the
rest of us is very sensitive, responding to environmental and hormonal
signals with the appropriate (!) reaction.

It's a single-user computer, purpose-developed (teleology rules, OK!) to
help satisfy a biological individual's social (food, shelter) and sexual
needs in a species framework. This is not what desktop computers were
designed for originally. Of course, they did represent extensions of our
biological capabilities, and the extensions they and their descendants will
provide will fit more and more naturally into our way of solving individual
and species needs.

One of the brain's tasks is thinking, both as a routine operation (2+2=4)
and in the sense of a search for valid axioms, first principles and such
(what is a number? what is =?). Now, what Zeynep is pointing out is that
you can't apply the rules for the routine operations (what Aristotle called
analytical logic, didn't he?) to the creative search for principles
(dialectical logic) without getting into trouble. Goedel's theorem
formalizes this.

I think there's a parallel to Marx's aphorism about each epoch only setting
itself problems it can solve. If you like, each epoch has its own
principles which are thrashed out in practice and eventually formalized
(bit like the Reformation and the Enlightenment, for instance). The
concrete advances and solutions fall within this practical-theoretical
framework. (I should imagine this is roughly what Sartre is saying in his
Critique of Dialectical Reason, but I haven't read it, so I'm probably way
off target.) And the working out of the various aspects of practice in the
epoch can more and more be left to routine operations as it more and more
'runs itself'.

We are in the final stages of the Capitalist Age, and the earliest stages
of the  Socialist Age, and our present epoch of transition from the one to
the other offers the new spectacle of a necessary fusion of principles and
practice.

One question which arises given this fusion, is whether logic will see a
breakthrough in the sense of finally arriving at a formalization of the
search for and evaluation of first principles -- values, priorities and
all. Hegel talks somewhere of Apodeictic Logic, the logic of demonstration.
My guess is that it won't come anywhere near it in general terms -- though
it might get very close in one or two specialized areas -- until socialism
is established as the dominant mode of production worldwide, removing the
antagonistic class interests that bedevil all discussion of (and based on)
principles in the biological (nature/nurture) and social sciences
(economics and sociology).

Another guess is that the removal of capitalism and its antagonisms will
remove a lot more obstacles in the way of really human development of
'hard' science (maths, including its foundations, physics etc) than most of
us thought existed. For starters, the universal mastery of maths
fundamentals (including calculus of course) that will follow on the freeing
of education from fear ('mind-forged manacles' that a lot of us
internalize) and compartmentalization, will make possible a completely new
level and relevance of public debate in relation to selecting alternative
courses of action. And this discourse in turn will sharpen the questions
being investigated in front-line research.

Most people know the amazing impact on the depth and breadth of knowledge
in a subject that can be made if it is studied in the company of even a
single kindred spirit. Just imagine a world of kindred spirits!

Finally, all the Goedel stuff is very self-reflective, and capitalism
cannot tolerate anything self-reflective. So a lot of the investigations
being pursued in basic maths and natural science today are in fact
transcending the bounds of capitalist society. Behaviourism (capitalist
science par excellence) dogmatically proclaims an invisible, abstract,
unknowable Thing-In-Itself subject. Socialist science takes the real, live,
flesh-and-blood human subject as its starting point.

My friend and teacher, Goeran Printz-Paahlson, wrote a poem in Swedish
about this kind of thing in the early sixties, called Turing Machine.
(Might well have been when he was at Berkeley). There's a good translation
he did himself, but I haven't got it to hand right now, so I'll hack one.
It ends:

        De imiterar i spraaket. I oaendliga
        slingor, laengre and laengre tillbaka i sin retraett
           mot subtilare
        algoritmer, mera rekursiva funktioner.

        De aer konsekventa och beskriver sig sjaelva.
        Som naer en man med en handspegel tryckt mot sin naesa
           framfoer en spegel
        ser i oaendlig rad samma bild maangfaldigas

        i en krympande, moerknande korridor av glas.
        Det aer en Goedel-teorem lika gott som naagot.
           Han ser oaendlig-
        heten, men det han inte ser aer sitt ansikte.


        (They imitate in language. In endless
        loops, farther and farther back, retreating
           towards more subtle
        algorithms, more recursive functions.

        They are consistent and describe themselves.
        As when a man with a handmirror pressed to his nose
           in front of a mirror
        sees the same image multiplied in a row without end

        in a shrinking, darkening corridor of glass.
        That's a Goedel theorem as good as any.
           He sees eternity,
        but what he doesn't see is his own face.)


Anyone read (reread?) Phenomenology of the Spirit recently?

Cheers,

Hugh




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