[Marxism] Re: godel etc (was ...)

[Les Schaffer <schaffer@xxxxxxxxxxxxx>]

Carlos A. Rivera wrote:

> I argue that  incompleteness in mathematics and uncertainity in quantum
> mechanics actually point to materialist dialectics. As dynamic,
> never-ending systems, they exhibit the same continous struggle that
> dialectics call, while firmly footed on a materialist grasp on reality.
>
> Yeah, postmodernists eat your heart out!!!


the thing that impresses me this time around is expressed nicely by R B
Braithwaite in his Introduction in the Dover edition of Godel's paper:

"... Godel, in this paper which established his two great theorems by
methods which are constructive in a precise sense, on the one hand
showed the essential limitations imposed upon constructivist formal
systems (which include all systems basing a calculus for arithmetic upon
"mathematical induction"), and on the other hand displayed the power of
constructivist methods for establishing metamathematical truths."

Godel's efforts in 1930 depended on a clever arithmetization of
metamathematical statements (so-called Godel numbering), which
Braithwaite likens to Descartes handling of problems in geometry by the
introduction of coordinate systems and a reduction to algebra. by
(crudely speaking) mirroring arithmetic in statements about arithmetic,
he was able to establish limitations in the formal program of Hilbert
and others. In some fundamental sense human (mathematical) activity
cannot be reduced to formalism alone, such  formal systems are incomplete.

Somehow this incompleteness was interpreted by philosophers and
popularizers as a fundamental attack on the structure of mathematics
itself. but is any marxist here surprised to learn there is more to
sensuous (mathematical) activity than formal reasoning? i doubt it. in
some way what Godel really demonstrated was that the formal
metamathematical systems created at the turn of the 20-th century were
not so meta and outside of the system they were judging, so-to-speak, as
had been supposed. it takes more than arithmetic to practice true
arithmetic.

> "The question whether objective truth can be attributed to human
> thinking is not a question of theory but is a practical question. Man
> must prove the truth - i.e. the reality and power, the this-sidedness of
> his thinking in practice. The dispute over the reality or non-reality of
> thinking that is isolated from practice is a purely scholastic question."

one of the intriguiging aspects of Godel's paper was the construction of
a __formally__ undecidable proposition that was demonstrably __true__.
Braithwaite again:

"The undecidability of some arithmetical propositions within the
deductive system S may be classed among the syntactical metamathematical
characteristics of the system S (represented by the calculus P [Les: the
formal system P]) for the reason that this undecidability derives from
the undecidability of some formulae within the calculus which represents
S. Deductive systems, unlike calculi [Les: formal systems] have also
semantical metamathematical characteristics; in particular their
propositions have or lack the semantical property of being true -- what
Godel in his introductory Section 1 calls being "correct as regards
content" (inhaltlich richtig). Connecting the syntactical property of
being provable with the semantical property of being true ... gives an
additional kick to the undecidability in S of g {Les: g is the formally
undecidable but true proposition] -- by adding that g is true.  ... This
metamathematical argument, which combines semantical with syntactical
considerations, establishes the truth  of an arithmetical proposition
which cannot be proved within S.

In his introductory Section 1 Godel intermingles semantical with
syntactical considerations in sketching a proof of the undecidability of
g ... The distinction between what is syntactical and what semantical
was not made explictly until a year or two later (by Tarski, whose work
included rigorously establishing unprovability theorems that were
semantical) ... "

while thinking about Godel's work this afternoon i was reminded of long
lunches i had with a computer science professor at Cornell back in the
mid-80's. he had a ton of $$$$$ from the US military for investigating
programming systems that could be proven errorless via machine. this was
at the time of Reagan's Star Wars and the (rather mushy) criticism that
it could never work in practice because it was too complex a system to
be trusted to work without testing, but almost by definition could never
be properly tested short of a real nuclear attack, an event the system
purportedly was designed to prevent. I checked recently and it seems my
former lunchtime companion is still receiving millions of military
dollars to develop automated procedures for producing error-free and
secure software/firmware for military applications.

if you think Godel should have put a stop to all this, think again.

The military-industrial complex aside, far from undermining the
foundations of mathematics, Godel succeeding in opening a whole new
avenue of investigation. via the work of Tarski, Barkeley Rosser (father
of marxmail alumni J Barkeley Rosser), Church/Turing, and Gregory
Chaitin, we now have deep connections between mathematics, computational
systems, and, more recently, physics.

Les Schaffer



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