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[Marxism] Re: Marxism Digest, Vol 17, Issue 58

Re: Jim Farmelant <farmelantj@xxxxxxxx>: Re: [Marxism] Re: [PEN-L] More Godel

On Fri, 18 Mar 2005 11:15:15 -0800 Rod Holt <rholt@xxxxxxxxxxxxxx>
writes:


Friends one & all:
Any body who says "... Gödel kicked the whole [formalist program] over," and then goes on to Hofsteader make me stop reading.
I have a question: Why don't people read GÖDEL in stead of reading about him?



Kurt Gödel's paper, "On formally undecidable propositions
of Principia Mathematica and related systems," can
be found online at: http://home.ddc.net/ygg/etext/godel/godel3.htm
It probably won't hurt to have a good grasp of
predicate logic, set theory, and the theory of recursive
functions to understand Gödel's proof, although
the underlying idea is not that hard to grasp, which
is that just as Russell and Whitehead were able to
show that the axioms of Arithmetic could be formulated
in the notation of mathematical logic, so the propositions
of Russell & Whitehead's mathematical logic could
be formulated in terms of the arithmetic of natural
numbers. Given this, it is possible to formulate
in Arithmetic a proposition of the G, which asserts
that it is not provable in terms of the axioms of
the arithmetic of natural numbers. This leads
to a liars-type of paradox, since if G is provable
then it must be false, but since we know it is
true, then it cannot be provable. Therefore,
the axiom set of arithmetic cannot be complete
since there are propositions in arithmetic that
are true but are not provable in terms of those
axioms. And since we are always free to add
these Gödelian propositions to the axioms
of arithmetic and since we can prove that even
with the addition of those propositions, the axiom
set remains incomplete, therefore there must
be an infinite number of Gödelian propositions.

Also, see:
http://www.math.hawaii.edu/~dale/godel/godel.html
http://planetmath.org/encyclopedia/GodelsIncompletenessTheorems.html
http://www.apronus.com/math/goedel.htm



Why don't Marxists take a little time off and learn a little mathematics, or at least enough to know the difference between "True" and "derivable," between logic and metalogic??
Does anybody here know what a Gödel number is? If not, then avoid propounding on Gödel, the dialectic, etc. ad nausium.
--rod
_______________________________________________


We had some material on the Thaxis list . If somebody is concerned with philosophical aspects of Marxism then it is essential that they deal with the formal logic too. I am repeating my posting to the Thaxis list.

Re : "Charles Brown" <cbrown@xxxxxxxxxxxxxxxxx> Does Gödel Matter?

The romantic's favorite mathematician didn't prove what you think he did.

By Jordan Ellenberg

the Washington Post's SLATE/Posted Thursday, March 10, 2005, at 4:27 AM PT

The reticent and relentlessly abstract logician Kurt Gödel might seem an
unlikely candidate for popular appreciation. But that's what Rebecca
Goldstein aims for in her new book _Incompleteness_, an account of Gödel's
most famous theorem, which was announced 75 years ago this October.
Goldstein calls Gödel's incompleteness theorem "the third leg, together with
Heisenberg's uncertainty principle and Einstein's relativity, of that tripod
of theoretical cataclysms that have been felt to force disturbances deep
down in the foundations of the 'exact sciences.' "

What is this great theorem? And what difference does it really make?

Mathematicians, like other scientists, strive for simplicity; we want to
boil messy phenomena down to some short list of first principles called
axioms, akin to basic physical laws, from which everything we see can be
derived. This tendency goes back as far as Euclid, who used just five
postulates to deduce his geometrical theorems.

But plane geometry isn't all of mathematics, and other fields proved
surprisingly resistant to axiomatization; irritating paradoxes kept
springing up, to be knocked down again by more refined axiomatic systems.
The so-called "formalist program" aimed to find a master list of axioms,
from which all of mathematics could be derived by rigid logical deduction.
Goldstein cleverly compares this objective to a "Communist takeover of
mathematics" in which individuality and intuition would be subjugated, for
the common good, to logical rules. By the early 20th century, this outcome
was understood to be the condition toward which mathematics must strive.

Then Gödel kicked the whole thing over.

Gödel's incompleteness theorem says:

Given any system of axioms that produces no paradoxes, there exist
statements about numbers which are true, but which cannot be proved using
the given axioms.
In other words, there is no hope of reducing even mere arithmetic, the
starting point of mathematics, to axioms; any such system will miss out on
some truths. And Gödel not only shows that true-but-unprovable statements
exist -- he produces one! His method is a marvel of ingenuity; he encodes
the notion of "provability" itself into arithmetic and thereby devises an
arithmetic statement P that, when decoded, reads:

P is not provable using the given axioms.

So a proof of P would imply that P was false -- in other words, the proof of
P would itself constitute a disproof of P, and we have found a paradox. So
we're forced to concede that P is not provable -- which is precisely what P
claims. So P is a true statement that cannot be proved with the given
axioms. (The dizzy-making self-reference inherent in this argument is the
subject of Douglas Hofstadter's Pulitzer Prize-winning _Gödel, Escher,
Bach_, a mathematical exposition of clarity, liveliness, and scope
unequalled since its publication in 1979.)

One way to understand Gödel's theorem (in combination with his 1929
"completeness theorem") is that no system of logical axioms can produce all
truths about numbers because no system of logical axioms can pin down
exactly what numbers are. My fourth-grade teacher used to ask the class to
define a peanut butter sandwich, with comic results. Whatever definition you
propose (say, "two slices of bread with peanut butter in between"), there
are still lots of non-peanut-butter-sandwiches that fall within its scope
(say, two pieces of bread laid side by side with a stripe of peanut butter
spread on the table between them). Mathematics, post-Gödel, is very similar:
There are many different things we could mean by the word "number," all of
which will be perfectly compatible with our axioms. Now Gödel's undecidable
statement P doesn't seem so paradoxical. Under some interpretations of the
word "number," it is true; under others, it is false.

In his recent New York _Times_ review of _Incompleteness_, Edward Rothstein
wrote that it's "difficult to overstate the impact of Gödel's theorem." But
actually, it's easy to overstate it: Goldstein does it when she likens the
impact of Gödel's incompleteness theorem to that of relativity and quantum
mechanics and calls him "the most famous mathematician that you have most
likely never heard of." But what's most startling about Gödel's theorem,
given its conceptual importance, is not how much it's changed mathematics,
but how little. No theoretical physicist could start a career today without
a thorough understanding of Einstein's and Heisenberg's contributions. But
most pure mathematicians can easily go through life with only a vague
acquaintance with Gödel's work. So far, I've done it myself.

How can this be, when Gödel cuts the very definition of "number" out from
under us? Well, don't forget that just as there are some statements that are
true under any definition of "peanut butter sandwich" -- for instance,
"peanut butter sandwiches contain peanut butter" -- there are some
statements that are true under any definition of "number" -- for instance,
"2 + 2 = 4." It turns out that, at least so far, interesting statements
about number theory are much more likely to resemble "2 + 2 = 4" than
Gödel's vexing "P." Gödel's theorem, for most working mathematicians, is
like a sign warning us away from logical terrain we'd never visit anyway.

What is it about Gödel's theorem that so captures the imagination? Probably
that its oversimplified plain-English form -- "There are true things which
cannot be proved" -- is naturally appealing to anyone with a remotely
romantic sensibility. Call it "the curse of the slogan": Any scientific
result that can be approximated by an aphorism is ripe for misappropriation.
The precise mathematical formulation that is Gödel's theorem doesn't really
say "there are true things which cannot be proved" any more than Einstein's
theory means "everything is relative, dude, it just depends on your point of
view." And it certainly doesn't say anything directly about the world
outside mathematics, though the physicist Roger Penrose does use the
incompleteness theorem in making his controversial case for the role of
quantum mechanics in human consciousness. Yet, Gödel is routinely deployed
by people with antirationalist agendas as a stick to whack any offending
piece of science that happens by. A typical recent article, "Why
Evolutionary Theories Are Unbelievable," claims, "Basically, Gödel's
theorems prove the Doctrine of Original Sin, the need for the sacrament of
penance, and that there is a future eternity." If Gödel's theorems could
prove that, he'd be even more important than Einstein and Heisenberg!

One person who would not have been surprised about the relative
inconsequence of Gödel's theorem is Gödel himself. He believed that
mathematical objects, like numbers, were not human constructions but real
things, as real as peanut butter sandwiches. Goldstein, whose training is in
philosophy, is at her strongest when tracing the relation between Gödel's
mathematical results and his philosophical commitments. If numbers are real
things, independent of our minds, they don't care whether or not we can
define them; we apprehend them through some intuitive faculty whose nature
remains a mystery. From this point of view, it's not at all strange that the
mathematics we do today is very much like the mathematics we'd be doing if
Gödel had never knocked out the possibility of axiomatic foundations. For
Gödel, axiomatic foundations, however useful, were never truly necessary in
the first place. His work was revolutionary, yes, but it was a revolution of
the most unusual kind: one that abolished the constitution while leaving the
material circumstances of the citizens more or less unchanged



The problem with Godel's theorem is because it is based on a first-order logic with deep Fregean features. There are many logics like IF-first order logic which avoid the theorem. There are many papers on it including Hintikka's joint paper. They correctly say the problem starts at the syntax of the underlying language itself. Most mathematicians actually work in higher order logics with the axiom of choice.

Godel's theorem does mean something for those working in some types of constructivism. It's significance is in the enormous debates that it has generated.


A. Mani
Member, Cal. Math. Soc

Re:

1. Re: Does G?del Matter? (Oudeyis)
2. Re: Les Shaffer on Kurt G?del (Ralph Dumain)
3. Re: Re: [Marxism-Thaxis] Les Shaffer on Kurt G?del
(Jim Farmelant)
4. Significance of incompleteness and uncertainity in science
for dialectics (Oudeyis)
6. Les Shaffer on Heisenberg etc. (Jim Farmelant)
7. godel etc (Charles Brown)
8. godel etc (Charles Brown)
9. Les Shaffer on Kurt G?del (Charles Brown)
10. Re: Les Shaffer on Kurt G?del (Ralph Dumain)
11. More Godel (Charles Brown)
12. godel etc (was ...) (Charles Brown)
13. Re: More Godel (Ralph Dumain)
14. Does G?del Matter? (Charles Brown)
15. Re: Does G?del Matter? (Ralph Dumain)
18. More Godel (Charles Brown)
19. Re: More Godel (Ralph Dumain)
20. More Godel (Charles Brown)
23. Re: Does G?del Matter? (Oudeyis)
24. Re: More Godel (Oudeyis)
25. Re: More Godel (Waistline2@xxxxxxx)


-

There were lots of points spread out. I would like to comment on some.

It is obvious that theories of knowledge in general are not intended to be axiomatic systems. For first order theories we have a duality available between model-theory (semantics) and proof theory (syntax). One can always be transformed into the other formally. Marxist theories of knowledge include both semantic and methodological aspects. All mathematical systems are dialectical to different degrees. These are yet to be formalized as of now.

In physics they try all kinds of philosophy and so much non-physics tends to be done. Godel's theorem concerns AI and implementations thereof. Theoretically it can be written as a physical law in different physical systems ! Now if they have not done it as yet that can be a topic for a M.Phil theses.



A. Mani
Member, Cal. Math. Soc













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