[Marxism] Re: Marxism Digest, Vol 17, Issue 58
A. Mani
a_mani_sc_gs at vsnl.net
Sat Mar 19 15:22:53 MST 2005
Re: Jim Farmelant <farmelantj at juno.com>: Re: [Marxism] Re: [PEN-L] More Godel
>On Fri, 18 Mar 2005 11:15:15 -0800 Rod Holt <rholt at planeteria.net>
>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 at michiganlegal.org> 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 at aol.com)
>
>
> -
>
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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