__?__

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[Eduardo J. Dubuc]

Is the following nonsense ?

http://arxiv.org/abs/1112.2141

A new computational method that uses polynomial equations and dynamical
systems to evaluate logical propositions is introduced and applied to
G\"odel's
incompleteness theorems. The truth value of a logical formula subject to
a set
of axioms is computed from the solution to the corresponding system of
polynomial equations. A reference by a formula to its own provability is
shown
to be a recurrence relation, which can be either interpreted as such to
generate a discrete dynamical system, or interpreted in a static way to
create
an additional simultaneous equation. In this framework the truth values of
logical formulas and other polynomial objectives have complex data
structures:
sets of elementary values, or dynamical systems that generate sets of
infinite
sequences of such solution-value sets. Besides the routine result that a
formula has a definite elementary value, these data structures encode
several
exceptions: formulas that are ambiguous, unsatisfiable, unsteady, or
contingent. These exceptions represent several semantically different
types of
undecidability; none causes any fundamental problem for mathematics. It is
simple to calculate that G\"odel's formula, which asserts that it cannot be
proven, is exceptional in specific ways: interpreted statically, the formula
defines an inconsistent system of equations (thus it is called
unsatisfiable);
interpreted dynamically, it defines a dynamical system that has a periodic
orbit and no fixed point (thus it is called unsteady). These exceptions
are not
catastrophic failures of logic; they are accurate mathematical
descriptions of
G\"odel's self-referential construction. G\"odel's analysis does not
reveal any
essential incompleteness in formal reasoning systems, nor any barrier to
proving the consistency of such systems by ordinary mathematical means.
\\ ( http://arxiv.org/abs/1112.2141 ,  60kb)



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Re: __?__

[Mike Stay]

On Wed, Dec 14, 2011 at 1:35 PM, Eduardo J. Dubuc <edubuc@dm.uba.ar> wrote:
> Is the following nonsense ?
>
> http://arxiv.org/abs/1112.2141

He thinks that Godel made an error, namely failing to generalize from
two-valued logic to sequence-valued logic.  He then shows that
predicate logic can be encoded as polynomials, which is not new, and
sweeps quantification under the rug since it's too hard.

So while not strictly nonsense, it's not the earth-shattering wonder
the author thinks it is.
-- 
Mike Stay - metaweta@gmail.com
http://www.cs.auckland.ac.nz/~mike
http://reperiendi.wordpress.com


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Re: __?__

[Andrej Bauer]

On Wed, Dec 14, 2011 at 10:35 PM, Eduardo J. Dubuc <edubuc@dm.uba.ar> wrote:
> Is the following nonsense ?
>
> http://arxiv.org/abs/1112.2141

The connection between propositional logic and algebra has been known
for a long time and exploited in computational complexity, the
buzzword to search for is "algebraic proof complexity".

At a first glance the present paper rediscovers the fact that Boolean
algebras correspond to Boolean rings, and that therefore one may turn
propositional logic into systems of polynomial equations. Because the
paper aims to explain something about Gödel's theorems, one see what
it says about Peano axioms, in particular the principle of induction,
and about _predicate_ logic. It says nothing about the former, and it
has a short section 4.6 about the latter. This section states that
"quantifiers are not a problem because there is Tarski's quantifier
elimination for real-closed fields". I think this is a rash
conclusion. It is not clear what quantifier elimination over
real-closed fields has to do with Boolean rings. If the author wants
to take a system of polynomial equations over a Boolean ring and
pretend that it is over the reals, then he should argue very carefully
why that makes any sense (which it does not, as far as I can see).

With kind regards,

Andrej


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Re: __?__

[Vaughan Pratt]

On 12/14/2011 1:35 PM, Eduardo J. Dubuc wrote:
> Is the following nonsense ?
>
> http://arxiv.org/abs/1112.2141

The concluding paragraph of the conclusion, on page 37, reads as follows.

"Kurt Gödel provided innovative but convoluted proofs of some normal 
properties of a simple recurrence relation, accompanied by spectacular 
misinterpretations. Gödel’s incompleteness conjectures have beguiled 
generations of readers lost in the sea of his complex proofs; his 
unwarranted conclusions have smashed mathematical reason against the 
petrified relics of ancient misunderstandings. But other thinkers have 
seen more clearly; most importantly George Boole, whose brilliant 
synthesis of algebra and logic has shown the way to modern computer 
science. Perhaps it is time to raise the lamp in Boole’s lighthouse, and 
to let the beacon of unified mathematics and logic guide a renewed quest 
for the rational understanding of our world."

Let's go through this point by point.

1.  "...convoluted proofs..."  Gödel's 1931 paper was 25 pages. 
Norman's is 48.  Sounds like a pot calling the kettle black.

2.  "Gödel’s incompleteness conjectures have beguiled generations of 
readers lost in the sea of his complex proofs..."  Evidently Norman is 
one of those readers, since he appears to have mistaken a legitimate 
proof for a conjecture.

3.  "But other thinkers have seen more clearly; most importantly George 
Boole, whose brilliant synthesis of algebra and logic has shown the way 
to modern computer science."  Since Boole's book predated Gödel's proof 
by 77 years, Norman's implication would seem to be that logicians 
including Gödel had failed to grasp Boole's deep insights throughout 
that period, and furthermore throughout the following 80 years up to the 
present, obliging Norman to bring them to light.

Similar logic would show that Newton invented quantum mechanics on the 
basis of his corpuscular theory of light.

If there is any legitimate point in Norman's article, it would be along 
the lines of Kripke's semantics of the liar paradox (to which Haim 
Gaifman had a nice follow-up).  Norman may well have rediscovered 
Kripke's insight, but in that case he should either discuss the 
connection or explain why his references make a reference to Kripke 
unnecessary.  Boole himself is not sufficient for that purpose because 
arithmetic mod 2 did not occur to him as a model of his axioms.

Vaughan Pratt


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