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) [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On Wed, Dec 14, 2011 at 1:35 PM, Eduardo J. Dubuc <edubuc@dm.uba.ar> wrote:
Is the following nonsense ?
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On Wed, Dec 14, 2011 at 10:35 PM, Eduardo J. Dubuc <edubuc@dm.uba.ar> wrote:
Is the following nonsense ?
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On 12/14/2011 1:35 PM, Eduardo J. Dubuc wrote:
Is the following nonsense ?
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (4)
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Andrej Bauer -
Eduardo J. Dubuc -
Mike Stay -
Vaughan Pratt