The thesis is:
The Gödel incompleteness can be modeled on the alleged incompleteness of quantum mechanics. Then the proved completeness of quantum mechanics can be reversely interpreted as a strategy of completeness as to the foundation of mathematics.
That argument supposes that the Gödel incompleteness originates from the deficiency of the mathematical structure, on which it is grounded. Furthermore, one can point out that generalized structure, on which completeness is provable and thus it can serve as a reliable fundament of mathematics.
Set theory and arithmetic were what was put as the base of mathematics. However, it is a random historical fact appealing to intuition or to intellectual authorities such as Cantor, Frege, Russell, Hilbert, "Nicolas Bourbaki", etc. rather than to a mathematical proof. Even more, the so-called Godel incompleteness theorems demonstrated that set theory and arithmetic are irrelevant as the ground of mathematics rather than no relevant branch of mathematics allowing of self-grounding though the orthodox view.
One can utilize an analogy to the so-called fundamental theorem of algebra:
It needs a more general structure than the real numbers, within which it can be proved.
Analogically, the self-foundation of mathematics needs some more general structure than the positive integers in order to be provable.
The key for a relevant structure is Einstein's failure to show that quantum mechanics is incomplete. The incompleteness of set theory and arithmetic and the alleged incompleteness of quantum mechanics can be linked. The close friendship of the Princeton refugees Godel and Einstein might address that fact. However, Godel came to Princeton in 1940 much after the beginning of Einstein's attempts to reveal that and how quantum mechanics was incomplete. In particular, the famous triple article of Einstein, Podolsky, and Rosen "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" pointed out as a kind of theoretical forecast as to the phenomena of entanglement and thus of quantum information was published in 1935. So, there should exist a common mathematical structure underlying both \incompletenesses" and in turn interpretable as each of them.
The mathematical formalism of quantum mechanics is based on the complex Hilbert space featuring by a few important properties relevant to that structure apt to underlie mathematics:
1. It is a generalization of positive integers: Thus it involves innity.
2. It is both discrete and continuous (even smooth): Thus it can unify arithmetic and geometry.
3. It is invariant to the axiom of choice: Thus it can unify as the externality and internality of an innite set as the probabilistic and deterministic consideration of the modeled reality as well as even model and reality in general.
The target of the presentation is:
I. Those three properties of the complex Hilbert space to be demonstrated.
II. A simple mathematical structure underlying both the Godel incompleteness and the alleged incompleteness of quantum mechanics to be described explicitly.
III. The undecidable statements according to the Godel incompleteness theorems to be demonstrated as decidable in that generalized structure of Hilbert space.
IV. The so-called Gödel first incompleteness theorem to be interpreted as allowing of the self-foundation of mathematics.
The Gödel incompleteness can be modeled on the alleged incompleteness of quantum mechanics. Then the proved completeness of quantum mechanics can be reversely interpreted as a strategy of completeness as to the foundation of mathematics.
That argument supposes that the Gödel incompleteness originates from the deficiency of the mathematical structure, on which it is grounded. Furthermore, one can point out that generalized structure, on which completeness is provable and thus it can serve as a reliable fundament of mathematics.
Set theory and arithmetic were what was put as the base of mathematics. However, it is a random historical fact appealing to intuition or to intellectual authorities such as Cantor, Frege, Russell, Hilbert, "Nicolas Bourbaki", etc. rather than to a mathematical proof. Even more, the so-called Godel incompleteness theorems demonstrated that set theory and arithmetic are irrelevant as the ground of mathematics rather than no relevant branch of mathematics allowing of self-grounding though the orthodox view.
One can utilize an analogy to the so-called fundamental theorem of algebra:
It needs a more general structure than the real numbers, within which it can be proved.
Analogically, the self-foundation of mathematics needs some more general structure than the positive integers in order to be provable.
The key for a relevant structure is Einstein's failure to show that quantum mechanics is incomplete. The incompleteness of set theory and arithmetic and the alleged incompleteness of quantum mechanics can be linked. The close friendship of the Princeton refugees Godel and Einstein might address that fact. However, Godel came to Princeton in 1940 much after the beginning of Einstein's attempts to reveal that and how quantum mechanics was incomplete. In particular, the famous triple article of Einstein, Podolsky, and Rosen "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" pointed out as a kind of theoretical forecast as to the phenomena of entanglement and thus of quantum information was published in 1935. So, there should exist a common mathematical structure underlying both \incompletenesses" and in turn interpretable as each of them.
The mathematical formalism of quantum mechanics is based on the complex Hilbert space featuring by a few important properties relevant to that structure apt to underlie mathematics:
1. It is a generalization of positive integers: Thus it involves innity.
2. It is both discrete and continuous (even smooth): Thus it can unify arithmetic and geometry.
3. It is invariant to the axiom of choice: Thus it can unify as the externality and internality of an innite set as the probabilistic and deterministic consideration of the modeled reality as well as even model and reality in general.
The target of the presentation is:
I. Those three properties of the complex Hilbert space to be demonstrated.
II. A simple mathematical structure underlying both the Godel incompleteness and the alleged incompleteness of quantum mechanics to be described explicitly.
III. The undecidable statements according to the Godel incompleteness theorems to be demonstrated as decidable in that generalized structure of Hilbert space.
IV. The so-called Gödel first incompleteness theorem to be interpreted as allowing of the self-foundation of mathematics.
The presentation as a PDF, a video, or as slides @ EasyChair
Links to the paper @ Blogger or @ Wordpress
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