Penchev, Vasil (2022) Gentzen’s “cut rule” and quantum measurement in terms of Hilbert arithmetic. Metaphor and understanding modeled formally. [Preprint]
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Abstract
Hilbert arithmetic in a wide sense, including Hilbert arithmetic in a narrow sense consisting by two dual and anti-isometric Peano arithmetics, on the one hand, and the qubit Hilbert space (originating for the standard separable complex Hilbert space of quantum mechanics), on the other hand, allows for an arithmetic version of Gentzen’s cut elimination and quantum measurement to be described uniformly as two processes occurring accordingly in those two branches. A philosophical reflection also justifying that unity by quantum neo-Pythagoreanism links it to the opposition of propositional logic, to which Gentzen’s cut rule refers immediately, on the one hand, and the linguistic and mathematical theory of metaphor therefore sharing the same structure borrowed from Hilbert arithmetic in a wide sense. An example by hermeneutical circle modeled as a dual pair of a syllogism (accomplishable also by a Turing machine) and a relevant metaphor (being a formal and logical mistake and thus fundamentally inaccessible to any Turing machine) visualizes human understanding corresponding also to Gentzen’s cut elimination and the Gödel dichotomy about the relation of arithmetic to set theory: either incompleteness or contradiction. The metaphor as the complementing “half” of any understanding of hermeneutical circle is what allows for that Gödel-like incompleteness to be overcome in human thought.
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| Item Type: | Preprint | ||||||
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| Keywords: | cut-elimination, Hilbert arithmetic, metaphor, proposition, propositional logic, quantum measurement | ||||||
| Subjects: | Specific Sciences > Mathematics > Logic General Issues > History of Philosophy of Science General Issues > History of Science Case Studies |
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| Depositing User: | Prof. Vasil Penchev | ||||||
| Date Deposited: | 08 Mar 2023 14:30 | ||||||
| Last Modified: | 08 Mar 2023 14:30 | ||||||
| Item ID: | 21822 | ||||||
| Official URL: | https://www.cambridge.org/engage/coe/article-detai... | ||||||
| DOI or Unique Handle: | https://doi.org/10.33774/coe-2022-157v6 | ||||||
| Subjects: | Specific Sciences > Mathematics > Logic General Issues > History of Philosophy of Science General Issues > History of Science Case Studies |
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| Date: | 30 July 2022 | ||||||
| URI: | http://philsci-archive.pitt.edu/id/eprint/21822 |
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