109 research outputs found
Do Goedel's incompleteness theorems set absolute limits on the ability of the brain to express and communicate mental concepts verifiably?
Classical interpretations of Goedel's formal reasoning imply that the truth
of some arithmetical propositions of any formal mathematical language, under
any interpretation, is essentially unverifiable. However, a language of
general, scientific, discourse cannot allow its mathematical propositions to be
interpreted ambiguously. Such a language must, therefore, define mathematical
truth verifiably. We consider a constructive interpretation of classical,
Tarskian, truth, and of Goedel's reasoning, under which any formal system of
Peano Arithmetic is verifiably complete. We show how some paradoxical concepts
of Quantum mechanics can be expressed, and interpreted, naturally under a
constructive definition of mathematical truth.Comment: 73 pages; this is an updated version of the NQ essay; an HTML version
is available at http://alixcomsi.com/Do_Goedel_incompleteness_theorems.ht
Computational Natural Philosophy: A Thread from Presocratics through Turing to ChatGPT
Modern computational natural philosophy conceptualizes the universe in terms
of information and computation, establishing a framework for the study of
cognition and intelligence. Despite some critiques, this computational
perspective has significantly influenced our understanding of the natural
world, leading to the development of AI systems like ChatGPT based on deep
neural networks. Advancements in this domain have been facilitated by
interdisciplinary research, integrating knowledge from multiple fields to
simulate complex systems. Large Language Models (LLMs), such as ChatGPT,
represent this approach's capabilities, utilizing reinforcement learning with
human feedback (RLHF). Current research initiatives aim to integrate neural
networks with symbolic computing, introducing a new generation of hybrid
computational models.Comment: 17 page
Fermatâs last theorem proved in Hilbert arithmetic. I. From the proof by induction to the viewpoint of Hilbert arithmetic
In a previous paper, an elementary and thoroughly arithmetical proof of Fermatâs last theorem by induction has been demonstrated if the case for ân = 3â is granted as proved only arithmetically (which is a fact a long time ago), furthermore in a way accessible to Fermat himself though without being absolutely and precisely correct. The present paper elucidates the contemporary mathematical background, from which an inductive proof of FLT can be inferred since its proof for the case for ân = 3â has been known for a long time. It needs âHilbert mathematicsâ, which is inherently complete unlike the usual âGödel mathematicsâ, and based on âHilbert arithmeticâ to generalize Peano arithmetic in a way to unify it with the qubit Hilbert space of quantum information. An âepochĂ© to infinityâ (similar to Husserlâs âepochĂ© to realityâ) is necessary to map Hilbert arithmetic into Peano arithmetic in order to be relevant to Fermatâs age. Furthermore, the two linked semigroups originating from addition and multiplication and from the Peano axioms in the final analysis can be postulated algebraically as independent of each other in a âHamiltonâ modification of arithmetic supposedly equivalent to Peano arithmetic. The inductive proof of FLT can be deduced absolutely precisely in that Hamilton arithmetic and the pransfered as a corollary in the standard Peano arithmetic furthermore in a way accessible in Fermatâs epoch and thus, to himself in principle. A future, second part of the paper is outlined, getting directed to an eventual proof of the case ân=3â based on the qubit Hilbert space and the Kochen-Specker theorem inferable from it
Gödel Mathematics Versus Hilbert Mathematics. II Logicism and Hilbert Mathematics, the Identification of Logic and Set Theory, and Gödelâs 'Completeness Paper' (1930)
The previous Part I of the paper discusses the option of the Gödel incompleteness statement (1931: whether âSatz VIâ or âSatz Xâ) to be an axiom due to the pair of the axiom of induction in arithmetic and the axiom of infinity in set theory after interpreting them as logical negations to each other. The present Part II considers the previous Gödelâs paper (1930) (and more precisely, the negation of âSatz VIIâ, or âthe completeness theoremâ) as a necessary condition for granting the Gödel incompleteness statement to be a theorem just as the statement itself, to be an axiom. Then, the âcompleteness paperâ can be interpreted as relevant to Hilbert mathematics, according to which mathematics and reality as well as arithmetic and set theory are rather entangled or complementary rather than mathematics to obey reality able only to create models of the latter. According to that, both papers (1930; 1931) can be seen as advocating Russellâs logicism or the intensional propositional logic versus both extensional arithmetic and set theory. Reconstructing history of philosophy, Aristotleâs logic and doctrine can be opposed to those of Plato or the pre-Socratic schools as establishing ontology or intensionality versus extensionality. Husserlâs phenomenology can be analogically realized including and particularly as philosophy of mathematics. One can identify propositional logic and set theory by virtue of Gödelâs completeness theorem (1930: âSatz VIIâ) and even both and arithmetic in the sense of the âcompactness theoremâ (1930: âSatz Xâ) therefore opposing the latter to the âincompleteness paperâ (1931). An approach identifying homomorphically propositional logic and set theory as the same structure of Boolean algebra, and arithmetic as the âhalfâ of it in a rigorous construction involving information and its unit of a bit. Propositional logic and set theory are correspondingly identified as the shared zero-order logic of the class of all first-order logics and the class at issue correspondingly. Then, quantum mechanics does not need any quantum logics, but only the relation of propositional logic, set theory, arithmetic, and information: rather a change of the attitude into more mathematical, philosophical, and speculative than physical, empirical and experimental. Hilbertâs epsilon calculus can be situated in the same framework of the relation of propositional logic and the class of all mathematical theories. The horizon of Part III investigating Hilbert mathematics (i.e. according to the Pythagorean viewpoint about the world as mathematical) versus Gödel mathematics (i.e. the usual understanding of mathematics as all mathematical models of the world external to it) is outlined
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