The umbilic theory of entanglement and quantum gravity

 Abstract: The central contribution is the construction of a multisegment, nonorientable geometric framework that treats quantum entanglement and quantum gravity as dual manifestations of a common topological field, with Weyl information providing the proposed conserved bridge between their quantum, gravitational, and thermodynamic descriptions developing a multisegment umbilic theory of quantum entanglement and quantum gravity based on a nonorientable topological manifold and a unified projective geometric algebra framework. The theory interprets quantum states and spacetime geometry as dual projections of a common multivector field, and proposes that their apparent separation arises from the use of local orientable descriptions for globally connected phenomena. Locally two sided segments are globally coupled through a Mobius strip structure, providing a topological interpretation of quantum entanglement as global locality and of gravitational geometry as its macroscopic dual. The theory introduces Weyl information as a globally conserved, scale invariant quantity and formulates a generalized hierarchy in which the unified field equation degenerates into the Einstein field equation, the Dirac equation, and the Schrodinger equation under appropriate limiting conditions. Its multisegment form is further reformulated as a thermodynamic equation of state describing the relationship between quantum information and gravitational energy. Four components are identified: local entanglement, global gravitational geometry, information to energy conversion, and energy to information conversion. A central element is the Higgs-Maxwell demon, interpreted as a thermodynamic and topological interface between local quantum phase space and global gravitational structure. Through a Legendre duality between bundle and co-bundle descriptions, recording and erasing quantum states are associated respectively with the absorption and generation of physical geometric degrees of freedom. The framework therefore proposes an intrinsic connection between quantum mechanics, thermodynamics, entanglement, and gravity. The paper derives terrestrial and weak gravity limits in which the generalized theory reduces to conventional physics and proposes astronomical and laboratory experiments capable of falsifying its predicted information energy transduction and polytropic behavior.

Keywords: Einstein field equation; Higgs mechanism; Legendre duality; information-energy transduction; Majorana neutrinos; Maxwell demon; Möbius strip manifold; multi-segment theory of entanglement and quantum gravity; nonorientable topology; polytropic equation of state; projective geometric algebra (PGA); quantum entanglement; quantum gravity; quantum information spacetime geometry; Schrodinger equation; thermodynamics;  topological locality; umbilic theory; Weyl information

https://papers.ssrn.com/sol3/papers.cfm?abstract_id=7394178

Umbilic Thermodynamics

 https://vasil7penchev.wordpress.com/wp-content/uploads/2026/08/umbilic-thermodynamics-2.pdf

Abstract:This paper develops umbilic thermodynamics, a generalized thermodynamic framework based on the conservation

of dimensionless, scale-invariant Weyl information across a non-orientable Möbius topology. Within a conformal

projective geometric algebra (PGA) formulation, probability-distribution shapes and pseudo-Riemannian metric

shapes are interpreted as grade-dual projections of a single underlying structure, linking microscopic phase

entropy with macroscopic metric stress-energy. A corresponding Weyl temperature is introduced to characterize

the conversion between these informational and geometric manifestations. The framework formulates four

principles of umbilic thermodynamics concerning symplectic-capacity equilibrium, global Weyl-information

conservation, topological entropy directionality, and the unattainability of global absolute zero. A singular

Legendre transformation at the Gromov symplectic boundary provides a local “backdoor” through which

phase-space compression and negative informational temperatures can produce localized metric stress-energy,

while global Weyl-information conservation prohibits net creation ex nihilo. The resulting generalized umbilic

ergodic theorem identifies local “creatio ex nihilo” events as recurrent, measure-preserving transfers between

dual phase sheets. Classical thermodynamics and quantum thermodynamics are recovered as limiting projections

of the generalized framework, while a statistical formulation is developed in Boltzmannian, Gibbsian, and

Einsteinian editions. The paper further proposes falsifiable experimental tests involving entangled quantum

systems, cryogenic cavities, superconducting qubits, and non-orientable photonic structures. Thus, umbilic

thermodynamics is presented as a unified topodynamical framework connecting information, entropy, quantum

phase structure, spacetime curvature, and thermodynamic evolution.

Keywords:umbilic thermodynamics; Weyl information; Weyl temperature; conformal projective geometric algebra;

Möbius topology; non-orientable manifold; Legendre transformation; Gromov symplectic capacity;

phase entropy; negative temperature; informational temperature; creatio ex nihilo; metric stress-energy;

generalized ergodic theorem; topodynamical dynamics; quantum thermodynamics; statistical thermodynamics;

quantum entanglement; spacetime curvature; thermodynamic duality.

The Umbilic Totality I:

 One proposes a scale-invariant, non-orientable topological model of the universe, called the umbilic totality, which unifies quantum mechanical dualities (bra/ket, state/apparatus, or derivatively, entanglement) and pseudo-Riemannian general relativity within a single mathematical (claiming to be ontomathematical) framework. By replacing standard 2 complex phase trigonometry via a 4 non-orientable Möbius strip (MS) topology,  one shows that quantum entanglement and covariant metric curvature emerge as grade-dual projections of a single projective geometric algebra (PGA) multivector field. One generalizes Gromov non-squeezing, Birkhoff ergodicity, and Noether’s theorem to non-orientable phase spaces, demonstrating that local metric energy fluctuations are balanced by a globally conserved topological action. One demonstrates that Occam’s razor strictly favors Majorana neutrinos over Dirac neutrinos under umbilic Fermi transformations, offering a clear empirical test for the model. The ontomathematical framework is designed to resolve the long-standing incompatibility between quantum mechanics and general relativity and  argues that this incompatibility stems from a fundamental topological misconception: using orientable, locally Euclidean structures rather than a global, scale-invariant, non-orientable topology. Using conformal projective geometric algebra (PGA), the framework replaces the conventional complex phase with a Möbius topology. Within this single multivector structure, physical concepts traditionally treated as distinct (such as particle versus field, metric curvature versus quantum state, bra versus ket vectors, and spacetime geometry versus phase space) are reinterpreted as complementary local projections of a single underlying geometric totality. The proposed generalized PGA-Einstein field equation establishes a mathematical hierarchy whose limiting cases naturally degenerate into the core equations of established physics: general relativity, the Dirac equation, and the Schrödinger equation. Crucially, a multi-segment degeneration simultaneously accounts for spacetime connectivity and non-local quantum correlations. Quantum entanglement is thus reinterpreted not as instantaneous action at a distance, but as continuous global locality across a non-orientable manifold, providing a topological explanation for Bell inequality violations. Beyond postdictions such as CPT-invariance and cosmological flatness, the model offers a series of experimentally or observably falsifiable predictions, providing concrete empirical criteria to validate or falsify the proposed framework.

Keywords: 4 non-orientable holonomy and topology; Bell - Tsirelson bound; bra-ket unification; chiral parity inversion (P-flip); conformal projective geometric algebra (PGA); contravariant-covariant duality; generalized PGA-Einstein field equation(s); Majorana neutrinos; Möbius strip topology; non-orientable phase space; ontomathematics; quantum gravity and entanglement correspondence; quantum mechanics foundations; real projective geometry (RP2); scale-invariant Weyl information conservation; spacetime Clifford algebra (Cl3,1); spinor-torsion coupling; topological entanglement; umbilic Fermi transformation; Umbilic Totality

https://dx.doi.org/10.2139/ssrn.7280720        https://papers.ssrn.com/sol3/papers.cfm?abstract_id=7280720

 

 

Hilbert Mathematics Versus Gödel Mathematics. IV. The New Approach of Hilbert Mathematics Easily Resolving the Most Difficult Problems of Gödel Mathematics

 

The paper continues the consideration of Hilbert mathematics to mathematics itself as an additional “dimension” allowing for the most difficult and fundamental problems to be attacked in a new general and universal way shareable between all of them. That dimension consists in the parameter of the “distance between finiteness and infinity”, particularly able to interpret standard mathematics as a particular case, the basis of which are arithmetic, set theory and propositional logic: that is as a special “flat” case of Hilbert mathematics. The following four essential problems are considered for the idea to be elucidated: Fermat’s last theorem proved by Andrew Wiles; Poincaré’s conjecture proved by Grigori Perelman and the only resolved from the seven Millennium problems offered by the Clay Mathematics Institute (CMI); the four-color theorem proved “machine-likely” by enumerating all cases and the crucial software assistance; the Yang-Mills existence and mass gap problem also suggested by CMI and yet unresolved. They are intentionally chosen to belong to quite different mathematical areas (number theory, topology, mathematical physics) just to demonstrate the power of the approach able to unite and even unify them from the viewpoint of Hilbert mathematics. Also, specific ideas relevant to each of them are considered. Fermat’s last theorem is shown as a Gödel insoluble statement by means of Yablo’s paradox. Thus, Wiles’s proof as a corollary from the modularity theorem and thus needing both arithmetic and set theory involves necessarily an inverse Grothendieck universe. On the contrary, its proof in “Fermat arithmetic” introduced by “epoché to infinity” (following the pattern of Husserl’s original “epoché to reality”) can be suggested by Hilbert arithmetic relevant to Hilbert mathematics, the mediation of which can be removed in the final analysis as a “Wittgenstein ladder”. Poincaré’s conjecture can be reinterpreted physically by Minkowski space and thus reduced to the “nonstandard homeomorphism” of a bit of information mathematically. Perelman’s proof can be accordingly reinterpreted. However, it is valid in Gödel (or Gödelian) mathematics, but not in Hilbert mathematics in general, where the question of whether it holds remains open. The four-color theorem can be also deduced from the nonstandard homeomorphism at issue, but the available proof by enumerating a finite set of all possible cases is more general and relevant to Hilbert mathematics as well, therefore being an indirect argument in favor of the validity of Poincaré’s conjecture in Hilbert mathematics. The Yang-Mills existence and mass gap problem furthermore suggests the most general viewpoint to the relation of Hilbert and Gödel mathematics justifying the qubit Hilbert space as the dual counterpart of Hilbert arithmetic in a narrow sense, in turn being inferable from Hilbert arithmetic in a wide sense. The conjecture that many if not almost all great problems in contemporary mathematics rely on (or at least relate to) the Gödel incompleteness is suggested. It implies that Hilbert mathematics is the natural medium for their discussion or eventual solutions.

Keywords: Fermat’s last theorem (FLT), four-color theorem, Gödel mathematics, Hilbert arithmetic, Hilbert mathematics, Perelman’s proof, Poincaré’s conjecture, qubit Hilbert space, quantum information, Wiles’s proof, Yang-Mills existence and mass gap problem

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Hilbert mathematics versus Gödel mathematics III. Hilbert mathematics by itself, and Gödel mathematics versus the physical world within it: both as its particular cases

 Abstract. The paper discusses Hilbert mathematics, a kind of Pythagorean mathematics, to which the physical world is a particular case. The parameter of the “distance between finiteness and infinity” is crucial. Any nonzero finite value of it features the particular case in the frameworks of Hilbert mathematics where the physical world appears “ex nihilo” by virtue of an only mathematical necessity or quantum information conservation physically. One does not need the mythical Big Bang which serves to concentrate all the violations of energy conservation in a “safe”, maximally remote point in the alleged “beginning of the universe”. On the contrary, an omnipresent and omnitemporal medium obeying quantum information conservation rather than energy conservation permanently generates action and thus the physical world. The utilization of that creation “ex nihilo” is accessible to humankind, at least theoretically, as long as one observes the physical laws, which admit it in their new and wider generalization. One can oppose Hilbert mathematics to Gödel mathematics, which can be identified as all the standard mathematics until now featureable by the Gödel dichotomy of arithmetic to set theory: and then, “dialectic”, “intuitionistic”, and “Gödelian” mathematics within the former, according to a negative, positive, or zero value of the distance between finiteness and infinity. A mapping of Hilbert mathematics into pseudo-Riemannian space corresponds, therefore allowing for gravitation to be interpreted purely mathematically and ontologically in a Pythagorean sense. Information and quantum information can be involved in the foundations of mathematics and linked to the axiom of choice or alternatively, to the field of all rational numbers, from which the pair of both dual and anti-isometric Peano arithmetics featuring Hilbert arithmetic are immediately inferable. Noether’s theorems (1918) imply quantum information conservation as the maximally possible generalization of the pair of the conservation of a physical quantity and the corresponding Lie group of its conjugate. Hilbert mathematics can be interpreted from their viewpoint also after an algebraic generalization of them. Following the ideas of Noether’s theorem (1918), locality and nonlocality can be realized both physically and mathematically. The “light phase of the universe” can be linked to the gap of mathematics and physics in the Cartesian organization of cognition in Modernity and then opposed to its “dark phase”, in which physics and mathematics are merged. All physical quantities can be deduced from only mathematical premises by the mediation of the most fundamental physical constants such as the speed of light in a vacuum, the Planck and gravitational constants once they have been interpreted by the relation of locality and nonlocality.

Keywords: energy conservation, Gödel mathematics, Hilbert mathematics, light and dark phases of the universe, locality and nonlocality, Noether theorems of conservation, Pythagoreanism, quantum information conservation

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The 2022 Nobel Prize in physics for entanglement and quantum information: the new revolution in quantum mechanics and science

 

Abstract. The paper discusses the 2022 Nobel Prize in physics for experiments of entanglement “establishing the violation of Bell inequalities and pioneering quantum information science” in a much wider, including philosophical context legitimizing by the authority of the Nobel Prize a new scientific area out of “classical” quantum mechanics relevant to Pauli’s “particle” paradigm of energy conservation and thus to the Standard model obeying it. One justifies the eventual future theory of quantum gravitation as belonging to the newly established quantum information science. Entanglement, involving non-Hermitian operators for its rigorous description, non-unitarity as well as nonlocal and superluminal physical signals “spookily” (by Einstein’s flowery epithet) synchronizing and transferring some nonzero action at a distance, can be considered to be quantum gravity so that its local counterpart to be Einstein’s gravitation according to general relativity therefore pioneering an alternative pathway to quantum gravitation different from the “secondary quantization” of the Standard model. So, the experiments of entanglement once they have been awarded by the Nobel Prize launch particularly the relevant theory of quantum gravitation grounded on “quantum information science” thus granted to be nonclassical quantum mechanics in the shared framework of the generalized quantum mechanics obeying rather quantum-information conservation than only energy conservation. The concept of “dark phase” of the universe naturally linked to the very well confirmed “dark matter” and “dark energy” and opposed to its “light phase” inherent to classical quantum mechanics and the Standard model obeys quantum-information conservation, after which reversible causality or the mutual transformation of energy and information are valid. The mythical Big Bang after which energy conservation holds universally is to be replaced by an omnipresent and omnitemporal medium of decoherence of the dark and nonlocal phase into the light and local phase. The former is only an integral image of the latter and borrowed in fact rather from religion than from science. Physical, methodological and proper philosophical conclusions follow from that paradigm shift heralded by the 2022 Nobel Prize in physics. For example, the scientific theory of thinking should originate from the dark phase of the universe, as well: probably only approximately modeled by neural networks physically belonging to the light phase thoroughly. A few crucial philosophical sequences follow from the break of Pauli’s paradigm: (1) the establishment of the “dark” phase of the universe as opposed to its “light” phase, only to which the Cartesian dichotomy of “body” and “mind” is valid; (2) quantum information conservation as relevant to the dark phase, furthermore generalizing energy conservation as to its light phase, productively allowing for physical entities to appear “ex nihilo”, i.e., from the dark phase, in which energy and time are yet inseparable from each other; (3) reversible causality as inherent to the dark phase; (4) the interpretation of gravitation only mathematically: as an interpretation of the incompleteness of finiteness to infinity, for example, following the Gödel dichotomy (“either contradiction or incompleteness”) about the relation of arithmetic to set theory; (5) the restriction of the concept of hierarchy only to the light phase; (6) the commensurability of both physical extremes of a quantum and the universe as a whole in the dark phase obeying quantum information conservation and akin to Nicholas of Cusa’s philosophical and theological worldview.

Keywords: classical quantum mechanics, dark and light phases of the universe, dark energy and dark matter, Einstein, energy conservation, entanglement, general relativity, Hermitian and non-Hermitian quantities in quantum mechanics, locality and nonlocality, Pauli’s particle paradigm, quantum gravity, quantum information, quantum information conservation, qubit,
the Standard model, unitarity and non-unitarity

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Gödel mathematics versus Hilbert mathematics. II Logicism and Hilbert mathematics, the identification of logic and set theory, and Gödel’s “completeness paper” (1930)

The previous Part I of the paper (https://doi.org/10.33774/coe-2022-wlr02) discusses the option of the Gödel incompleteness statement (1931: whether “Satz VI” or “Satz X”) 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 Gödel’s paper (1930) (and more precisely, the negation of “Satz VII”, or “the completeness theorem”) 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 “completeness paper” 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 Russell’s logicism or the intensional propositional logic versus both extensional arithmetic and set theory. Reconstructing history of philosophy, Aristotle’s logic and doctrine can be opposed to those of Plato or the pre-Socratic schools as establishing ontology or intensionality versus extensionality. Husserl’s phenomenology can be analogically realized including and particularly as philosophy of mathematics. One can identify propositional logic and set theory by virtue of Gödel’s completeness theorem (1930: “Satz VII”) and even both and arithmetic in the sense of the “compactness theorem” (1930: “Satz X”) therefore opposing the latter to the “incompleteness paper” (1931). An approach identifying homomorphically propositional logic and set theory as the same structure of Boolean algebra, and arithmetic as the “half” 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. Hilbert’s 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.

Keywords: arithmetic, Aristotle, bit of information, Boolean algebra, first-order logic, Gödel, epsilon calculus, Husserl, logicism, propositional logic, ontology, Pythagoreanism, quantum logic, Russell, set theory

The paper as a PDF or @ repositories: @ SocArxiv, @ Easychair, @ SSRN, @ CambridgeOpenEngage, @ HAL, @ PhilPapers. @ PhilSci (Pittsburgh)

Gödel mathematics versus Hilbert mathematics. I The Gödel incompleteness (1931) statement: axiom or theorem?

 The present first part about the eventual completeness of mathematics (called “Hilbert mathematics”) is concentrated on the Gödel incompleteness (1931) statement: if it is an axiom rather than a theorem inferable from the axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic, this would pioneer the pathway to Hilbert mathematics. One of the main arguments that it is an axiom consists in the direct contradiction of the axiom of induction in arithmetic and the axiom of infinity in set theory. Thus, the pair of arithmetic and set are to be similar to Euclidean and non-Euclidean geometries distinguishably only by the Fifth postulate now, i.e. after replacing it and its negation correspondingly by the axiom of finiteness (induction) versus that of finiteness being idempotent negations to each other. Indeed, the axiom of choice, as far as it is equivalent to the well-ordering “theorem”, transforms any set in a well-ordering either necessarily finite according to the axiom of induction or also optionally infinite according to the axiom of infinity. So, the Gödel incompleteness statement relies on the logical contradiction of the axiom of induction and the axiom of infinity in the final analysis. Nonetheless, both can be considered as two idempotent versions of the same axiom (analogically to the Fifth postulate) and then unified after logicism and its inherent intensionality since the opposition of finiteness and infinity can be only extensional (i.e., relevant to the elements of any set rather than to the set by itself or its characteristic property being a proposition). So, the pathway for interpreting the Gödel incompleteness statement as an axiom and the originating from that assumption for “Hilbert mathematics” accepting its negation is pioneered. A much wider context relevant to realizing the Gödel incompleteness statement as a metamathematical axiom is consistently built step by step. The horizon of Hilbert mathematics is the proper subject in the third part of the paper, and a reinterpretation of Gödel’s papers (1930; 1931) as an apology of logicism as the only consistent foundations of mathematics is the topic of the next second part.

Keywords: Boolean algebra, completeness, dual axiomatics, Euclidean and non-Euclidean geometries, Gödel, the Fifth postulate, finitism, foundations of mathematics, Hilbert arithmetic, Hilbert program, Husserl, incompleteness, logicism, Peano arithmetic, phenomenology, Principia mathematica, propositional logic, Riemann’s “space curvature”, Russell, set theory

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@ HAL, @ PhilPapers, @ SocArxiv, @ SSRN, @ Authorea. @ PhilSci (Pittsburgh), @ International Journal of Pure and Applied Mathematics Research

Gentzen’s “cut rule” and quantum measurement in terms of Hilbert arithmetic. Metaphor and understanding modeled formally

 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. 

Keywords: cut-elimination, Hilbert arithmetic, metaphor, proposition, propositional logic, quantum measurement 

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Fermat’s last theorem proved in Hilbert arithmetic. III. The quantum-information unification of Fermat’s last theorem and Gleason’s theorem

Abstract. The previous two parts of the paper (correspondingly, https://philpapers.org/rec/PENFLT-2  and https://philpapers.org/rec/PENFLT-3) demonstrate that the interpretation of Fermat’s last theorem (FLT) in Hilbert arithmetic meant both in a narrow sense and in a wide sense can suggest a proof by induction in Part I and by means of the Kochen - Specker theorem in Part II. The same interpretation can serve also for a proof FLT based on Gleason’s theorem and partly similar to that in Part II. The concept of (probabilistic) measure of a subspace of Hilbert space and especially its uniqueness can be unambiguously linked to that of partial algebra or incommensurability, or interpreted as a relation of the two dual branches of Hilbert arithmetic in a wide sense. The investigation of the last relation allows for FLT and Gleason’s theorem to be equated in a sense, as two dual counterparts, and the former to be inferred from the latter, as well as vice versa under an additional condition relevant to the Gödel incompleteness of arithmetic to set theory. The qubit Hilbert space itself in turn can be interpreted by the unity of FLT and Gleason’s theorem. The proof of such a fundamental result in number theory as FLT by means of  Hilbert arithmetic in a wide sense can be generalized to an idea about “quantum number theory”. It is able to research mathematically the origin of Peano arithmetic from Hilbert arithmetic by mediation of the “nonstandard bijection” and its two dual branches inherently linking it to information theory. Then, infinitesimal analysis and its revolutionary application to physics can be also re-realized in that wider context, for example, as an exploration of the way for physical quantity of time (respectively, for time derivative in any temporal process considered in physics) to appear at all. Finally, the result admits a philosophical reflection of how any hierarchy arises or changes itself only thanks to its dual and idempotent counterpart.     

Keywords: completeness, Gleason’s theorem, Fermat’s last theorem, Hilbert arithmetic, idempotency and hierarchy, Kochen and Specker theorem, nonstandard bijection, Peano arithmetic, quantum information 

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Fermat’s last theorem proved in Hilbert arithmetic. II. Its proof in Hilbert arithmetic by the Kochen-Specker theorem with or without induction

 

The paper is a continuation of another paper (https://philpapers.org/rec/PENFLT-2) published as Part I. Now, the case of “n=3” is inferred as a corollary from the Kochen and Specker theorem (1967): the eventual solutions of Fermat’s equation for “n=3” would correspond to an admissible disjunctive division of  qubit into two absolutely independent parts therefore versus the contextuality of any qubit, implied by the Kochen – Specker theorem. Incommensurability (implied by the absence of hidden variables) is considered as dual to quantum contextuality. The relevant mathematical structure is Hilbert arithmetic in a wide sense (https://dx.doi.org/10.2139/ssrn.3656179), in the framework of which Hilbert arithmetic in a narrow sense and the qubit Hilbert space are dual to each other. A few cases involving set theory are possible: (1) only within the case “n=3” and implicitly, within any next level  of “n” in Fermat’s equation; (2) the identification of the case “n=3” and the general case utilizing the axiom of choice rather than the axiom of induction. If the former is the case, the application of set theory and arithmetic can remain disjunctively divided: set theory, “locally”, within any level; and arithmetic, “globally”, to all levels. If the latter is the case, the proof is thoroughly within set theory. Thus, the relevance of Yablo’s paradox to the statement of Fermat’s last theorem is avoided in both cases. The idea of “arithmetic mechanics” is sketched: it might deduce the basic physical dimensions of mechanics (mass, time, distance) from the axioms of arithmetic after a relevant generalization, Furthermore, a future Part III of the paper is suggested: FLT by mediation of Hilbert arithmetic in a wide sense can be considered as another expression of Gleason’s theorem in quantum mechanics: the exclusions about (n = 1, 2) in both theorems as well as the validity for all the rest values of “n” can be unified after the theory of quantum information. The availability (respectively, non-availability) of solutions of Fermat’s equation can be proved as equivalent to the non-availability (respectively, availability) of a single probabilistic measure as to Gleason’s theorem.

Keywords: arithmetic mechanics, Gleason’s theorem, Fermat’s last theorem (FLT), Hilbert arithmetic, Kochen and Specker’s theorem, Peano arithmetic, quantum information

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Fermat’s last theorem proved in Hilbert arithmetic. I. From the proof by induction to the viewpoint of Hilbert arithmetic

 Abstract. In a previous paper (https://dx.doi.org/10.2139/ssrn.3648127 ), an elementary and thoroughly arithmetical proof of Fermat’s last theorem by induction has been demonstrated if the case for “n = 3” 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 “n = 3” has been known for a long time. It needs “Hilbert mathematics”, which is inherently complete unlike the usual “Gödel mathematics”, and based on “Hilbert arithmetic” to generalize Peano arithmetic in a way to unify it with the qubit Hilbert space of quantum information. An “epoché to infinity” (similar to Husserl’s “epoché to reality”) is necessary to map Hilbert arithmetic into Peano arithmetic in order to be relevant to Fermat’s 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 “Hamilton” 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 Fermat’s 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 “n=3” based on the qubit Hilbert space and the Kochen-Specker theorem inferable from it.   

Keywords: Fermat’s last theorem, Hilbert arithmetic, Kochen and Specker’s theorem, Peano arithmetic, quantum information, qubit Hilbert space

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What the Tortoise Said to Achilles: Lewis Carroll’s paradox in terms of Hilbert arithmetic

 Lewis Carroll, both logician and writer, suggested a logical paradox containing furthermore two connotations (connotations or metaphors are inherent in literature rather than in mathematics or logics). The paradox itself refers to implication demonstrating that an intermediate implication can be always inserted in an implication therefore postponing its ultimate conclusion for the next step and those insertions can be iteratively and indefinitely added ad lib, as if ad infinitum. Both connotations clear up links due to the shared formal structure with other well-known mathematical observations: (1) the paradox of Achilles and the Turtle; (2) the transitivity of the relation of equality. Analogically to (1), one can juxtapose the paradox of the Liar (for Lewis Carroll’s paradox) and that of the arrow (for “Achilles and the Turtle”), i.e. a logical paradox, on the one hand, and an aporia of motion, on the other hand, suggesting a shared formal structure of both, which can be called “ontological”, on which basis “motion” studied by physics and “conclusion” studied by logic can be unified being able to bridge logic and physics philosophically in a Hegelian manner: even more, the bridge can be continued to mathematics in virtue of (2), which forces the equality (for its property of transitivity) of any two quantities to be postponed analogically ad lib and ad infinitum. The paper shows that Hilbert arithmetic underlies naturally Lewis Carroll’s paradox admitting at least three interpretations linked to each other by it: mathematical, physical and logical. Thus, it can be considered as both generalization and solution of his paradox therefore naturally unifying the completeness of quantum mechanics (i.e. the absence of hidden variables) and eventual completeness of mathematics as the same and isomorphic to the completeness of propositional logic in relation to set theory as a first-order logic (in the sense of Gödel (1930)’s completeness theorems). 

Keywords: equality, Lewis Carroll’s paradox, Liar’s paradox, paradox of the arrow, “Achilles and the Turtle”, Hilbert arithmetic, qubit Hilbert space

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The homeomorphism of Minkowski space and the separable complex Hilbert space: the physical, mathematical and philosophical interpretations

 A homeomorphism is built between the separable complex Hilbert space (quantum mechanics) and Minkowski space (special relativity) by meditation of quantum information (i.e. qubit by qubit). That homeomorphism can be interpreted physically as the invariance to a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincaré’s conjecture (proved by G. Perelman) hinting at another way for proving it, more concise and meaningful physically. Furthermore, the conjecture can be generalized and interpreted in relation to the pseudo-Riemannian space of general relativity therefore allowing for both mathematical and philosophical interpretations of the force of gravitation due to the mismatch of choice and ordering and resulting into the “curving of information” (e.g. entanglement). Mathematically, that homeomorphism means the invariance to choice, the axiom of choice, well-ordering, and well-ordering “theorem” (or “principle”) and can be defined generally as “information invariance”. Philosophically, the same homeomorphism implies transcendentalism once the philosophical category of the totality is defined formally. The fundamental concepts of “choice”, “ordering” and “information” unify physics, mathematics, and philosophy and should be related to their shared foundations.

Keywords: axiom of choice, choice, general relativity, gravitation, Hilbert space, information, Minkowski space, ordering, Poincaré’s conjecture, pseudo-Riemannian space, quantum information, qubit, well-ordering

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“Noema” and “Noesis” by Information after Husserl’s Phenomenology Interpreted Formally

Along with “epoché” or his “reductions”, Husserl’s “noema” and “noesis”, being neologisms invented by him, are main concepts in phenomenology able to represent its originality. Following the trace of a recent paper (Penchev 2021 July 23), its formal and philosophical approach is extended to both correlative notions, in the present article. They are able to reveal the genesis of the world from consciousness in a transcendental method relevant to Husserl, but furthermore described formally as a process of how subjective temporality appears being isomorphic to objective temporality of the “world by itself” (an abstraction meaning it out of consciousness or transcendental consciousness): thus, it shares the same mathematical structure, which is embodied in the physical process of decoherence by the physical quantity of quantum information. The temporal world is able to appear naturally (rather as a ridiculous effect of the mythical “Big Bang”). The same process translated by formal and mathematical tools as interpreted in terms of “noema”, “noesis”, or transcendental consciousness is isomorphic to how “Self” (including in an individual and psychological sense) appears in virtue of transcendental consciousness.  

Keywords: consciousness, formal phenomenology, noema, noesis, the Self, transcendental consciousness.

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Hilbert arithmetic as a Pythagorean arithmetic: arithmetic as transcendental

The paper considers a generalization of Peano arithmetic, Hilbert arithmetic as the basis of the world in a Pythagorean manner. Hilbert arithmetic unifies the foundations of mathematics (Peano arithmetic and set theory), foundations of physics (quantum mechanics and information), and philosophical transcendentalism (Husserl’s phenomenology) into a formal theory and mathematical structure  literally following Husserl’s tracе of “philosophy as a rigorous science”. In the pathway to that objective, Hilbert arithmetic identifies by itself information related to finite sets and series and quantum information referring to infinite one as both appearing in three “hypostases”: correspondingly, mathematical, physical and ontological, each of which is able to generate a relevant science and area of cognition. Scientific transcendentalism is a falsifiable counterpart of philosophical transcendentalism. The underlying concept of the totality can be interpreted accordingly also mathematically, as consistent completeness, and physically, as the universe defined not empirically or experimentally, but as that ultimate wholeness containing its externality into itself.

Keywords: eidetic, phenomenological and transcendental reductions; Hilbert arithmetic; Peano arithmetic; set theory and logic as Boolean algebra; qubit Hilbert space

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Quantum phenomenology as a “rigorous science”: the triad of epoché and the symmetries of information

 Abstract. Husserl (a mathematician by education) remained a few famous and notable philosophical “slogans” along with his innovative doctrine of phenomenology directed to transcend “reality” in a more general essence underlying both “body” and “mind” (after Descartes) and called sometimes “ontology” (terminologically following his notorious assistant Heidegger). Then, Husserl’s tradition can be tracked as an idea for philosophy to be reinterpreted in a way to be both generalized and mathenatizable in the final analysis. The paper offers a pattern borrowed from the theory of information and quantum information (therefore relating philosophy to both mathematics and physics) to formalize logically  a few key concepts of Husserl’s phenomenology such as “epoché” “eidetic, phenomenological, and transcendental reductions” as well as the identification of “phenomenological, transcendental, and psychological reductions” in a way allowing for that identification to be continued to “eidetic reduction” (and thus to mathematics). The approach is tested by an independent and earlier idea of Husserl, “logical arithmetic” (parallelly implemented in mathematics by Whitehead and Russell’s Principia) as what “Hilbert arithmetic” generalizing Peano arithmetics is interpreted. A basic conclusion states for the unification of philosophy, mathematics, and physics in their foundations and fundamentals to be the Husserl tradition both tracked to its origin (in the being itself after Hidegger or after Husserl’s “zu Sache selbst”)  and embodied in the development of human cognition in the third millennium.  Keywords: epoché, Hilbert arithmetic, Husserl reductions, information and quantum information, qubit Hulbert space

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Quantity in Quantum Mechanics and the Quantity of Quantum Information

The paper interprets the concept “operator in the separable complex Hilbert space” (particalry, “Hermitian operator” as “quantity” is defined in the “classical” quantum mechanics) by that of “quantum information”. As far as wave function is the characteristic function of the probability (density) distribution for all possible values of a certain quantity to be measured, the definition of quantity in quantum mechanics means any unitary change of the probability (density) distribution. It can be represented as a particular case of “unitary” qubits. The converse interpretation of any qubits as referring to a certain physical quantity implies its generalization to non-Hermitian operators, thus neither unitary, nor conserving energy. Their physical sense, speaking loosely, consists in exchanging temporal moments therefore being implemented out of the space-time “screen”. “Dark matter” and “dark energy” can be explained by the same generalization of “quantity” to non-Hermitian operators only secondarily projected on the pseudo-Riemannian space-time “screen” of general relativity according to Einstein's “Mach’s principle” and his field equation.

Key words: quality, quantity, quantum information, qubit Hilbert space, space-time 

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“Two bits less” after quantum-information conservation and their interpretation as “distinguishability / indistinguishability” and “classical / quantum”

The paper investigates the understanding of quantum indistinguishability after quantum information in comparison with the “classical” quantum mechanics based on the separable complex Hilbert space. The two oppositions, correspondingly   “distinguishability / indistinguishability” and “classical / quantum”, available implicitly in the concept of quantum indistinguishability can be interpreted as two “missing” bits of classical information, which are to be added after teleportation of quantum information to be restored the initial state unambiguously. That new understanding of quantum indistinguishability is linked to the distinction of classical (Maxwell-Boltzmann) versus quantum (either Fermi-Dirac or Bose-Einstein) statistics. The latter can be generalized to classes of wave functions (“empty” qubits) and represented exhaustively in Hilbert arithmetic therefore connectible to the foundations of mathematics, more precisely, to the interrelations of propositional logic and set theory sharing the structure of Boolean algebra and two anti-isometric copies of Peano arithmetic.

Key words: Bose-Einstein statistics, Fermi-Dirac statistics, Hilbert arithmetic, Maxwell-Boltzmann statistics, qubit Hilbert space, quantum indistinguishability, quantum-information conservation, teleportation

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Both classical & quantum information: both bit & qubit. Both physical & transcendental time

Information can be considered as the most fundamental, philosophical, physical and mathematical concept originating from the totality by means of physical and mathematical transcendentalism (the counterpart of philosophical transcendentalism). Classical and quantum information, particularly by their units, bit and qubit, correspond and unify the finite and infinite. As classical information is relevant to finite series and sets, as quantum information, to infinite ones. A fundamental joint relativity of the finite and infinite, of the external and internal is to be investigated. The corresponding invariance is able to define physical action and its quantity only on the basis of information and especially: on the relativity of classical and quantum information. The concept of transcendental time, an epoché in relation to the direction of time arrow can be defined. Its correlate is that information invariant to the finite and infinite, therefore unifying both classical and quantum information. 

Key words: bit and qubit, classical and quantum information, epoché, physical and mathematical transcendentalism, qubit space (qubit Hilbert space), separable complex Hilbert space, the totality, transcendental time

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