ARTICLE | doi:10.20944/preprints202205.0112.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: arithmetic mechanics; Gleason’s theorem; Fermat’s last theorem (FLT); Hilbert arithmetic; Kochen and Specker’s theorem; Peano arithmetic; quantum information
Online: 9 May 2022 (09:54:41 CEST)
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.
ARTICLE | doi:10.20944/preprints202103.0281.v2
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Goldbach's conjecture; numbers prime; Arithmetic Theorem
Online: 12 March 2021 (11:29:49 CET)
Since 1742, the year in which the Prussian Christian Goldbach wrote a letter to Leonhard Euler with his Conjecture in the weak version, mathematicians have been working on the problem. The tools in number theory become the most sophisticated thanks to the resolution solutions. Euler himself said he was unable to prove it. The weak guess in the modern version states the following: any odd number greater than 5 can be written as the sum of 3 primes. In response to Goldbach's letter, Euler reminded him of a conversation in which he proposed what is now known as Goldbach's strong conjecture: any even number greater than 2 can be written as a sum of 2 prime numbers. The most interesting result came in 2013, with proof of weak version by the Peruvian Mathematician Harald Helfgott, however the strong version remained without a definitive proof. The weak version can be demonstrated without major difficulties and will not be described in this article, as it becomes a corollary of the strong version. Despite the enormous intellectual baggage that great mathematicians have had over the centuries, the Conjecture in question has not been validated or refuted until today.
ARTICLE | doi:10.20944/preprints202203.0183.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Fermat’s last theorem; Hilbert arithmetic; Kochen and Specker’s theorem; Peano arithmetic; quantum information; qubit Hilbert space
Online: 14 March 2022 (10:54:07 CET)
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.
ARTICLE | doi:10.20944/preprints202206.0276.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: completeness; Gleason’s theorem; Fermat’s last theorem; Hilbert arithmetic; idempotency and hi-erarchy; Kochen and Specker theorem; nonstandard bijection; Peano arithmetic; quantum information
Online: 21 June 2022 (03:22:50 CEST)
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.
ARTICLE | doi:10.20944/preprints202007.0415.v4
Subject: Mathematics & Computer Science, Logic Keywords: Structuralism; Set Theory; Type Theory; Arithmetic Model; Data Type; Tree, Group
Online: 15 August 2022 (04:55:03 CEST)
A construction for the systems of natural and real numbers is presented in Zermelo-Fraenkel Set Thoery, that allows for simple proofs of the properties of these systems, and practical and mathematical applications. A practical application is discussed, in the form of a Simple and Linear Fast Adder (Patent Pending). Applications to finite group theory and analysis are also presented. A method is illustrated for finding the automorphisms of any finite group $G$, which consists of defining a canonical block form for finite groups. Examples are given, to illustrate the procedure for finding all groups of $n$ elements along with their automorphisms. The canonical block form of the symmetry group $\Delta_4$ is provided along with its automorphisms. The construction of natural numbers is naturally generalized to provide a simple and sound construction of the continuum with order and addition properties, and where a real number is an infinite set of natural numbers. A basic outline of analysis is proposed with a fast derivative algorithm. Under this representation, a countable sequence of real numbers is represented by a single real number. Furthermore, an infinite $\infty\times\infty$ real-valued matrix is represented with a single real number. A real function is represented by a set of real numbers, and a countable sequence of real functions is also represented by a set of real numbers. In general, mathematical objects can be represented using the smallest possible data type and these representations are calculable. In the last section, mathematical objects of all types are well assigned to tree structures in a proposed type hierarchy.
ARTICLE | doi:10.20944/preprints202207.0429.v1
Subject: Arts & Humanities, Philosophy Keywords: cut-elimination; Hilbert arithmetic; metaphor; proposition; propositional logic; quantum meas-urement
Online: 28 July 2022 (06:27:31 CEST)
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 uniformy 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.
ARTICLE | doi:10.20944/preprints202108.0176.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: consecutive sum of the digits; algebraic equations; diophantine equations; arithmetic functions
Online: 9 August 2021 (07:52:49 CEST)
In this paper defines the consecutive sum of the digits of a natural number, so far as it becomes less than ten, as an arithmetic function called and then introduces some important properties of this function by proving a few theorems in a way that they can be used as a powerful tool in many cases. As an instance, by introducing a test called test, it has been shown that we are able to examine many algebraic equalities in the form of in which and are arithmetic functions and to easily study many of the algebraic and diophantine equations in the domain of whole numbers. The importance of test for algebraic equalities can be considered equivalent to dimensional equation in physics relations and formulas. Additionally, this arithmetic function can also be useful in factorizing the composite odd numbers.
Subject: Keywords: arithmetic figures; Pascal’s triangle; geometrization of brain; central nervous system; nonlinear
Online: 4 December 2019 (04:43:17 CET)
The brain, rather than being homogeneous, displays an almost infinite topological genus, because is punctured with a very high number of topological vortexes, i.e., .e., nesting, non-concentric brain signal cycles resulting from inhibitory neurons devoid of excitatory oscillations. Starting from this observation, we show that the occurrence of topological vortexes is constrained by random walks taking place during self-organized brain activity. We introduce a visual model, based on the Pascal’s triangle and linear and nonlinear arithmetic octahedrons, that describes three-dimensional random walks of excitatory spike activity propagating throughout the brain tissue. In case of nonlinear 3D paths, the trajectories in brains crossed by spiking oscillations can be depicted as the operation of filling the numbers of the octahedrons in the form of “islands of numbers”: this leads to excitatory neuronal assemblies, spaced out by empty area of inhibitory neuronal assemblies. These procedures allow us to describe the topology of a brain of infinite genus, to assess inhibitory neurons in terms of Betti numbers, and to highlight how non-linear random walks cause spike diffusion in neural tissues when tiny groups of excitatory neurons start to fire.
ARTICLE | doi:10.20944/preprints201908.0045.v1
Subject: Mathematics & Computer Science, General & Theoretical Computer Science Keywords: floating-point arithmetic; inverse square root; magic constant; Newton-Raphson method
Online: 5 August 2019 (05:09:06 CEST)
We present an improved algorithm for fast calculation of the inverse square root for single-precision floating-point numbers. The algorithm is much more accurate than the famous fast inverse square root algorithm and has a similar computational cost. The presented modification concern Newton-Raphson corrections and can be applied when the distribution of these corrections is not symmetric (for instance, in our case they are always negative).
Subject: Physical Sciences, Astronomy & Astrophysics Keywords: arithmetic figures, black hole, deterministic model, geometrization of physics, random walk
Online: 26 March 2019 (10:23:11 CET)
The Universe, rather than being homogeneous, displays an almost infinite topological genus, because it is punctured with a countless number of gravitational vortexes, i.e., black holes. Starting from this view, we aim to show that the occurrence of black holes is constrained by geometric random walks taking place during cosmic inflationary expansion. At first, we introduce a visual model, based on the Pascal’s triangle and linear and nonlinear arithmetic octahedrons, which describes three-dimensional cosmic random walks. In case of nonlinear 3D paths, trajectories in an expanding Universe can be depicted as the operation of filling the numbers of the octahedrons in the form of “islands of numbers”: this leads to separate cosmic structures (standing for matter/energy), spaced out by empty areas (constituted by black holes and dark matter). These procedures allow us to describe the topology of an universe of infinite genus, to assess black hole formation in terms of infinite Betti numbers, to highlight how non-linear random walks might provoke gravitational effects also in absence of mass/energy, and to propose a novel interpretation of Beckenstein-Hawking entropy: it is proportional to the surface, rather than the volume, of a black hole, because the latter does not contain information.
ARTICLE | doi:10.20944/preprints202112.0507.v1
Subject: Mathematics & Computer Science, Logic Keywords: equality; Lewis Carroll’s paradox; Liar’s paradox; paradox of the arrow; “Achilles and the Turtle”; Hilbert arithmetic; qubit Hilbert space
Online: 31 December 2021 (11:02:32 CET)
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).
ARTICLE | doi:10.20944/preprints202012.0195.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: consistent interval probabilities; generalized probability intervals; interval probabilities; Kaucher arithmetic; permissible interval probabilities; probabilistic inference; probability trees; reachable interval probabilities
Online: 8 December 2020 (10:01:39 CET)
Probabilistic inference problems have very broad practical applications. To solve this kind of problems under conditions of certainty, an effective mathematical apparatus has been developed. In real situations, obtaining deterministic estimates of relevant probabilities is often difficult; therefore, problems with handling uncertain estimates of probabilities appear. This paper examines the problem of probabilistic inference with probability trees provided that the initial probabilities are given in the form of intervals of their possible values.