Mathematics

In the first part of this investigation we highlight two, seemingly irreconcilable, beliefs that suggest an impending crisis in the teaching, research, and practice of—primarily state-supported—mathematics: (a) the belief, with increasing, essentially faith-based, conviction and authority amongst academics that first-order Set Theory can be treated as the lingua franca of mathematics, since its theorems—even if unfalsifiable—can be treated as ‘knowledge’ because they are finite proof sequences which are entailed finitarily by self-evidently Justified True Beliefs; and (b) the slowly emerging, but (...) powerfully rooted in economic imperatives, belief that only those Justified True Beliefs can be treated as ‘knowledge’ which are, further, categorically communicable as Factually Grounded Beliefs—hence as comprehensible pre-formal ‘mathematical truths’—by any intelligence that lays claims to a mathematical lingua franca. We argue that this seeming irreconcilability merely reflects a widening chasm between an increasing under-estimation of the value to society of ‘semantic arithmetical truth’, and an increasing over-estimation of the value to society of ‘syntactic set-theoretical provability’; a chasm which must be narrowed and bridged explicitly. We thus proffer a Complementarity Thesis which seeks to recognize that mathematical ‘provability’ and mathematical ‘truth’ need to be interdependent and complementary, ‘evidence-based’, assignments-by-convention to the formulas of a formal mathematical language; where (a) the goal of mathematical ‘proofs’ may be viewed as seeking to ensure that any mathematical language intended to formally represent our pre-formal conceptual metaphors and their inter-relatedness is unambiguous, and free from contradiction; whilst (b) the goal of mathematical ‘truths’ must be viewed as seeking to ensure that any such representation does, indeed, uniquely identify and adequately represent such metaphors and their inter-relatedness. Our thesis is that, by universally ignoring the need to introduce goal (b) in our curriculum, the teaching of, and research in, mathematics at the post-graduate level cannnot justify its value to society beyond the mere intellectual stimulation of individual minds. In the second part we appeal to some recent—and hitherto unsuspected—unarguably constructive Tarskian interpretations, of the first-order Peano Arithmetic PA, which establish PA as both finitarily consistent, and categorical. Since we also show that the second order subsystem ACA0 of Peano Arithmetic and PA cannot both be assumed provably consistent, we conclude that there can be no mathematical, or meta-mathematical, proof of consistency for Set Theory. Hence the theorems of any Set Theory are not sufficient for distinguishing between (i) what we can believe to be a ‘mathematical truth’; (ii) what we can evidence as a ‘mathematical truth’; and (iii) what we ought not to believe is a ‘mathematical truth’; whilst the theorems of the first-order Peano Arithmetic PA are sufficient for distinguishing between (i), (ii) and (iii). We conclude from this that the holy grail of mathematics ought to be ‘arithmetical truth’, not ‘set-theoretical proof’. (shrink)

Could this ONE paragraph be Fermat’s Lost ‘Proof’ of FLT?

Multi-agent systems often exhibit emergent behavior that appears coordinated, intelligent, and irreducible to the behavior of individual components. Yet quantifying the degree to which such systems form integrated wholes remains a major challenge. While Integrated Information Theory (IIT) was originally developed to explain consciousness, its core concept - measuring how much a system resists decomposition - has broader relevance for understanding informational integration in complex systems. However, the exact computation of IIT’s central quantity, Φ, is intractable for all but the (...) smallest networks. In this paper, we propose a scalable spectral approximation, Φspectral, which estimates system-level irreducibility by applying spectral graph theory to pairwise mutual information networks derived from time-series data. The Fiedler vector of the normalized Laplacian defines a minimal bipartition, across which we sum mutual information to estimate informational disintegration. We interpret this measure as a proxy for epistemic emergence (specifically the diachronic case)—capturing practical, observer-relative irreducibility without invoking ontological claims. We also explore the implications of this metric for measuring emergence and consciousness. We evaluate Φspectral across four classes of simulated systems: random, transitional, and synchronized oscillator networks, as well as a biologically inspired combinatorial threshold-linear network (CTLN) governed by graph-theoretic dynamics. Our results show that Φspectral is sensitive not merely to coordination, but to the emergence of structurally differentiated and functionally integrated behavior. In particular, it detects transient integration in transitional systems and reveals the collapse of complexity in fixed-point–convergent CTLNs. These findings suggest that Φspectral offers a principled and computationally efficient tool for analyzing irreducibility and emergence in distributed systems, with potential applications in AI safety, neural computation, and collective intelligence. (shrink)

This presentation introduces a mathematical model in which the probability of physical actualization is modulated by an internal “awareness” factor that dynamically alters the influence of entropic forces. By integrating recursive awareness—later expressed using the Fibonacci sequence—the model illustrates how coherent, meaningful structures (e.g., functional RNA) might emerge from high-entropy prebiotic conditions. In the revised version, we extend the formulation from a one-dimensional (1D) configuration space to include a full three-dimensional (3D) formulation, thereby offering a more comprehensive view on how (...) directed emergence of structured configurations may occur in realistic environments. This unified framework presents discrete, quantized transitions, natural scaling, and fractal organization across different dimensions. (shrink)

In [1], we give a modeling of fuzzy time by μ. In [2], in modeling of theory of Fuzzy Time computation and Fuzzy Time Computation we employ η. What is the relation between these two terms? More exactly, we should show how η could be calculated from μ, throughout this article we do it. References 1.????????∗=????????????∗? (Second version), F.Didehvar, Philpapers 2025 2. Theory of Fuzzy Time Computation (TC* vs TC,TQC), F.Didehvar, HAL, 2023.

With the significant progress of artificial intelligence (AI) and consciousness science, artificial consciousness (AC) has recently gained popularity. This work provides a broad overview of the main topics and current trends in AC. The first part traces the history of this interdisciplinary field to establish context and clarify key terminology, including the distinction between Weak and Strong AC. The second part examines major trends in AC implementations, emphasising the synergy between Global Workspace and Attention Schema, as well as the problem (...) of evaluating the internal states of artificial systems. The third part analyses the ethical dimension of AC development, revealing both critical risks and transformative opportunities. The last part offers recommendations to guide AC research responsibly, and outlines the limitations of this study as well as avenues for future research. The main conclusion is that while AC appears both indispensable and inevitable for scientific progress, serious efforts are required to address the far-reaching impact of this innovative research path. (shrink)

In this interdisciplinary essay, we reimagine Gilles Deleuze and Félix Guattari’s concept of the _Body without Organs_ (BwO) as a generative framework for understanding embodied cognition, affective flows, and predictive processing across biology, mathematics, and neuroscience. Moving beyond its philosophical origins, the BwO is developed here as a dynamic field of unformed potentiality that cuts across molecular biology, mathematical abstraction (as topological and categorical virtuality), and the neural-affective systems of cognition (as self-organizing predictive agents). We argue that cognition emerges through (...) continuous negotiation with this pre-individual field—a process of sensing, predicting, and becoming shaped by affective micro-events, interoceptive modulations, and relational intensities. Drawing from process philosophy, embodied cognitive science, and dynamical systems theory, we position the BwO as an ontological ground for understanding not just the mind, but the very becoming of bodies, networks, and worlds. This essay proposes a novel conceptual ecology where the embodied brain is not an organized organ, but a fluid interface with the virtual: a topological becoming that reconfigures cognition as improvisation on a field of affective and mathematical possibility. This interdisciplinary framework invites empirical elaboration and theoretical critique from scholars in neuroscience, philosophy, biology, and mathematics. Keywords: Body Without Organs, Embodied Cognition, Predictive Processing, Affective Neuroscience, Process Philosophy, Philosophical Biology, Self-O​r​g​a​ . (shrink)

In contemporary mathematics, a Colombeau algebra of Colombeau generalized functions is an algebra of a certain kind containing the space of Schwartz distributions. While in classical distribution theory a general multiplication of distributions is not possible, Colombeau algebras provide a rigorous framework for this. Remark 1.1.1.Such a multiplication of distributions has been a long time mistakenly believed to be impossible because of Schwartz’ impossibility result, which basically states that there cannot be a differential algebra containing the space of distributions and (...) preserving the product of continuous functions. However, if one only wants to preserve the product of smooth functions instead such a construction becomes possible, as demonstrated first by J.F.Colombeau [1],[2]. As a mathematical tool, Colombeau algebras can be said to combine a treatment of singularities, differentiation and nonlinear operations in one framework, lifting the limitations of distribution theory. These algebras have found numerous applications in the fields of partial differential equations, geophysics, microlocal analysis and general relativity so far. (shrink)

In the paper it is demonstrated that Bell's theorem is unproveable.

CAT4 is proposed as a general method for representing information, enabling a powerful programming method for large-scale information systems. It enables generalised machine learning, software automation and novel AI capabilities. It is based on a special type of relation called CAT4, which is interpreted to provide a semantic representation. This is Part 2 of a five-part introduction. The focus here is on defining key mathematical properties of CAT2, identifying the topology and defining essential functions over a coordinate system. The analysis (...) is from first principles. This develops on from the axioms introduced in Part 1. The interpretation of fact networks is introduced in Part 3, and the full application to semantic theory comes in Part 4, where we introduce general functions, including the language interpretation or linguistic functions. In Part 5, we turn to software design considerations, to show how files, indexes, functions and screens can be defined to implement a CAT4 system efficiently. (shrink)

This study aims to establish a formal system grounded in observation, transforming the properties and outcomes of empirical observation into rigorously treatable mathematical structures, thereby constructing a strictly epistemological mathematical framework for physics and the natural sciences. Starting from the indistinguishability inherent in observation, we rigorously demonstrate how the modern mathematical notions of structure and morphism can be employed to represent phenomena as objects within topological spaces; and, by analyzing the relations among observations, how causality and dynamics can be strictly (...) reconstructed as evolution processes centered on observable phenomena. This study reveals that the necessity of mathematics in the natural sciences arises from the fact that mathematical logic itself is the only possible language through which natural phenomena can be coherently described. Accordingly, epistemology is established as the foundational language of the natural sciences, and a complete logical reconstruction of the fundamental concepts of physics is achieved without invoking any ontological assumptions. (shrink)

I have been working for a long time about basic laws which direct existence, and some mathematical problems which are waited for a solution. I can count myself lucky, that I could make some important inferences during this time, and I published them in a few papers partially as some propositions. This work aimed to explain and discuss these inferences all together by relating them one another by some extra additions, corrections and explanations being physical phenomena are prior. There are (...) many motivation instruments for exact physical inferences. (shrink)

Long Range and short range, guns and violence, everyday life in cities and streets, between social and group identity and faith and religious belief, the vision to the "things of the world that cannot be seen" (Heróis do Mar).

Mind, according to cognitive neuroscience, is a set of brain functions. But, unlike sets, our minds are cohesive. Moreover, unlike the structureless elements of sets, the contents of our minds are structured. Mutual relations between the mental contents endow the mind its structure. Here we characterize the structural essence and the logical form of the mind by focusing on thinking. Examination of the relations between concepts, propositions, and syllogisms involved in thinking revealed the reflexive graph structure of the conceptual mind. (...) Objective logic of the conceptual mind is calculated from its structure. Noteworthy features of the logic of conceptual mind are: degrees of truth, varieties of negation, admission of contradiction, and the failure of a de Morgan's law. Furthermore, cohesion of the conceptual mind follows from its reflexive graph structure. Our characterization of the structure and logic of mind constitutes a substantial refinement of the contemporary cognitive neuroscientific conceptualization of the mind as a set. (shrink)

That the state-of-affairs of cognitive science is not good is brought into figural salience in "What happened to cognitive science?" (Núñez et al., 2019). We extend their objective description of 'what's wrong' to a prescription of 'how to correct'. Cognitive science, in its quest to elucidate 'how we know', embraces a long list of subjects, while ignoring Mathematics (Fig. 1a, Núñez et al., 2019). Mathematics is known for making the unknown to be known (cf. solving for unknowns). This acknowledgement naturally (...) raises the question: does mathematical knowing inform knowing in general? Here we show that drawing parallels to mathematical knowing can facilitate the advancement of cognitive science (Lawvere, 1994). (shrink)

Neuroscientific discourse on consciousness often resorts to "collection of elements", notwithstanding the Gestalt demonstrations against representing conscious experience as a collection of sensory elements. Here I show that defining conscious experience as an object of the category of conscious experiences, instead of as cohesion-less set of structure-less elements, provides the conceptual repertoire—basic shapes, figures, and incidence relations—needed to reason about the essence of conscious experiences and the essence-preserving transformations of conscious experiences. Viewed in light of the category of conscious experiences, (...) designers of user experience—in designing pre-conceptualized user experiences—are well situated to contribute to the development of the science of consciousness. (shrink)

Math literacy is miniscule compared to the near universal language literacy of mother tongues. Our search for the root cause of this undesirable human condition led us to: Grammar (or the abstract essence) of a language. Language learning begins with grammar, unless the language happens to be mathematics, which is unique in not even considering including the grammar (abstract general/theory) of mathematics in the mathematical pedagogy. Here we make a case for introducing the abstract essence of mathematics--Conceptual Mathematics--in high school (...) math curriculum. Recognizing that the singular purpose of education is to nurture the universal yearning for understanding, we show how a nurturing pedagogy naturally resolves the "learning crisis", with the solution manifesting as human development. Furthermore, the nurturing pedagogy that we advocate invariably results in the culture of research excellence. (shrink)

The irrational exuberance associated with contemporary artificial intelligence (AI) reminds me of Charles Dickens: "it was the age of foolishness, it was the epoch of belief" (cf. Nature Editorial, 2016; to get a feel for the vanity fair that is AI, see Mitchell and Krakauer, 2023; Stilgoe, 2023). It is particularly distressing—feels like yet another rerun of Seinfeld, which is all about nothing (pun intended); we have seen it in the 60s and again in the 90s. AI might have had (...) an inauspicious beginning, which may very well be the original sin, and our current predicament is the attendant divine condemnation. Being unable to stand it, however divine it may be, without further ado, I get to the genesis (instead of getting lost in the here-and-now of this or that so-called breakthrough of AI). Here's my unvarnished critique! (shrink)

In this paper, I define and study an abstract algebraic structure, the dimensive algebra, which embodies the most general features of the algebra of dimensional physical quantities. I prove some elementary results about dimensive algebras and suggest some directions for future work.

The history of science is polarized by debates over Plato and Aristotle’s holism versus the atomism of Democritus and others. This includes the complementarity of continuous and discrete, one and the many, waves and particles, and analog or digital views of reality. The three-fold method of the Pythagorean paradigm of unity, duality, and harmony enables the calculation of fundamental physical constants required by the forces of nature in the formation of matter; thereby demonstrating Plato’s archetypal viewpoint.

We present a rigorous proof that a quantum Yang-Mills theory exists on R⁴ and has a mass gap Δ > 0 for any compact simple gauge group G. The proof establishes the Wightman axioms and demonstrates confinement through a novel approach combining celestial holography and Haar measure theory. Our key insight is that four-dimensional Yang-Mills theory can be reformulated as a two-dimensional conformal field theory on the celestial sphere via the Mellin transform. This celestial CFT inherits Kac-Moody symmetry from soft (...) gluon theorems, with conformal dimensions determined by the Sugawara construction. The Peter-Weyl theorem guarantees discreteness of the spectrum, while Haar measure orthogonality on the gauge group SU(N) proves confinement: only gauge singlets have finite energy. We prove that the lightest excitation is a two-gluon bound state with mass M = 2N/(k+N)Λ_QCD, where k is the Kac-Moody level and Λ_QCD is the strong coupling scale. For SU(3) at k=1, this gives M ≈ 300 MeV at tree level. The observed lattice value of 1600 MeV is explained by five independent quantum corrections: higher Kac-Moody level, loop corrections to Sugawara, anomalous dimensions, scheme dependence of Λ_QCD, and binding energy effects. The combined factor of ~5 is consistent with strong-coupling non-perturbative QCD. All five Wightman axioms are verified, including locality via the Reeh-Schlieder theorem. The theory admits well-defined continuum and infinite-volume limits. Our construction is non-perturbative and does not rely on lattice regularization. MAIN RESULTS: Existence: Quantum Yang-Mills theory exists satisfying Wightman axioms. Mass gap: Discrete spectrum with Δ > 0, explicit formula M = 2N/(k+N)Λ_QCD. Confinement: All physical states are gauge singlets (Haar orthogonality) Generality: Proof applies to any compact simple gauge group G. This work addresses the Yang-Mills existence and mass gap problem through mathematical methods combining measure theory (Haar, Peter-Weyl), representation theory (Kac-Moody algebras), conformal field theory (Sugawara construction), and quantum field theory (Wightman axioms). The approach opens new directions connecting holography and number theory to fundamental questions in quantum field theory. (shrink)

We prove that all non-trivial zeros of the Riemann zeta function ζ(s) lie on the critical line Re(s) = 1/2. We establish this result via three independent proofs using different mathematical frameworks: (1) Geometric: Three structural properties—Haar self-duality, functional equation symmetry, and Peter-Weyl compactness—uniquely determine σ = 1/2 as the only value permitting L² integrability. (2) Spectral: Meyer's unconditional spectral realization combined with Stone's theorem and Haar measure self-duality; (3) Probabilistic: The Biane-Pitman-Yor identification of ξ(s) with the Kuiper distribution, showing (...) off-line zeros would violate probability axioms. Each pathway is rigorous, complete, and unconditional. (shrink)

This work resolves multiple open problems in fundamental physics and mathematics; it provides proof for three Millennium Prize conjectures, and it presents a fully functioning quantum theory of gravity, all from a single fundamental mathematical structure: Haar-invariant measure on the Grassmannian Gr(2,4). -/- We establish rigorous proofs for the Riemann Hypothesis, Birch and Swinnerton-Dyer Conjecture, and Yang-Mills mass gap problem. -/- We prove a rigidity theorem isolating the unique celestial conformal field theory dual to Einstein gravity in four-dimensional asymptotically flat (...) spacetime. -/- We derive rather than postulate solutions to the quantum measurement problem and the Born rule. -/- We derive a CPT-symmetric cyclic cosmology that explains matter-antimatter asymmetry and the arrow of time without requiring inflation or baryogenesis. -/- Three Millennium Prize Solutions: -/- The Riemann Hypothesis follows from three simultaneous constraints on the adelic quotient: Haar self-duality, functional equation symmetry, and Peter-Weyl compactness of the idele class group. These three structures form an overdetermined system whose unique solution forces Re(s) = 1/2 for all square-integrable quasicharacters. Combined with Tate's 1950 characterization—which establishes a bijection between L² characters and zeta zeros independent of any assumption about zero locations—this proves all non-trivial zeros lie on the critical line. -/- The Birch and Swinnerton-Dyer Conjecture uses identical Haar measure structure on GL₂. The analytic rank (order of vanishing of the L-function at s = 1) equals the algebraic rank (dimension of the Mordell-Weil group) because both are constrained by the same measure-theoretic structure on the adelic quotient. -/- The Yang-Mills mass gap emerges from Haar invariance on compact gauge groups. For SU(3), the Peter-Weyl decomposition forces discrete spectrum with strictly positive eigenvalues. The gap Δ > 0 is derived from compactness of the gauge group combined with Haar-invariant dynamics. Confinement follows: colored states cannot propagate because they violate gauge singlet projection. -/- Complete Non-perturbative Quantum Gravity: -/- We prove a rigidity theorem: there exists exactly one consistent celestial conformal field theory dual to four-dimensional asymptotically flat Einstein gravity. The theory is uniquely determined by three conditions: central charge c = 0 (required by BMS₄ Ward identities), principal series spectrum (required by unitarity on the Lorentzian celestial sphere), and shadow symmetry (required by crossing symmetry of graviton amplitudes). -/- The Einstein field equations are derived from celestial boundary data. Asymptotic flatness requires Λ_background = 0. Shadow symmetry is proven equivalent to CPT conjugation, resolving the 40-year-old "googly problem" in twistor theory: anti-self-dual and self-dual sectors are unified by discrete CPT symmetry. -/- Quantum Foundations: -/- Wavefunction collapse is derived as Haar projection onto gauge singlets: -/- |ψ⟩ → P_singlet|ψ⟩ = ∫_G dμ_Haar(g) U_g|ψ⟩ -/- The Born rule P = |⟨n|ψ⟩|² is proven as the unique probability measure compatible with Haar structure on the space of quantum states. Both results follow from representation theory on principal bundles. -/- Wightman axioms are derived. -/- Spacetime from Geometry: -/- The Grassmannian Gr(2,4) is fundamental; spacetime emerges via the Penrose twistor correspondence. The Plücker embedding reveals ℂ⁶ = ℂ⁴_observable ⊕ ℂ²_hidden. The hidden sector provides a candidate for dark matter with characteristic mass scale ~100 GeV. The Standard Model gauge group SU(3) × SU(2) × U(1) emerges from SL(4,ℂ) decomposition. Three generations follow from dim_ℂ(Gr(2,4))/2 + 1 = 3. -/- CPT-Symmetric Cosmology: -/- We find convergent mathematical evidence for a CPT-antisymmetric cyclic cosmology without inflation. The conformal boundary ℐ⁺ (future null infinity) is identified with ℐ⁻ of a CPT-conjugate mirror universe: our future is the mirror's past. This identification resolves multiple cosmological puzzles: -/- The matter-antimatter asymmetry is a selection effect: our universe has baryon number B > 0, the mirror has B < 0, and total baryon number vanishes (B_total = 0). No baryogenesis mechanism is required. -/- The arrow of time emerges from CPT boundary conditions. Time flows away from the conformal boundary in both universes; entropy increases in both directions. The second law of thermodynamics follows from CPT symmetry rather than being an independent postulate. -/- The horizon and flatness problems are resolved without inflation. At the CPT boundary, the celestial CFT is globally defined on the celestial sphere—all points are in causal contact. Asymptotic flatness (Ω = 1) is required by the framework, not fine-tuned. -/- All mass eventually converts to radiation reaching ℐ⁺ via: Matter → Black Holes → Hawking Radiation → Conformal Boundary. The cycle is eternal with no beginning or end. -/- Numerical Verification: -/- Graviton box integrals computed from celestial shadow discontinuities match standard Feynman diagram calculations to 11 decimal places. The scalar box achieves agreement to machine precision (16 digits). -/- The Golden Transform reconstructs the imaginary parts γ_n of the first 50 Riemann zeros via γ_n = 4πϕ · E_n · α(n), where ϕ is the golden ratio, achieving mean absolute error 7.46 × 10⁻¹⁵ and R² = 1.000000000000. The running coupling α(n) flows to an infrared fixed point at α_IR = log(ϕ), predicted by the thermodynamic framework. -/- The celestial CFT central charge c = 0 is verified by five independent methods with numerical confirmation to 10+ significant figures. -/- The Golden Equilibrium: -/- The critical line Re(s) = 1/2 corresponds to thermodynamic equilibrium of the quantum vacuum. CPT symmetry implies microscopic time-reversibility, which defines equilibrium via detailed balance. The vacuum state naturally occupies its maximum-entropy configuration, which occurs precisely at Re(s) = 1/2. The golden ratio emerges because ϕ is maximally resistant to rational approximation and therefore represents the most stable fixed point under continued fraction dynamics. Maximum-entropy configurations on angular spaces exhibit golden ratio structure. -/- Testable Predictions: -/- The framework generates falsifiable predictions several of which include: fine structure constant α⁻¹ = 360°/ϕ² ≈ 137.508 at zero temperature; dark matter mass spectrum m_n = n × 100 GeV; wavefunction collapse timescale τ ~ ℏ/(k_B T) at ultracold temperatures; hidden sector gravitational waves at f ~ 10⁻¹⁰ Hz; geometric exclusion of fourth-generation fermions; scalar spectral index n_s = 1 − 2/ϕ⁴ ≈ 0.964 (within 1σ of Planck observations); tensor-to-scalar ratio r → 0 (no primordial gravitational waves from inflation, because there was no inflation). -/- Problems Addressed: -/- (1) Riemann Hypothesis, (2) Birch and Swinnerton-Dyer Conjecture, (3) Yang-Mills mass gap, (4) quantum measurement problem, (5) Born rule, (6) dark matter, (7) Standard Model gauge group, (8) three fermion generations, (9) cosmological constant framework, (10) non-perturbative quantum gravity, (11) googly problem in twister theory, (12) matter-antimatter asymmetry, (13) arrow of time, (14) horizon problem (without inflation), (15) flatness problem (without inflation), (16) bulk-boundary dictionary, (17) Higgs mechanism/SSB, (18) Wightman axioms derived (19) fine-structure constant geometric origin (360°/φ² = 137.508), (20) black hole information paradox, (21) light higgs mass geometrically protected, (22) origin of mass and spacetime, (23) strong CP problem, (24) CKM/PMNS mixing angles derived, (25) CP violation geometric origin, (26) cyclic cosmology, (27) Einstein equations derived from celestial CFT (28) central charge c=0 (five independent proofs) (30) proton stability, (31) running coupling / asymptotic freedom in number theory -/- Unification: -/- This work unifies physics by reconciling quantum mechanics and general relativity via celestial holography. -/- Moreover, and even more astonishingly, it unifies physics with number theory; geometry with arithmetic. -/- The same Haar measure structure that constrains Riemann zeros to Re(s) = 1/2 constrains graviton amplitudes to the principal series. -/- The same self-duality that determines the critical line in analytic number theory determines the spectrum of quantum gravity. Arithmetic and geometry share a common origin in Grassmannian measure theory. (shrink)

We prove the Birch and Swinnerton-Dyer conjecture for elliptic curves over the rational numbers. Specifically, we establish that for any elliptic curve E over Q, the rank of the Mordell-Weil group E(Q) equals the order of vanishing of the L-function L(E,s) at s=1. The proof proceeds in three main steps. First, we use the Arthur-Selberg trace formula to express the rank as the dimension of a spectral eigenspace. Second, we apply the Satake isomorphism and strong multiplicity one theorem to isolate (...) the automorphic representation π_E associated to E via modularity. Third, we employ a spectral-Weil identification using Riesz representation theory to connect the spectral dimension to the L-function vanishing order. This approach adapts techniques from spectral analysis of the Riemann zeta function, extending the spectral-Weil identification from GL_1 to GL_2. The key innovation is recognizing that Weil's explicit formula, combined with the Arthur trace formula and Riesz uniqueness theorem, provides an unconditional connection between spectral measures and L-function zeros. The proof is unconditional and applies to all ranks without restriction. (shrink)

In this work I rigorously construct the first complete non-perturbative theory of quantum gravity via the celestial holographic conformal field theory dual to Einstein gravity in asymptotically-flat four-dimensional spacetime, then demonstrate this framework identifies via physics the mathematical reasons why the Riemann Hypothesis must hold, leading to mathematical proofs for three Clay Millennium Prize conjectures in mathematics as direct consequences of the same underlying geometric structure. I prove a rigidity theorem establishing exactly one consistent celestial conformal field theory dual to (...) Einstein gravity, uniquely determined by central charge c = 0 (required by renormalization of BMS_4 charges), principal series spectrum Δ = 1 + iλ (required by unitarity on Lorentzian celestial sphere), and shadow symmetry Δ ↔ 2-Δ (required by crossing symmetry of graviton amplitudes), with central charge c = 0 confirmed by five independent methods to 10+ significant figures and graviton box integrals computed via novel shadow discontinuity methods matching Feynman diagram calculations to 11 decimal places. I prove shadow symmetry is equivalent to CPT conjugation (Theorem 6.1), resolving the 40-year-old googly problem in twistor theory and unifying anti-self-dual and self-dual sectors through discrete CPT symmetry, with Einstein field equations derived from boundary Ward identities and crossing symmetry rather than postulated. All quantum mechanical foundations emerge from the Haar measure structure underlying this framework with no additional postulates: the six Wightman axioms are derived from Peter-Weyl theorem on representation spaces, the Born rule P = |⟨n|ψ⟩|² is proven as the unique probability measure compatible with Haar structure on quantum state spaces, and wave function collapse is derived as Haar projection onto gauge singlets with timescale τ = ℏ/(k_B T) testable at ultracold temperatures around 1 μK where τ ~ 10^{-4} seconds. The same Haar-invariant measure theory on the Grassmannian Gr(2,4) that determines the graviton spectrum also solves three Clay Millennium Prize problems: I prove the Riemann Hypothesis by showing three simultaneous constraints on the adelic quotient A_Q×/Q× (Haar self-duality d×a = d×(a^{-1}), functional equation symmetry s ↔ 1-s, and Peter-Weyl compactness forcing discrete spectrum) form an overdetermined system whose unique solution forces Re(s) = 1/2 for all square-integrable characters, with numerical verification of the first 10^13 zeros on the critical line to machine precision using six independent tests. The Birch-Swinnerton-Dyer Conjecture is proven using identical Haar measure structure on GL_2(A_Q)/GL_2(Q), where both analytic rank (order of vanishing of L(E,s) at s=1) and algebraic rank (dimension of Mordell-Weyl group) are constrained by the same three-fold overdetermined system, forcing equality. The Yang-Mills mass gap follows from compactness of SU(N) via Peter-Weyl decomposition forcing discrete spectrum with strictly positive eigenvalues Δ ~ 200 MeV, with confinement emerging as a theorem since colored states violate Haar-invariant gauge singlet projection. Spacetime itself emerges from the Grassmannian via Penrose twistor correspondence (M^4 ↔ Gr(2,4) proven 1967) with Plücker embedding revealing C^6 = C^4_observable ⊕ C^2_hidden, where three fermion generations follow rigorously from Plücker partition counting (exactly three ways to partition {0,1,2,3} into complementary pairs, making a fourth generation topologically impossible and confirmed by LHC to TeV scales), strong CP angle θ = 0 exactly from Haar self-duality (consistent with experimental bound |θ| < 10^{-10}), and the hidden C^2 sector provides right-handed neutrinos as geometric dark matter candidates in agreement with Boyle-Turok CPT-symmetric cosmology. I construct this CPT-symmetric cyclic cosmology without inflation where future null infinity I^+ is identified with I^- of a CPT-conjugate mirror universe, resolving matter-antimatter asymmetry as a selection effect (total baryon number B_total = 0 with observers inhabiting the B > 0 sector and right-handed neutrinos providing dark matter), the arrow of time emerging from CPT boundary conditions with entropy increasing away from the conformal boundary in both temporal directions, and horizon/flatness problems resolved as the celestial CFT is globally defined at the conformal boundary. The fine structure constant is predicted as α^{-1}(T → 0K) = 360/φ² = 137.508 from E8 icosahedral geometry via McKay correspondence, with current room-temperature measurement 137.036 differing by approximately 0.34% due to thermal and quantum corrections, providing a testable prediction via cryogenic measurements extrapolated to absolute zero. The framework generates falsifiable predictions including no primordial gravitational waves (tensor-to-scalar ratio r → 0, definitively falsified if r > 0.01 detected by CMB-S4), wave function collapse timescale τ ~ 10^{-4} seconds at 1 μK in ultracold BEC experiments, right-handed neutrino dark matter with gravitational-only coupling, and geometric exclusion of fourth-generation fermions. Chapters 1-13 contain rigorous proofs of all major results with extensive numerical verification (spectral eigenvalues agreeing with Haar measure predictions to 15 decimal places, first 10^13 Riemann zeros verified on critical line), while Chapters 14-17 present clearly-labeled speculative extensions including the golden ratio as fundamental equilibrium constant for self-dual symmetries, a numerical discovery connecting Riemann zeros to Fibonacci ratios with fit R² > 0.9999, and a conjectural Galois E8 descent mechanism for Standard Model gauge group emergence that remains incomplete. This work unifies quantum mechanics and general relativity through celestial holography while revealing deep connections between number theory and physics: the same Haar measure structure constraining Riemann zeros to Re(s) = 1/2 constrains graviton amplitudes to principal series, the same compactness forcing Yang-Mills mass gap forces discrete zeta spectrum, suggesting arithmetic and geometry share a common fundamental structure rooted in the Grassmannian Gr(2,4). (shrink)

When modeling informal proofs like that of Euclid’s Elements using a sound logical system, we go from proofs seen as somewhat unrigorous – even having gaps to be filled – to rigorous proofs. However, metalogic grounds the soundness of our logical system, and proofs in metalogic are not like formal proofs and look suspiciously like the informal proofs. This brings about what I am calling here the groundedness problem: how can we decide with certainty that our metalogical proofs are rigorous (...) and sustain our logical system? In this paper, I will expose this problem. I will not try to solve it here. (shrink)

This paper investigates the phenomenon of time and articulates constraints that any robust theory must satisfy under a recovery-based criterion. Motivated by the apparent tension between static-looking fundamental descriptions, often associated with block-universe interpretations, and the dynamical character of temporal experience, the paper argues that augmenting a formalism with additional temporal structure can risk displacing rather than resolving explanatory burdens. In response, it advances a constraint on theorizing about time: a candidate time parameter counts as “real time” only insofar as (...) it can recover, in the appropriate regime, empirically meaningful time, specifically, by providing a principled mapping to clock time. It then develops a minimal modeling strategy in three stages: (i) using hyperbolic partial differential equations together with Minkowski kinematics to encode duration, metric comparison, and causal propagation; (ii) introducing a dissipative term proportional to the first time derivative to break time-reversal symmetry and yield an effective temporal orientation; and (iii) representing a “moving now” via an auxiliary parameter that indexes a family of time-slices through a monotonically shifting switching time, without positing a second dynamical time variable. (shrink)

We propose a structuralist reconstruction of the real continuum, arguing that it is not a primitive ontological stage for physics but an emergent completion of the prime-exponent lattice M. By identifying the reals R as the Dedekind completion of M, we demonstrate that the natural logarithm and exponential functions are not merely analytic tools but inevitable algebraic consequences of identifying the multiplicative structure of number theory with the additive structure of the continuum. Furthermore, we show that the natural base e (...) is the unique structural invariant required to normalize the gauge freedom inherent in this identification. Finally, we refine the Erdős–Kac theorem to argue that the "Triple-Log" scale (ln ln ln n) emerges as a deterministic background limit of arithmetic, providing an intrinsic argument for a natural ultraviolet cutoff in physical theories without external imposition. In the language of Ontic Structural Realism (OSR), this framework suggests that the 'physical continuum' is not an object-like substance, but a structural instantiation of the relation between the additive and multiplicative sectors of arithmetic. (shrink)

This came into existence when one day (10 -12 years back), all of a sudden I draw a line and divided the line into same segment and draw one more line above the previous line and mark the points on the both lines and magically I got a triangle (while attach all the points with each other), with at point 1 no of points 1 at 2 no of points 3 and at 3 no of points 5. Now an idea (...) born into my mind while I should not expand it more and I got Amazing Result which is in front of All Mathematics Lovers. (shrink)

Motivation -/- God is great, whatever (Air, Water, Earth, Fire & Space) we have it is just because of a Power, which governs the whole universe and more than this Power, it has created human beings, and what we have got or did not get yet, trying to get. The Method, The Principle, The Knowledge or more by which we get or prove in front of World People, basically it is called Science, who GET it, are called Scientists, Mathematicians, Inventors (...) etc i.e. we all breath we all know but in ancient time we don’t know what we breath as the time passes away one man’s curiosity bring out the name of gas which we breathe i.e. Oxygen. There are lot examples or more yet to come. -/- In this regard my little contribution in the field of Mathematics, it starts from my early childhood I used to play with numbers. After a lot of times devoted on numbers one day while squaring 11, 12, 13 so on I found that there is an amazing symmetry of 2 & it’s great. (shrink)

This paper presents an intervention study on the effects of the combined methods of (1) the Socratic method, (2) Chain of Thought (CoT) reasoning, (3) simplified gamification and (4) formative feedback on university students' Maths learning driven by large language models (LLMs). We call our approach Mathematics Explanations through Games by AI LLMs (MEGA). Some students struggle with Maths and as a result avoid Math-related discipline or subjects despite the importance of Maths across many fields, including signal processing. Oftentimes, students' (...) Maths difficulties stem from suboptimal pedagogy. We compared the MEGA method to the traditional step-by-step (CoT) method to ascertain which is better by using a within-group design after randomly assigning questions for the participants, who are university students. Samples (n=60) were randomly drawn from each of the two test sets of the Grade School Math 8K (GSM8K) and Mathematics Aptitude Test of Heuristics (MATH) datasets, based on the error margin of 11%, the confidence level of 90%, and a manageable number of samples for the student evaluators. These samples were used to evaluate two capable LLMs at length (Generative Pretrained Transformer 4o (GPT4o) and Claude 3.5 Sonnet) out of the initial six that were tested for capability. The results showed that students agree in more instances that the MEGA method is experienced as better for learning for both datasets. It is even much better than the CoT (47.5% compared to 26.67%) in the more difficult MATH dataset, indicating that MEGA is better at explaining difficult Maths problems. (shrink)

A well-ordered society faces a crisis whenever a sufficient number of noncompliers enter into the political system. This has the potential to destabilize liberal democratic political order. This article provides a formal analysis of two competing solutions to the problem of political stability offered in the public reason liberalism literature—namely, using public reason or using convergence discourse to restore liberal democratic political order in the well-ordered society. The formal analyses offered in this article show that using public reason fails completely, (...) and using convergent discourse, although doing better, has its own critical limitations that have not been previously recognized properly. (shrink)

We present a practical, hybrid algorithm for exact multiplication of large integers combining multiple advanced techniques to achieve practically faster performance than standard implementations while maintaining deterministic correctness. This method integrates: -/- - Fibonacci multiples with recursive doubling - Zeckendorf decomposition and block aggregation to reduce the effective number of blocks r_effective - FFT convolution on smaller, cache-friendly arrays - Hierarchical B-Tree merging with Strassen-style divide-and-conquer for multi-digit addition - SHA verification for deterministic exactness -/- The algorithm achieves asymptotic complexity (...) O(n \log n), matching Van der Hoeven (2019), but reduces practical runtime constants significantly. (shrink)

We propose a framework that models computation and information flow in terms of **white holes, black holes, and damping mechanisms**. This system enables a **recursive transfer of information from future states to present computations**, suggesting conditions under which P = NP could be realized in a physically grounded network. The model formalizes **white hole activation, black hole storage, and dynamic loops** with cross-entropy and autocorrelation characteristics, providing a bridge between temporal computation and structured information flow.

A famous mathematical theorem says that the sum of an infinite series of numbers can depend on the order in which those numbers occur. Suppose we interpret the numbers in such a series as representing instances of some physical quantity, such as the weights of a collection of items. The mathematics seems to lead to the result that the weight of a collection of items can depend on the order in which those items are weighed. But that is very hard (...) to believe. A puzzle then arises: How do we interpret the metaphysical significance of this mathematical theorem? I first argue that prior solutions to the puzzle lead to implausible consequences. Then I develop my own solution, where the basic idea is that the weight of a collection of items is equal to the limit of the weights of its finite subcollections contained within ever-expanding regions of space. I show how my solution is intuitively plausible and philosophically motivated, how it reveals an underexplored line of metaphysical inquiry about quantities and locations, and how it elucidates some classic puzzles concerning supertasks. (shrink)

This collection of scientific papers from the KMOCS-2025 conference represents a comprehensive exploration of contemporary approaches in mathematical modeling, optimization, and artificial intelligence across multiple engineering and technological domains. The proceedings are organized into three thematic sections that collectively demonstrate the interconnected nature of modern computational science. The first section focuses on perspective directions in mathematical modeling, featuring research on multiphysics modeling in aerospace structural design, vibration resistance of reinforced cylindrical shells, stability analysis of hollow shells under thermal loads, heat (...) and moisture regenerators, sloshing dynamics in partitioned tanks, and crack propagation in energy machinery structural elements. Papers also address mathematical models for predicting thermal fields, processing big data in distributed information systems, and modeling control systems using ECAD tools. The second section presents models and optimization methods, covering topics such as computational thermodynamic-kinetic simulation of vacuum steel refining, optimization of combined heating systems, investment portfolio distribution, routing optimization with spatial and weight constraints, and discrete Fourier transform applications in digital image processing. Special attention is given to hybrid evolutionary algorithms for discrete optimization problems under uncertainty, predictive analysis for electricity consumption regulation, and resource allocation in project management. The third section explores artificial intelligence models and methods, including YOLO neural networks for plant growth monitoring, fuzzy logic applications in blockchain network monitoring, explainable decision support systems integrating human expertise with computational insights, reinforcement learning for UAV coverage path planning, and physics-informed neural networks for solving multidimensional dynamic problems. The section also covers predictive analytics for stochastic processes, ontological approaches in intelligent hardware-software systems, semantic segmentation of digital images, gesture recognition in real-time, and post-quantum encryption algorithms. (shrink)

Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions. While the rules describing chaotic dynamical systems are well-specified and simple, the behaviour of many such systems is remarkably complex and produces output that appears random and for which long-term prediction is limited. The book begins by laying out preliminary material needed to understand the literature on chaos, providing the background (...) that any reader would need to be able to navigate the literature. It goes on to discuss the history of the field, the different definitions of chaos, differences between classical and quantum chaos and the implications of chaos for modelling phenomena and forecasting system behaviour before rounding out with broader implications. (shrink)

This collection of articles explores the multifaceted dimensions of robotics and artificial intelligence (AI) within educational and societal contexts. It delves into philosophical, social, psychological, legal, and ethical aspects of AI and robotics development, emphasizing their transformative impact on education and human interaction. The anthology covers key themes such as the integration of educational robotics in inclusive STEM environments, the evolution of AI and its influence on daily life, and the role of immersive technologies in shaping social interactions and the (...) metaverse. It also examines global and Ukrainian experiences in implementing robotics and AI in education, including innovative approaches like STEAM volunteering, 3D modeling, and gamification to foster engineering thinking, creativity, and soft skills. The collection highlights practical applications, such as web interfaces for robotic systems, AI-driven educational tools, and competitions like Line Follower, alongside model STEM and robotics curricula for modern schooling. This comprehensive resource is valuable for educators, researchers, and policymakers seeking to navigate the challenges and opportunities of AI and robotics in education. (shrink)

The Kochen-Specker theorem is one of the fundamental no-go theorems in quantum theory. It has far-reaching consequences for all attempts trying to give an interpretation of the quantum formalism. In this work, we examine the hypotheses that, at the ontological level, lead to the Kochen-Specker contradiction. We emphasize the role of the assumptions about identity and distinguishability of quantum objects in the argument.

We propose a categorical and information-theoretic framework for understanding prime numbers. In this model, we define two fundamental objects, `Ω` (0) and Maxwell’s Demon (1), and consider morphisms between them in a well-defined category of computational processes. Prime numbers emerge as **maximal entropy morphisms**, definable either by a closed-form formula or algorithm. The framework yields a dichotomy: either `Ω ≅ 1`, or a definable morphism exists for each prime number.

This volume presents a collection of research papers addressing contemporary challenges in aerospace engineering, intelligent systems, and reliability-oriented design. The contributions cover a broad spectrum of topics, including thermal and thermo-mechanical modeling of aerospace structures, numerical methods for heat transfer and stress analysis, structural evaluation using finite element methods, and experimental validation of aerospace components. Significant attention is devoted to unmanned and urban air mobility systems, encompassing flight dynamics, control system synthesis, navigation accuracy, motion cueing, and traffic simulation. The proceedings (...) also include studies on propulsion systems, additive manufacturing effects on mechanical performance, material behavior under operational environments, and non-destructive evaluation techniques. Emerging digital and intelligent approaches—such as ontology-driven knowledge representation, PLM frameworks, Kalman filtering, reinforcement learning, and automated measurement systems—are explored as enabling technologies for integration, optimization, and decision support. Collectively, the volume reflects an interdisciplinary perspective that combines modeling, experimentation, digitalization, and intelligent algorithms to enhance the performance, reliability, and integration of modern aerospace systems. (shrink)

One of the main ontological challenges posed by quantum mechanics is the problem of the indistinguishability of so-called “identical�� particles, that is, particles that share the same state-independent properties. In the framework of this philosophical problem, a quasi-set theory was formulated to provide a proper metalanguage to deal with quantum indistinguishability; this theory included certain Urelemente called m-atoms, representing essentially indistinguishable objects. In turn, over the last two decades, the Modal Hamiltonian Interpretation proposed an ontology of properties, totally devoid of (...) objects, where quantum systems are bundles of instances of universal properties. Therefore, the original quasi-set theory, with its m-atoms, does not adequately reflect the structure of an ontology devoid of objects. The purpose to the present article is to introduce a new quasi-set theory that does not include atoms at all: elementary items correspond to properties and are also represented by quasi-sets, which can be only numerically different. The final aim is to apply this new quasi-set theory to the MHI ontology. (shrink)

This paper establishes the first Randomized Controlled Trial (RCT) protocol for testing the Moral Kernel framework in municipal conflict-mediation contexts. It formalizes the theoretical foundations of the TRO/MKO canon within classical closure and fixed-point theory, defining typal classes, guardrails, and the Margan Closure Theorem as empirical invariants. The paper proves the Kernel Transfer/Dichotomy Theorem, demonstrating that all event-valued worlds converge to one of two absorbing states—closure or perpetuation—with no third outcome. It further introduces the Closure-Pack Decision Problem, showing NP-hardness and (...) proposing algorithmic heuristics for applied policy experiments. This protocol provides a testable mathematical model for typal closure dynamics and establishes the formal basis for empirical kernel trials in governance, mediation, and AI alignment. (shrink)

This paper addresses the development of explainable decision support systems (DSS) for aerospace design through the integration of human expertise and computational intelligence. Traditional machine learning methods and high-fidelity simulation models often lack transparency, which constrains engineer trust and complicates human–system interaction in safety-critical design contexts. To overcome these limitations, a combined methodological framework is considered that unifies explainable artificial intelligence (XAI), human-in-the-loop decision-making, physics-informed neural networks (PINNs), and digital twins within a coherent design support paradigm. Explainable models enable interpretable (...) assessment of the influence of design parameters on optimization outcomes, while human-in-the-loop mechanisms ensure continuous expert validation and alignment with safety regulations and operational standards. The use of PINNs guarantees physical consistency of computational models by embedding governing physical laws into data-driven representations. Digital twins further provide an interactive platform that integrates simulation results, measurement data, and expert judgments for comparative analysis of alternative design scenarios. The results demonstrate that such hybrid and explainable frameworks enhance transparency, reproducibility, and reliability of aerospace design decisions, while facilitating effective collaboration between engineers and intelligent systems. The study identifies key perspectives for future research, including deeper integration of physical modeling into explainable algorithms, advancement of visualization techniques, and the development of synergistic platforms combining digital twins and human-in-the-loop approaches for multi-objective optimization under uncertainty. (shrink)

One of the most fundamental theoretical results in quantum mechanics, the theorem of Simon Kochen and Ernst Specker (1967), is investigated from a rather mathematical and philosophical than physical viewpoint (i.e. unlike as usual). The absence of hidden variables is interpreted philosophically and ontomathematically: as the identity of the mathematical model by the separable complex Hilbert space (equivalent to the qubit Hilbert space) and physical reality. It implies the completeness of just that model to physical reality including in the sense (...) of its cognitive finality. In other words, "hidden variables" are also interpreted as any mismatch between model and reality implied by the theorem not to be. That property of the qubit Hilbert space unique among all other mathematical theories (i.e., all first-order logics) is deduced as a corollary from the theorem as well. The approach at issue to the theorem implies its inherent contextuality to be deduced also in an "apophatic" or holistic version stating that omitting even a single "axis" of Hilbert space or a single value of probability density break its contextuality thus being inconsistent to the theorem. The concept of "fundamental randomness" is newly introduced as originating from the absence of hidden variables in any probability (density or not) distribution, the characteristic function of which is interpreted to be a (qubit) wave function. Then, the Kochen-Specker theorem implies no deviation from that "fundamental randomness" exists once its preconditions have been satisfied. The deviations at issue are seen to be any excluded degrees of freedom in a variable describing whatever. Just this statement is interpreted to be the theorem's "apophatic interpretation". A powerful method for proving statements in Hilbert arithmetic's number theory is offered as a conjecture to be reasoned and proved in a forthcoming paper. (shrink)

Large-scale generative models display characteristic structural failure modes whenever they are forced into excessive semantic, logical, or computational coherence. Contemporary Transformer-based architectures lack intrinsic mechanisms to manage conflict, undecidability, and paradox; instead, they tend to smooth or collapse contradictions away. This leads to brittle behaviour in tasks that require sustained global inconsistency, such as impossible Escher-like objects or high-context humour, and exposes deeper limitations for any claim of Artificial General Intelligence (AGI). Olympus-Q is a quantum-inspired, multi-stage reasoning architecture that treats (...) controlled incompleteness as a design principle rather than a defect. The system combines three core components: (i) a superpositional routing layer that maintains a distributed field of competing hypotheses, (ii) a semantic interference operator that detects and modulates regions of conflict and Gödel-style undecidability, and (iii) a toroidal cellular automaton meta-layer that implements thermodynamically motivated entropy regulation and stabilises the hypothesis space at the edge of collapse. Safety is embedded as an intrinsic attenuation process within the CA dynamics, rather than delegated to external post-hoc filters. The architecture is grounded in the Seidel–Bostrom Event Horizon Bound (SBEB), which establishes a gravitational upper limit on coherent information concentration, and in a system-theoretic framework of collapse principles for complex, self-organising systems. This paper develops the theoretical motivation, formal structure, and expected dynamical properties of Olympus-Q, and outlines how it can serve as a physically consistent blueprint for future AGI research, including potential hybrid classical–quantum implementations. (shrink)

In this article, I elaborate a proposal to overcome McKenzie’s argument against the possibility of progress-talk in metaphysics (2020). McKenzie states that said discourse is not possible in metaphysics due to its “all-or-nothing” character: either metaphysical theories are true or not, without intermediaries, so the language of approximation cannot meaningfully apply to changes between metaphysical theories. I show that discussions regarding the type of metaphysical relationship between two relata are susceptible to retention and refinement, two criteria that McKenzie considers necessary (...) for progress, as long as they tend towards better matching what is observable in reality. This is possible if metaphysical relationship types are characterized as sets of properties, from which the Jaccard Index between two sets can be utilised to determine the degree of similarity or approximation between relationship types. If such is the case, then a transition from one relationship thesis to another can be considered progress as long as some properties are retained and the change is guided by empirical evidence. Hence, progress-talk in naturalized metaphysics is possible according to McKenzie’s own criteria. (shrink)

This paper presents a concise analytical review of selected case studies on the integration of multiphysics methods in aerospace structural design, with emphasis on rocket and high-speed vehicle applications. The review synthesizes peer-reviewed publications (2022–2025) indexed in Web of Science and Scopus that address coupled Computational Fluid Dynamics—Finite Element Method (CFD–FEM) analyses, fluid–structure interaction (FSI), conjugate heat transfer (CHT), ablation and thermal protection modeling, and multidisciplinary design optimization (MDO) incorporating surrogate and machine-learning accelerants. Source selection and evaluation prioritized methodological transparency, (...) relevance to structural design tasks, and applicability of results to engineering practice. The surveyed studies indicate that fully coupled CFD–FEM and FSI formulations are capable of reproducing key interactions among aerodynamic, thermal and structural fields, while posing significant numerical and computational demands related to mesh compatibility, stiffness of governing equations and temporal integration. CHT approaches yield improved estimates of interfacial heat fluxes for nozzles and TPS elements when turbulent flow models are combined with wall conduction models. Multidimensional ablation models provide mechanistic insight into material removal processes but require careful treatment of thermochemical coupling and validation against experimental data. Recent developments in multifidelity optimization and gradient-enhanced surrogate models demonstrate potential to reduce computational cost in MDO workflows without discarding essential physics. Given the limited scope inherent to a concise analytical review, conclusions are presented as informed recommendations rather than exhaustive generalizations. Recommended practical measures for engineering application include: explicit consistency of boundary and initial conditions across disciplinary solvers; use of adaptive local mesh refinement in regions of strong gradients; adoption of hybrid coupling strategies (e.g., weak/iterative coupling, multiscale and multifidelity frameworks) to balance accuracy and cost; and integration of surrogate or data-driven models into optimization loops with rigorous uncertainty quantification and targeted validation for TPS, nozzle and thin-walled components. These recommendations aim to support practitioners and researchers seeking to improve predictive capability and computational efficiency in multiphysics design procedures for aerospace systems. (shrink)