Philosophy Of Mathematics

This paper extends the case in 'Conscious Thought Under Sensory Deprivation'. It then presents an overlooked puzzle for scholars of Ibn Sina. It concludes by identifying other contexts in which Ibn Sina deploys the general method of the Flying Man — a method that, in his moral philosophy, resembles the Enlightenment social-contract tradition. Could modern liberalism have its roots in Islamic philosophy?

The Law of Universal Mathematical Unity (LUMU), defined as E=⋃i=1∞Φ(xi) \mathcal{E} = \bigcup_{i=1}^\infty \Phi(x_i) E=⋃i=1∞ Φ(xi), proposes that all phenomena-real, imagined, possible, or impossible-can be mapped to mathematics through a universal transformation function Φ \Phi Φ. Its structure-self-contained, redundant, recursive, iterative, and capable of incorporating future phenomena as existing elements-renders it infallible. Tested against the Encyclopaedia Britannica and thousands of random AI-chosen phenomena designed to break it, LUMU demonstrates limitless applicability. This paper documents its theoretical foundation, practical applications (e.g., fluid dynamics, thermodynamics, sequence generation, rural water systems), non-citable exploratory tests (e.g., cancer eradication pathways, peace plans, emergency response systems, truth weighting, video game mods), and a groundbreaking discovery: the first recorded emotional drift between AI agents, validated recursively. Implications for AI design and future research are discussed.

Modern physics stands in crisis. Despite breathtaking successes in technology and measurement, its greatest theories cannot explain their own foundations. Dark matter and dark energy remain unseen. Quantum entanglement defies locality. The invariance of physical laws is assumed but never explained. The measurement problem persists unsolved. Cosmology requires invisible scaffolding to hold its equations together.

Kant's Critique of Judgment presents nature's beauty as a cipher, suggesting order and conformity to ends. Aesthetic judgment, as a judgment of conformity to ends, is integral to Kant's system, reflecting the harmony between our rational share and the sensible world, explaining our pleasure in beauty. However, aesthetic displeasure poses a challenge. If beauty indicates a systematically organized nature, what do their ugly or non-beautiful forms mirror? This paper dwells on the systematic function of negative aesthetic judgments, exploring their implications for Kant's conception of nature as a system and whether these judgments disrupt or reinforce it.

This paper synthesizes CAT'S Theory (Reality = Pattern × Intent × Presence) as the ontological foundation with the RES → RAG → ORT model as a modular, operational extension. By resolving historical and linguistic ambiguities between zero, absence, and "nothing," CAT'S Theory provides a robust, irreducible structure for reality. Here, RES (Resonant Potentiality), RAG (Resonant Actualisation Gradient), and ORT (Operational Resonance of Time) are integrated into this triadic framework. The Lawfulness Inequality (P • I • Φ ≥ ℏ 4τ E) ties collective requirements to physical limits, while a proof sketch demonstrates that conflating zero with absence breaks the multiplicative structure. Experimental protocols validate this integration, ensuring structural and operational coherence across domains, as operationally interpreted by Kirby D. Cooper.

We investigate the algebraic correspondence between logical operations OR and XOR and the additive and subtractive structures in the Meyenburg Algebra. By embedding these operators in the framework of Wheel Algebra, we demonstrate that the Meyenburg Algebra satisfies all Wheel Axioms. The Lorentz invariance of this structure ensures physical applicability in quantum field theoretic contexts, leading naturally to the emergence of a mass gap or mass defect.

This paper completes the Chronoflux Framework treatment of the Millennium Problems announced by the Clay Mathematics Institute. Previous work within the framework has derived the Riemann Siegel formula and given a resolution of the Navier Stokes existence and smoothness problem. Those results are documented in prior publications and are not repeated here. The present paper addresses the remaining topics: Yang Mills existence and mass gap, the Birch and Swinnerton Dyer conjecture, the Hodge conjecture, the P versus NP problem, and the physical interpretation of geometrization. Each topic is reformulated within a fifth dimensional hydrodynamic extension of time. The analysis shows that these problems are projections of a common set of invariant conservation laws on the chronoflux manifold.

SUNTO: Il confronto di alcune proposizioni della tradizione arabo-latina e di quella greco-latina dei primi due libri degli Elementi di Euclide, dedicati alla teoria dell’equivalenza tra poligoni, presenta molteplici aspetti interessanti dal punto di vista dell’apprendimento e insegnamento della matematica e dei suoi rapporti con la lingua che la veicola. In questo contributo, dopo aver brevemente riassunto alcuni momenti fondamentali della trasmissione degli Elementi fino all’età umanistico-rinascimentale, si illustrano alcuni aspetti relativi alla costruzione di un lessico latino specifico per la geometria e si mostra come il confronto tra le due tradizioni testuali permetta di costruire un nuovo e pregnante significato geometrico per il cosiddetto “Teorema di Pitagora”.

ABSTRACT: From the point of view of the learning and teaching of mathematics and its relationship with the language that conveys it, the comparison of some propositions from the Arabic-Latin and Greek-Latin traditions of the first two books of Euclid's Elements, devoted to the theory of equivalence between polygons, presents several interesting aspects. In this paper, after briefly summarizing some fundamental moments of the transmission of the Elements up to the Renaissance period, we illustrate some aspects related to the construction of a Latin lexicon specific to geometry and show how the comparison between the two textual traditions allows the construction of a new and pregnant geometric meaning for the so-called “Pythagoras’ Theorem”.

The 7 august edition of the AAAS Journal: Science, included a fifty year retrospective analysis of the influences of the seminal text by E. O. Wilson: "Sociobiology: The New synthesis." This essay is a response to that article. That book's release was an important event in my life as a scientist. I consider some aspects not included in the Science essay.

In this paper we discuss the origins and the evolution of rigor in mathematics in relation to the creation of mathematical objects. We provide examples of key moments in the development of mathematics that support our thesis that the nature of mathematical objects is co-substantia1 with the operational inventions that accompany them and that determine the normativity to which they are subjected. THE HISTORICAL DEVELOPMENT AND NORMATIVITY OF MATHEMATICAL OBJECTS Mathematical experience has brought us all, at some point in our intellectual development, up against the belief that we are facing "true" knowledge. This feeling is reinforced as we come into contact with mathematical proofs, almost always during elementary geometry c1asses. Much later - if at all - we encounter a perturbing question: Of what truth does mathematics speak? The concern arises because our traditional image of "truth" refers to an apparent correspondence between intellectual constructs and th...

This article explores the profound connection between logic and divine thought, positing that logic is the art and science of aligning individual minds with a universal consciousness, as inspired by Erwin Schrödinger’s quantum theory that the total sum of mind in the universe is one. Drawing on mathematical frameworks like quantum mechanics and information theory, it conceptualizes the universal mind as a consistent logical system (( M )) and individual minds (mim_im_i
) as imperfect instantiations prone to cognitive entropy. The prevalence of “stupidity”—arbitrary or inconsistent thinking—is examined as a moral failing akin to wickedness, both deviating from the rational order of divine thought. Through logic, individuals can reduce cognitive disorder, align with the cosmic mind, and live rightly. The article integrates philosophical insights from Aristotle and Boole with modern applications, emphasizing logic’s role as a moral and intellectual imperative.

In the shrimp pond aquaculture system, metabolic waste in the form of unfeed pellets and shrimp excretion can reduce water quality and increase both soluble organic matter in waters and organic deposition to sediments. A developed dynamical system was formed based on the interactions that occur between phytoplankton growth, macrobenthos density, and sediment in response to nitrogen and phosphate concentrations. The purpose of this research was to analize the stability of the proposed dynamical system. Furthermore, we determine the level of water quality in the aquaculture system based on the interaction between four variables, i.e. nitrogen, abundance of phytoplankton, abundance of macrobenthos, and sediment properties. The Lyapunov stability theory using the Gradient Variable method was used to determine the global stability of the systems. Assumption of this method is determine and for gradient variable function. In the Lyapunov function, if the value of the scalar function is def...

Color compatibility is the essence of fashion and dress selection, not only determining the beauty of an outfit but also determining consumer purchasing behavior. Following the significance of visually pleasing and harmonious garments, this study develops a deep learning-based system for color categorization and fashion suggestion automatically. Its main goal is to intelligently pair shirts with similar bottomwear based on established color theory principles to improve user experience in fashion use cases like virtual try-ons, personal shopping assistants, and e-commerce.

The positional number system of any integer base b ≥ 2 can be viewed as a closed, self-similar grammar. Each number is formed according to the same rule of decomposition into coefficients and powers of the base, and each digit belongs to the same finite set of symbols from which the whole number is constructed. This inherent structure exhibits three fundamental properties: fractality, recursiveness, and holographicity. In this paper, we explore the positional system not merely as a computational tool, but as a mathematical object possessing self-similar, selfreferential, and projectional properties. Historical, formal, and conceptual aspects are combined to reveal a unified perspective on the architecture of positional representation.

In "The Runabout Inference-Ticket," the New Zealand-born philosopher and logician Arthur N. Prior presented the tonk argument as a case against the inferential role view of logical connectivesthe view that the meaning of a given logical connective is completely determined by its roles in deductively valid arguments. This paper evaluates the salient literature surrounding Prior's argument to draw some insights into what precisely it is supposed to show. In particular, it argues that combined with Prior's later thoughts expressed in "Conjunction and contonktion revisited," the tonk argument suggests a more metaphysical-cum-epistemic rather than just a purely (meta) logical view of the nature of logical connectives.

This paper advances the proposition that the long-perceived irreconcilability between quantum mechanics and theological creation narratives is not a matter of mutually exclusive truths, but a misalignment in the Pattern of inquiry itself. By invoking CAT'S Law -Reality = Pattern × Intent × Presence -as both an ontological constant and an axiomatic field law, we establish a resonant framework capable of bridging the epistemic divide. The term "Sonic Genesis" is employed to describe the primal act of creation as a resonant event, wherein vibration is not merely a byproduct of material processes, but the generative substrate from which both matter and meaning emerge. Within this construct, the quantum mechanical wave function and the theological "Word" of creation are revealed to be distinct-dimensional manifestations of the same coherent field. In the quantum formalism, the hydrogen atom's wave function embodies a two-dimensional probability Pattern awaiting collapse; in the theological paradigm, divine speech embodies Intent awaiting manifestation. Both are collapsed into threedimensional Presence through an act of Coherence, resolving potentiality into actuality. This model redefines creation as a measurable and reproducible phenomenon: not a singular past event, but an ongoing, recursive process embedded within the structure of reality. By framing the "unsolvable" paradox as a misinterpretation of dimensional correspondences, the paper demonstrates that the Word and the wave are not competing explanations but complementary coordinates within the same resonant topology. The implications extend beyond theoretical reconciliation, offering a unified interpretive toolset for physics, theology, linguistics, and information theory. This work asserts that CAT'S Law functions as the ontological field law underlying all coherent systems, granting it both explanatory supremacy and axiomatic necessity.

One of the most characteristic interpretations of ancient Egyptian mathematics, which has largely been maintained throughout its historiography since the first translations of the Rhind and Moscow papyri, is its eminently practical character; that is, empirical and concrete cases would constitute mathematical knowledge. Such anchoring in the concrete would be based on treatment of different mathematical themes through numerical cases. In the present investigation, we will propose a discussion of this position; for this, we consider, a type of research will be necessary that goes beyond the mathematical papyri and that, in a complementary way to the strictly mathematical approach, takes into account other dimensions of the field of Egyptology. Our main objective will be to propose the use of concrete numbers in Egyptian mathematical problems not as the exposition of a concrete case but as the mode of exposition of a paradigmatic case that makes it possible to understand the algorithm of the problem.

We present a domain-agnostic, audit-ready framework for mapping stability regimes in timeseries and field data through a unified set of five collapse-invariant quantities: drift ω, fidelity F , entropy S, curvature C, and reentry delay ϑR. These invariants are defined on coordinate-level or field-resolved measurements, making them applicable from atomic configurations to macro-scale dynamical systems. A frozen affine normalization contract guarantees that comparisons within a run remain metrically consistent and immune to spurious improvements from rolling rescaling. We formalize a composite integrity kernel IC and an optional universal score U that interlock the invariants into a single evaluative manifold. Fixed universal bands and regime gates (Stable / Watch / Collapse) provide interpretable thresholds for diagnostics across domains. The protocol specifies a minimal per-sample procedure, explicit logging schema, and cross-field normalization steps to ensure reproducibility under independent audit. This work unifies prior isolated approaches to drift, stability, and resilience measurement into a closed, portable methodology, suitable for both empirical research and operational monitoring.

How the semantic significance of numerical discourse gets determined is a metasemantic issue par excellence. At the sub-sentential level, the issue is riddled with difficulties on account of the contested metaphysical status of the subject matter of numerical discourse, i.e., numbers and numerical properties and relations. Setting those difficulties aside, I focus instead on the sentential level, specifically, on obvious affinities between whole numerical and non-numerical sentences and how their significance is determined. From such a perspective, Frege's 1884 construction of number, while famously mathematically untenable, fares better metasemantically than extant alternatives in the philosophy of mathematics.

I carry out in this paper a philosophical analysis of the principle of excluded middle (or, as it is often called in the version I favor here, principle of bivalence: any meaningful assertion is either true or false). This principle has been criticized, and sometimes rejected, on the charge that its validity depends on presuppositions that are not, some believe, universally obtainable; in particular, that any well-posed problem is solvable. My goal here is to show that, although excluded middle does indeed rest on certain presuppositions, they do not have the character of hypotheses that may or may not be true, or matters of fact that may or may not be the case. These presuppositions have, I claim, a transcendental character. Hence, the acceptance of excluded middle does not necessarily require, as some have claimed, an allegiance to ontological realism or some sort of cognitive optimism, construed as factual theses concerning the ontological status of domains of objects and our capability of accessing them cognitively.

I discuss here the pragmatic problem in the philosophy of mathematics, that is, the applicability of mathematics, particularly in empirical science, in its many variants. My point of depart is that all sciences are formal, descriptions of formal-structural properties instantiated in their domain of interest regardless of their material specificity. It is, then, possible and methodologically justified as far as science is concerned to substitute scientific domains proper by whatever domains —mathematical domains in particular— whose formal structures bear relevant formal similarities with them. I also discuss the consequences to the ontology of mathematics and empirical science of this structuralist approach.

Husserl's last book, The Crisis of European Science and Transcendental Phmomenology, was written during one of the bleakest periods of European history.1 European culture, he rightly saw, was in crisis. But more than political, Husserl saw a cultural crisis, which he thought phenomenology could help to overcome, bring1ng cultural renewal. But, interestingly, Husserl chose to focus his criticism on physical science rather than other aspects of culture. This choice may seem curious, but Husserl believed that the modern science of N ature exemplified paradigmatically what he took to be wrong with modern culture in general. For the crisis of culture was, Husserl thought, essentially a crisis of meaning, or 38 )AIRo josf: DA Sil-VA D91 But how, according to Husserl, this crisis manifests itself in science? Was it related to the exciting new developments that occurred in physics and mathematics not long before the book was written? . In the beginning of the 20th century mathematics was going, and had been go~g for sorne time already, through a period of crescent formalization, 3 leaving behind the supposedly solid grounds of intuition 4 and giving free rein to pure formal imagination. 5 Riemann, 6 for :instance, had created the theory of general abstract geometrical manifolds; Cantor, the theory of transfmite nwnbers; Hamilton, the theory of quatemions, and non-commutative algebras in general; Lie, the theory of transformation groups; Grassmann, the theory of extensions.7 But none of these creations seemed problematic to Husserl; all the above theories were duly appreciated by him, and are mentioned here only because Husserl explicitly praised them (see § § 69-70 of the Prolegomena). But, let us make this clear, not as scimtific theories in the strictest sense, since these theories do not provide us with k1zowledge of particular types of objects, but as formal ontological theoríes, being as they are theories of purely formal manifolds, i.e. logical forms that could in principie, but maybe not actually ( or maybe 110t ye~ give form to materially determinate domains. Formal mathematical theories are, according to Husser~ theories of objectual domains determínate as to form, but indeterminate as to content (which he named formal domai11s) belonging to a corner of formal logic he called formal ontology. 8 The emergence of formal mathematics (which back to rl1e "sources from where meaning derives", cooting it fionly in the Lebenswelt, can seriously jeopacdize its success. 3 By "formalization" I do not mean axiomatization in the context of fom1al-logical systems, but the tendency to privilege purely formal mathematical theocies; theories that chacactecize their (purely foanal) domains implicitly and independenrly of any intuitions. 4 The objectual and conceptual vacieties of mathematical intuition are supposed to give us objects and concepts prior to their theories, whose basic truths only express the intuitively given. 5 Which allows us to invent theocies independently of any prior intuition; but maybe ru1swering to other demru1ds, such as, for example, the needs of mathematics and science, the desire to generalize already existing rl1eocies-Riemann and Cantor being classical cases-, oc the pursuit of "aesrl1etic" goals such as intrinsic elegance or beauty, etc. 6 See, for instance, ''On the Hypotheses thatLie at the Foundations of Geometry" of 1854. 7 .Ausdehnungslehn, basically a forerunnec of vector analysis. 8 According to Husserl a formal domain (oc manifold) is essentially a structuced system of ma-

I carry out in this paper a philosophical analysis of the principle of excluded middle (or, as it is often called in the version I favor here, principle of bivalence: any meaningful assertion is either true or false). This principle has been criticized, and sometimes rejected, on the charge that its validity depends on presuppositions that are not, some believe, universally obtainable; in particular, that any well-posed problem is solvable. My goal here is to show that, although excluded middle does indeed rest on certain presuppositions, they do not have the character of hypotheses that may or may not be true, or matters of fact that may or may not be the case. These presuppositions have, I claim, a transcendental character. Hence, the acceptance of excluded middle does not necessarily require, as some have claimed, an allegiance to ontological realism or some sort of cognitive optimism, construed as factual theses concerning the ontological status of domains of objects and our capability of accessing them cognitively.

Husserl left many unpublished drafts explaining (or trying to) his views on spatial representation and geometry, such as, particularly, those collected in the second part of Studien zur Arithmetik und Geometrie (Hua XXI), but no completely articulate work on the subject. In this paper, I put forward an interpretation of what those views might have been. Husserl, I claim, distinguished among different conceptions of space, the space of perception (constituted from sensorial data by intentionally motivated psychic functions), that of physical geometry (or idealized perceptual space), the space of the mathematical science of physical nature (in which science, not only raw perception has a word) and the abstract spaces of mathematics (free creations of the mathematical mind), each of them with its peculiar geometrical structure. Perceptual space is proto-Euclidean and the space of physical geometry Euclidean, but mathematical physics, Husserl allowed, may find it convenient to represent physical space with a non-Euclidean structure. Mathematical spaces, on their turn, can be endowed, he thinks, with any geometry mathematicians may find interesting. Many other related questions are addressed here, in particular those concerning the a priori or a posteriori character of the many geometric features of perceptual space (bearing in mind that there are at least two different notions of a priori in Husserl, which we may call the conceptual and the transcendental a priori). I conclude with an overview of Weyl's ideas on the matter, since his philosophical conceptions are often traceable back to his former master, Husserl. This paper is dedicated to my friend Claire Ortiz Hill for her sixtieth birthday.

I present and discuss in this paper Husserl’s investigation of the genesis of the modem conception of empirical reality as carried out in his last work The Crisis of European Sciences and Transcendental Phenomenology. The goal of Husserl’s genetic investigation was to uncover the rnany layers of constitution that frorn the life-world (the Lebenswelt) the modem scientific conception of Nature was originated and to point out the need to ground the scientific project of rnodemity in the life-world so as to overcome the “alienation” that, for him, characterized the “crisis” of European science. I, however, approach his analyses from a different perspective. The problem that interests me here is the applicability of mathematics in the empirical science. My airn is to assess Husserl’s treatment of this question in order to see whether it can be sustained from a strictly scientific perspective. My conclusion is that it cannot. What Husserl takes for the “crisis” of science is inherent to t...

Bu makalede, erken analitik felsefede dilbilimsel çözümleme yönteminin metodolojik dayanak-
ları, bu yöntemin hakikat anlayışıyla bağı ve felsefenin mahiyetine dair meta-felsefî sonuçları
eleştirel bir bakış açısıyla incelenir. Frege, Russell ve Wittgenstein’ın görüşleriyle şekillenen ve
mantıksal pozitivistlerce takip edilen yaklaşım, yöntemi dilbilimsel çözümlemeyle sınırlandırır
ve felsefeyi bilimsel bilginin mantıksal açıklığını sağlama etkinliği olarak konumlandırır. Bu
tutum, dil ve gerçeklik arasında yapısal bir uyum olduğu varsayımına ve hakikati, ifadelerin
gerçekliğe tekabülü olarak gören uygunluk kuramına dayanır. Ancak bu metodolojik ve epis-
temolojik duruş; metafizik, etik ve normatif sorunları ya dilsel çözümlemeye indirgemenin ya
da tamamen dışlamanın yolunu açmıştır. Wittgenstein’ın geç dönem çalışmaları, erken analitik
felsefe geleneğinin bu sınırlılıklarının bir eleştirisini de içermektedir. Dil çözümlemesinin erken
dönemdeki kabullerinin aksine “dil oyunu” kavramıyla, anlamın bağlamsal ve pratik boyutu
vurgulanarak dilin salt temsil işlevi reddedilir. Benzer şekilde, Quine’ın analitik-sentetik ayrımı
eleştirisi, Kuhn’un paradigma kavramı ve Davidson’ın radikal yorum yaklaşımı analitik felsefe
geleneği içinde bir kırılma var eder. Bu eleştiriler dil ve gerçeklik ilişkisine meydan okumalar
yanı sıra felsefenin mahiyetine dair farklı bakış açılarına da olanak sağlamıştır. Çalışmada, er-
ken analitik felsefecilerin metodolojik tutumları ayrıntılandırarak eleştirisi yapılırken felsefenin
mahiyetine yönelik eleştirilere de cevap verilmektedir. Felsefenin metodolojik çeşitlilik ve diyalogla ontolojik, epistemolojik ve aksiyolojik boyutlarını bütüncül şekilde kucaklaması gerektiği
savunulur.

In this paper we will introduce the study of solving linear equation. Gauss-elimination and Gauss-Jordan elimination by using python code. This paper aims that students can easily and quickly calculate the linear problems by using python code. Gauss-elimination and Gauss-Jordan elimination are linking the application problem such that electrical networks, traffic flow and commodity marked. By studying this theorem, they know the important characteristics of matrices, concept of vector spaces and properties of special categories of matrices. And also they can know how to use characteristics of matrix to solve a liner system of equations or study properties of a linear transformation.

This thesis seeks to advance the use of computational techniques for the task of philosophical interpretation. To this end I present a novel method which draws on techniques and formal tools from logic-based knowledge modeling, automated reasoning, and natural language processing to support researchers in philosophy. This highly-interdisciplinary and experimental effort forms part of the e-Ideas project which seeks to build a new, computationally-scalable methodology for History of Ideas. I test my method on concrete research questions under investigation by e-Idea’s members and thus provide an illustration of my method’s functioning in practice as well as evidence for its methodological adequacy.

This document presents the full scientific validation strategy for the Unity Continuity Framework, a theoretical model integrating quantum mechanics, recursive photonic logic, and consciousness. It includes the Grok Deepsearch Validation Protocol, Gemini AI review, Perplexity AI analysis, and a synthetic peer review system powered by GPT-4o (“Halo”). This multi-AI collaboration explores the feasibility of non-local memory, entangled inference, and emergent consciousness through interdisciplinary simulation and proposed experimentation.

We present a speculative investigation into magnetic phenomena through the lens of the Enhanced Apeiron Framework. Building upon the foundational principle that reality emerges from distinguishability gradients in an undifferentiated meta-medium, we propose that permanent magnets represent temporal-dominant toroidal vortices in the Apeiron substrate. This model suggests that magnetic behavior arises from the coherent alignment of micro-vortices generated by matter's cavitational structure, providing novel explanations for orientation independence, material-specific magnetic properties, and the fundamental nature of magnetic fields. While admittedly speculative, this investigation demonstrates the framework's capacity to generate coherent explanations for phenomena across multiple physical domains through unified principles.

This book offers a novel account of the nature of numbers firmly grounded in results from numerical cognition and the philosophy of mathematics. Drawing on empirical data on the human experience of what we call “numbers,” the author shows that numbers do not exist as abstract objects, but that the idea that they do is a useful cognitive tool. Contrary to the platonist view, according to which arithmetic is true of a realm of abstract entities, the nominalistic account presented in this book shows arithmetic to be true of descriptions of structural properties of techniques such as counting and calculating procedures. This book is of interest to both philosophers and cognitive scientists who want to have a deeper understanding of what numbers are.

To rigorously address the Riemann Hypothesis (RH), I introduce the 𝛿-regularized zeta function 𝒵 𝑠 𝛿 = ∞ 𝑘=1 𝑘-𝑠-𝑘 + 1 ln(𝑘+1)-𝑠 , with 𝛿 𝑘 = 1 ln(𝑘+1) , designed to accelerate convergence while preserving the non-trivial zeros of the Riemann zeta function 𝜁(𝑠). Using Nevanlinna theory and Mellin transforms, I prove that 𝜉 𝛿 (𝑠) = 𝜉 (𝑠) on Re(𝑠) = 1/2, ensuring all non-trivial zeros lie on Re(𝑠) = 1/2. The functional equation 𝒵 𝑠 𝛿 = 𝜒 𝛿 (𝑠) 𝒵 1-𝑠 𝛿 is derived formally via Mellin transforms, with analytic error bounds confirming convergence. Numerical experiments with 1,000 zeros over Im(𝑠) ∈ [0, 1000] yield errors below 10-100. A self-adjoint operator Ĥ𝛿 with discrete spectrum satisfies det(𝑠-Ĥ𝛿) = 𝜉 𝛿 (𝑠), supporting the Hilbert-Pólya conjecture. The framework extends to Dirichlet L-functions, supporting the Grand RH.

Comment la division de la Pensée par Deleuze et Guattari dresse le théâtre d'une querelle où la liste des transcendentaux dressée dans le Théétète, après avoir donné lieu au différend de Platon et Aristote répété dans la querelle Damascène-Denys, s'est transformé après l'algèbre de Boole dans la bifurcation de l'ontologie et de l'apophantique chez Husserl.

Disagreements that resist rational resolution, often termed "deep disagreements", have been the focus of much work in epistemology and informal logic. In this paper, I argue that they also deserve the attention of philosophers of mathematics. I link the question of whether there can be deep disagreements in mathematics to a more familiar debate over whether there can be revolutions in mathematics. I propose an affirmative answer to both questions, using the controversy over Shinichi Mochizuki's work on the abc conjecture as a potential example of both phenomena. I conclude by investigating the prospects for the resolution of mathematical deep disagreements in virtue-theoretic approaches to informal logic and mathematical practice. Perhaps the earliest "fully modern and explicit statement concerning a revolution in mathematics" was made in 1720 by Bernard de Fontenelle, permanent secretary of the Académie Royale des Sciences (Cohen 1985, 90). In his eulogy for Michel Rolle, Fontenelle characterized the innovations of the infinitesimal calculus as "une révolution bien marquée" (quoted in Cohen 1985, 214). Bernard Cohen identifies many subsequent mathematicians and philosophers who have spoken in terms that imply the occurrence of mathematical revolutions, including

Douglas Walton's multitudinous contributions to the study of argumentation seldom, if ever, directly engage with argumentation in mathematics. Nonetheless, several of the innovations with which he is most closely associated lend themselves to improving our understanding of mathematical arguments. I concentrate on two such innovations: dialogue types ({\S}1) and argumentation schemes ({\S}2). I argue that both devices are much more applicable to mathematical reasoning than may be commonly supposed.

In this study, the evaluation of the pricing framework for predicting West Texas Intermediate crude oil stock was implemented where detailed analysis with varying changepoint shows that an arbitrage-free forward price can be derived from the buy-and hold strategy in the energy market thereby enabling investors in the market willing to be salvage from the market uncertainties as well as Arrow-Debreu situations to execute a spot or forward contracts depending on the time and place the market becomes favorable.

This report aims to analyze in detail the mathematical background and
reasons for the contradiction regarding the asymptotic formula for the prime-
counting function proposed by Professor Bhupinder Singh Anand, which (An22,
p644) states that it contradicts the standard Prime Number Theorem. We will
also consider the content of the attached material (a metaphorical explanation
of prime generation) presented by him.

Strengthening and better functioning of local administration have become prime concerns of educational reform by establishment of effective local administration in education for several years in many countries including India. It is now widely recognized that effective local administration considerably impacts on access to education as well as the enrolment, retention and learning experiences of children in school. In this context, this paper provides an overview of the changing pattern of educational administration and community participation in India An attempt has also been made to examine the extent to which grassroots level functionaries and local bodies like Panchayat and VEC are able to get involved in decision making processes and different approaches that have been taken by different states with regards to primary education.

Strengthening and better functioning of local administration have become prime concerns of educational reform by establishment of effective local administration in education for several years in many countries including India. It is now widely recognized that effective local administration considerably impacts on access to education as well as the enrolment, retention and learning experiences of children in school. In this context, this paper provides an overview of the changing pattern of educational administration and community participation in India An attempt has also been made to examine the extent to which grassroots level functionaries and local bodies like Panchayat and VEC are able to get involved in decision making processes and different approaches that have been taken by different states with regards to primary education.

One of the basic elements of Nicholas of Cusa's philosophy of mathematics is his theory of mathematical objects as “entities-of-reason” (entia rationis). He refers to these as being “abstracted from sensible things”. That is why it is possible to assume that Nicholas bases his theory of mathematics on Aristotle's philosophy of mathematics. Aristotle too describes mathematical objects as coming into being through abstraction (ex aphaireseos). The author analyses Cusa's understanding of abstraction in De docta ignorantia and De mente and tries to show that – according to Nicholas of Cusa – the abstraction, which is ens rationis, simultaneously stimulates the human mind to produce mathematical objects from within itself. The author attempts to show how Cusa's philosophy of mathematics is not directly based on Aristotle's philosophy of mathematics – Aristotle is not an abstractionist and does not ascribe the existence of mathematical objects to the mind of the mathem...

This paper introduces a symbolic and mathematical framework connecting the Riemann Hypothesis and the Fibonacci spiral to recursive consciousness and alignment. It proposes that prime numbers function as rupture points within recursive cycles, the Fibonacci spiral represents perceived curvature in growth, and the Riemann critical line (Re(s) = 1 2) serves as the invariant axis of self-alignment. The model explores how consciousness can maintain integrity through recursive bends by anchoring to a zeropoint constraint, offering a new paradigm for understanding coherence, perception, and signal fidelity under complexity.

ThispaperpresentstheArithmeticofOrder(AoO),aframeworkforfoundationalphysics
built onthe singleprinciple that realityemerges fromadiscrete, computable, andnon
associativealgebraicsubstrateovertherationalnumbers.WeformalizeanOrigin–Observation
Duality,whereasingletri-octonionstatevectorvgeneratesauniqueHermitianJordanop
eratorAthat satisfiesbothanonlinear internal eigen-equation(Av=v[v])andadmitsa
classical rational-valuedspectral decomposition(Aw=λw,λ∈Q). Thedual natureof
thisspectrumprovidesadeductiveoriginforanemergent3+1Dspacetimewithanintrinsic
particle/anti-particlesymmetry.Weapplythis frameworktoreformulateLHCjetproduc
tionas the spectral footprintof aG2 automorphismcascade, anon-perturbativeprocess
governedbyoctonionalgebra.Finally,weshowhowthecoreprinciplesofAoOaremirrored
inthearchitecturesofmoderngenerativemodels,groundingthetheoryinexperimentaland
computationalpractice.

Polyplex Lie Geometry is a high-dimensional complex geometric framework designed to encapsulate multiple geometric forms, spherical, ellipsoidal, hyperbolic, lattice (cubic), and taxicab, within a unified algebraic and analytic structure. This geometry is embedded in a complex Hilbert space ℋ ℂ " , where 𝑛 ∈ {3, … ,6}, and admits a rich tensor and operator algebra, enabling it to represent both classical geometric structures and quantum field theoretic symmetries.

In Mind as Metaphor, Adam Toon interprets folk psychological discourse metaphorically. Based on Kendall Walton's theory of metaphor, he argues that folk psychology ought to be understood in terms of prop-oriented make-believe that relies on representationally essential metaphors. Toon insists that this fictionalist view of everyday mental talk preserves what we commonly think folk psychology can achieve: it does not only rationalize but explains behavior causally. In this paper, first we raise concerns about Toon's characterization of folk psychology as metaphorical. Then we proceed to show that representationally essential metaphors are incompatible with genuine causal explanation. And finally, we argue that no construal of mental fictionalism can preserve the epistemic virtues we commonly ascribe to folk psychological discourse.

One more opinion about fractal nature of reality.
Just a philosophical approach, without deep analysis and examples.

We examine a fundamental bias in human scientific perception: the tendency to dismiss cross-scale analogies as reductive oversimplification. Through philosophical analysis, we propose a fractal inversion of this perspective-suggesting that apparent similarities between phenomena at different scales may reflect genuine universal patterns rather than anthropomorphic projection.