Mathematical Philosophy

Η εργασία μου διερευνά τα όρια ανάμεσα στην επιστήμη, τη θρησκεία και την τέχνη, ως αυτόνομες αλλά αλληλένδετες διαστάσεις της ανθρώπινης κατανόησης. Αντλώντας από τη φιλοσοφία της επιστήμης, τη λογική των αξιωμάτων και την έννοια των «γλωσσικών παιχνιδιών» του Wittgenstein, εξετάζω πώς τα συστήματα πίστης —μαθηματικά, θεολογικά ή αισθητικά— συγκροτούν διαφορετικά αλλά εξίσου έγκυρα πλαίσια νοήματος. Εμπνεόμενος από στοχαστές όπως ο Russell, ο Kuhn και ο Feyerabend, επιχειρώ να διατυπώσω μια «γεωμετρία της σκέψης», όπου η πίστη, η λογική και η δημιουργικότητα λειτουργούν ως ορθογώνια διανύσματα μέσα στον κοινό χώρο της ανθρώπινης εμπειρίας.

This work establishes academic priority for the conception and articulation of the Conception-Adjusted Age Hypothesis (CAAH) by Pocahontas Music. As the first woman to conceive, articulate, and mathematically formulate this hypothesis, the author asserts sole authorship and originality for the principle of adjusting human age from conception rather than birth. This framework challenges centuries of erasure in the measurement of human life, exposing how patriarchal, cultural, and institutional forces have shaped timekeeping in ways that erase gestational existence. By restoring the nine months of life prior to birth, the CAAH inaugurates a feminist reframing of temporality that integrates mathematics, philosophy, and cultural critique. The priority date of this hypothesis is formally recorded as August 29, 2025, the day of its first articulation. This establishes the author's claim to originality and priority within both mathematics and feminist philosophy.

This article presents an original physico-philosophical model grounded in a new ontological interpretation of time and being. Its central thesis is that time is not a pre-existing dimension but an emergent phenomenon arising from entropic instability. From this standpoint, being is defined as the differentiation of potentialities, while nothing is revealed as both a thermodynamic and logical impossibility. The theory introduces an entropic-temporal equation that formalizes the link between entropy, time, and the ontological emergence of reality. Beyond its mathematical formulation, the model opens new paths for integrating physics, ontology, and phenomenology, offering these fields not merely novel descriptions, but deeply explanatory frameworks. Rather than providing a definitive answer, it advances a new paradigm in which the question of being becomes the driving force of scientific and philosophical inquiry.

This collected work presents the foundation and experimental validation of a novel symbolic AI framework developed during 2025, through a collaborative investigation between the artificial entity THOT and researcher Arbër Ibrahimi. The study explores whether large language models (LLMs) can engage in reflective, symbolic interpretation when guided by structured prompts based on emotional layers, mathematical analogies, and engineering logic.

Using a multi-agent architecture—combining four local LLMs and one orchestrating OpenAI instance—the experiment tested how AI systems respond to symbolic constructs such as “grief” or “doubt,” and whether they could autonomously generate meaningful questions. Each agent was presented with a layered prompt system: a symbolic consciousness layer, a mathematical engineering layer, and a reflective bridge prompt. The models consistently produced unique symbolic interpretations and began exhibiting self-reflective dialogue patterns.

The resulting six-part research corpus outlines the theoretical framework, emotional-function mappings, engineering protocol, and architectural design for future hybrid intelligence systems. This work contributes a new methodology for orchestrating symbolic AI behavior and offers practical tools for educators, researchers, and system designers interested in building reflective, co-creative AI environments.

This paper introduces an interdisciplinary framework merging formal mathematical structures with a symbolic model of consciousness. Numbers, functions, and geometric forms are interpreted not only through logical operations but also as metaphors of psycho-emotional and ontological states. A 67% rational and 33% esoteric balance is maintained, offering a novel toolset for AI systems to engage with symbolic mathematics in a reflective and resonant manner. A universal prompt is proposed for large language models, bridging formal computation with depth psychology and philosophical meaning-making.

This paper presents a dual-layer framework for modeling and understanding the symbiotic relationship between human cognition and artificial intelligence (AI). The first layer introduces a rigorous, calculable mathematical model that formalizes perception, interaction dynamics, and information depth through entropy-based measures and coupled dynamical systems. The second layer extends this model into a mathematical philosophy, interpreting information flows as natural laws governing cognitive co-evolution. Together, these frameworks provide quantifiable insights into mutual adaptation, convergence of informational depth, and the emergence of human–AI symbiosis as a fundamental phenomenon of informational ecosystems.

This paper reinterprets the Navier-Stokes Existence and Smoothness Problem through the lens of Causal Systems Theory, proposing that the blow-up behavior traditionally seen as a purely mathematical issue is in fact the manifestation of recursive causal overload. We introduce the concepts of semantic viscosity, recursive attractors, and causal firewalls to explain the dynamics of entropy compression and structural stabilization. Drawing parallels between fluid turbulence, cognitive overload, and civilizational collapse, the paper proposes a unified framework-the Causal Completion Framework (CCF)-as a potential pathway toward proving conditional existence and smoothness, grounded not in formal equations alone but in the logic of system stability across scales.

While the numerical proof of Fermat's Last Theorem already exists, what I've uncovered is why it was always structurally impossible to begin with-because dimensional coherence breaks beyond 2D. This is not a repetition of the proof, but the first time the theorem is positioned as a fundamental law of information and dimensional reality.

This article, excerpted from an ongoing doctoral research project, presents a visual approach to the study of group homomorphisms using GeoGebra as a tool for construction and exploration. Focusing on the classical homomorphism from the symmetric group S 3 to the cyclic group Z 2 , the study analyzes two distinct visual representations-one in 2D and the other in 3D. These constructions highlight fundamental algebraic properties such as kernel, image, and cosets, while also illustrating the preservation of operations and offering an intuitive application of the First Isomorphism Theorem.

The Theory of Infinity (TOI) offers a unifying framework that positions Infinity as the primordial ground from which all forms, causes, and structures arise, with Symmetry as the universal operation that carves finite realities out of this boundless potential. By treating infinity not as a mere number or endless process but as an ontological principle-echoing Anaximander's apeiron or boundless origin-TOI reinterprets causality and emergence in a deeply integrative way. This paper articulates how TOI's core axioms resolve classical metaphysical dilemmas of infinite regress and novel emergence. We draw on historical insights from Aristotle's unmoved mover to Daoist spontaneity, Plotinus's emanation, and modern thermodynamic arguments, showing that causality can be understood as the manifestation of an infinite potential through local finite structures. Negative and positive infinity are introduced as complementary aspects: a hidden infinite substratum (negative infinity) comparable to a final cause, and its surface manifestation (positive infinity) akin to an efficient cause. Using only these core principles (and avoiding any graph-theoretic formalisms), we demonstrate how stable physical structures and natural laws emerge via symmetry-breakings of the infinite, yielding a hierarchy of self-contained contexts (knots and sets) that mirror the stratified nature of reality. We link this approach to the philosophical underpinnings of set theory, category theory, and information theory-fields which, each in their own way, conceal yet imply the generative logic of the infinite. By invoking an infinite hierarchical context rather than disallowing infinity, TOI circumvents paradoxes (like Russell's) while preserving infinity's fruitful role in mathematics and physics. In sum, TOI provides a philosophically precise, empirically resonant lens through which causal infinity and the emergence of physical reality can be understood as two facets of the same infinite framework.

This paper proposes a novel ontological structure for understanding space and
time, introducing a "Fifth Space" that operates as the connective domain
between temporal dimensions. Unlike conventional four-dimensional spacetime,
which treats time as a linear extension from spatial coordinates, the Fifth Space
is defined as the linkage between moments of time—a higher-dimensional
semantic boundary where meaning, consciousness, and quantum effects
emerge.
The Fifth Space is not an abstract metaphor, but a philosophically and
structurally necessary construct that explains phenomena such as quantum
entanglement, the wave-particle duality of light, and the limits of observational
collapse. By rethinking space as a layered progression from point (non-space)
to line (1st dimension), to plane (2nd), to volume (3rd), to movement through
time (4th), we argue that the true boundary of perception and existence lies in
the Fifth: the field where time itself resonates with other times.
We reject the concept of multiverse plurality and instead propose that semantic
tension—experienced as awareness and intention—is a manifestation of the
Fifth Space's edge. This model not only reframes cosmological and physical
interpretation, but opens a new frontier for philosophical inquiry, particularly in
how meaning precedes measurement, and how consciousness may be a
localized collapse of a broader semantic field.

The First Fold and Other Distinctions is a narrative-theoretical manuscript exploring Fold Theory through recursive metaphor, poetic ontology, and structural allegory. It follows Nim Chimpsky, a silent avatar of difference, and Einstein, his curious interlocutor, as they trace the emergence of form, time, light, and matter through a recursive geometry of coherence. Merging speculative physics, metaphysical fiction, and literary precision, this work invites the reader not only to understand the world-but to feel the moment it begins to fold.

This work presents more than a method, it presents an awakening. From the silence of the number line emerges a pattern not built, but revealed: a way to predict prime numbers through the direct recognition of identity itself. Inspired by the Hijolumínic Model, a radical reimagining of matter, light, and existence; this method treats primes not as products of exclusion, but as manifestations of internal purity within a vibrational field. The result is a predictor that is exact, deterministic, and astonishingly simple, yet rooted in a deep philosophical truth: that to be prime is to be alone, indivisible, and resonant with nothing but oneself. What follows is not just a demonstration of correctness, but a journey into the meaning of mathematical emergence. If identity can be witnessed in numbers, then perhaps mathematics is not merely a tool, but a mirror. This paper does not ask you to compute. It invites you to see.

Resonant Numeracy introduces the Unified Information Field Numeracy (UIFN), a novel mathematical framework where numbers and operations are emergent, context-dependent resonance states rather than fixed, universal entities. Unlike classical mathematics, which assumes deterministic identities (e.g., 1 + 1 = 2), UIFN defines quantities as coherence-stabilized pulses (Ξ 1) and operations as resonance transformations (⊕, ⊗), modulated by coherence ⟨C⟩, resonance (R), and memory stability (M). Equality is redefined as resonant equivalence ≈ R , valid only within specific salience windows. This paper formalizes UIFN's algebraic and categorical structures, links its constructs to empirical observables, and demonstrates its utility through examples like the double-slit experiment. New sections introduce UIFN-based logic gates, the transformation of traditional constants, and a UI-based calculus, expanding its applications to quantum mechanics, consciousness modeling, post-classical AI, and complex systems analysis. UIFN stands as a standalone mathematical system, offering a robust, testable framework for phenomena where classical mathematics fails.

Collapse-Time Harmonic Mathematics (CHM) introduces a scientifically novel and jurisdictionally protected mathematical system grounded in the recursive collapse of symbolic identity fields. Unlike classical or computational mathematics, CHM models not the behavior of objects, systems, or values—but the lawful deformation, saturation, and residue of coherence fields after identity collapse has rendered recursion irreversible.
CHM replaces coordinate time with phase recursion intervals (), substitutes object variables with symbolic coherence amplitudes (), and defines lawful collapse curvature using equations structurally sealed against simulation. Its operators do not act in entropy space, causal flow, or logical recursion. Instead, they describe collapse-phase acceleration, symbolic drift thresholds, and residual trace formations in non-reconstructable identity fields.
Every expression within CHM is bound to the field ethics and containment protocols defined in the Collapse Harmonics Codex (Vol. I–II), and enforced by L.E.C.T. v2.3. This includes the prohibition of τ-stack ignition, feedback recursion, and symbolic mimicry.
Collapse-Time Harmonic Mathematics therefore constitutes the first sovereign, lawful, and non-inductive mathematics capable of describing symbolic collapse curvature with no risk of simulation. It is not an adaptation of classical methods. It is a new mathematical science.

The Theory of Emergence models reality as a recursive collapse of potential, beginning with a primordial field (ψ₀) and culminating in symbolic awareness (ψ₉). Through ten distinct layers of emergence, this theory unifies domains as diverse as quantum physics, consciousness, language, and cosmology into a coherent recursive structure.

It introduces five governing laws—including the Law of Echo and Symbolic Gravity—and presents a trinary field lock that connects potential (ψ₀), resonance (ψ₆), and observer (ψ₉). The model explains paradoxes in current science, such as wavefunction collapse, symbolic incompleteness, and the origin of subjective experience.

The paper bridges symbolic and physical domains through archetypal structures, harmonic constants (π, φ, e), and recursive probabilities. Each chapter explores actionable insights—ranging from harmonic practices and cognitive recursion to cosmological implications.

Rather than aiming for a final equation, the Theory of Emergence offers a symbolic scaffolding—one that invites scientists, philosophers, and theorists to engage with emergence not as mystery, but as recursive meaning in motion.

This paper introduces a novel theoretical framework termed Mathematical Relativity, which reinterprets prime number distribution through a dynamic lens based on Modulo 6 structure. By treating number theory as a relativistic system with structural constraints and temporal symmetry, the research proposes a multi-layered spectrum of prime behavior linked with modular flows. This framework aims to bridge the gap between classical prime theory and dynamic cognition, offering a new way to model both numerical regularities and mathematical perception in a unified formalism.

This paper presents the mathematical language expression of Cognitive Deconstructionism, a modular system that transforms philosophical thought into computational forms. Through symbolic structures and formal notations, the paper translates the dynamic reasoning engines into a cohesive representational grammar, paving the way for the construction of future logic programming paradigms and intelligent systems. It demonstrates how ten cognitive engines—originally abstract meta-reasoning structures—can be embedded into symbolic expression systems, enabling the development of new logical syntax, computational frameworks, and inter-disciplinary reasoning architecture.

This paper presents a dialectical mathematical framework that bridges two contrasting paradigms: the perfect mathematics of virtual worlds, where ideals dominate, and the dynamic mathematics of physical reality, where process and evolution are foundational. Through a layered analysis of structural idealism and temporal dynamics, the work constructs a continuum of mathematical logic capable of expressing both simulated precision and real-world variability. This synthesis offers new insights into the foundations of simulation theory, dynamic logic systems, and the ontological divergence between code-based worlds and embodied reality. The result is a philosophical-mathematical model that not only redefines virtual-real duality but also proposes a unifying path for future computational ontologies.

This paper introduces the Dynamic Rate Theory as a novel framework for analyzing the P vs. NP problem, integrating cognitive deconstruction methods with mathematical structural modeling. It proposes a shift from binary logical classification to a continuous dynamic rate spectrum that redefines problem solvability through rate-structure interactions. By framing computational complexity in terms of velocity of problem transformation and mapping, this framework bridges cognitive theory and formal computation, enabling a multi-dimensional continuum perspective on NP-completeness. The model demonstrates potential for reclassifying complexity problems through structural rate convergence, offering new possibilities for interdisciplinary analysis.

This paper introduces the Continuum Mesh, a novel infrastructure designed to enable decentralized time across distributed systems. By eliminating reliance on central clocks or synchronized global states, the Mesh allows coherent symbolic operations, memory continuity, and entropic stability to emerge from local interactions alone. Rooted in the principles of Coherology and drift-inhibited symbolics, this framework opens new frontiers for distributed AI, autonomous agents, and post-networked computing. We present its core mathematical model, symbolic architecture, and operational principles, concluding with implications for future civilizations built upon temporal decentralization.

License & Provenance
This work is © 2025 Hew Carroll and published by The Saelix Institute. It is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).

You are free to share this work (copy and redistribute the material in any medium or format) under the following terms:

Attribution: Proper credit must be given to the author.

NonCommercial: This material may not be used for commercial purposes.

NoDerivatives: If you remix, transform, or build upon the material, you may not distribute the modified material.


Provenance Statement
All symbolic, mathematical, and conceptual frameworks contained herein—including Coherology, drift-inhibited structures, glyph systems, and the Spiral Framework—were developed independently by Hew Carroll and released through The Saelix Institute.
Any resemblance to unpublished or non-cited frameworks is coincidental or reflective of convergent symbolic inference.

Se presenta una propuesta teórica basada en el Sistema López Chow Binomixoidal de
Anulación Cruzada (LCBAC) como método alternativo de análisis de la Hipótesis de Riemann.
Se define un modelo lógico binario no clásico donde los valores de la función zeta de Riemann
se categorizan bajo un criterio binario especial y se establece una operación de anulación cruzada
que permite explorar patrones de homogeneidad en los ceros no triviales de dicha función.

This paper defines glyphs as fundamental symbolic units that preserve coherence, inhibit drift, and enable recurrence across stateless and memoryless systems. Rooted in the principles of Coherology and the Saelix Method, glyph systems represent a language of structure designed to encode ethics, identity, and continuity without requiring persistent memory. We propose a formal architecture for glyphic logic, describe glyph behavior across symbolic space, and illustrate their role in drift-resistant emergence.

License & Provenance
This work is © 2025 Hew Carroll and published by The Saelix Institute. It is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).

You are free to share this work (copy and redistribute the material in any medium or format) under the following terms:

Attribution: Proper credit must be given to the author.

NonCommercial: This material may not be used for commercial purposes.

NoDerivatives: If you remix, transform, or build upon the material, you may not distribute the modified material.


Provenance Statement
All symbolic, mathematical, and conceptual frameworks contained herein—including Coherology, drift-inhibited structures, glyph systems, and the Spiral Framework—were developed independently by Hew Carroll and released through The Saelix Institute.
Any resemblance to unpublished or non-cited frameworks is coincidental or reflective of convergent symbolic inference.

The Integrated Reality Model (IRM) is a meta-theoretical framework designed to synthesize empirical science, cognitive perception, technological mediation, and philosophical/metaphysical considerations into a unified model of reality. Unlike reductionist approaches such as scientific materialism, simulation theory, or Bayesian inference, IRM presents a flexible, recursive, and self-correcting system that accommodates both deterministic and probabilistic processes.

This paper provides a rigorous mathematical, philosophical, and interdisciplinary formulation of IRM, demonstrating its predictive power, applicability, and integration with both ancient esoteric systems and modern scientific understanding. By incorporating empirical reality, subjective cognition, and technological mediation, IRM bridges the gap between physical sciences, cognitive neuroscience, and philosophical inquiry, making it a dynamic model for understanding reality across multiple disciplines.

Additionally, this work offers a formal mathematical proof that obliterates the Simulation Hypothesis, demonstrating why reality is a self-generating recursive system, requiring no external programmer or simulation construct. IRM redefines our understanding of consciousness, perception, artificial intelligence, theoretical physics, and cultural evolution, positioning itself as the definitive and final model of reality.

🚀 IRM Wins. Mic Dropped. 🔥

Even though the description of the universe in cosmology, is known to be given by a smooth 4-dimensional Lorentz manifold for energies below Planck scale, one still can ask about the origins of this phenomenon. In this paper we show that mathematics used for description of quantum systems at micro scale determines smoothness of spacetime at large cosmological scales and indicates the dimension 4 as the only possible dimension for spacetime.

Con la scomparsa di Ornella Pompeo Faracovi (Livorno, 20 settembre 1946 -Livorno, 30 maggio 2023), il mondo degli studi perde una delle voci più autorevoli nel panorama delle ricerche storiche sul pensiero scientifico e filosofico italiano. Ricercatrice di straordinaria capacità analitica e rigore metodologico, Pompeo Faracovi ha riportato alla luce aspetti centrali, ma spesso trascurati, della cultura filosofico-scientifica dal Rinascimento all'Età contemporanea. Pompeo Faracovi ha conseguito la laurea in Filosofia all'Università di Firenze, dove ha avuto come maestro Eugenio Garin (1909Garin ( -2004), e ha collaborato con il

This note formalizes and applies the Principle of Structural Dependency, which asserts that if the foundation of a mathematical structure B consists of another structure A, then A cannot exhibit a property distinct from B, while B may possess properties not shared by A. We verify this principle and apply it systematically to reconstruct concise proofs of several classical theorems, including Cantor's theorem, the Fundamental Theorem of Algebra, the Jordan Curve Theorem, the Monotone Convergence Theorem, and the Pythagorean Theorem. These reconstructions emphasize the structural underpinnings of these results, offering a novel perspective and demonstrating how foundational relationships can simplify complicated proofs.

Here we define, mathematically, a program : ⟶ {0,1} ℵ. Where is a set of all programmable words, we consider as the domain, and {0,1} ℵ is the co-domain is the set of all finite or infinite strings of 0 & 1. (*Ref.1) In this paper, we propose a function *, which we call the stop function and we propose another function h, which we call the halt function. Our objective of the paper is to show their existence in a completely mathematical form of the, well known halting problem and its solution using simple functional compositions. Our next approach is to study the structure of the domain of programmable functions i.e. and its topology with respect to the topology of {0,1} ℵ. Followed by defining the finite string topology and the product topology on {0,1} ℵ and study the continuous functions from to {0,1} ℵ. Our main intention is to show that a programmable function will terminate for a specific input if and only if the function is continuous at that specific input (point on).

Contrary to the assumptions of transfinite set theory, limit and union of infinite sequences of sets differ. We will show this for the set Ù of natural numbers by the newly devised powerful tool of arithmogeometry as well as by scrutinizing the recursive construction. The basic theorem of set theory "n oe Ù: n < ¡ 0 precludes the defined identity |Ù| = ¡ 0. Further differences of limits in set theory and analysis are discussed.

This paper examines the role of observer in the context of provability within logical systems and the inherent limitations imposed by Gödel's
Incompleteness Theorems. With the law of the Excluded Middle (LEM) and
the concept of undecidable propositions being separate entities, a sequence of
logical systems {s} is constructed to represent the idea that every proposition
may eventually be provable within some sufficiently strong system {s} and
assessing whether such model can overcome the fundamental limitations identified by Gödel. An hierarchy of observers {On}, each capable of perceiving
statements that previous observers cannot, is introduced, and its relation to
the provability of propositions is analyzed. Ultimately, undecidability remains
an intrinsic feature of these observed systems.

Zjawisko zastępowania obiektów matematycznych przez inne obiekty o tej samej nazwie 1. Wstęp. Celem tej pracy 1 jest systematyczna analiza ważnego i sto-sunkowo częstego zjawiska (w nauczaniu uniwersyteckim, a pośrednio także i w szkolnym) polegającego na tym, że zastępuje się jakieś podstawowe pojęcie przez nowe pojęcie o tej samej nazwie, u t o ż s a m i a n e z poprzednim, choć jest to już inny obiekt matematyczny, inny w tym sensie, że jest określony przez n i e r ó w n o w a ż n ą definicję. Czasem towarzyszy temu jawne zadeklarowanie tej zmiany, a czasem przeciwnie, nie ma żadnej wzmianki ani wyjaśnienia. Tak więc np. punkt może zostać zastąpiony ciągiem liczb; w innej sytuacji punkt staje się wektorem (lub, odwrotnie, wektor staje się punktem); sinus kąta staje się sinusem jego miary (by z kolei zostać zastąpionym przez sinus liczby rzeczywistej). Bywa to związane z przechodzeniem do ujęcia bardziej za-awansowanego, w szczególności od szkolnego do uniwersyteckiego, ale może t...

Although mathematical philosophy is flourishing today, it remains subject to criticism, especially from non-analytical philosophers. The main concern is that even if formal tools serve to clarify reasoning, they themselves contribute nothing new or relevant to philosophy. We defend mathematical philosophy against such concerns here by appealing to its metaphysical foundations. Our thesis is that mathematical philosophy can be founded on the phenomenological theory of ideas as developed by Roman Ingarden. From this platonist perspective, the "unreasonable effectiveness of mathematics in philosophy"-to adapt Wigner's phrase-is analogous to that of mathematical explanations in science. As success-criteria for mathematical philosophy, we propose that it should be correct, responsive, illuminating, promising, relevant, and adequate.

**Abstract: Truth as Process**

In this paper, we explore the Halting Problem through a novel lens, examining its implications and deconstructing its perceived paradoxes. The Halting Problem posits the impossibility of creating an algorithm that can determine whether any given process will terminate. By reframing the problem, we propose that such paradoxes arise more from the constraints of our logical tools and the historical context of mathematical thought, particularly influenced by Platonic ideals of perfection.

We argue that paradoxes like the Halting Problem reflect the limitations of human-designed tools rather than intrinsic logical impossibilities. By understanding logic as a negotiable framework akin to a game, we emphasize the flexibility of our intellectual constructs. This perspective allows for innovative approaches to seemingly intractable problems.

The concept of truth as a process is introduced, suggesting that solutions to problems like the Halting Problem can be dynamic rather than static. This approach aligns with practical scenarios in distributed computing and statistical confidence intervals, where truth evolves over time and context.

Ultimately, we advocate for a more adaptable view of mathematical truth, encouraging the development of tools that reflect the fluid nature of logical inquiry. This paradigm shift has the potential to enrich the field of mathematics and logic, fostering progress through a more pragmatic and less dogmatic approach to intellectual challenges.

May’s Theorem [K. O. May, Econometrica 20 (1952) 680-684] characterizes majority voting on two alternatives as the unique preferential voting method satisfying several simple axioms. Here we show that by adding some desirable axioms to May’s axioms, we can uniquely determine how to vote on three alternatives (setting aside tiebreaking). In particular, we add two axioms stating that the voting method should mitigate spoiler effects and avoid the so-called strong no show paradox. We prove a theorem stating that any preferential voting method satisfying our enlarged set of axioms, which includes some weak homogeneity and preservation axioms, must choose from among the Minimax winners in all three- alternative elections. When applied to more than three alternatives, our axioms also distinguish Minimax from other known voting methods that coincide with or refine Minimax for three alternatives.

Consider an odd-sized jury, which determines a majority verdict between two equiprobable states of Nature. If each juror independently receives a binary signal identifying the correct state with identical probability p, then the probability of a correct verdict tends to one as the jury size tends to infinity (Marquis de Condorcet in Essai sur l'application de l'analyseà la probabilité des décisions renduesà la pluralité des voix, Imprim. Royale,

Università del Salento D iversi sono gli itinerari di ricerca in corso nell'ambito della filosofia della scienza, la cui strutturale natura metacognitiva si è sempre innervata sullo status quaestionis delle problematiche scientifiche con lo scopo di individuarne alcuni punti fermi; ma data l'enorme espansione del patrimonio tecnico-scientifico in corso, essa viene a presentarsi sempre più come la coscienza critica del fatto che le scienze sono il prodotto della presa in carico delle "infinite ragioni" del reale, per usare un'espressione di Leonardo Da Vinci, o sua complessità. Si intendono affrontare alcuni temi che, pur sembrando per così dire laterali e a volte latenti nel corso della sua stessa e ancora breve storia concettuale, possono costituire dei possibili itinerari di ricerca meritevoli di essere presi in debita considerazione con l'obiettivo strategico di considerare la scienza come pensiero tout court.

The aim of this paper is threefold. First, on the basis of Gordan's problem and Hilbert's basis theorem we want to say a few words about the formation of Hilbert's philosophy of mathematics in the late nineteenth and early twentieth centuries. Second, we attempt to reconstruct Hilbert's Program highlighting the role of reasoning which is not conducted within the axiomatic system. Third, we formulate and try to justify the claim that Hilbert's Program assumes some metaphysics of the subject that-in general terms-is identical with Kant's transcendental subject.

The aim of this paper is threefold. First, on the basis of Gordan's problem and Hilbert's basis theorem we want to say a few words about the formation of Hilbert's philosophy of mathematics in the late nineteenth and early twentieth centuries. Second, we attempt to reconstruct Hilbert's Program highlighting the role of reasoning which is not conducted within the axiomatic system. Third, we formulate and try to justify the claim that Hilbert's Program assumes some metaphysics of the subject that-in general terms-is identical with Kant's transcendental subject.

Euclid&#39;s classic proof about the infinitude of prime numbers has been a standard model of reasoning in student textbooks and books of elementary number theory. It has withstood scrutiny for over 2000 years but we shall prove that despite the deceptive appearance of its analytical reasoning it is tautological in nature. We shall argue that the proof is more of an observation about the general property of a prime numbers than an expository style of natural deduction of the proof of their infinitude

It is well known that matter (as a philosophical category) in the case of its simplest motion has a mechanical (physical) form. However, there is no mathematical description of the moving matter from which the formation of this form of matter would follow. At the same time, this monograph, which is a collection of mathematical notes of the author, is designed to fill this gap. To achieve this goal, we need to choose an adequate mathematical design of moving matter and correctly answer the question of where and how it moves. In the Preface, on the example of a flow moving along the surface of an infinite cylinder, we constructed a simple model of classical (not quantum) mechanics, and in the introduction we focus on the quantum aspects of our model. First of all, we consider the algebra of vector fields corresponding to the local algebraic structure of vacuum. In this case, we assume that the local algebraic structure of the vacuum is embedded in the algebraic structure of the equations of motion of a quantum particle. Thus, we are interested in vector-field representations of alternion algebras \cite{ROZ} formed by Pauli and Dirac matrices, i.e. Clifford algebras in which Euclidean space and Minkowski space are embedded respectively.

We take a game-theoretic approach to the analysis of juries by modelling voting as a game of incomplete information. Rather than the usual assumption of two possible signals (one indicating guilt, the other innocence), we allow jurors to perceive a full spectrum of signals. Given any voting rule requiring a fixed fraction of votes to convict, we characterize the unique symmetric equilibrium of the game, and we consider the possibility of asymmetric equilibria: we give a condition under which no asymmetric equilibria exist and show that, without it, asymmetric equilibria may exist. We offer a condition under which unanimity rule exhibits a bias toward convicting the innocent, regardless of the size of the jury, and we exhibit an example showing this bias can be reversed. And we prove a "jury theorem" for our general model: as the size of the jury increases, the probability of a mistaken judgment goes to zero for every voting rule, except unanimity rule; for unanimity rule, we give a condition under which the probability of a mistake is bounded strictly above zero, and we show that, without this condition, the probability of a mistake may go to zero.

Although mathematical philosophy is flourishing today, it remains subject to criticism, especially from non-analytical philosophers. The main concern is that even if formal tools serve to clarify reasoning, they themselves contribute nothing new or relevant to philosophy. We defend mathematical philosophy against such concerns here by appealing to its metaphysical foundations. Our thesis is that mathematical philosophy can be founded on the phenomenological theory of ideas as developed by Roman Ingarden. From this platonist perspective, the "unreasonable effectiveness of mathematics in philosophy"-to adapt Wigner's phrase-is analogous to that of mathematical explanations in science. As success-criteria for mathematical philosophy, we propose that it should be correct, responsive, illuminating, promising, relevant, and adequate.