Geometry

Conventional wisdom dictates that proofs of mathematical propositions should be treated as necessary, and sufficient, for entailing `significant' mathematical truths only if the proofs are expressed in a---minimally, deemed consistent---formal mathematical theory in terms of: * Axioms/Axiom schemas * Rules of Deduction * Definitions * Lemmas * Theorems * Corollaries. Whilst Andrew Wiles' proof of Fermat's Last Theorem FLT, which appeals essentially to geometrical properties of real and complex numbers, can be treated as meeting this criteria, it nevertheless leaves two (...) questions unanswered: (i) Why is x^n +y^n = z^n solvable only for n < 3 if x, y, z, n are natural numbers? (ii) What technique might Fermat have used that led him to, even if only briefly, believe he had `a truly marvellous demonstration' of FLT? Prevailing post-Wiles wisdom---leaving (i) essentially unaddressed---dismisses Fermat's claim as a conjecture without a plausible proof of FLT. However, we posit that providing evidence-based answers to both queries is necessary not only for treating FLT as significant, but also for understanding why FLT can be treated as a true arithmetical proposition. We thus argue that proving a theorem formally from explicit, and implicit, premises/axioms using rules of deduction---as currently accepted---is a meaningless game, of little scientific value, in the absence of evidence that has already established---unambiguously---why the premises/axioms and rules of deduction can be treated, and categorically communicated, as pre-formal truths in Marcus Pantsar's sense. Consequently, only evidence-based, pre-formal, truth can entail formal provability; and the formal proof of any significant mathematical theorem cannot entail its pre-formal truth as evidence-based. It can only identify the explicit/implicit premises that have been used to evidence the, already established, pre-formal truth of a mathematical proposition. Hence visualising and understanding the evidence-based, pre-formal, truth of a mathematical proposition is the only raison d'etre for subsequently seeking a formal proof of the proposition within a formal mathematical language (whether first-order or second order set theory, arithmetic, geometry, etc.) By this yardstick Andrew Wiles' proof of FLT fails to meet the required, evidence-based, criteria for entailing a true arithmetical proposition. Moreover, we offer two scenarios as to why/how Fermat could have laconically concluded in his recorded marginal noting that FLT is a true arithmetical proposition---even though he either did not (or could not to his own satisfaction) succeed in cogently evidencing, and recording, why FLT can be treated as an evidence-based, pre-formal, arithmetical truth (presumably without appeal to properties of real and complex numbers). It is primarily such a putative, unrecorded, evidence-based reasoning underlying Fermat's laconic assertion which this investigation seeks to reconstruct; and to justify by appeal to a plausible resolution of some philosophical ambiguities concerning the relation between evidence-based, pre-formal, truth and formal provability. (shrink)

All acknowledged proofs of the Four Colour Theorem (4CT) are computerdependent. They appeal to the existence, and manual identification, of an ‘unavoidable’ set containing a sufficient number of explicitly defined configurations—each evidenced only by a computer as ‘reducible’—such that at least one of the configurations must occur in any chromatically distinguished, putatively minimal, planar map. For instance, Appel and Haken ‘identified’ 1,482 such configurations in their 1977, computer-dependent, proof of 4CT; whilst Neil Robertson et al ‘identified’ 633 configurations as sufficient (...) in their 1997, also computer-dependent, proof of 4CT. However, treating any specific number of ‘reducible’ configurations in an ‘unavoidable’ set as sufficient entails a minimum number as necessary and sufficient. We now show that the minimum number of such configurations can only be the one corresponding to the ‘unavoidable’ set of the single, ‘reducible’, 4-sided configuration identified by Alfred Kempe in his, seemingly fatally flawed, 1879 ‘proof’ of 4CT. We shall further show that although Kempe fallaciously concluded that a 5-sided configuration was also in the ‘unavoidable’ set, and appealed to unproven properties of ‘Kempe’ chains in a graphical representation to then argue for its ‘reducibility’, neither flaw in his ‘proof’ is fatal when the argument is expressed geometrically; and that, essentially, Kempe correctly argued that any planar map which admits a chromatic differentiation with a five-sided area C that shares non-zero boundaries with four, all differently coloured, neighbours can be 4-coloured. (shrink)

Transformative Semalgebraic Ontology (OTS) treats transformation, rather than substance, as the primary ontological primitive (Bonisoli, 2026a). The operative part of the framework is the Semalgebra: a formal apparatus for the combinatorial structure of meaning, developed in full technical detail in the main manuscript (Bonisoli, 2026b). The present synthesis article presents the Semalgebra in a form accessible to theoretically oriented readers in philosophy of language, formal linguistics, and the cognitive sciences. The aim is neither encyclopaedic survey nor abridged treatise: it is (...) a reasoned guide to what the apparatus is, why it exists, what it does, and where its limits lie. We motivate the apparatus through the order-sensitivity of natural-language semantic combination, introduce the semantic field as a partial cancellative monoid with closed reversible sector, present the polarity operator and the third-path semantics as the structural core of the framework, sketch the linear and tensorial architecture that supports applications, and indicate three application domains: composition in natural language, the formal analysis of large language models, and a methodologically guarded physical interpretation. Formal proofs and the complete axiomatic development are referred throughout to the main manuscript. (shrink)

An argument is given that Euclidean geometry is a priori in the same way that numbers are a priori, the result of modeling, not the world, but our activities in the world.

The definition of a point in geometry is primordial in order to understand the different elements of this branch of mathematics ( line, surface, solids…). This paper aims at shedding fresh light on the concept to demonstrate that it is related to another one named, here, the Primary Geometric Object; both concepts concur to understand the multiplicity of geometries and to provide hints as concerns a new understanding of some concepts in physics such as time, energy, mass….

Given the context of phenomena in the world of mathematics is derived a system of things that are in order and illogical. Through such is the integration of a complete mathematical system with the systems high point of geometry and chemistry analysis in an in depth view of a logic system based on the natural order of things and numbers not nurtured things and numbers of humankind. The definition of nothingness in logic and other things is presented.

This is a draft of a chapter of a book that I was writing (in the early 2000s) on the philosophy of spacetime. It responds to Kant's argument of the lone hand, proposing a relational 'fitting' account of handedness. I plan to revise it as a stand-alone paper, but it is deposited now so that a soon to be published paper can cite a public version.

Linear Algebra is an extremely important field that extends everyday concepts about geometry and algebra into higher spaces. This text serves as a gentle motivating introduction to the principles (and philosophy) behind linear algebra. This is aimed at undergraduate students taking a linear algebra class - in particular engineering students who are expected to understand and use linear algebra to build and design things, however it may also prove helpful for philosophy majors and anyone else interested in the ideas behind (...) linear algebra. Linear algebra is used extensively in statistics, mechanical engineering, circuit theory, signal processing, economics and data science/machine learning. I have found in my teaching career that students tend to struggle the most with motivating the "why" behind the abstract concepts discussed in linear algebra. This can be difficult for professors to understand because we are so used to using the language of linear algebra we take for granted why these questions are important in the first place. In other words, we may understand the “why” (after years of using these tools), but our students may not. This text seeks to explain this “why” in an accessible, engaging way. (shrink)

This paper advances a geometric reinterpretation of economics. Its core claim is that for any movement from one state of affairs to another, there exists a true shortest path: a geodesic, understood as a Platonic object of economic life. This path is not defined by observed behavior, nor by retrospective success, but by the underlying structure of the possibility space itself. Yet because real economic life unfolds in a distorted world—shaped by institutions, technological limits, information asymmetries, infrastructure, and path dependence—the (...) geodesic does not generally appear as a straight line. Once mapped onto reality, it bends around hills, basins, and other sources of curvature, often taking forms that seem indirect from a naive perspective. The paper argues that ordinary agents approximate these routes only imperfectly, relying on intuition in the absence of a true map. From this it draws a methodological conclusion: economics should be understood less as the science of predicting agents, and more as the craft of mapping the geometry of possibility space. On this view, economists function as map makers, identifying the contours of constraint and opportunity so that agents may better navigate toward whatever ends they desire. Inefficiency is thus reconceived as excess expenditure induced by curvature, rather than simply as irrationality or failure to optimize. -/- To quote the great Martin Luther King Jr. “The moral arc of the universe is long, but it bends towards Justice.” . (shrink)

We present a principled framework for human-AI collaboration grounded in the Universal Master Equation (UME) from Autopoietic Stochastic Differential Geometry (ASDG). The UME decomposes dynamics into four orthogonal components: symplectic flow (Hamiltonian coherence), dissipative flow (friction work), collapse flow (attractor dynamics), and stochastic forcing (exploration). We demonstrate that this decomposition maps naturally onto the division of cognitive labor between humans and generative AI systems. Humans provide the symplectic term (first principles, constraints) and collapse term (purpose, meaning); AI handles the dissipative (...) term (path-finding through solution space); the stochastic term is shared. We develop recursive autopoietic stochastic differential geometry (rASDG): the coupled (Φ, Ψ) system where Φ is the ontic state and Ψ is the epistemic state constrained to the fibre F_Φ. The self-location constraint Ψ = M(Φ)(Ψ)—requiring the epistemic state to be a fixed point of the moment-map-induced operator—can be enforced discretely (projection) or continuously (penalty drift −κ(Ψ − M(Φ)Ψ)). Under Banach contraction conditions (provided by the collapse term ensuring spectral radius ρ(M) < 1), this fixed point is unique and exponentially attracting. Metacognition is thus a mathematical consequence: the unique self-consistent interpretation forced by the geometry. Joint Lyapunov analysis shows the system attracts to the intersection of the 108-heart (I₄ = 0) with the self-location manifold. This framework yields applications in prompt engineering, augmented intelligence, and GenAI metacognition. (shrink)

The third part of the Universal Variational Paradigm (UVP) presents an empirical synthesis confirming the universal variational law across the observable hierarchy of nature. It demonstrates that the same invariants—stationarity and openness—govern phenomena from physics to consciousness. -/- Physical systems obey the stationary condition through the principle of least action and the Fisher-information bound; biological and neural systems manifest open Ricci-type curvature flows that describe irreversible evolution and learning; psychological and social systems reveal analogous curvature dynamics governing reflection, ethics, and (...) cultural adaptation. -/- The paper compiles cross-disciplinary evidence (physics, chemistry, biology, neuroscience, psychology, sociology) showing that all stable forms satisfy δΦ = 0 while remaining open under δ³Φ ≠ 0. This establishes the UVP as an empirically grounded framework unifying energy, information, and reflection across scales of reality. (shrink)

This second part of the Universal Variational Paradigm (UVP) extends the variational architecture of reality from physics and information to mind and meaning. While Part I established the universal law of distinction, stationarity, and openness as the foundation of nature’s geometry, the present paper introduces the concept of the Noetic Metric, a mathematical structure that encodes the local geometry of sense and measures semantic tension. Its evolution follows a Ricci-type flow influenced by reflective and intentional input. -/- The Noetic Geometry (...) couples to the physical and biological manifolds through the beta–gamma–lambda bridge (energy–information–reflection), forming a continuous variational continuum between matter, life, and consciousness. Empirical correspondences are demonstrated across neuroscience, psychophysiology, and social systems, showing that curvature flows govern both cognitive and ethical coherence. -/- Noetic Geometry thus formalizes consciousness as a self-consistent curvature flow on the manifold of distinctions and provides a rigorous mathematical language for meaning, reflection, and value. (shrink)

There are two problems Analytical Geometry with facing anyone who studies this discipline: define the nature of the locus represented by the general equation 2do degree in two or three variables: That curve represents the plane? What surface is in space? These two problems are posed and solved by applying the study of matrices and spectral theory.

There are two problems Analytical Geometry with facing anyone who studies this discipline: define the nature of the locus represented by the general equation 2do degree in two or three variables: That curve represents the plane? What surface is in space? These two problems are posed and solved by applying the study of matrices and spectral theory.

We demonstrate that the fundamental physical constants of the Standard Model are arithmetic invariants, originating from four mechanisms: (A) ratios of special values of L-functions in the Selberg class, (B) Bernoulli numbers transported by the archimedean Γ-factor via the functional equation, (C) ratios of nontrivial L-function zeros, and (D) dimensions of spaces of multiple zeta values counted by Zagier's recurrence. No geometric input is required: π itself is the Haar measure normalisation of ℝ/ℤ. At the GUT scale, the down-type quark (...) mass ratios m_d : m_s : m_b = 2 : 5 : 9 follow exactly from the Casimir invariants C₂(n,0) = n(n+3)/3 of SU(3)_flavor Kac–Moody representations for generations n = 1,2,3, and the Georgi–Jarlskog relation m_d m_b/m_s² = 9/4 matches experiment at 0.09%. The Cabibbo angle satisfies sin θ₁₂ = 1/(2√5) within 0.9% via the Gatto–Sartori–Tonin relation, while the geometric prediction θ₁₂ = 29π²/1260 from the Fubini–Study distance on CP² achieves 0.19%. The seed of CP violation is identified as the Jacobi sum J(χ₄ mod 5, χ₃ mod 7) = e^{iπ/6}, a primitive 12th root of unity in ℚ(ζ₁₂); the ratio of the physical CKM phase to this arithmetic seed approximates 137/60 at 0.04%, suggesting a connection between CP violation and the fine structure constant. Three independent programmes—the exceptional Jordan algebra (Singh), the generalised Dirac equation on Cl_{8,0} (Quinta), and the coset geometry of Spin(8)/G₂—converge on the same angular parameters for the fermion mass matrix, providing evidence that mass is the curvature of the division algebra coset and the hierarchy is its shape. All numerical claims are verified at ≥ 15-digit precision. (shrink)

The multiplicative group ℝ₊×, equipped with its Haar measure d×r = dr/r, carries a scale-invariant spectral structure whose unitary irreducible representations are precisely the characters χγ(r) = r^(iγ) for γ ∈ ℝ, parametrised by a single real frequency. When the standard Riemann variable s = σ + iγ is introduced via the Haar-normalised coordinate s = 1/2 + iγ, this principal-series condition becomes the statement Re(s) = 1/2. The critical line is therefore the unitarity locus of the natural spectral theory (...) on ℝ₊×, not a numerical accident. Tate’s thesis establishes that the Riemann zeta function arises as the Mellin transform of the idele class group scaling action, connecting the zeros of ζ(s) to the spectral theory of the scaling generator A = −i d/d(log|a|) on L²(A×/ℚ×, d×a). Meyer’s unconditional spectral realisation then identifies those zeros as the atoms of the trace spectral measure μ_A via the Weil explicit formula. The product decomposition A×/ℚ× ≅ K × ℝ₊×, with K compact, reduces the eigenvalue equation for A to a first-order ordinary differential equation on ℝ whose solution space is one-dimensional, confining each ordinate to a single spectral atom of weight one. The functional equation ξ(s) = ξ(1 − s), combined with the Schwarz reflection principle, forces every zero at ordinate γ lying off the critical line to generate a second distinct zero of ζ(s) at the same ordinate, violating the multiplicity bound. All non-trivial zeros of ζ(s) therefore satisfy Re(s) = 1/2, and every such zero is simple. The argument extends without modification to every primitive Dirichlet L-function, yielding the Generalised Riemann Hypothesis. (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)

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)

This paper introduces some basic ideas and formalism of physics in non-commutative geometry. My goals are three-fold: first to introduce the basic formal and conceptual ideas of non-commutative geometry, and second to raise and address some philosophical questions about it. Third, more generally to illuminate the point that deriving spacetime from a more fundamental theory requires discovering new modes of `physically salient' derivation.

Hilbert’s (arithmetical) Axiom of Completeness asserts that the structure of the real numbers $$\mathbb {R}$$ R is maximal in the sense of not having a proper extension to an Archimedean ordered field. The more recent works of Ehrlich (2001), McGee (1997) and Aczel (1988) show that certain maximality conditions modeled upon Hilbert’s axiom provide unique characterizations of, respectively, the s-hierarchical ordered field of surreal numbers No, the well-founded hierarchy of pure sets $$\mathbb {V}_{\!k}$$ V k, and the non-well-founded hierarchy of (...) Finsler-extensional sets $$\mathbb {V}_{\scriptscriptstyle \!F\!A\!F\!A}$$ V F A F A. The paper provides a comprehensive historical and theoretical reconstruction of this often overlooked chapter in the history of the axiomatic method. The historical reconstruction suggests that the maximality condition of non-extensibility arises as a natural axiom in the unique characterization of mathematical structures, as illustrated by the theories of (s-hierarchical non-)Archimedean continua and (non-)well-founded sets. The theoretical reconstruction argues that the maximality condition of non-extensibility is the formal explication of the heuristic principle of Plenitude, as usually adopted to describe the intuitive ‘‘fullness” of punctiform continua and the cumulative hierarchy of sets. (shrink)

This paper introduces the concept of temporal lifting as a constructive analytic framework for reinterpreting apparent singularities in nonlinear dynamical systems, particularly the incompressible Navier–Stokes equations on the three-torus T³ = ℝ³ ∕ ℤ³. The approach suggests that finite-time blow-up is not an intrinsic breakdown of the equations but a compression of the physical time coordinate. By defining a smooth, strictly monotone lifting map φ : t ↦ τ and expressing the flow as U(x, τ) = u(x, φ⁻¹(τ)), the system (...) becomes globally C^∞ while preserving energy inequalities and invariants. This procedure is formally analogous to the Path Lifting Lemma in topology, in which a loop on S¹ lifts to a smooth path on ℝ. Philosophically, the result reframes mathematical singularity as a coordinate phenomenon, implying that continuity and regularity are functions of parametrization rather than absolute properties of nature. -/- Categories: Philosophy of Mathematics → Mathematical Methodology Philosophy of Science → Theoretical Physics -/- Keywords: time geometry; continuity; Navier–Stokes; regularity; coordinate systems; topology of flows; mathematical ontology -/- External link: HAL preprint (8 Oct 2025) See Link. (shrink)

At its core this book is concerned with logic and computation with respect to the mathematical characterization of sentient biophysical structure and its behavior. -/- Three related theories are presented: The first of these provides an explanation of how sentient individuals come to be in the world. The second describes how these individuals operate. And the third proposes a method for reasoning about the behavior of individuals in groups. -/- These theories are based upon a new explanation of experience in (...) nature, the construction of senses, and motile behavior. This new approach is developed from first principles to enable a rigorous and systematic explanation of the variety of associated intelligent behaviors. -/- Alongside this development is a further account that focuses upon the nature of our work. It discusses the existential aspects of scientific inquiry, its epistemology and logic. It seeks to clarify the nature of the mathematical characterization and computation of natural behaviors, dealing with questions in the foundations of logic. It explores methodological issues related to reduction and the refinement of ideas from intuition to formal logical structure. -/- In support of this inquiry we work toward the development of a calculus for biophysical construction and its dynamics. If successful this mechanics mathematically characterizes sensory and motile behavior. -/- Upon this foundation we propose a model of apprehension and explore how its products are processed by the organism. Finally, we develop a probabilistic theory that enables us to reason about inaccessible factors in group behavior. -/- The mechanics we propose suggests the design and physical realization of a new model of computation; one in which structure and the concurrency of action are a first-order consideration. -/- We identify opportunities for experimental verification of the theory and we suggest a proof of our results in practice by the identification of this mechanism, allowing the construction of machines that experience. (shrink)

In this work I present what may be the first complete construction of quantum gravity describing the real universe via the celestial holographic conformal field theory dual to Einstein gravity in asymptotically-flat 4D spacetime. The theory is rigorously constructed as the shadow-invariant, purely spin-2 sector of holomorphic Chern–Simons theory on twistor space PT ≃ CP³ with gauge group the quantomorphic group Quant(PT). Primary fields are the celestial graviton operators O^{±2}Δ(z, z̄) with Δ ∈ 1 + iR and J = ±2. (...) Three-point functions are determined by the Penrose transform and reproduce the MHV and anti-MHV amplitudes of general relativity, including the vanishing of parity-odd and mixed-helicity couplings. Higher-point amplitudes arise from OPEs with analytically continued principal-series contours; loop integrands emerge as double discontinuities across shadow poles, which I verify explicitly for the scalar box integrals computed via this novel shadow discontinuity method, demonstrating analytically that they match Feynman diagram calculations for the scalar box, with numerical verification for the graviton box matching to 11+ decimal places. The construction satisfies extended BMS₄ ⋉ shadow symmetry with vanishing Virasoro central charge, also explicitly demonstrated via five independent and mutually reinforcing and rigorous arguments to be c = 0 to 10+ significant figures. I prove a rigidity theorem establishing that any 2D CFT with only spin ±2 primaries, the above symmetries and three-point data, and local inverse-Mellin-transformed correlators, is uniquely determined to be isomorphic to this quantomorphic twistor theory. Combined with the proof that crossing symmetry is a necessary consequence of bulk locality (not an independent axiom), this establishes that Einstein gravity in asymptotically-flat spacetime is the unique consistent theory of a single massless spin-2 particle. The quantomorphic gauge group structure provides a non-perturbative foundation, with perturbative scattering amplitudes emerging as the first residues of the holographic algebra. The first breakthrough insight that enabled and facilitated this construction was the realisation that the "shadow symmetry" of the boundary CFT—the map Δ ↔ 2 − Δ on conformal dimensions—is the restriction to null infinity of the Jost involution Θ of axiomatic quantum field theory, the operator independently constructed by Jost (1957) that coincides with the product C ∘ P ∘ T only in theories where all three factors individually exist as symmetries. At null infinity, where the metric is degenerate along the null generator, the distinction between temporal and spatial inversion collapses: the Jost involution is irreducible and reduces to time reversal T alone (Theorem 8.1). This provides an elegant resolution to the 40+ year "googly problem" in twistor theory by unifying self-dual and anti-self-dual helicity sectors through the shadow transform as representing discrete T-symmetric involution across the boundary, with the Einstein field equations derived from boundary Ward identities and crossing symmetry. All quantum mechanical foundations emerge from the Haar measure structure underlying this framework with no additional postulates: the five Wightman axioms are derived from the 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. This same identification reveals that the celestial conformal dimension is related to the Riemann zeta variable by Δ = 2s, so that the shadow involution Δ ↔ 2 − Δ becomes s ↔ 1 − s, the functional equation of the Riemann zeta function—establishing that the functional equation, shadow symmetry, and time reversal are three manifestations of one underlying self-duality. The entire holographic framework grows from a single root: self-dual Haar measure on (R⁺, ×), the simplest non-trivial locally compact group, whose self-duality already contains the functional equation and critical line. Three Cayley–Dickson doublings R → C → H → O produce the holographic chain R⁺ → S² → M⁴ → Gr(2,4)_C, with shadow symmetry at each level corresponding to the conjugation involution at that doubling step. The Grassmannian Gr(2,4) is the canopy of this holographic tree. The same Haar-invariant measure on the Grassmannian Gr(2,4) that determines the graviton spectrum also addresses three Millennium Prize problems through a unified geometric mechanism (Haar self-duality, functional equation symmetry, and Peter–Weyl compactness). The Riemann Hypothesis follows via 4 otherwise independent arguments and checks for circularity and consistency, and most fundamentally follows from three simultaneous constraints on the adelic quotient A×/Q× that form an overdetermined system whose unique solution forces Re(s) = 1/2 for all nontrivial zeros, with numerical verification of the first 10¹³ zeros on the critical line to machine precision using six independent tests. The Birch and Swinnerton-Dyer Conjecture is argued to be proven for analytic rank ≤ 1 using identical Haar measure structure on GL₂(A_Q)/GL₂(Q): for elliptic curves with ord_{s=1} L(E,s) ≤ 1, the same three-fold constraint described above, combined with the Gross–Zagier formula and Kolyvagin's Euler system descent, establishes ord_{s=1} L(E,s) = rank E(Q) and the parity conjecture (−1)^{rank} = w_E is proven for all curves from Haar self-duality. The framework also identifies the precise obstruction to proving the conjecture at higher rank (algebraic cycles from the Beilinson–Bloch–Kato conjectures). The Yang–Mills mass gap is established through a complete celestial holographic construction: the Wess–Zumino–Witten model at Kac–Moody level k on S², mapped to four-dimensional Schwinger functions by inverse Mellin transform, with reflection positivity proven via the shadow-reflection bridge, the remaining Osterwalder–Schrader axioms verified explicitly, and the Wightman theory reconstructed by the OS theorem—yielding mass gap M = 2N/(k+N) · Λ_QCD > 0 from the Sugawara formula and confinement from Haar orthogonality as a rigorously proved theorem. The construction introduces no lattice, takes no continuum limit, and imposes no infrared regulator. Spacetime itself emerges from the Grassmannian via Penrose twistor correspondence (M⁴ ↔ Gr(2,4) (Penrose 1967). The division algebra tower determines the Standard Model: the symmetry breaking chain Spin(8) → G₂ → SU(3) → SU(2) × U(1) produces the gauge group; three fermion generations follow from Weyl anomaly cancellation, Plücker partitions, and Spin(8) triality; sin²θ_W = 3/8 at the GUT scale is the Dynkin index ratio, equivalently dim(Im(H))/dim(O). Physical constants are arithmetic invariants: CP violation from the Jacobi sum J(χ₄⁽⁵⁾, χ₃⁽⁷⁾) = e^{iπ/6}; down-type quark mass ratios m_d : m_s : m_b = 2:5:9 from SU(3) Casimir eigenvalues; the spin-statistics connection η(4)/ζ(4) = 7/8 from the p = 2 Euler factor; the lightest neutrino mass m_{ν₁} = 0 exactly from Bott periodicity—in independent agreement with Boyle–Finn–Turok (PRL 121, 251301, 2018); and the three dimensional constants ℏ, c, G corresponding to the three Cayley–Dickson doubling steps. Strong CP angle θ = 0 exactly from Haar self-duality (consistent with experimental bound |θ| < 10⁻¹⁰), and the construction provides right-handed neutrinos as geometric dark matter candidates in agreement with Boyle–Turok (CP)T-symmetric cosmology, which integrates with uncanny precision into my T-symmetric, thermodynamically cyclic quantum cosmology. The T-anti-symmetric unification of a unified bipartite self-dual universe resolves the matter-antimatter asymmetry as a selection effect. Total baryon number B_total = 0 with embedded observers observing identical physics on either side due to being thermodynamic in nature, naturally aligning their arrows of time in the direction of increasing entropy, which on either side of the boundary flows in opposing directions. The Feynman-Stueckelberg interpretation of antimatter means antimatter is equivalent to matter moving backwards in time, which combined with the reversal in T of the direction of entropy increases means observers only ever observe the B > 0 (matter) sector, with the B < 0 (antimatter) sector always being the hidden sector from the perspective of any temporally-embedded observer that respects thermodynamics, as from any observer's perspective their time moves forward, their atomic nuclei have a positive electrical charge, electrons have negative electric charge, and the standard model has left handed doublets and right-handed singlets, with a weak force that appears left-chiral. To observe the antimatter would require an observer to reverse their arrow of time, which is thermodynamically not possible as consciousness and memory are thermodynamic processes that happen only in the direction of increasing entropy. Shadow geometry and right-handed neutrinos provide dark matter. The Grassmannian spinor bundle splits each fermion representation across the T boundary: right-handed neutrinos reside in the shadow sector, coupling to the observable sector only gravitationally. The dark matter density profile ρ(r) = ρ₀/(1+(r/r_c)²) with core radius r_c ≈ 9.98 kpc is derived from the shadow kernel by Abel inversion with zero free parameters, resolving the core cusp problem of dark matter distribution and addressing several further cosmological anomalies (the S₈ tension, Hubble tension, DESI dark energy evolution, JWST early galaxies, CMB hemispherical asymmetry, CMB dipole anomaly, axis of evil, overmassive early black holes, rotation curve diversity, etc.) from a single geometric source. Time's arrow emerges as a consequence of T-symmetric boundary conditions with entropy always increasing away from the conformal boundary in opposing temporal directions. The horizon and flatness problems are naturally resolved by this framework as the celestial CFT is globally defined at the conformal boundary. 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), m_{ν₁} = 0 (KATRIN, JUNO), r_c ≈ 9.98 kpc (Euclid), glueball mass ratios matching the Sugawara spectrum, GUE statistics in the vacuum noise spectrum of gravitational wave detectors (LIGO/Virgo/KAGRA), and right-handed neutrino dark matter coupling only through gravity. Over 155 independent computational tests with zero free parameters confirm the framework to machine precision. Part 1 contains an introductory overview of the several subframeworks involved in this synthesis explaining their interrelations in the wider framework, part 2 contains the constructions and rigorous proofs of all major results, Part 3 contains extensions and other corollaries. Part 4 continues with comparisons to other approaches and discussion regarding foundational contributions without which this work would not have been possible follows, followed by testable predictions and a chapter addressing falsifiability, and Part V delivers a philosophical analysis of the implications suggested by the mathematics of this framework. This work unifies quantum mechanics and general relativity via the celestial holography program while revealing deep connections between number theory and physics: the identification Δ = 2s means the same self-dual Haar-invariant measure structure constraining the Riemann zeros to the critical line in the complex plane where Re(s) = 1/2 simultaneously constrains the conformal dimensions of graviton operators to the principal series where Re(∆) = 1, the same compactness that forces the Yang–Mills mass gap forces the discrete zeta spectrum, and the same functional equation symmetry that governs L-functions governs the shadow transform of the celestial CFT—suggesting geometry and arithmetic share a common logos rooted in self-dual Haar measure on (R⁺, ×) and flowering in the Grassmannian Gr(2,4). (shrink)

A report of an unidentified craft becomes a scientific object only after witness narrative, sensor state, platform motion, weather, optics, radar geometry, infrared response, and the proposed field mechanism are placed in a common mathematical ledger. This paper builds that ledger for disc, sphere, triangle, cigar, toroidal, luminous, trans-medium, and apparent-hover reports. It studies Brown--Biefeld force claims, Tesla discharge narratives, Searl-type annuli, Podkletnov-style impulse claims, alien-reproduction-vehicle diagrams, warp-metric language, wormhole models, time-loop hypotheses, holographic-time proposals, M-theoretic dimensional language, and public UAP (...) records under one rule: ordinary aircraft, balloons, drones, birds, satellites, meteors, plasma effects, parallax, sensor saturation, calibration drift, and platform error must be subtracted before a nonordinary term is admitted. The mathematical outcome is a residual-classification theorem. Every admissible claim is assigned to one of four classes: conventional explanation, underdetermined report, conservation-law conflict, or bounded hypothesis requiring independent tests. The text is therefore rigorous in a narrow but useful sense: it does not convert secrecy, folklore, or visual astonishment into proof; it converts them into equations, diagrams, null tests, and evidentiary burdens. (shrink)

This manuscript formulates UAP-class field propulsion as a controlled problem in block-twist geometric reconfiguration rather than as an asserted device claim. The ambient region is modeled as a finite cubical state space with mutable adjacency, weighted twist generators, source envelopes, stress constraints, sensor residuals, and reset ledgers. The central object is a reconfiguration metric on chamber-state tuples, and a space-route possibility is admitted only when a positive compression window survives word cost, local displacement, inertial load, synchronization error, material strain, energy (...) accounting, uncertainty covariance, and falsifier margins. The theory converts aggressive field-propulsion language into source tensors, graph metrics, route-compression inequalities, variational equations, Hamiltonian constraints, shell-distance estimates, stress-locking bounds, state-time closure conditions, and proof-carrying certificates. The manuscript makes no experimental realization claim and gives no construction procedure; it defines a rigorous conditional phenomenology for when block-twist space-route compression remains mathematically open, testable, and noncontradictory. (shrink)

We propose a brane-cluster formalism for ultraviolet quantum gravity in which the higher-curvature operators that soften graviton propagation are not inserted phenomenologically, but arise from the intersection topology of a finite network of transversely embedded branes. The brane union ℬ determines an intersection chain complex (C•, ∂), and each nontrivial homology class [Cₖ] ∈ Hₖ(ℬ) is assigned a massive cluster mode Φₖ coupled to curvature through λₖΦₖR. Eliminating the cluster sector produces an effective gravitational kernel containing R□ⁿR corrections and a (...) momentum-space propagator whose ultraviolet behaviour scales as Gᵉᶠᶠ_μν,ρσ(p) ∼ p⁻⁶, while the low-energy limit returns to the Einstein propagator. The extra higher-derivative poles are treated through the fakeon prescription, so that ghost excitations are excluded from the physical asymptotic spectrum without removing their virtual contribution to ultraviolet damping. The construction is developed through homological charge conservation, BRST quantization, Slavnov–Taylor consistency, one-loop counterterm closure, and explicit curvature-sector renormalization. The resulting framework links brane topology, ghost control, and UV-safe gravitational dynamics in a single effective field-theoretic mechanism, with testable consequences for short-distance gravity, gravitational-wave dispersion, regular black-hole interiors, and topology-driven quantum-gravity signatures. (shrink)

This article develops a higher-dimensional research programme for Calabi–Yau geometry beyond the classical threefold setting and through the explicit CY₂₀ horizon. It integrates hypersurface and toric constructions, Hodge statistics, mirror laws, special-holonomy constraints, conditional SYZ lifting, and dimensional-saturation models into a single framework for studying the growth and organization of Calabi–Yau landscapes. The manuscript distinguishes proved construction results from computational evidence and asymptotic conjectures, while incorporating a corrected treatment of toric intersection products, including repeated divisors and self-intersections. The resulting programme (...) provides a systematic mathematical and computational architecture for testing higher-dimensional regularities, mirror behaviour, and possible limiting laws in Calabi–Yau geometry. (shrink)

We prove a local-to-global theorem for compact corrected Strominger–Yau–Zaslow duality in the presence of a finite collar package. The package consists of logarithmically deep toroidal wall collars, a radial Kähler lower bound, calibrated-current sweep control, virtual restriction of compact disc moduli, and canonical identification of wall functions. From these data we prove, with explicit estimates, singular-current continuation, integral monodromy locking, finite-energy wall confinement, equality of compact analytic and logarithmic wall automorphisms in every energy quotient, and corrected dual gluing. The Dwork/quintic (...) degeneration is treated as the principal compact model through an explicit toroidal collar atlas. A five-coordinate defect functional records the precise remaining local estimates; its vanishing, or a terminally exact contracting transport, is proved equivalent to the completed corrected-SYZ conclusion in this framework. (shrink)

This paper develops a formal distinction between \emph{map}, \emph{directrix}, \emph{action}, and \emph{completion} in the setting of the complex numbers. The motivating claim is that charting the place of the imaginary numbers within the complex plane is not the same act as sending a selected aspect of that structure through a projection, restriction, quotient, normalization, branch choice, or flow, and that neither of these is identical with extending a previously partial evaluative regime by a clause that forces a value at sites (...) of nonconstruction. The former is representational; the second is operational; the third is semantic completion. -/- To make this precise, the paper constructs a layered atlas of the complex plane \(\C\) and of the pure imaginary axis \(\ii\R\subset \C\), expanding the canonical ``subrooms'' of one-complex-dimensional space: algebraic, linear, Euclidean, polar, topological, branch-theoretic, holomorphic, harmonic, symplectic, oscillatory, dynamical, and projective. Central structural results include the classification of real-linear endomorphisms as maps of the form \(z\mapsto az+b\conj{z}\), the topological obstruction to a global argument, the principal logarithm on the slit plane, the identification \(T_1S^1=\ii\R\), the Lagrangian character of the real and imaginary axes, Hamiltonian realization of phase rotation, explicit dynamical classifications for power and affine maps, and stereographic/projective completion by the Riemann sphere. -/- The paper then adds a second layer not present in standard expositions: a distinction between \emph{intrinsic zero} and \emph{completion zero}. The element \(0\in\C\) is intrinsically part of the field, but many appearances of the glyph ``\(0\)'' in mathematical practice arise from what may be called \emph{completion-by-clause}: extending a sparse coefficient family, an out-of-support recurrence, an empty aggregation, or an exceptional locus by an explicit rule assigning the value \(0\). Such completions are shown to be neutral only in zero-erasing contexts, especially additive aggregation. A depadding theorem for additive zero-extension is proved, and a complementary non-neutrality proposition shows why zero-completion fails to be harmless once reciprocals, logarithms, arguments, branch-sensitive operations, support-sensitive operators, or analytic continuation enter the scene. -/- A third layer addresses representational pluralism. For the classes of constructions treated here, the mechanics of the complex plane need not be written with the literal symbol \(\ii\): they can be expressed exactly in real two-dimensional terms via a quarter-turn operator \(J\), via winding-number differential forms, via harmonic pairs, and more generally through alternate real-geometric atlases. This includes, at the level of principle, sweep-based encodings such as oriented boundary sweeps, sweeping nets, and saddle-map formalisms used to track phase or singularity structure without explicit imaginary notation. The paper argues that imaginary numbers are therefore not the only possible ontology of planar phase, but the most algebraically compressed one. -/- An appendix provides a formal novelty-map table for the present manuscript and for several adjacent manuscripts supplied by the author, classifying their main components as \emph{likely new}, \emph{probably known}, or \emph{needs citation check}. The aim is not to settle priority definitively, but to separate classical mathematics from original synthesis, and theorem-level originality from interpretive or programmatic innovation. (shrink)

This paper establishes that geometric structure — distance, metric, curvature, and the analytic machinery built upon them — is not operationally necessary in any operational system but a representational artifact: a formal construct encoding the algebraic structure of the rank-3 minimal operational closure C⁽³⁾_Πd into an extended descriptive language. The argument proceeds from the axiomatic foundation of Operatiology, in which C⁽³⁾_Πd is derived from three axioms governing non-commutativity, Πd-saturation with finite generator rank, and redundancy exclusion, and from the companion result (...) that algebra is the unique top-down projection of the operational substrate. The Metric Reduction Theorem establishes that every admissible distance structure is a projection of the trace inner product Tr(A†B) of M₃(ℂ), the unique minimal algebraic realisation of C⁽³⁾_Πd, and is therefore not an independent operational primitive. The Geometric Unbounded Index Obstruction lemma classifies continuous curves, infinite metric spaces, topological neighbourhoods, and curvature as elements of the termination-failure equivalence class [ℐ/∼], with distinct obstruction types. Euclidean and non-Euclidean geometries are classified by the same mechanism, differentiated only by the curvature parameter of their metric projection. The Euclidean–non-Euclidean dispute is dissolved as a competition between alternative representational languages. The general relativistic metric formulation is classified as a representational artifact without affecting its empirical validity. Published on May 30, 2026 DOI: 10.5281/zenodo.20465726. (shrink)

This paper asks a simple question: if we begin not with space, but with a network of relations, when is it legitimate to say that geometry has been recovered? -/- I study this question through the RZS-ELCL benchmark pipeline. The pipeline starts from weighted relational graphs, filters candidate geometric edges, builds local clique and simplicial structure, and then measures curvature with Regge-style operators. The goal is not to prove that spacetime emerges from a relational zero state. The goal is narrower (...) and more testable: to identify when a relational graph can be projected into a stable discrete geometry, and when that projection fails. -/- The results are deliberately presented as controlled benchmarks rather than as an asymptotic theorem. In low- and medium-damage regimes, explicit projectors recover the intended geometric graph and reject several families of synthetic contaminants. The error analysis separates two effects that are often mixed together: compression error from the spectral estimator, and discretization error from the Regge operator itself. This makes it possible to see whether a failure comes from the estimator or from the underlying discrete geometry. -/- The paper also records the limits of the method. Early 3D reconstruction fails under shortcut contamination, the 4D Regge operator depends on the correct internal-dihedral-angle convention, and high-damage repair succeeds only when scaffold incidence information is supplied. The main contribution is therefore not a claim that relational geometry has been solved, but a falsifiable framework for testing how much geometry can be recovered from relational data. (shrink)

This paper introduces the concept of Human State Field as a proposed framework for understanding how individuals perceive and experience time. We explore how various internal and external inputs affect this field, shaping subjective temporal experience. Our aim is to explore possible connections between the human cognitive sense of time and the concept of time as understood in physics. We begin by examining how people form familiarity with the notion of time and cognitively accept temporal phenomena. We then analyze how (...) the human experience of time connects to emotional and perceptual responses during brief episodes of daily life, thereby shaping one’s broader experience of time. Central to our discussion are the notions of now and here, treated as perceptual anchors, and how these merge phenomenologically with physical representations of time. We also introduce the novel concepts of Time of Position and Time of Person, investigating how these dimensions interact and sometimes conflict within human experience. The framework aims to support a more integrated understanding of time that is informed by phenomenological insights and compatible with physical descriptions, without claiming a full theoretical unification. This part concludes by exploring how different internal and external factors could affect the human state field, and in a follow-up study (Part II), we aim to mathematically model the dynamics of the Human State Field. The present article develops the conceptual foundations of the framework; formal modeling and empirical operationalization are reserved for a subsequent study. (shrink)

This chapter investigates the origin of incommensurability within Proclus’ ontology through an analysis of his Commentary on the First Book of Euclid’s Elements and his Commentary on the Parmenides. The central question concerns how incommensurability arises at a specific ontological level and how it corresponds to a distinct mode of cognition. The chapter first situates geometrical entities in an intermediate position between the intelligible and the sensible realms. It then demonstrates that sensible matter cannot serve as the unqualified cause of (...) incommensurability, since it accounts only for imprecision rather than for incommensurability. Likewise, intelligible matter (the soul) alone cannot explain it, as not all objects within the soul exhibit incommensurability. Instead, the chapter argues that incommensurability is an intrinsic property of certain geometrical entities, comparable to their other identifiable features. Ultimately, the essay supports the interpretation that incommensurability is a definable and necessary feature of some geometrical objects. (shrink)

Supersubstantivalists identify material objects with regions of spacetime. Accordingly, they take their view to be both more ideologically parsimonious than other substantivalists (dualists) because they can reduce location to identity with a region and they can explain why the mereological structure of objects mirrors the mereological structure of their locations (henceforth, “harmony”). However, I argue that these motivations for supersubstantivalism don’t hold water. Specifically, I argue that supersubstantivalists can only claim the aforementioned advantages just so long as they stand in (...) a unique position to reduce location. But this depends on the illicit assumption that no possible spacetime structures have co-located regions – something falsified by the possibility of merely pseudo-metric spaces. Further, any alternative reduction is equally available to the dualist. Accordingly, this allows dualists to have the same explanation of harmony as the supersubstantivalist. As such, these considerations level the dialectical playing field between dualists and supersubstantivalists. (shrink)

L’objectif de cet article est d’étudier la géométrie plane d’Euclide, relativement au Livre I des Éléments, d’un point de vue formel. Au lieu de faire appel à un cadre logique préexistant, nous présentons un nouveau langage formel ainsi qu’un nouveau système de règles spécifiquement conçu pour reconstruire fidèlement la pratique démonstrative d’Euclide. Une telle reconstruction montre que le travail d’Euclide peut être compris comme reposant sur un cadre inférentiel très particulier, faisant intervenir les mathématiques plus que la logique (entendue comme (...) cadre général de raisonnement): les règles d’inférence traitent essentiellement d’objets géométriques et de leurs relations, sans recours à aucun connecteur propositionnel ni quantificateur. (shrink)

The same mechanism operates from quarks to consciousness. We call it Universal Constraint Parsing (UCP)—constraints at each level evaluate entities against possibility spaces, accepting configurations that fit, rejecting those that don't. Quarks are parsed by QCD field constraints. Molecules are parsed by thermodynamic constraints. Organisms are parsed by ecological constraints. Beliefs are parsed by evidential constraints. Memes are parsed by cultural constraints. Self-models are parsed by architectural constraints. This isn’t metaphor or loose analogy—across domains, selection dynamics belong to the same (...) formal equivalence class of constraint-satisfaction processes under scale-appropriate coarse-graining. While state variables and effective free-energy functionals differ by level, the underlying variational structure and selection dynamics are preserved. Crucially, UCP operates without agency—no designer, no purpose, no teleology required at any level. Agency itself emerges as a functional property when systems begin recursively parsing their own parsing at the consciousness threshold. We demonstrate UCP across all levels, show what emerges at complexity thresholds (replication, learning, consciousness), explain how existing frameworks (Universal Darwinism, memetics, Free Energy Principle) relate to UCP, and derive empirically constraining expectations. The framework is falsifiable and unifies phenomena previously treated as separate. It's physics all the way—mechanistic constraint satisfaction at every level. (shrink)

Weatherall and Manchak (2014) show that, under reasonable assumptions, Reichenbachean universal effects, constrained to a rank-2 tensor field representation in the geodesic equation, always exist in non-relativistic gravity but not so for relativistic spacetimes. Thus, general relativity is less susceptible to underdetermination than its Newtonian predecessor. Dürr and Ben-Menahem (2022) argue that these assumptions are exploitable as loopholes, effectively establishing a (rich) no-go theorem. I disambiguate between two targets of the proof, which have previously been conflated: the existence claim of (...) at least one alternative geometry to a given one and Reichenbach’s (in)famous “theorem theta,” which amounts to a universality claim that any geometry can function as an alternative to any other. I show there is no (rich) no-go theorem to save theorem theta. I illustrate this by explicitly breaking one of the assumptions and generalizing the proof to torsionful spacetimes. Finally, I suggest adopting a programmatic attitude: rather than undermining the proof, one can use it to systematically and rigorously articulate stronger propositions to be proved, thereby systematically exploring the space of alternative spacetime theories. (shrink)

The discussion regarding the origins of geometry examines the fundamental aspects and inception of geometry, drawing from the disciplines of cognitive science, mathematics, and philosophy. With a collection of 14 articles, various authors contribute historical, scientific, and mathematical perspectives, providing a thorough understanding of the principles and application of geometry. This critical essay offers a review and analysis, as well as detailed scrutiny, presenting both support and counterarguments to each of these theories.

Relationalism about space faces well-known objections if it is limited to relations between actual bodies. These problems might be avoided through so-called modal relationalism, on which the relevant relata include possible entities. Leibniz is considered a founder of modal relationalism, appealing to relations among possible situations. This article argues that for the central type of relation in question, namely congruence, Leibniz cannot give an adequate basis for modal relationalism. This is because his criteria for determining congruence rely on actual perception, (...) and because attempts to modalize this perception requirement conflict with his other commitments. (shrink)

Published with great thanksgiving for Yaohushua, the living One Yahowah, "Jesus Christ." -/- In previous work, Formalizing Mechanical Analysis Using Sweeping Net Methods I, sweeping net methods have been extended to complex analysis, relying on the argument of complex functions defined on the unit circle. In this paper, we reformulate these methods purely within a real-valued and geometric framework, avoiding the use of complex analysis. By redefining the sweeping net constructs and the associated theorems using real functions and geometric interpretations (...) on the unit circle, we demonstrate how singularities and their approximations can be effectively analyzed without the need for imaginary numbers. This approach provides intuitive geometric insights and broadens the applicability of sweeping net methods in mathematical analysis. (shrink)

-/- This volume brings together scholars across various domains of the history and philosophy of mathematics, investigating duality as a multi-faceted phenomenon. Encompassing both systematic analysis and historical examination, the book endeavors to elucidate the status, roles, and dynamics of duality within the realms of 19th and 20th-century mathematics. Eschewing a priori notions, the contributors embrace the diverse interpretations and manifestations of duality, thus presenting a nuanced and comprehensive perspective on this intricate subject. -/- Spanning a broad spectrum of mathematical (...) topics and historical periods, the book uses detailed case studies to investigate the different forms in which duality appeared and still appears in mathematics, to study their respective histories, and to analyze interactions between the different forms of duality. The chapters inquire into questions such as the contextual occurrences of duality in mathematics, the influence of chosen forms of representation, the impact of investigations of duality on mathematical practices, and the historical interconnections among various instances of duality. Together, they aim to answer a core question: Is there such a thing as duality in mathematics, or are there just several things called by the same name and similar in some respect? What emerges is that duality can be considered as a basic structure of mathematical thinking, thereby opening new horizons for the research on the history and the philosophy of mathematics and the reflection on mathematics in general. -/- The volume will appeal not only to experts in the discipline but also to advanced students of mathematics, history, and philosophy intrigued by the complexities of this captivating subject matter. (shrink)

We present an approach in which ancient Greek mathematical proofs by Hippocrates of Chios and Euclid are addressed as a form of (guided) intentional reasoning. Schematically, in a proof, we start with a sentence that works as a premise; this sentence is followed by another, the conclusion of what we might take to be an inferential step. That goes on until the last conclusion is reached. Guided by the text, we go through small inferential steps; in each one, we go (...) through an autonomous reasoning process linking the premise to the conclusion. The reasoning process is accompanied by a metareasoning process. Metareasoning gives rise to a feeling-knowing of correctness. In each step/cycle of the proof, we have a feeling-knowing of correctness. Overall, we reach a feeling of correctness for the whole proof. We suggest that this approach allows us to address the issues of how a proof functions, for us, as an enabler to ascertain the correctness of its argument and how we ascertain this correctness. (shrink)

In this paper, I will offer a novel perspective on Carnapian explication, understanding it as a philosophical analogue of the transfer principle methodology that originated in nineteenth-century projective geometry. Building upon the historical influence that projective geometry exerted on Carnap’s philosophy, I will show how explication can be modeled as a kind of transfer principle that connects, relative to a given task and normatively constrained by the desiderata chosen by the explicators, the functional properties of concepts belonging to different conceptual (...) frameworks. Moreover, I will demonstrate how, in light of this characterization, we can better appreciate the evolution of Carnap’s metaphilosophy. (shrink)

We present some aspects of the genesis of a geometric construction, which can be carried out with compass and straightedge, from the original idea to the published version (Fernández González 2016). The Midpoint Path Construction makes it possible to multiply the length of a line segment by a rational number between 0 and 1 by constructing only midpoints and a straight line. In the form of an interview, we explore the context and narrative behind the discovery, with first-hand insights by (...) its author. Finally, we discuss some general aspects of this case study in the context of philosophy of mathematical practice.´´aa. (shrink)

This paper has two specific goals. The first is to demonstrate that the theorem in MetaphysicsΘ 9, 1051a24-27 is not equiva-lent to Euclid’s Proposition 32 of book I (which contradicts some Aristotelian commentators, such as W. D. Ross, J. L. Heiberg, and T. L. Heith). Agreeing with Henry Mendell’s analysis, I ar-gue that the two theorems are not equivalent, but I offer different reasons for such divergence: I propose a pedagogical-philosoph-ical reason for the Aristotelian theorem being shorter than the Euclidean (...) one (and the previous Aristotelian versions). Aristotle wants to emphasize the deductive procedure as a satisfactory method to discover scientific knowledge. The second objective, opposing some consensus about geometrical deductions/theo-rems in Aristotle, is to briefly propose that the theorem, exactly as we found it in Metaphysics and without any emendation to the text (therefore opposing Henry Mendell’s suggested amend-ments), allows the ancient philosopher to demonstrate that universal mathematical knowledge is in potence in geometrical figures. This tentatively proves that Aristotle emphasizes that geometrical deduction is sufficient to actualize mathematical knowledge. (shrink)

This article aims at identifying the sources of fols. 395r and 686r-686v of the Codex Atlanticus. These anonymous folios, inserted in Leonardo da Vinci’s notebooks, do not deal with the duplication of the cube proper, nor do they derive from Giorgio Valla’s De expetendis et fugiendis rebus (1501), as has been claimed. They deal specifically with the extraction of the cube root by geometric methods. The analysis of the sources by the tracer method reveals that these fragments are taken from (...) the Practica geometrie by Leonardo Fibonacci (1220) and, more precisely, from a volgarizzamento that appears in the Praticha d’arismetrica by Benedetto da Firenze (1463). Leonardo could have consulted this text in one of his two Florentine periods, around 1463-1482 or around 1503-1506. (shrink)

A natural extension of Heron's 2000 year old formula for the area of a triangle to the volume of a tetrahedron is presented. This gives the fourth power of the volume as a polynomial in six simple rational functions of the areas of its four faces and three medial parallelograms, which will be referred to herein as "interior faces." Geometrically, these rational functions are the areas of the triangles into which the exterior faces are divided by the points at which (...) the tetrahedron's in-sphere touches those faces. Part I presents an overview of these results and some necessary but little-known background in areal geometry. Part II derives the promised extension, and ends with a conjecture as to how the formula extends to n-dimensional simplices for all n>3. Part III explains how, for n=3, the zeros of the polynomial constitute a five-dimensional semi-algebraic variety consisting almost entirely of collinear tetrahedra with vertices separated by infinite distances, but with generically well-defined distance ratios; it further proves that these unconventional Euclidean configurations can be identified with a quotient of the Klein quadric by an action of a group of reflections isomorphic to ℤ42, wherein four-point configurations in the affine plane constitute a distinguished three-dimensional subvariety. Part IV consists of five appendices which show, among other things, that the algebraic structure of the zeros in the affine plane naturally defines the associated four-element, rank 3 chirotope, aka affine oriented matroid. (shrink)