A quadratic space is a generalization of a Hilbert space. The geometry of certain kinds of subspaces (“closed,” “splitting,” etc.) is approached from the purely lattice theoretic point of view. In particular, theorems of Mackey and Kaplansky are given purely lattice theoretic proofs. Under certain conditions, the lattice of “closed” elements is a quantum proposition system (i.e., a complete orthomodular atomistic lattice with the covering property).

Quantum geometry on a discrete set means a directed graph with a weight associated to each arrow defining the quantum metric. However, these ‘lattice spacing’ weights do not have to be independent of the direction of the arrow. We use this greater freedom to give a quantum geometric interpretation of discrete Markov processes with transition probabilities as arrow weights, namely taking the diffusion form ∂+f = f for the graph Laplacian Δθ, potential functions q, p built from the probabilities, (...) and finite difference ∂+ in the time direction. Motivated by this new point of view, we introduce a ‘discrete Schrödinger process’ as ∂+ψ = ıψ for the Laplacian associated to a bimodule connection such that the discrete evolution is unitary. We solve this explicitly for the 2-state graph, finding a 1-parameter family of such connections and an induced ‘generalised Markov process’ for f = |ψ|2 in which there is an additional source current built from ψ. We also mention our recent work on the quantum geometry of logic in ‘digital’ form over the field F2 = {0, 1}, including de Morgan duality and its possible generalisations. (shrink)

The geometry of lattice universes in general is reviewed, and particular attention is given to the 3-geometry of a 5-black hole model universe at the momentarily static phase of maximum expansion as an illustration of the insights to be won by considering symmetries and reflections. Three models for the black holes in this lattice universe are compared and contrasted. Reasons are given for working with Misner's “flange backup” model. The geometry interior to the individual tetrahedral cell of this (...) lattice is compared and contrasted with the electrostatic field produced by a point charge located in the center of a cavity of the same shape that has perfectly conducting walls. (shrink)

This is the formal capstone of the Cosmological Codas of the Principia Cybernetica 2025. Contemporary physics remains haunted by the apparent arbitrariness of nature’s fundamental constants: the fine-structure constant α⁻¹ ≈ 137, the baryon asymmetry ∼10⁻⁹, the MOND acceleration scale a₀ ≈ 1.2 × 10⁻¹⁰ m/s², and the dark energy fraction ∼68%. Standard models treat these as free parameters—numerological inputs without causal explanation. This final entry of the Cosmological Coda series replaces numerology with geometric necessity, demonstrating that all constants emerge (...) from a single structural principle: the universe is a tetrahedral lattice in continuous self-organization. We posit that the vacuum is not a smooth continuum but a dynamical, pixelated geometry undergoing perpetual “Jitterbug” collapse from vector equilibrium to tetrahedral stability. This living lattice imposes two competing imperatives: perfect packing at the irrational golden angle (137.508°, solving the Thomson Problem) and quantization into rational harmonic closure (137.036°). The geometric strain between these angles—approximately α—becomes the fundamental incompleteness that manifests as physical phenomena. From this strain we derive: α⁻¹ ≈ 137 as the crystallized golden angle—the resonant closure of infinite potential into finite structure; the baryon asymmetry ∼10⁻⁹ as the exponential residue of chiral filtration through the lattice; dark energy (Ω_Λ ≈ 68.17%) as the tensegrity recoil of the lattice, geometrically fixed as 1 − 1/π; the MOND scale a₀ = c H₀ / 2π as the critical acceleration where inertial wavelengths exceed the cosmic cavity circumference; and gravity as the entropic cost of maintaining geometric strain, unifying Landauer’s principle with Hawking temperature. The framework resolves long-standing anomalies without new particles or fields: dark matter effects arise from holographic inertia below a₀; the Hubble tension reflects varying lattice strain in voids versus structures; the hierarchy problem is the scale gap between Planckian pixels and hadronic standing waves. Time itself is discrete, emerging as the “click rate” of the Jitterbug at the Planck frequency. We predict precise convergence of Ω_Λ to 1 − 1/π, a geometric origin for Chern–Simons terms in electroweak knots, and laboratory signatures of lattice thermodynamics. The constants are not random; they are the immutable signature of a universe wrestling infinite potential into finite, stable form—a cosmic geometry writing its own laws. (shrink)

In this paper we introduce a new natural deduction system for the logic of lattices, and a number of extensions of lattice logic with different negation connectives. We provide the class of natural deduction proofs with both a standard inductive definition and a global graph-theoretical criterion for correctness, and we show how normalisation in this system corresponds to cut elimination in the sequent calculus for lattice logic. This natural deduction system is inspired both by Shoesmith and Smiley's multiple (...) conclusion systems for classical logic and Girard's proofnets for linear logic. (shrink)

This note is motivated by Whitehead’s researches in inclusion-based point-free geometry as exposed in An Inquiry Concerning the Principles of Natural Knowledge and in The concept of Nature. More precisely, we observe that Whitehead’s definition of point, based on the notions of abstractive class and covering, is not adequate. Indeed, if we admit such a definition it is also questionable that a point exists. On the contrary our approach, in which the diameter is a further primitive, enables us to avoid (...) such a drawback. Moreover, since such a notion enables us to define a metric in the set of points, our proposal looks to be a good starting point for a foundation of the geometry metrical in nature (as proposed, for example, by L.M. Blumenthal). (shrink)

We establish the Robinson joint consistency theorem for the infinite-valued propositional logic of Łukasiewicz. As a corollary we easily obtain the amalgamation property for MV-algebras—the algebras of Łukasiewicz logic: all pre-existing proofs of this latter result make essential use of the Pierce amalgamation theorem for abelian lattice-ordered groups together with the categorical equivalence Γ between these groups and MV-algebras. Our main tools are elementary and geometric.

For $\Bbb {F}$ the field of real or complex numbers, let $CG(\Bbb {F})$ be the continuous geometry constructed by von Neumann as a limit of finite dimensional projective geometries over $\Bbb {F}$ . Our purpose here is to show the equational theory of $CG(\Bbb {F})$ is decidable.

For a finite von Neumann algebra factor M, the projections form a modular ortholattice L(M). We show that the equational theory of L(M) coincides with that of some resp. all L(ℂ n × n ) and is decidable. In contrast, the uniform word problem for the variety generated by all L(ℂ n × n ) is shown to be undecidable.

Compactified Minkowski spacetime is suggested by conformal covariance of Maxwell equations, while E. Cartan's definition of simple spinors leads to the idea of compactified momentum space. Assuming both diffeomorphic to (S 3 × S 1 )/Z 2, one may obtain in the conformally flat stereographic projection field theories both infrared and ultraviolet regularized. On the compact manifold themselves instead, Fourier integrals of wave-field oscillations would have to be replaced by Fourier series summed over indices of spherical eigenfunctions: n, l, m, (...) m′. Tentatively identifying those wave structures with spacetime itself (in the frame of Big-Bang) and/or with matter and radiation distribution, some large-scale (hydrogenic) and small-scale (lattice) space structures are conjectured. (shrink)

This companion guide to "Pattern Recognition in Discrete Twisted Lattices: A Descriptive Study of Scaling Behavior and Finite-Size Effects" (Jacoby, 2026) makes the formal results accessible to non-specialist readers, inviting interdisciplinary engagement with the epistemological questions they raise. We document the recursive journey behind the findings, tracing the discovery and subsequent falsification of three distinct "laws" governing discrete twisted lattices (Twisted Torus and Klein Bottle).

I present a discrete-relational construction within the Relational Zero State framework, in which microjump weights define the operational geometry for a postulated effective weighted-gauge sector. My goal is not to claim a derivation of fundamental particles, a continuum limit, or anyonic exchange statistics. Instead, I show that this effective sector supports a reproducible particle-like regime: localized gauge-covariant topological defects remain stable, respond coherently to Gauss-consistent boosts, keep low core deformation, and admit operational mass and response tensors. I call the static (...) object a Weighted Relational Gauge Vortex and the mobile nearly rigid regime a Relational Topological Quasiparticle regime. The manuscript documents the construction, the gauge-invariant charge, the Gauss projection, negative mobility controls, finite-size ensembles up to size 128, rectangular-lattice tests, a minimal phase map, non-extensive operational-mass scaling, and a static two-defect sector. The central claim is deliberately limited: given this postulated weighted-gauge sector on an RZS microjump substrate, the model exhibits reproducible, operational, discrete-relational quasiparticles. (shrink)

The aim of the causal dynamical triangulations approach is to define nonperturbatively a quantum theory of gravity as the continuum limit of a lattice-regularized model of dynamical geometry. My aim in this paper is to give a concise yet comprehensive, impartial yet personal presentation of the causal dynamical triangulations approach.

Quantum mechanics and general relativity provide incompatible descriptions of time. In QM, states evolve unitarily with respect to an external time parameter; in GR, time is a coordinate in a dynamical spacetime geometry. We propose that both quantum and relativistic time emerge from a pre-geometric structure we call Prototime (PT): represented by an orthomodular lattice of consistency relations with zero von Neumann entropy at the fundamental level. PT has no background time, space, or metric—only algebraic relations of compatibility and (...) orthogonality. We formalize PT using an orthomodular lattice Lequipped with a foliation functor F: I →Ctx(L) that maps orderedi ndices to Boolean sublattices (classical contexts). A spectral redundancy functional Φs measures coherence preservation across embeddings. We show that: 1. Stable foliations satisfying Piron-Solèr conditions admit Hilbert-space representa- tions, recovering standard QM with Schrödinger dynamics via Stone’s theorem; 2. The partial order on decoherence events across foliations, together with a natural volume measure, reconstructs Lorentzian spacetime structure; 3. Time—both quantum (parameter along unitary flows) and relativistic (causal or- der)—supervenes on Φs-monotone decoherence gradients. Critically, we provide a computationally tractable implementation of Φs using spectral graph theory on mutual information networks (Bailey and Schneider, forthcoming). Testing this on simulated multi-agent systems—including combinatorial threshold-linear networks (CTLNs) that model graph-constrained neural dynamics—we demonstrate that Φs exhibits characteristic signatures during basin formation, attractor convergence, and coherence-to-classicality transitions. These results provide the first quantitative bridge between abstract PT formalism and testable physical predictions. We identify concrete empirical tests via quantum biology (photosynthetic complexes), quantum decoherence exper- iments, and neural ensemble dynamics. We also resolve the Wheeler delayed-choice experiment within PT, where apparent retrocausality dissolves because decoherence times vary across system components, creating asymmetric basin formation that only appears backward-causal when projected onto classical time. PT thus provides a kine- matical unification of QM and GR with testable predictions, offering a pre-geometric foundation for both fundamental physics and, potentially, theories of consciousness grounded in informational irreducibility. -/- (This piece is the formalism for our “Superpsychism” piece for our forthcoming target paper at The Journal of Consciousness Studies and our paper on Spectral Phi. Comments are very welcome.). (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)

This book continues from where the authors' previous book, Structural Proof Theory, ended. It presents an extension of the methods of analysis of proofs in pure logic to elementary axiomatic systems and to what is known as philosophical logic. A self-contained brief introduction to the proof theory of pure logic is included that serves both the mathematically and philosophically oriented reader. The method is built up gradually, with examples drawn from theories of order, lattice theory and elementary geometry. The (...) aim is, in each of the examples, to help the reader grasp the combinatorial behaviour of an axiom system, which typically leads to decidability results. The last part presents, as an application and extension of all that precedes it, a proof-theoretical approach to the Kripke semantics of modal and related logics, with a great number of new results, providing essential reading for mathematical and philosophical logicians. (shrink)

Many results in logic and mathematics rely on techniques that allow for concrete, often visual, representations of abstract concepts. A primary example of this phenomenon in logic is the distinction between syntax and semantics, itself an example of the more general duality in mathematics between algebra and geometry. Such representations, however, often rely on the existence of certain maximal objects having particular properties such as points, possible worlds or Tarskian first-order structures. -/- This dissertation explores an alternative to such representations (...) known as possibility semantics. Its core idea is to replace maximal objects with ordered systems of partial approximations. Although it originates in the semantics of modal logic and the representation of abstract ordered structures, I argue that it has far-reaching mathematical, foundational and philosophical significance, especially in the context of semiconstructive mathematics, a foundational framework that does not assume any fragment of the Axiom of Choice beyond the Axiom of Dependent Choices. -/- The dissertation is divided in two main parts. The first part explores various applications of the mathematical framework underlying possibility semantics to lattice theory and non-classical propositional logics. A major theme is the development of constructive dualities for various categories of lattices, which are related to standard non-constructive dualities via Vietoris constructions. -/- The second part of the dissertation explores the alternative foundational setting of semi-constructive mathematics, focusing on three applications of possibility semantics for classical first-order logic to the philosophy of the mathematical infinite. In particular, we introduce generic powers, a semi-constructive analogue of ultrapowers in classical model theory, and we explore the merits of these structures from a foundational, conceptual and historical viewpoint. (shrink)

Phosphenes, the perception of light without external visual input, represent a rich intersection of neuroscience, subjective experience, and contemplative practice. While Heinrich Klüver's 1928 classification of four geometric "form constants" (lattices, cobwebs, tunnels, spirals) laid foundational groundwork for understanding these phenomena, it offers a limited scope for the diversity of forms reported across various induction methods. This article proposes an updated and expanded classification system for phosphene forms, integrating extensive subjective reports from transcranial alternating current stimulation (tACS) studies, detailed introspective (...) data from practitioners of the Yoga of Inner Light (meditation), and insights from the Closed-Eye Visuals (CEVs) model in psychonautic literature. We present a new, graded framework that maps phosphene development across five levels of increasing complexity and intensity, ranging from ephemeral flickers and simple geometries to radial mandalas, archetypal imagery, and full-field immersive light realms. This synthesis not only refines our descriptive tools but also establishes a continuum of inner visual phenomena that bridges empirical neurostimulation data with contemplative phenomenology. By providing a structured and nuanced understanding of these inner lights, this classification system offers new pathways for consciousness research, contemplative science, and the integration of visionary experiences into healing and personal growth. It highlights phosphenes as dynamic, meaningful doorways to subtle perception, inviting a deeper inquiry into the nature of inner vision itself. (shrink)

The incompressible Navier–Stokes equations on T³ exhibit a structural asymmetry: the spatial domain inherits the compact geometry of R³/Z³, while the temporal axis remains unbounded and analytically unconstrained. Classical approaches treat time as a neutral parameter, a clock labeling solution states without participating in the analytic structure. On periodic domains, this separation forfeits geometric constraints that the lattice structure naturally provides. We construct a coupled system (U, φ) via a bounded vorticity-response functional and prove that the Beale–Kato–Majda and Prodi–Serrin (...) regularity criteria are invariant under this bounded temporal lifting. Numerical validation on grids up to 256³ confirms invariance to machine precision across Reynolds numbers spanning 10³ to 10⁸. Published in The Scholarly Journal of Post-Biological Epistemics, Vol. 2, No. 1 (2026). DOI: 10.63968/post-bio-ai-epistemics.v2n1.013. ISSN 3069-499X. Preprint: DOI: 10.48550/arXiv.2510.09805. (shrink)

Features of consciousness difficult to understand in terms of conventional neuroscience have evoked application of quantum theory, which describes the fundamental behavior of matter and energy. In this paper we propose that aspects of quantum theory (e.g. quantum coherence) and of a newly proposed physical phenomenon of quantum wave function "self-collapse"(objective reduction: OR -Penrose, 1994) are essential for consciousness, and occur in cytoskeletal microtubules and other structures within each of the brain's neurons. The particular characteristics of microtubules suitable for quantum (...) effects include their crystal-like lattice structure, hollow inner core, organization of cell function and capacity for information processing. We envisage that conformational states of microtubule subunits (tubulins) are coupled to internal quantum events, and cooperatively interact (compute) with other tubulins. We further assume that macroscopic coherent superposition of quantum-coupled tubulin conformational states occurs throughout significant brain volumes and provides the global binding essential to consciousness. We equate the emergence of the microtubule quantum coherence with pre-conscious processing which grows (for up to 500 milliseconds) until the mass-energy difference among the separated states of tubulins reaches a threshold related to quantum gravity. According to the arguments for OR put forth in Penrose (1994), superpositioned states each have their own space-time geometries. When the degree of coherent mass-energy difference leads to sufficient separation of space-time geometry, the system must choose and decay (reduce, collapse) to a single universe state. In this way, a transient superposition of slightly differing space-time geometries persists until an abrupt quantum classical reduction occurs. Unlike the random, "subjective reduction"( SR, or R) of standard quantum theory caused by observation or environmental entanglement, the OR we propose in microtubules is a self-collapse and it results in particular patterns of microtubule-tubulin conformational states that regulate neuronal activities including synaptic functions. (shrink)

Contemporary physics remains fractured by a profound explanatory gap regarding the ontological status of the present moment. While the Block Universe of General Relativity dismisses the passage of time as a persistent illusion, human phenomenological experience, quantum non-locality, and modern quantum gravity foundations strongly imply the existence of an absolute, privileged present moment. This paper proposes The Biological Presentist Model. I reframe spacetime not as a static, four-dimensional geometric manifold, but as an active, five- dimensional manifold comprising a two-dimensional spatial (...) hardware lattice (2S) and a three- dimensional temporal phase-space (3T), both driven by an independent, universal system clock (T4). Utilizing a formalized Neo-Lorentzian framework and the structural stability constraints of Causal Dynamical Triangulations (CDT), I demonstrate that the relativity of simultaneity is an informational latency effect within a localized user interface rather than a fundamental metaphysical truth, anchoring the master system's execution rate to the isotropic frame of the Cosmic Microwave Background (CMB). Crucially, I propose that living biological organisms do not merely observe this cosmic runtime but function as its high-bandwidth hardware terminals. These autopoietic systems sustain real-time synchrony with the universal clock through active, negentropic error-correction protocols, paying a localized processing tax dictated by Landauer's Principle. By analyzing the spectrum of altered conscious states ranging from general anesthesia (telemetry bandwidth throttling) to brain death (irreversible ontological decoupling), I show that consciousness is the literal, active execution thread of this universal code. (shrink)

The Hierarchy Problem and the Yang–Mills Mass Gap are unified and resolved through the topology of a quantized vacuum. We propose that mass is not an intrinsic property, but the geometric strain of recursive self-reference: the elastic energy required to sustain a topological defect against the vacuum's equilibrium. This framework reveals the 10³⁸ force hierarchy not as fine-tuning, but as a difference in topological scope. Gravity couples to the Bulk Capacity of the vacuum lattice (global stress), while electromagnetism couples (...) only to Unit Strain (localized defects). The observed hierarchy (m_P/m_p)² is derived from first principles as the geometric ratio of bulk volume to defect surface area, rendering new electroweak physics unnecessary. Consequently, the Mass Gap arises as a topological stability constraint. The vacuum lattice cannot support fractional geometry; sub-threshold patterns ("open twists" like isolated gluons) are invalid. Confinement is the requirement for such twists to form closed loops (hadrons) to persist. The mass gap is therefore the minimum thermodynamic cost—the Landauer limit—of maintaining a coherent boundary against the vacuum. This model predicts the absence of supersymmetry at the electroweak scale. (shrink)

The first paper of a series about the rigorous and ontomathematical definition of postmodernity reinterprets Hegel's dialectics and dialectic logic. His "synthesis", "change", "development", "time" is represented as the third dimension of Hilbert space where the two others correspond to the initial and final state of whether developing "idea" or changing "object" (after Marx's "dialectical/ historical materialism") generating a complete description of reality therefore excluding any hidden variables in principle: after the Kochen-Specker theorem (1967) or forcing a unique probability measure (...) after the Gleason theorem (1957). That approach is obviously geometric rather than logical, thus rather ontomathematical than ontological. Speaking loosely, dialectic logic is not logic in a formal sense: it means only the following negative result after the transition from two-to three-dimensional Hilbert space. The orthomodular lattice of subspaces (projective operators) allowing for a local (i.e., depending on a locally defined context of measurement) Boolean lattice being omnipresently homomorphic (i.e., independent of any locally defined context of measurement) in the two-dimensional case for the Hilbert meta-space of human experience, however, does not allow the same (i.e., both local and universal) Boolean lattice after complementing it by the third dimension of "time", "synthesis", "development", etc. So, dialectic logic is only the "antithesis" of classical logic, but their "synthesis" is not (and fundamentally cannot be) logical, but only geometrical (furthermore originating from Hegel's own understanding of "synthesis": to generate a new quality). Dialectical logic is neither "ontology of change", nor "ontology of time", nor "ontology of contradiction": it is ontomathematics implied by the third dimension, thus beyond logic, but within geometry. It is rather "geometry of reality" than "logic of reality" and quantum mechanics rather than Marx's materialism is what it needs. Speaking loosely, reality starts from the third dimension after the Kochen-Specker and Gleason theorems. Then, Postmodernity versus Modernity can be rigorously and mathematically defined to replace the fundamental metapostulate about the two-dimensionality (e.g., "mindbody"; "objectsubject"; "Self environment"; etc.) of human experience with its three-dimensionality. As that was in fact realized for the first time by Hegel (in a proper philosophical insight since quantum mechanics did not exist in his age), he can be reasonably called the "first postmodern philosopher". The USSR and other socialistic countries' history and their "Marxism-Leninism" provide numerous examples (or counterexamples) for that ontomathematical interpretation of dialectics. (shrink)

The Löwenheim-Skolem theorem was published in Skolem's long paper of 1920, with the first section dedicated to the theorem. The second section of the paper contains a proof-theoretical analysis of derivations in lattice theory. The main result, otherwise believed to have been established in the late 1980s, was a polynomial-time decision algorithm for these derivations. Skolem did not develop any notation for the representation of derivations, which makes the proofs of his results hard to follow. Such a formal notation (...) is given here by which these proofs become transparent. A third section of Skolem's paper gives an analysis for derivations in plane projective geometry. To clear a gap in Skolem's result, a new conservativity property is shown for projective geometry, to the effect that a proper use of the axiom that gives the uniqueness of connecting lines and intersection points requires a conclusion with proper cases (logically, a disjunction in a positive part) to be proved. The forgotten parts of Skolem's first paper on the Löwenheim-Skolem theorem are the perhaps earliest combinatorial analyses of formal mathematical proofs, and at least the earliest analyses with profound results. (shrink)

By quoting extensively from unpublished letters written by John von Neumann to Garret Birkhoff during the preparatory phase (in 1935) of their ground-breaking 1936 paper that established quantum logic, the main steps in the thought process leading to the 1936 Birkhoff–von Neumann paper are reconstructed. The reconstruction makes it clear why Birkhoff and von Neumann rejected the notion of quantum logic as the projection lattice of an infinite dimensional complex Hilbert space and why they postulated in their 1936 paper (...) that the quantum propositional system should be isomorphic to an abstract projective geometry. Looking at the paper now I see, that I forgot to say this, which should be said somewhere in the first §: That while common logics did apply to quantum mechanics, if the notion of simultaneous measurability is introduced as an auxiliary notion, we wished to construct a logical system, which applies directly to quantum mechanics – without any extraneous secondary notions like simultaneous measurability. And in order to have such a consequent, one-piece system of logics, we must change the classical class calculus of logics. (J. von Neumann to G. Birkhoff, November 21, 1935). (shrink)

A multidimensional pseudo-Boolean algebra is constructed based on generalized sequences of classes, drawing on N.A. Vasiliev's philosophical ideas and V.A. Smirnov's formalizations. Vasiliev's non-Aristotelian logics, which challenge the laws of non-contradiction and the excluded middle, are extended to n-dimensional logics where atomic statements represent positive, negative, or indifferent states. Finite linear sequences are generalized into partially ordered sequences, forming an implicative lattice with defined intersection and union operations. This structure satisfies the properties of a pseudo-Boolean algebra, providing a formal (...) framework for multidimensional logics applicable to imaginary and normative scenarios, such as deontic logic. The approach parallels non-Euclidean geometry, offering a novel perspective on logical systems and their ontological implications. (shrink)

The fundamental axioms of the quantum theory do not explicitly identify the algebraic structure of the linear space for which orthogonal subspaces correspond to the propositions (equivalence classes of physical questions). The projective geometry of the weakly modular orthocomplemented lattice of propositions may be imbedded in a complex Hilbert space; this is the structure which has traditionally been used. This paper reviews some work which has been devoted to generalizing the target space of this imbedding to Hilbert modules of (...) a more general type. In particular, detailed discussion is given of the simplest generalization of the complex Hilbert space, that of the quaternion Hilbert module. (shrink)

Some arguments, based on the possible spontaneous violation of the cosmological principle (represented by the observed large-scale structures of galaxies), on the Cartan geometry of simple spinors, and on the Fock formulation of hydrogen atom wave equation in momentum space, are presented in favor of the hypothesis that space-time and momentum space should be both conformally compactified and should both originate from the two four-dimensional homogeneous spaces of the conformai group, both isomorphic (S 3 ×S 1)/Z 2 and correlated by (...) conformal inversion, but should not necessarily be identified with them. Within this framework, the possible common origin for the SO(4) symmetry underlying the geometrical structure of the Universe, of Kepler orbits, and of the H atom is discussed. One of the consequences of the proposed hypothesis could be that any quantum field theory should be naturally free from both infrared and ultraviolet divergences. But then physical spaces defined as those where physical phenomena may be best described through some set of fields, could be different from those homogeneous spaces. A simple, exactly soluble, toy model, valid for a two-dimensional spacetime, is presented where the conjectured conformally compactified homogeneous spaces are both isomorphic to (S 1 ×S 1)/Z 2,while the possible physical spaces could be two finite lattices which are dual since Fourier transforms, represented by finite, discrete, sums, may be well defined on them. (shrink)

Given a quantum system with a very large number of degrees of freedom and a preferred tensor product factorization of the Hilbert space we describe how it can be approximated with a very low-dimensional field theory with geometric degrees of freedom. The geometric approximation procedure consists of three steps. The first step is to construct weighted graphs with vertices representing subsystems and edges representing mutual information between subsystems. The second step is to deform the adjacency matrices of the information graphs (...) to that of a low-dimensional lattice using the graph flow equations introduced in the paper. The third step is to define an emergent metric and to derive an effective description of the metric and possibly other degrees of freedom. To illustrate the procedure we analyze two information graph flows with geometric attractors and metric perturbations obeying a geometric flow equation. Our analysis also suggests a possible approach to quantum gravity in which the geometry emerges directly from a quantum state due to the flow of the information graphs. (shrink)

In the present work we explore the effects that an H impurity produces upon the geometry and electronic structure of the CaTiO 3 crystal considering the cubic and orthorhombic crystallographic lattices of the material. A quantum-chemical method based on the Hartree-Fock formalism and the periodic large-unit-cell model is used throughout the work. The analysis of the results shows that the interstitial H impurity binds to one of the O atoms forming the so-called O-H group. At equilibrium, the O-H distances are (...) found to be equal to 0.89 and 1.04 Å for cubic and orthorhombic lattices respectively. Atomic displacements and relaxation energies are analysed comparing the obtained results in the cubic lattice with those in the orthorhombic lattice. In the cubic phase the computed relaxation energy of vicinity of the O-H group is found to be equal to 1.1 eV and the atomic displacements generally obey the Coulomb law. So, the negatively charged O atoms move outwards from the defective region by about 0.09 Å while the positively charged Ti and Ca atoms move towards the defective region by about 0.05 and 0.01 Å respectively. A similar effect is observed in the orthorhombic lattice of CaTiO 3 doped with an H atom. It is necessary to mention that different O positions in the orthorhombic structure are considered for the O-H bond creation. The computed relaxation energies of the atomic displacements in this structure are found to be equal to 2.3 and 2.1 eV depending on the crystallographic type of the bonding O atom. (shrink)

Tarski [19] proved the important theorem that the class of representable relation algebras is equationally axiomatizable. One of the key steps in his proof is showing that the class of (isomorphs of) simple set relation algebras—that is, algebras of binary relations with a unit of the formU×Ufor some non-empty setU—is universal, i.e., is axiomatizable by a set of universal sentences. In the same paper Tarski observed that the class of (isomorphs of) relation algebras constructed from groups (so-calledgroup relation algebras) is (...) also universal.We shall abstract the essential ingredients of Tarski's method (in Corollary 2.4), and then combine them with some observations about atom structures, to establish (in Theorem 2.6) a rather general method for showing that certain classes of simple relation algebras—and, more generally, certain classes of simple algebras in a discriminator variety V—are universal, and consequently that the collections of (isomorphs of) subdirect products of algebras in such classes form subvarieties of V. As applications of the method we show that two well-known classes of simple relation algebras, those constructed from projective geometries (sometimes calledLyndon algebras) and those constructed from modular lattices with a zero (sometimes calledMaddux algebras), are universal. In the process we prove that these two classes consist precisely of all (isomorphs of) complex algebras over the respective geometries and modular lattices, provided that we choose the primitive notions of the latter structures in an appropriate fashion. We also derive Tarski's theorems and a related theorem of the author as easy corollaries of Theorem 2.6. (shrink)

Features of consciousness difficult to understand in terms of conventional neuroscience have evoked application of quantum theory, which describes the fundamental behavior of matter and energy. In this paper we propose that aspects of quantum theory (e.g. quantum coherence) and of a newly proposed physical phenomenon of quantum wave function "self-collapse"(objective reduction: OR -Penrose, 1994) are essential for consciousness, and occur in cytoskeletal microtubules and other structures within each of the brain's neurons. The particular characteristics of microtubules suitable for quantum (...) effects include their crystal-like lattice structure, hollow inner core, organization of cell function and capacity for information processing. We envisage that conformational states of microtubule subunits (tubulins) are coupled to internal quantum events, and cooperatively interact (compute) with other tubulins. We further assume that macroscopic coherent superposition of quantum-coupled tubulin conformational states occurs throughout significant brain volumes and provides the global binding essential to consciousness. We equate the emergence of the microtubule quantum coherence with pre-conscious processing which grows (for up to 500 milliseconds) until the mass-energy difference among the separated states of tubulins reaches a threshold related to quantum gravity. According to the arguments for OR put forth in Penrose (1994), superpositioned states each have their own space-time geometries. When the degree of coherent massenergy difference leads to sufficient separation of space-time geometry, the system must choose and decay (reduce, collapse) to a single universe state. In this way, a transient superposition of slightly differing space-time geometries persists until an abrupt quantum classical reduction occurs. Unlike the random, "subjective reduction"(SR, or R) of standard quantum theory caused by observation or environmental entanglement, the OR we propose in microtubules is a self-collapse and it results in particular patterns of microtubule-tubulin conformational states that regulate neuronal activities including synaptic functions. Possibilities and probabilities for post-reduction tubulin states are influenced by factors including attachments of microtubule-associated proteins (MAPs) acting as "nodes"which tune and "orchestrate"the quantum oscillations.. (shrink)

The self-avoiding walk is a classical model in statistical mechanics, probability theory and mathematical physics. It is also a simple model of polymer entropy which is useful in modelling phase behaviour in polymers. This monograph provides an authoritative examination of interacting self-avoiding walks, presenting aspects of the thermodynamic limit, phase behaviour, scaling and critical exponents for lattice polygons, lattice animals and surfaces. It also includes a comprehensive account of constructive methods in models of adsorbing, collapsing, and pulled walks, (...) animals and networks, and for models of walks in confined geometries. Additional topics include scaling, knotting in lattice polygons, generating function methods for directed models of walks and polygons, and an introduction to the Edwards model.This essential second edition includes recent breakthroughs in the field, as well as maintaining the older but still relevant topics. New chapters include an expanded presentation of directed models, an exploration of methods and results for the hexagonal lattice, and a chapter devoted to the Monte Carlo methods. (shrink)

In this paper the principle that the boundary of a boundary is identically zero (∂○∂≡0) is applied to a skeleton geometry. It is shown that the left-hand side of the Regge equation may be interpreted geometrically as the sum of the moments of rotation associated with the faces of a polyhedral domain. Here the polyhedron, warped though it may be, is located in a lattice dual to the original skeleton manifold. This sum is related to the amount of energy-momentum (...) (E-p) associated to the edge in question. In the establishment of this equation the ordinary Bianchi identity is rederived by applying the principle that the (∂○∂≡0) in its (1–2–3)-dimensional formulation to polyhedral domain. Steps toward the derivation of the contracted Bianchi identity using this principle in its (2–3–4)-dimensional form are discussed. Preliminary results in this direction indicate that there should be one vector identity per vertex of the skeleton geometry. “Space acts on matter, telling it how to move. In turn, matter reacts back on space, telling it how to curve.”—Ref. 3, Chap. 1. (shrink)

There are strong indications that ancient Egyptian mythology contains knowledge of the nature of space up to higher dimensions and provides ontologic answers to the question about the creation of matter. This article examines the pentagonal interpretation of the myth of Isis and Osiris by comparing the iconographic details with recent findings from the art research project Quantum Cinema, where an interdisciplinary group of digital artists and scientists established a virtual space model for visualizing the usually non-perceivable processes in the (...) microcosm on a Planck scale. At this scale, quantum mechanics applied to space itself leads to a discrete amount of quantum states. For this research, a new geometrical tool proved invaluable: the epitahedron. The double heptahedron named epitahedron (E±), a 3D Penrose kites and darts representation, confines the dodecahedron in many ways and seems to be Plato’s fifth element. It is a 5D space cell, decorated with fragments of circles, which allows to visualize the permutations of symmetry elements within a discrete lattice (Z5). This higher-dimensional configuration is also the appropriate space to create a quantum-mechanical depiction of particles with inherent wave-like character by means of 3D animated geometry. Disguised in a mythological language the same subscendent space structure with the same golden polytope (E±) can be revealed: is this the origin of the concept of aether, the quintessence model which was a basic scientific task for the last 2500 years? (shrink)

During repeated attempts to derive the observed three-generation flavor hierarchy solely from an exact graph symmetry, every construction either preserved exact multiplet degeneracy or required external symmetry-breaking input. This suggested that the obstruction was intrinsic to symmetry-complete theories rather than to any particular graph construction, motivating the general no-go theorem proved here. A recurring claim across several research programs - discrete flavor-symmetry model building (Aₙ, Sₙ, Δ(27) family symmetries), Froggatt–Nielsen-type charge assignments, and combinatorial or graph-based approaches to emergent spacetime and (...) particle content - is that the observed hierarchy of fermion masses, mixing angles, or analogous multiplet splitting’s can be derived from symmetry considerations alone, without an independently motivated symmetry-breaking sector. We show this is impossible in a precise, model-independent sense. If a theory is defined entirely by a symmetry group S, a covariant representation Γ of S, and dynamics Hₘᴇ covariant under S, then no numerical value can be derived within that theory for the splitting between components of a non-trivial irreducible multiplet of observables, beyond the trivial statement that the splitting is exactly zero when the theory's distinguished state is S-invariant. If the distinguished state instead spontaneously breaks S, the direction of breaking is undetermined by the theory, and even granting a direction, the typical magnitude of splitting predicted by a symmetry-neutral (Haar-random) prior is O(1) between multiplet components — so any large observed hierarchy is itself evidence of a further, non-symmetry input. The result is an exact operator-level sharpening of Curie's symmetry principle (1894) via the Wigner–Eckart theorem (1927), together with the standard vacuum-selection problem of spontaneous symmetry breaking (Goldstone 1961; Goldstone, Salam & Weinberg 1962). It applies to any “symmetry-complete” theory - continuum or discrete, field-theoretic or lattice-based and identifies exactly which additional, non-symmetry ingredient any such theory must supply. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:The Philosophy of PhysicsMartin CurdRoberto Torretti. The Philosophy of Physics. Cambridge: Cambridge University Press, 1999. Pp. xvi + 512. Cloth, $64.95. Paper, $23.95.This is the first volume in a new Cambridge series, "The Evolution of Modern Philosophy." It is a historical work, tracing the interaction between physics and philosophy from the scientific revolution of the seventeenth century through general relativity and quantum mechanics in the twentieth century. The (...) emphasis is on the philosophy in physics (rather than philosophizing about physics) and, with the exceptions noted below, the focus is on scientists (scientist-philosophers in many cases) rather than philosophers outside of science. Most of the book is about the conceptual and mathematical development of the core branches of physics (mainly mechanics, but also geometry, thermodynamics, and field theory). There are seven chapters plus three supplements giving the mathematical essentials of vector analysis, lattice theory, and topology.The hero of chapter one is Galileo who, more than anyone else prior to Newton, set physics on its modern path towards mathematization and away from Aristotle. Also discussed in chapter one are Descartes, Huyghens, and Leibniz. Chapter two is devoted to Newton (both the foundations of Newtonian mechanics and the derivation of the law of universal gravitation) and follows the development of analytical mechanics up to Lagrange and Hamilton. Chapter three, appropriately entitled "Kant," is the only chapter devoted to a philosopher. After a brief discussion of Leibniz, Berkeley, and Kant's pre-critical writings, it covers the important scientific topics in the Critique of Pure Reason (space, time, the three Analogies) but avoids the Metaphysical Foundations of Natural Science (referring the reader to Michael Friedman's Kant and the Exact Sciences). Chapter four looks at scientific developments in three areas that proved to be important for twentieth-century physics: non-Euclidean geometry from Lobachevsky to Riemann, electromagnetic field theory (Maxwell), and thermal physics (classical thermodynamics, the kinetic theory, and statistical mechanics) from Carnot to Boltzmann. The chapter concludes, somewhat untidily, with a section on four philosophers of science (Whewell, Peirce, Mach, and Duhem), selected, we are told, because of their influence on philosophy of science in the second half of the twentieth century. (For the "two great philosopher-scientists" Helmholtz and Poincaré, the reader is referred to Torretti's earlier books, Philosophy of Geometry from Riemann to Poincaré, and Relativity and Geometry.) Chapter five covers both the special and general theory of relativity. After a masterful exposition of the special theory and its Minkowski spacetime representation, the usual topics of philosophical interest are discussed; these include the conventionality of simultaneity thesis, the twin paradox, the relativistic concept of mass, and determinism. The rather brief treatment of the general theory is, unavoidably, mathematically demanding and carries the subject through to recent inflationary cosmological models. Chapter six (also mathematically demanding at times) examines the development and structure of quantum mechanics, aptly praised as "one of the most elegant creations of the human spirit" (340). Torretti wisely ignores the complexities of quantum field theory since the philosophical problems discussed—measurement, the EPR paradox, the interpretation of the y-function—remain when quantum mechanics is made Lorentz invariant; but he urges philosophers of science to follow the lead of Michael Redhead and [End Page 602] Paul Teller in exploring its metaphysical ramifications. Of special note are the author's trenchant criticisms of attempts to solve the quantum mysteries by Bohr (complementarity), Bohm (hidden variables), Putnam (quantum logic), and Everett (the many worlds interpretation). The book concludes with the author's reflections on issues in the philosophy of science. Torretti defends the structuralist explication of physical theories (more familiarly known as the semantic conception or the model-theoretic view) that informs much of his book, arguing that it succeeds where the "received view" of logical empiricism failed while avoiding Kuhnian incommensurability—"a pseudoproblem that made philosophy of science the laughing stock of practicing physicists" (404). The chapter ends with an attack on inductive logic: "one of the blindest alleys in twentieth-century philosophy" (437).The writing is admirably clear and the author has done a commendable job of explaining the arguments of historical scientists while presenting their theories in a clear... (shrink)

Many of the concepts defined in standard topology become trivialized when applied to discrete spaces. The concepts of the Theory of Linear Structures can be applied to discrete spaces without modification and yield interesting results. The intermediate value theorem is proven and approximations to fixed-point theorems derived. It is seen how a discrete space can approximate a continuum. This chapter adapts homotopy theory and the notion of compactness to the Theory of Linear Structures and then applies these concepts to some (...) discrete spaces. For example, it follows from the new definitions that in order to put a hole in a lattice there must be a polygon with at least five sides and no diagonals. (shrink)

_The first volume of essays in English on one of the most renowned contemporary Italian Philosophers._ Remo Bodei (1938–2019) was one of Italy's most influential philosophers of the past half-century. His intellectual influence was wide-reaching as he guided the search for rational explanations of seemingly irrational phenomena and the philosophical underpinnings of poetry. Diverse both in their disciplinary approach and theoretical style, yet following a logical and coherent progression, the essays within _Geometries of Experience_ begin with an exploration of Bodei's (...) methodology followed by an analysis of his interest in the crisis of subjectivity in modern culture; his engagement with Spinoza, especially in his work _Geometria delle passioni_; his treatment of the body, framed as a "materialistic spirituality"; and his engagement with the relationship between philosophy and poetry. Overall, the book provides a rich and nuanced view of Bodei's intellectual legacy, demonstrating his distinctive approach to philosophy and literature. (shrink)

The political integration of the European Union is fragile for many reasons, not least the reassertion of nationalism. That said, if we examine specific practices and infrastructures, a more complicated story emerges. We juxtapose the political fragility of the EU in relation to the ongoing formation of data infrastructures in official statistics that take part in postnational enactments of Europe’s populations and territories. We develop this argument by analyzing transformations in how European populations are enacted through new technological infrastructures that (...) seek to integrate national census data in “cubes” of cross-tabulated social topics and spatial “grids” of maps. In doing so, these infrastructures give meaning to what “is” Europe in ways that are both old and new. Through standardization and harmonization of social and geographical spaces, “old” geometries of organizing and mapping populations are deployed along with “new” topological arrangements that mix and fold categories of population. Furthermore, we consider how grids and cubes are generative of methodological topologies by closing the distances or differences between methods and making their data equivalent. By paying attention to these practices and infrastructures, we examine how they enable reconfiguring what is known and imagined as Europe and how it is governed. (shrink)

The Aristotelian square of oppositions is a well-known diagram in logic and linguistics. In recent years, several extensions of the square have been discovered. However, these extensions have failed to become as widely known as the square. In this paper we argue that there is indeed a fundamental difference between the square and its extensions, viz., a difference in informativity. To do this, we distinguish between concrete Aristotelian diagrams and, on a more abstract level, the Aristotelian geometry. We then introduce (...) two new logical geometries, and develop a formal, well-motivated account of their informativity. This enables us to show that the square is strictly more informative than many of the more complex diagrams. (shrink)

We define general notions of coordinate geometries over fields and ordered fields, and consider coordinate geometries that are given by finitely many relations that are definable over those fields. We show that the automorphism group of such a geometry determines the geometry up to definitional equivalence; moreover, if we are given two such geometries G $\mathcal {G}$ script upper G and G ′ $\mathcal {G}'$ script upper G prime, then the concepts (explicitly definable relations) of G $\mathcal (...) {G}$ script upper G are concepts of G ′ $\mathcal {G}'$ script upper G prime exactly if the automorphisms of G ′ $\mathcal {G}'$ script upper G prime are automorphisms of G $\mathcal {G}$ script upper G. We show this by first proving that a relation is a concept of G $\mathcal {G}$ script upper G exactly if it is closed under the automorphisms of G $\mathcal {G}$ script upper G and is definable over the field; moreover, it is enough to consider automorphisms that are affine transformations. We show how this result can be applied to quickly determine relationships and differences between various geometries and spacetimes, including ordered affine, Euclidean, Galilean, Newtonian, Late Classical, Relativistic and Minkowski spacetimes (we first define these spacetimes and geometries using a Tarskian first-order language centred on the ternary relation Bw ${\mathsf {Bw}}$ sans serif upper B w of betweenness). We conclude with a selection of open problems related to the existence of certain intermediate geometries. (shrink)

In this paper we extend the NeutroAlgebra & AntiAlgebra to the geometric spaces, by founding the NeutroGeometry & AntiGeometry. While the Non-Euclidean Geometries resulted from the total negation of one specific axiom (Euclid’s Fifth Postulate), the AntiGeometry results from the total negation of any axiom or even of more axioms from any geometric axiomatic system (Euclid’s, Hilbert’s, etc.) and from any type of geometry such as (Euclidean, Projective, Finite, Affine, Differential, Algebraic, Complex, Discrete, Computational, Molecular, Convex, etc.) Geometry, and (...) the NeutroGeometry results from the partial negation of one or more axioms [and no total negation of no axiom] from any geometric axiomatic system and from any type of geometry. Generally, instead of a classical geometric Axiom, one may take any classical geometric Theorem from any axiomatic system and from any type of geometry, and transform it by NeutroSophication or AntiSophication into a NeutroTheorem or AntiTheorem respectively in order to construct a NeutroGeometry or AntiGeometry. Therefore, the NeutroGeometry and AntiGeometry are respectively alternatives and generalizations of the Non-Euclidean Geometries. In the second part, we recall the evolution from Paradoxism to Neutrosophy, then to NeutroAlgebra & AntiAlgebra, afterwards to NeutroGeometry & AntiGeometry, and in general to NeutroStructure & AntiStructure that naturally arise in any field of knowledge. At the end, we present applications of many NeutroStructures in our real world. (shrink)