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.

What reasons can a physicist have to reject the principle of a mathematical method, which he nonetheless uses and which he used frequently in his unpublished works? We are concerned here with Galileo’s doubts and objections against Cavalieri’s “geometry of indivisibles.” One may be astonished by Galileo’s behaviour: Cavalieri’s principle is implied by the Galilean mathematization of naturally accelerated motion; some Galilean demonstrations in fact hinge on it. Yet, in the Discorsi Galileo seems to be opposed to this principle. (...) e fundamental reason of Galileo’s reluctance with respect to Cavalieri’s geometry is to be sought in Galileo’s ideal of intelligibility. It is true that Galilean physics, and more particularly Galileo’s theories of motion and matter, faces deep paradoxes, which Cavalieri’s geometry succeeds to avoid, thanks to a clear determination of the concept of “aggregatum.” But while avoiding these difficulties, Cavalieri does not furnish any solution for the problems raised by Galilean physics. (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)

Noncommutative geometry (NCG) is a branch of pure mathematics with a wide range of applications to spacetime physics. Stemming from the divergence problem in QFT, modern contributions conjecture that the fundamental structure of spacetime is noncommutative. This seemingly homogeneous picture is the result of almost a century of discontinuous interest in noncommutative spacetime (NCST). In this paper, I present a three-phase division of the development of NCST approaches. The initial phase (1930s–1950s) introduced noncommutativity as a means of addressing the (...) divergence problem while maintaining Lorentz-invariance. The second phase (1950s–early 1990s) initially dismissed noncominutativity and focused on the pressing problem of localisation at high energies. In this context, independent alternative approaches to QG identified the value of a fundamental length scale, which ultimately contributed to a reconsideration of the conjecture of spacetime noncommutativity. This was despite the fact that it undermined the original operationalist methodology. Finally, a third phase (1990s–today) has seen a renewed interest in NCST. I argue that this resurgence can be attributed to the discovery of new mathematics in the 1980s. Furthermore, I demonstrate how the third phase builds upon and continues instances from the previous attempts, but redirects the research question. In light of this, I finally argue that the history of NCG in physics can be understood as an example of “interlaced convergence.” This instance of theory convergence may prove usefill as a case study for the more general problem of theory building in QG. (shrink)

Richard Tieszen [Tieszen, R. (2005). Philosophy and Phenomenological Research, LXX(1), 153–173.] has argued that the group-theoretical approach to modern geometry can be seen as a realization of Edmund Husserl’s view of eidetic intuition. In support of Tieszen’s claim, the present article discusses Husserl’s approach to geometry in 1886–1902. Husserl’s first detailed discussion of the concept of group and invariants under transformations takes place in his notes on Hilbert’s Memoir Ueber die Grundlagen der Geometrie that Hilbert wrote during the (...) winter 1901–1902. Husserl’s interest in the Memoir is a continuation of his long-standing concern about analytic geometry and in particular Riemann and Helmholtz’s approach to geometry. Husserl favored a non-metrical approach to geometry; thus the topological nature of Hilbert’s Memoir must have been intriguing to him. The task of phenomenology is to describe the givenness of this logos, hence Husserl needed to develop the notion of eidetic intuition. (shrink)

Hume’s discussion of space, time, and mathematics at T 1.2 appeared to many earlier commentators as one of the weakest parts of his philosophy. From the point of view of pure mathematics, for example, Hume’s assumptions about the infinite may appear as crude misunderstandings of the continuum and infinite divisibility. I shall argue, on the contrary, that Hume’s views on this topic are deeply connected with his radically empiricist reliance on phenomenologically given sensory images. He insightfully shows that, working within (...) this epistemological model, we cannot attain complete certainty about the continuum but only at most about discrete quantity. Geometry, in contrast to arithmetic, cannot be a fully exact science. A number of more recent commentators have offered sympathetic interpretations of Hume’s discussion aiming to correct the older tendency to dismiss this part of the Treatise as weak and confused. Most of these commentators interpret Hume as anticipating the contemporary idea of a finite or discrete geometry. They view Hume’s conception that space is composed of simple indivisible minima as a forerunner of the conception that space is a discretely (rather than continuously) ordered set. This approach, in my view, is helpful as far as it goes, but there are several important features of Hume’s discussion that are not sufficiently appreciated. I go beyond these recent commentators by emphasizing three of Hume’s most original contributions. First, Hume’s epistemological model invokes the “confounding” of indivisible minima to explain the appearance of spatial continuity. Second, Hume’s sharp contrast between the perfect exactitude of arithmetic and the irremediable inexactitude of geometry reverses the more familiar conception of the early modern tradition in pure mathematics, according to which geometry (the science of continuous quantity) has its own standard of equality that is independent from and more exact than any corresponding standard supplied by algebra and arithmetic (the sciences of discrete quantity). Third, Hume has a developed explanation of how geometry (traditional Euclidean geometry) is nonetheless possible as an axiomatic demonstrative science possessing considerably more exactitude and certainty that the “loose judgements” of the vulgar. (shrink)

Causal–Continuity Stabilization Theory (CCST) characterizes macroscopic causal persistence as a physical stability property of finite substrates rather than as an ontological feature of the universal wavefunction. In its original formulation, CCST formalized descriptive admissibility: a coarse–grained macroscopic causal history remains well–posed only when thermodynamic dissipation stays below a substrate–dependent bound and when coarse–grained dynamical sensitivity preserves closure. This work develops an extension of CCST that targets a logically distinct stability question in Everettian dynamics: not only whether a macroscopic description remains (...) admissible within a branch, but whether a branch structure itself can remain dynamically coherent as an organized physical entity under sustained environmental coupling. Although unitary quantum mechanics preserves all components of the global quantum state [1, 2], organized branch structure may dynamically disperse into microscopic degrees of freedom under open–system dynamics, thereby losing the correlation patterns required for macroscopic organization [3, 4]. We distinguish structural dispersion (loss of branch–level organizational coherence) from Causal Exit (loss of description–level definability on a finite physical substrate), emphasizing that these correspond to stability failures at distinct physical layers and should not be conflated. We derive a dynamical weighting over coarse–grained path ensembles from open–system effective action and influence functional formalisms [5–7], showing how microscopic dynamics inducesanon–uniformstabilitygeometryinpathspacethroughirreversibledynamicalsuppression rather than probabilistic selection, without introducing collapse, selection rules, or inter–branch transport. Finally, a minimal toy model is introduced to visualize the qualitative sensitivity of branch structural persistence under controlled stability thresholds. The model is strictly illustrative and is accompanied by transparent parameter definitions and a reproducible numerical implementation in the appendix. (shrink)

The aim of the paper is to analyze Hölder’s understanding of geometry and measurement presented in Intuition and Reasoning [Hölder 1900], “The Axioms of Quantity and the Theory of Measurement” [Hölder 1901], and The Mathematical Method [Hölder 1924]. The paper explores the relations between a) Hölder’s demarcation of geometry from arithmetic based on the notion of given concepts, b) his philosophical stance towards Kantian apriorism and empiricism, and c) the choice of Dedekind’s continuity in the axiomatic formulation of (...) the theory of magnitudes. The paper shows that the choices made at the axiomatic level reflect Hölder’s epistemological framework, and individuates the originality of his approach in the case by case analysis of deductive procedures. (shrink)

El artículo documenta y analiza las vicisitudes en torno a la incorporación de Hilbert de su famoso axioma de completitud, en el sistema axiomático para la geometría euclídea. Esta tarea es emprendida sobre la base del material que aportan sus notas manuscritas para clases, correspondientes al período 1894–1905. Se argumenta que este análisis histórico y conceptual no sólo permite ganar claridad respecto de cómo Hilbert concibió originalmentela naturaleza y función del axioma de completitud en su versión geométrica, sino que además (...) permite disipar equívocos en cuanto a la relación de este axioma con la propiedad metalógica de completitud de un sistema axiomático, tal como fue concebida por Hilbert en esta etapa inicial.The paper reports and analyzes the vicissitudes around Hilbert’s inclusion of his famous axiom of completeness, into his axiomatic system for Euclidean geometry. This task is undertaken on the basis of his unpublished notes for lecture courses, corresponding to the period 1894–1905. It is argued that this historical and conceptual analysis not only sheds new light on how Hilbert conceived originally the nature of his geometrical axiom of completeness, but also it allows to clarify some misunderstandings regarding this axiom and the metalogical property of completeness of an axiomatic system, as it was understoodby Hilbert in this initial stage. (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)

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)

Identity has long been treated as a metaphysical primitive, defined by material continuity, causal chains, psychological persistence, narrative coherence, or sortal concepts. These domain-specific accounts succeed locally but fail to generalize across organisms, artificial systems, mathematical structures, and institutions. This work develops a unified, substrate-independent theory of identity based not on continuity or essence but on constraint geometry: identity is a region of invariance within a possibility space determined by structural constraints, dynamical constraints, and an interpretive frame. This framework (...) distinguishes three canonical forms—point identity (instantaneous coherence), field identity (episodic or bounded coherence), and flow identity (generative persistence)—showing that debates about personal identity, biological individuality, and computational unity are arguments about scale and invariance, not metaphysics. The theory subsumes and clarifies two recent approaches: PatternSense, which characterizes persistent generative identity, and ODEI, which characterizes episodic identity in artificial systems. It also explains identity drift, bifurcation, fusion, and extinction in geometric terms. The result is a general identity theory applicable to persons, organisms, AI systems, institutions, and formal structures. It replaces metaphysical disputes with a precise structural foundation: identity is not a substance or a story but a constraint-induced invariance pattern across scales. (shrink)

This paper develops a geometric semantics for metaphysics by showing how the Modal–Dependence Calculus (MDC) induces curvature, metric structure, and geodesic necessity across modal space. Dependence is interpreted as metaphysical curvature, invariance as a metric tensor, and necessity as the straightest (geodesic) trajectory in a governed modal field. The result is a field-theoretic account of grounding that unifies explanation, fundamentality, and modal stability under a single geometric framework. The paper also connects this structure to cognitive architecture via Path C and (...) the Semantic Artificial Intelligence Reactor (SAIR), illustrating how intelligence can be modeled as traversal within an externally governed semantic manifold. These developments collectively establish the conceptual foundation for Continuity Field Theory (CFT), a unified structural framework for metaphysics, logic, and cognition. (shrink)

This book offers a contribution to our understanding of Nicholas of Cusa’s theory of geometry. It is based not only on his—generally more famous—philosophical texts (e.g., De docta ignorantia, Idiota, etc.), but also, and more significantly, on the strictly speaking mathematical texts drafted between 1445 and 1459, where Cusanus attempts to provide a solution to the vexata quaestio of the squaring of the circle. First critically edited in 2010—and translated into Italian by the author in 2020—Cusanus’ Scripta mathematica are (...) here scrutinized as to their context, genesis, and content, in order to shed light, on the one hand, on the philosophical and theological motives behind Cusanus’ mathematical endeavours; and, on the other hand, on the scope and limits of Cusanus’ geometrical constructive attempts to reach “mathematical perfection”. In addition to providing a scrupulous survey of Cusanus’ sources on mathematics, this book focuses on a little explored aspect in Cusanus’ work, namely his original theory of geometrical space as men’s space, where the activity of mensurare takes place, establishing relations of continuous proportionality to the original unity. In conclusion, the author provides an extensive account on the reception of Cusanus’ mathematical texts, from their early dissemination to the reactions of several thinkers throughout history, ensuring a full and rounded overview of Cusanus’ work in geometry, mathematics, and philosophy. (shrink)

Following ideas elaborated by Hering in his celebrated analysis of color, the psychologist and gestalt theorist Otto Selz developed in the 1930s a theory of “natural space”, i.e., space as it is conceived by us. Selz’s thesis is that the geometric laws of natural space describe how the points of this space are related to each other by directions which are ordered in the same way as the points on a sphere. At the end of one of his articles, Selz (...) tries to derive within his framework two of Hilbert’s axioms for Euclidean geometry. Such derivations would, according to Selz, disclose the psychological origin and meaning of the geometric axioms and would thus contribute to a clarification of their epistemological status. In the present article Selz’s theory is explained and analyzed, his basic principles are amended, and implicit assumptions are made explicit. It is shown that the resulting system is one of ordered affine geometry in which all of Hilbert’s axioms except the axioms of congruence and those of continuity are derivable. The primary aim of the present paper is to make explicit the basic principles behind Selz’s geometry of natural space. The question of the logical independence of these principle is not investigated. (shrink)

Explications of the reconstruction of Leibniz’s metaphysics that Deleuze undertakes in 'The Fold: Leibniz and the Baroque' focus predominantly on the role of the infinitesimal calculus developed by Leibniz.1 While not underestimat- ing the importance of the infinitesimal calculus and the law of continuity as reflected in the calculus of infinite series to any understanding of Leibniz’s metaphysics and to Deleuze’s reconstruction of it in The Fold, what I propose to examine in this paper is the role played by other (...) developments in mathematics that Deleuze draws upon, including those made by a number of Leibniz’s near contemporaries – the projective geometry that has its roots in the work of Desargues (1591–1661) and the ‘proto-topology’ that appears in the work of Du ̈rer (1471–1528) – and a number of the subsequent developments in these fields of mathematics. Deleuze brings this elaborate conjunction of material together in order to set up a mathematical idealization of the system that he considers to be implicit in Leibniz’s work. The result is a thoroughly mathematical explication of the structure of Leibniz’s metaphysics. What is provided in this paper is an exposition of the very mathematical underpinnings of this Deleuzian account of the structure of Leibniz’s metaphysics, which, I maintain, subtends the entire text of The Fold. (shrink)

This book explores and articulates the concepts of the continuous and the infinitesimal from two points of view: the philosophical and the mathematical. The first section covers the history of these ideas in philosophy. Chapter one, entitled ‘The continuous and the discrete in Ancient Greece, the Orient and the European Middle Ages,’ reviews the work of Plato, Aristotle, Epicurus, and other Ancient Greeks; the elements of early Chinese, Indian and Islamic thought; and early Europeans including Henry of Harclay, (...) Nicholas of Autrecourt, Duns Scotus, William of Ockham, Thomas Bradwardine and Nicolas Oreme. The second chapter of the book covers European thinkers of the sixteenth and seventeenth centuries: Galileo, Newton, Leibniz, Descartes, Arnauld, Fermat, and more. Chapter three, 'The age of continuity,’ discusses eighteenth century mathematicians including Euler and Carnot, and philosophers, among them Hume, Kant and Hegel. Examining the nineteenth and early twentieth centuries, the fourth chapter describes the reduction of the continuous to the discrete, citing the contributions of Bolzano, Cauchy and Reimann. Part one of the book concludes with a chapter on divergent conceptions of the continuum, with the work of nineteenth and early twentieth century philosophers and mathematicians, including Veronese, Poincaré, Brouwer, and Weyl. Part two of this book covers contemporary mathematics, discussing topology and manifolds, categories, and functors, Grothendieck topologies, sheaves, and elementary topoi. Among the theories presented in detail are non-standard analysis, constructive and intuitionist analysis, and smooth infinitesimal analysis/synthetic differential geometry. No other book so thoroughly covers the history and development of the concepts of the continuous and the infinitesimal. (shrink)

An integrable version of the Weyl-Dirac geometry is presented. This framework is a natural generalization of the Riemannian geometry, the latter being the basis of the classical general relativity theory. The integrable Weyl-Dirac theory is both coordinate covariant and gauge covariant (in the Weyl sense), and the field equations and conservation laws are derived from an action integral. In this framework matter creation by geometry is considered. It is found that a spatially confined, spherically symmetric formation made (...) of pure geometric quantities is a massive entity. This may be treated either as a fundamental particle or as a cosmic body. In an F-R-W universe at the very beginning of the expansion phase the cosmic matter is created from an initial Planckian egg made of geometry, and during the following expansion geometric fields continue to stimulate the matter production. (shrink)

Aristotle assumes that continuity is found in geometry and in physical bodies and motions. Thomas Bradwardine’s _De continuo_, composed in the mid-fourteenth century, attempts to demonstrate that recent indivisibilist (or atomistic) theories such as those proposed by Henry of Harclay and Walter Chatton, are self-contradictory. The work ends, possibly incomplete, by concluding that, except for human souls, indivisibles do not exist in physical reality. This may lead to the conclusion that issues concerning the relations of continuity and indivisibles belong (...) to mathematics and therefore concern entities in the mind or imagination, rather than entities in the outside world. (shrink)

This article explores the changing relationships between geometric and arithmetic ideas in medieval Europe mathematics, as reflected via the propositions of Book II of Euclid’s Elements. Of particular interest is the way in which some medieval treatises organically incorporated into the body of arithmetic results that were formulated in Book II and originally conceived in a purely geometric context. Eventually, in the Campanus version of the Elements these results were reincorporated into the arithmetic books of the Euclidean treatise. Thus, while (...) most of the Latin versions of the Elements had duly preserved the purely geometric spirit of Euclid’s original, the specific text that played the most prominent role in the initial passage of the Elements from manuscript to print—i.e., Campanus’ version—followed a different approach. On the one hand, Book II itself continued to appear there as a purely geometric text. On the other hand, the first ten results of Book II could now be seen also as possiblytranslatable into arithmetic, and in many cases even as inseparably associated with their arithmetic representation. (shrink)

The distinction between the discrete and the continuous lies at the heart of mathematics. Discrete mathematics (arithmetic, algebra, combinatorics, graph theory, cryptography, logic) has a set of concepts, techniques, and application areas largely distinct from continuous mathematics (traditional geometry, calculus, most of functional analysis, differential equations, topology). The interaction between the two – for example in computer models of continuous systems such as fluid flow – is a central issue in the applicable mathematics of the last (...) hundred years. This article explains the distinction and why it has proved to be one of the great organizing themes of mathematics. (shrink)

This paper develops a conservative route from weighted microscopic jumps to an operational notion of geometry within the Relational Zero State (RZS) framework. The central claim is deliberately narrow. A scalar stability law may remain useful at the macroscopic level, but it does not carry enough structure to recover distance, anisotropy, effective dimension, or curvature. The relevant object is instead the weighted microscopic transition operator. I formulate the dynamics as a normalized continuous-time random walk on a weighted graph, (...) derive the associated random-walk Laplacian and heat semigroup, and use standard results from graph-Laplacian convergence, diffusion maps, and short-time heat-kernel asymptotics to motivate an operational RZS distance. No new theorem is claimed. The proposed bridge is then tested on two public benchmark networks with observed edge weights: Zachary’s Karate Club and the Les Misérables coappearance graph. The empirical protocol uses exact matrix exponentials on the observed graphs, descriptive bootstrap intervals, and Monte Carlo permutation tests. On Karate Club, the weighted operator yields a slightly larger faction-separation score than the unweighted baseline, but the gain is not decisive at the main diffusion scale. On Les Misérables, the weighted operator aligns much more strongly with observed coappearance intensities than the unweighted baseline. The conclusion is limited but useful: weighted microscopic jumps provide a rigorous first bridge from transition structure to emergent spatial geometry in RZS, but they do not by themselves derive the full Lorentzian space-time of physics. (shrink)

The Mellin transform is the Fourier transform of scale. This elementary sentence is the kinematical center of the shadow framework. A positive variable is not naturally additive; it is multiplicative. Its invariant measure is Haar measure d×x = dx/x, its time coordinate is u = log x, its translation group is dilation, and its spectral transform is Mellin analysis. This paper develops that statement as a precise spectral principle. The logarithm identifies L²(R⁺, d×x) with L²(R, du); after the half-density passage (...) to the classical Mellin convention on L²(R⁺, dx), the unitary spectral axis becomes Re(s) = 1/2, and the self-adjoint dilation generator is −i(x∂ₓ + 1/2). In celestial holography, the same positive scale appears as the boost energy ω, and conformal primary wavefunctions are Mellin transforms in ω with dimension Δ. The paper proves a uniqueness theorem for the scale bridge: among homogeneous positive scale laws x = ω^α (proved to be the unique continuous group homomorphisms of (R⁺, ×)), compatibility of the arithmetic half-density axis Re(s) = 1/2 with the four-dimensional celestial principal axis Re(Δ) = 1 forces α = 2. Thus x = ω², and the arithmetic kernel x^{s-1} dx transports to the celestial kernel ω^{Δ-1} dω precisely when Δ = 2s. Consequently the celestial principal series Δ = 1 + iλ is the critical line s = 1/2 + iλ/2, and the celestial shadow Δ → 2 - Δ is the arithmetic reflection s → 1 - s. The result is not a convention but a rigidity statement: once scale, Haar measure, half-densities, and four-dimensional celestial unitarity are fixed, the factor of two is forced. The paper also clarifies the relationship between Mellin spectral parameters (principal-series labels λ ∈ R) and physical conformal dimensions (Δ ∈ R⁺), which are distinct objects arising from the same dilation structure in different contexts. (shrink)

Minkowski geometry is axiomatized in terms of the asymmetric binary relation of optical connectibility, using ten first-order axioms and the second-order continuity axiom. An axiom system in terms of the symmetric binary optical connection relation is also presented. The present development is much simpler than the corresponding work of Robb, upon which it is modeled.

There is little agreement about Aristotle’s philosophy of geometry, partly due to the textual evidence and partly part to disagreement over what constitutes a plausible view. I keep separate the questions ‘What is Aristotle’s philosophy of geometry?’ and ‘Is Aristotle right?’, and consider the textual evidence in the context of Greek geometrical practice, and show that, for Aristotle, plane geometry is about properties of certain sensible objects—specifically, dimensional continuity—and certain properties possessed by actual and potential compass-and-straightedge drawings (...) qua quantitative and continuous. For their part, the objects of stereometry are potential sensible three-dimensional figures qua quantitative and continuous. (shrink)

In the Metaphysical Foundations of Natural Science, Kant attempts to argue a priori from the indefinite divisibility of space to the indefinite metaphysical divisibility of matter. This is one type of argument from the continuity of space – purportedly established by Euclidean geometry – to the continuity of matter. I compare Kant’s argument to parallel reasoning in Du Châtelet, whose work he knew. Both philosophers appeal to idealism about matter in their reasoning, yet also face difficulties in explaining why (...) continuity, though not some other properties from geometry, applies to matter. Both also risk inconsistency in adopting potentialist accounts of material parts, while also committing to realism about infinitesimals. An important difference between them is that Du Châtelet deploys at least three definitions of continuity; only one of these, amounting to indefinite divisibility, is shared with Kant. (shrink)

The essay records the changes in family organization for the importance of grandparents in these years of crisis. Their contribution is made in three areas: significant economic aid, organizational support, emotional support. It is an extraordinary contribution that has alleviated the consequences of the collapse, not just financial, of our country. But led by the generation that is usually defined as ‘lucky', a heavy existential commitment. The presence of grandparents, essential in cases of family separation to ensure security, continuity and (...) identity to the grandchildren, would entail the risk that their children, especially when they return to live in the family of origin, regress to a state of infantile dependence. The problems are not lacking, but the ethics of solidarity and generosity that are testifying is a heritage rich in possibilities and promises a view to a redefinition of "family form". (shrink)

In this paper, I attempt a reconstruction of the theory of intersections in the geometry of Euclid. It has been well known, at least since the time of Pasch onward, that in the Elements there are no explicit principles governing the existence of the points of intersections between lines, so that in several propositions of Euclid the simple crossing of two lines (two circles, for instance) is regarded as the actual meeting of such lines, it being simply assumed that (...) the point of their intersection exists. Such assumptions are labelled, today, as implicit claims about the continuity of the lines, or about the continuity of the underlying space. Euclid’s proofs, therefore, would seem to have some demonstrative gaps that need to be filled by a set of continuity axioms (as we do, in fact, find in modern axiomatizations).I show that Euclid’s theory of intersections was not in fact based on any notion of continuity at all. This is not only because Greek concepts of continuity (such as the Aristotelian ones) were largely insufficient to ground a geometrical theory of intersections, but also, at a deeper level, because continuity was simply not regarded as a notion that had any role to play in the latter theory. Had Euclid been asked to explain why the points of intersections of lines and circles should exist, it would have never occurred to him to mention continuity in this connection. Ancient geometry was very different from ours, and it is only our modern views on continuity that tend to give rise to the expectation that this latter must be included in the foundations of elementary mathematics.We may hope to reconstruct Euclid’s views on intersections if we undertake a critical examination both of the extension and of the limits of ancient diagrammatic practices. This is tantamount to attempting to understand to what extent the existence of an intersection point may be inferred just from the inspection of a diagram, and in which cases we need rather to supplement such inference by propositional rules. This leads us to discuss Euclid’s definition of a point, and to work out the details of the complex interaction between diagrams and text in ancient geometry. We will see, then, that Euclid may have possessed a theory of intersections that was sufficiently rigorous to dispel the main objections that could be advanced in antiquity. (shrink)

In the Anglophone world, the philosophical treatment of geometry has fallen on hard times. While in Germany as late as the 1920s there were vibrant discussions concerning the nature of geometry— especially in relation to the direction of its development, the role of intuition and the perception of physical space—the rise of logical empiricism largely brought these to a close. Accounts of mathematics in its totality as uniformly reducible to a language such as set theory have led the (...) recipients of logical empiricist doctrines to ignore the thematic contours of modern mathematics. I argue, however, that geometry in all of its many guises flourishes in contemporary mathematics, and that the term ‘geometry’ itself continues to have an important meaning. One way then for philosophy to come to a better understanding of mathematics is to provide an account of modern geometry, including its development of new forms of space, both for mathematical physics and for arithmetic. I argue that we can find a good starting point for this work by returning to the discussions of Weyl and Cassirer on geometry. With the help of modern interpreters, we see that issues salient to these thinkers are very much relevant to us today. I also propose that an effective way to encompass a great part of modern geometry is via (1, 1)-toposes, also known as homotopy toposes, an important development of category theory. Recently an internal language for such categories has been devised, known as ‘homotopy type theory’, a variant of which captures the ‘cohesiveness’ of geometric spaces. With these tools in place, we can now start to see what an adequate philosophical account of current geometry might look like. (shrink)

This chapter gives a brief account of the debate about the foundations of geometry after general relativity, with a special focus on Cassirer’s view in 1921. Cassirer emphasized that the geometrical hypotheses of general relativity differed completely from those of Newtonian mechanics and of special relativity. Therefore, in 1921, he revised his argument for the aprioricity of geometry as stated in 1910. Nevertheless, Cassirer argued for continuity across theory change with regard to the symbolic function of geometry (...) in its correlation with the empirical manifold of physical events. In this regard, he reaffirmed his conviction that neo-Kantian views were compatible with empiricist arguments such as Helmholtz’s. Not only did Helmholtz offer an argument for the generalization of the Kantian theory of space, but he influenced the view that the principles of measurement play some a priori role relative to physical theories and can be subject to revision for considerations of uniformity in the formulation of physical laws and empirical evidence. (shrink)

We suggest possible approaches to point-free geometry based on multi-valued logic. The idea is to assume as primitives the notion of a region together with suitable vague predicates whose meaning is geometrical in nature, e.g. ‘close’, ‘small’, ‘contained’. Accordingly, some first-order multi-valued theories are proposed. We show that, given a multi-valued model of one of these theories, by a suitable definition of point and distance we can construct a metrical space in a natural way. Taking into account that interesting (...) metrical approaches to geometry exist, this looks to be promising for a point-free foundation of the notion of space. We hope also that this way to face point-free geometry provides a tool to illustrate the passage from a naïve and ‘qualitative’ approach to geometry to the ‘quantitative’ approach of advanced science. (shrink)

The paper outlines an interpretation of one of the most important and original contributions of David Hilbert’s monograph Foundations of Geometry , namely his internal arithmetization of geometry. It is claimed that Hilbert’s profound interest in the problem of the introduction of numbers into geometry responded to certain epistemological aims and methodological concerns that were fundamental to his early axiomatic investigations into the foundations of elementary geometry. In particular, it is shown that a central concern that (...) motivated Hilbert’s axiomatic investigations from very early on was the aim of providing an independent basis for geometry. Accordingly, these concerns about an independent grounding for elementary geometry determined very clear methodological constraints in the process of embedding it into a formal axiomatic system. It is argued that Hilbert not only sought to show that geometry could be considered a pure mathematical theory, once it was presented as a formal axiomatic system; he also aimed at showing that in the construction of such an axiomatic system one could proceed purely geometrically, avoiding concept formations borrowed from other mathematical disciplines like arithmetic or analysis. (shrink)

After sketching the historical development of “emergence” and noting several recent problems relating to “emergent properties”, this essay proposes that properties may be either “emergent” or “mergent” and either “intrinsic” or “extrinsic”. These two distinctions define four basic types of change: stagnation, permanence, flux, and evolution. To illustrate how emergence can operate in a purely logical system, the Geometry of Logic is introduced. This new method of analyzing conceptual systems involves the mapping of logical relations onto geometrical figures, following (...) either an analytic or a synthetic pattern (or both together). Evolution is portrayed as a form of discontinuous change characterized by emergent properties that take on an intrinsic quality with respect to the object(s) or proposition(s) involved. Causal leaps, not continuous development, characterize the evolution of human life in a developing foetus, of a thought out of certain brain states, of a new idea (or insight) out of ordinary thoughts, and of a great person out of a set of historical experiences. The tendency to assume that understanding evolutionary change requires a step-by-step explanation of the historical development that led to the appearance of a certain emergent property is thereby discredited. (shrink)

A central claim of Deleuze's reading of Bergson is that Bergson's distinction between space as an extensive multiplicity and duration as an intensive multiplicity is inspired by the distinction between discrete and continuous manifolds found in Bernhard Riemann's 1854 thesis on the foundations of geometry. Yet there is no evidence from Bergson that Riemann influences his division, and the distinction between the discrete and continuous is hardly a Riemannian invention. Claiming Riemann's influence, however, allows Deleuze to argue (...) that quantity, in the form of ‘virtual number’, still pertains to continuous multiplicities. This not only supports Deleuze's attempt to redeem Bergson's argument against Einstein in Duration and Simultaneity, but also allows Deleuze to position Bergson against Hegelian dialectics. The use of Riemann is thereby an important element of the incorporation of Bergson into Deleuze's larger early project of developing an anti-Hegelian philosophy of difference. This article first reviews the role of discrete and continuous multiplicities or manifolds in Riemann's Habilitationsschrift, and how Riemann uses them to establish the foundations of an intrinsic geometry. It then outlines how Deleuze reinterprets Riemann's thesis to make it a credible resource for Deleuze's Bergsonism. Finally, it explores the limits of this move, and how Deleuze's later move away from Bergson turns on the rejection of an assumption of Riemann's thesis, that of ‘flatness in smallest parts’, which Deleuze challenges with the idea, taken from Riemann's contemporary, Richard Dedekind, of the irrational cut. (shrink)

In this paper, I reconstruct Edmund Husserl's view on the relationship between formal inquiry and the life-world, using the example of formal geometry. I first outline Husserl's account of geometry and then argue that he believed that the applicability of formal geometry to intuitive space (the space of everyday-experience) guarantees the conceptual continuity between different notions of space.

Although Hume's analysis of geometry continues to serve as a reference point for many contemporary discussions in the philosophy of science, the fact that the first Enquiry presents a radical revision of Hume's conception of geometry in the Treatise has never been explained. The present essay closely examines Hume's early and late discussions of geometry and proposes a reconstruction of the reasons behind the change in his views on the subject. Hume's early conception of geometry as (...) an inexact non-demonstrative science is argued to be a consequence of his attempt to discredit geometrical proofs of infinite divisibility of extension by anchoring the meaning of geometrical concepts in inherently inexact qualitative measurement procedures. This measurement-based attack on the exactness and certainty of geometry is analyzed and shown to be both self-refuting and inconsistent with the general epistemological framework of the Treatise. The revised conception of geometry as a demonstrative science in the first Enquiry is then interpreted as Hume's response to the failure of his earlier attempt to discredit geometrical proofs of infinite divisibility of extension. (shrink)

The Cognitive Foundations and Epistemology of Arithmetic and Geometry How is knowledge of arithmetic and geometry developed and acquired? In the tradition established by Plato and often associated with Kant, the epistemology of mathematics has been focused on a priori approaches, which take mathematical knowledge and its study to be essentially independent of sensory experience. … Continue reading The Cognitive Foundations and Epistemology of Arithmetic and Geometry →.

This paper discusses the development of integrated design-to-production frameworks for Robotic Concrete 3D Printing (RC3DP) of context-specific urban furniture projects. The study focuses on two main objectives: developing computational methods for continuous toolpath planning of bespoke components and examining integrated frameworks to make design-to-production systems more socio-environmentally inclusive and tailored to specific contexts. Following an introduction to outline the key challenges of continuous robotic concrete 3D printing, the paper is organized into two sets of case studies. The first (...) set explores curved-based continuous 3D printing for surface-based design strategies, and the second set investigates voxel-based approaches for volumetric design 3D printing. Curve-based projects consisted of three case studies. The first case uses an AI-enabled generative system to translate a porous, cellular structure to a continuous toolpath with variable heights. The second case study is based on a workflow using a controlled reaction-diffusion algorithm as a generative strategy to create continuous infill for complex geometries. The third project focuses on growing larger than the size of the production setup by testing the assembly of multiple printed components, all produced with continuous toolpaths. The workflows and projects are developed as an integral part of a graduate-level digital design and fabrication course focusing on inclusive automation and robotic concrete 3D printing. Therefore, in addition to sharing the details of the developed computational design to production methodologies, the paper discusses the research findings from pedagogical lenses. Building on these cases, the investigation advances to the development of a generative system centered on voxels, employing a tetrahedron structure. This volumetric approach simultaneously produces external surface tectonics and internal infill through a seamless toolpath. The paper concludes by presenting sets of guidelines and future directions for both the introduced curve and voxel-based approaches in Robotic Concrete 3D printing, emphasizing the early integration of computational and fabrication intelligence into the design process. (shrink)

When national borders in the modern sense first began to be established in early modern Europe, non-contiguous and perforated nations were a commonplace. According to the conception of the shapes of nations that is currently preferred, however, nations must conform to the topological model of circularity; their borders must guarantee contiguity and simple connectedness, and such borders must as far as possible conform to existing topographical features on the ground. The striving to conform to this model can be seen at (...) work today in Quebec and in Ireland, it underpins much of the rhetoric of the P.L.O., and was certainly to some degree involved as a motivating factor in much of the ethnic cleansing which took place in Bosnia in recent times. The question to be addressed in what follows is: to what extent could inter-group disputes be more peacefully resolved, and ethnic cleansing avoided, if political leaders, diplomats and others involved in the resolution of such disputes could be brought to accept weaker geometrical constraints on the shapes of nations? A number of associated questions then present themselves: What sorts of administrative and logistical problems have been encountered by existing non contiguous nations and by perforated nations, and by other nations deviating in different ways from the received geometrical ideal? To what degree is the desire for continuity and simple connectedness a rational desire, and to what degree does it rest on species of political rhetoric which might be countered by, for example, philosophical argument? These and a series of related questions will form the subject- matter of the present essay. (shrink)

This chapter continues the treatment of the relation between phenomenology and mathematics during the twenties began in Chapter 10. The investigation focuses on Hermann Weyl and Oskar Becker and their thoughts on the relation between phenomenology and the foundations of geometry and physics. After a description of the four letters from Becker to Weyl treating such matters, the chapter investigates in detail the related mathematical and phenomenological context of Weyl’s work on geometry and relativity theory that inspired Becker’s (...) work on the phenomenological foundation of physics and geometry. (shrink)

Bernard Bolzano (1781-1848) was a philosophical mathematician, especially interested in foundations and the analysis of important mathematical concepts. The notion of continuity was a subject of sustained reflection throughout his life. He deals with the notion in many settings: the theory of space (geometry), the theory of time (chronometry), the theory of functions (analysis), physics (continuous processes, matter), and numerical continuity (the theory of measurable numbers). One can also distinguish earlier and later versions of most of his work (...) in these areas. This chapter provides an overview of his thoughts on these matters, along with some indications of the historical and philosophical context of his work. (shrink)

Between 1873 and 1876, Felix Klein published a series of papers that he later placed under the rubric anschauliche Geometrie in the second volume of his collected works (1922). The present study attempts not only to follow the course of this work, but also to place it in a larger historical context. Methodologically, Klein’s approach had roots in Poncelet’s principle of continuity, though the more immediate influences on him came from his teachers, Plücker and Clebsch. In the 1860s, Clebsch reworked (...) some of the central ideas in Riemann’s theory of Abelian functions to obtain complicated results for systems of algebraic curves, most published earlier by Hesse and Steiner. These findings played a major role in enumerative geometry, whereas Plücker’s work had a strongly qualitative character that imbued Klein’s early studies. A leitmotif in these works can be seen in the interplay between real curves and surfaces as reflected by their transformational properties. During the early 1870s, Klein and Zeuthen began to explore the possibility of deriving all possible forms for real cubic surfaces as well as quartic curves. They did so using continuity methods reminiscent of Poncelet’s earlier approach. Both authors also relied on visual arguments, which Klein would later advance under the banner of intuitive geometry (anschauliche Geometrie). (shrink)

The present essay describes Leibniz’s foundational studies on continuity in geometry. In particular, the paper addresses the long-debated problem of grounding a theory of intersections in elementary geometry. In the early modern age, in fact, several mathematicians had claimed that Euclid’s Elements needed to be complemented with additional axioms in order to ground the existence of the intersection points between straight lines and circles. Leibniz was sensible to similar foundational issues in the Euclidean tradition, and dedicated several studies (...) to investigate a good definition of the continuity of space, in order to ground a general theory of intersections of curves and surfaces. While Leibniz’s researches on continuity in relation with the Calculus have been extensively studied in the past, the present essay deals with the less-known Leibnizian notion of a continuous space, as it is to be found in several unpublished writings preserved in Hannover. The subject widens as to encompass the relation between mereology and analysis situs, Leibniz’s studies on a geometrical characteristics, and Leibniz’s theory of space at large. (shrink)

I investigate the role of geometric intuition in Frege’s early mathematical works and the significance of his view of the role of intuition in geometry to properly understanding the aims of his logicist project. I critically evaluate the interpretations of Mark Wilson, Jamie Tappenden, and Michael Dummett. The final analysis that I provide clarifies the relationship of Frege’s restricted logicist project to dominant trends in German mathematical research, in particular to Weierstrassian arithmetization and to the Riemannian conceptual/geometrical tradition at (...) Göttingen. Concurring with Tappenden, I hold that Frege’s logicism should not be understood as a continuing a project of reductionist arithmetization. However, Frege does not quite take up the Riemannian banner either. His logicism supports a hierarchical understanding of the structure of mathematical knowledge, according to which arithmetic is applicable to geometry but not vice versa because the former is more general, as revealed by the strictly logical nature of its objects in comparison to the intuitional nature of geometric objects. I suggest, in particular, that Frege intended that foundational work would show the use of geometric intuition in complex analysis, a source of error for Riemann that Weierstrass was proud to have uncovered, to be inessential. (shrink)