The purported fact that geometric theories formulated in terms of points and geometric theories formulated in terms of lines are “equally correct” is often invoked in arguments for conceptual relativity, in particular by Putnam and Goodman. We discuss a few notions of equivalence between first-order theories, and we then demonstrate a precise sense in which this purported fact is true. We argue, however, that this fact does not undermine metaphysical realism.

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)

The preferred basis problem has resisted resolution because it is doubly ill-posed. First, it seeks an ontological solution to a structural problem: every proposed solution encounters the same difficulty, indicating that its source is architectural rather than interpretive. Second, it demands an answer in the language of orthonormal bases, presupposing that classical alternatives must correspond to orthogonal decompositions of a state space. They need not. Classical worlds require only logical incompatibility of stabilized distinctions, not linear-algebraic orthogonality. This paper develops refinement (...) geometry, a theory-neutral structural framework that reframes rather than solves the preferred basis problem by identifying its root cause. Physical histories are partially ordered by prefix extension. Stability predicates are existential claims over history prefixes, necessarily monotone since what has occurred cannot be uncertified by later extension, and admissible only if their truth sets are open under finite variation. Monotonicity and admissibility together induce a directed refinement structure admitting a canonical maximal partition. The root cause is then identifiable: purely synchronic refinement cannot reproduce maximal refinement structure because admissible stability predicates are defined over history prefixes and carry monotone certification semantics that no instantaneous state description can fully encode. A synchronic procedure powerful enough to recover the canonical maximal partition must implicitly reconstruct and traverse the prefix order, collapsing into diachronic refinement in disguise rather than achieving a genuinely synchronic result. In contrast, diachronic refinement, accumulating certified distinctions monotonically along histories, converges to the canonical maximal partition without basis selection or orthogonality. Significantly, quantum interference requires no special treatment: non-singleton partition blocks represent histories not yet separated by any certified distinction. Decoherence generates the stability predicates that resolve such blocks; classical worlds are the limit of that resolution. Persistence of distinctions, irreversibility, and absence of merging follow from predicate semantics alone, independent of dynamical law and without presupposing Hilbert-space structure or ontological commitments. (shrink)

Geometry, etymologically the “science of measuring the Earth”, is a mathematical formalization of space. Just as formal concepts of number may be rooted in an evolutionary ancient system for perceiving numerical quantity, the fathers of geometry may have been inspired by their perception of space. Is the spatial content of formal Euclidean geometry universally present in the way humans perceive space, or is Euclidean geometry a mental construction, specific to those who have received appropriate instruction? The (...) spatial content of the formal theories of geometry may depart from spatial perception for two reasons: first, because in geometry, only some of the features of spatial figures are theoretically relevant; and second, because some geometric concepts go beyond any possible perceptual experience. Focusing in turn on these two aspects of geometry, we will present several lines of research on US adults and children from the age of three years, and participants from an Amazonian culture, the Mundurucu. Almost all the aspects of geometry tested proved to be shared between these two cultures. Nevertheless, some aspects involve a process of mental construction where explicit instruction seem to play a role in the US, but that can still take place in the absence of instruction in geometry. (shrink)

I present an argument against a relational theory of spacetime that regards spacetime as a ‘structural quality of the field’. The argument takes the form of a trilemma. To make the argument, I focus on relativistic worlds in which there exist just two fields, an electromagnetic field and a gravitational field. Then there are three options: either spacetime is a structural quality of each field separately, both fields together, or one field but not the other. I argue that the first (...) option founders on a problem of geometric coordination and that the second and third options collapse into substantivalism. In particular, on the third option it becomes clear that the relationalist’s path to Leibniz equivalence is no simpler or more straightforward than the substantivalist’s. (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)

Does geometry constitues a core set of intuitions present in all humans, regarless of their language or schooling? We used two non verbal tests to probe the conceptual primitives of geometry in the Munduruku, an isolated Amazonian indigene group. Our results provide evidence for geometrical intuitions in the absence of schooling, experience with graphic symbols or maps, or a rich language of geometrical terms.

Weyl proposed a geometry that differed from Riemannian geometry, which underlies general relativity, in that it contained a vector that could be interpreted as describing the electromagnetic field. Dirac modified this geometry to remove certain difficulties and based it on a variational principle which gave satisfactory field equations for gravitation and electromagnetism. However, by changing the value of a parameter appearing in his variational principle one gets, instead of electromagnetism, a field of massive particles of spin 1, (...) which can be assumed to interact with ordinary matter only through gravitation. It is suggested that these bosons, called weylons, provide most of the dark matter in the universe. (shrink)

This book reconstructs, both from the historical and theoretical points of view, Leibniz's geometrical studies, focusing in particular on the research Leibniz...

Geometry and Chronometry in Philosophical Perspective was first published in 1968. Minnesota Archive Editions uses digital technology to make long-unavailable books once again accessible, and are published unaltered from the original University of Minnesota Press editions. In this volume Professor Grünbaum substantially extends and comments upon his essay "Geometry, Chronometry, and Empiricism," which was first published in Volume III of the Minnesota Studies in the Philosophy of Science. Commenting on the essay when it first appeared J. J. C. (...) Smart wrote in Mind (England): "In my opinion Adolf Grünbaum's paper ... supersedes nearly all that has been written on the logical status of physical geometry and chronometry." The full text of the essay is given here with the author's extension of it and his discussion of some of the critical comment it has evoked, particularly, a critique published by Hilary Putnam. Adolph Grünbaum is Andrew Mellon Professor of Philosophy at the University of Pittsburgh and the current president of the Philosophy of Science Association. (shrink)

The first edition of Newton’s Principia opens with a “Praefatio ad Lectorem.” The first lines of this Preface have received scant attention from historians, even though they contain the very first words addressed to the reader of one of the greatest classics of science. Instead, it is the second half of the Preface that historians have often referred to in connection with their treatments of Newton’s scientific methodology. Roughly in the middle of the Preface, Newton defines the purpose of philosophy (...) as a twofold task: to investigate the forces of the phenomena of nature and, once having established the forces, to demonstrate the remaining phenomena. Newton then introduces a distinction between the first two books, which deal with general propositions, and the third, where the propositions are applied to particular instances of celestial phenomena. From these phenomena, Newton claims, the force of gravity, thanks to which bodies tend toward the Sun, is derived. By assuming this force in mathematical propositions, other motions are deduced: the motions of the planets, the comets, the moon, and the sea. Newton then declares his hope that phenomena relative to small particles will also be explained, as per the celestial ones, thanks to the understanding of attractive forces hitherto unknown to philosophers. The Preface ends with a well-deserved laudatio of Edmond Halley and an apologetic passage concerning certain imperfections in the presentation of advanced subjects, such as the motions of the moon. It is these lines to which historians have given most attention in their various studies. (shrink)

Finsler geometry on the tangent bundle appears to be applicable to relativistic field theory, particularly, unified field theories. The physical motivation for Finsler structure is conveniently developed by the use of “gauge” transformations on the tangent space. In this context a remarkable correspondence of metrics, connections, and curvatures to, respectively, gauge potentials, fields, and energy-momentum emerges. Specific relativistic electromagnetic metrics such as Randers, Beil, and Weyl can be compared.

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)

Richard Jeffrey and Michael Goldstein have both introduced systematic approaches to the structure of opinion changes. For both approaches there are theorems which indicate great generality and width of scope. The main questions addressed here will be to what extent the basic forms of representation are intertranslatable, and how we can conceive of such programs in general.

The Ethics of Geometry is a study of the relationship between philosophy and mathematics. Essential differences in the ethos of mathematics, for example, the customary ways of undertaking and understanding mathematical procedures and their objects, provide insight into the fundamental issues in the quarrel of moderns with ancients. Two signal features of the modern ethos are the priority of problem-solving over theorem-proving, and the claim that constructability by human minds or instruments establishes the existence of relevant entities. These figures (...) are combined in the emblematic statement of Salomon Maimon, "In mathematical construction we are, as it were, gods." Construction is the mark of modernity. The disciplines of classical philology, literary interpretation and the history of philosophy and of mathematics are woven together in this volume. (shrink)

This thesis is directed towards a philosophy of music by attention to conceptions and perceptions of space. I focus on melody and harmony, and do not emphasise rhythm, which, as far as I can tell, concerns time rather than space. I seek a metaphysical account of Western Classical music in the diatonic tradition. More specifically, my interest is in wordless, untitled music, often called 'absolute' music. My aim is to elucidate a spatial approach to the world combined with a curiosity (...) about the nature of the Pythagorean intervals. Thus one question I seek to answer is: What do the Pythagorean intervals import to music? In their original formulation by Pythagoras, and further, by Plato, their import seemed mystical and analogous to a 'harmony of the spheres.' In this spatial approach, the Pythagorean intervals are indicative of infinite depth. The faculty of hearing alone does not ordinarily spatially locate sources. Hearing is ordinarily combined with sight in order to spatially locate a sound. Yet we talk about music as if it were spatially located, that is, as if it were a visible object with a surface and themes or 'objects'. I explore the faculty of vision and consider in what ways the processes of vision might be similar to and distinct from audition. My source for this approach is David Marr's book 'Vision' and his theory of the 2 1/2-dimensional sketch. In a medium in which there is no spatial location, it is important to situate the listener. Newton solved the problem of location by demonstrating the effects of gravity. P. F. Strawson claims that an analogy of distance is required in hearing to situate the listener, that is, a sense of nearer to and further from a source or object that permits a perception of distance. I claim that in music, the key signature and the scalar relations of musical themes provide this analogy of distance. The works of Plato, Aristotle, Descartes, Leibniz, and Newton are studied for their accounts of motion, bodies, and space. From these metaphysicians can be gleaned elements of music that include form, motion, timbre, dynamics, and attraction. By incorporating motion, I aim to show that although we may not know the true nature of space, we can learn from hearing music that space is perceptible in other than three dimensions. (shrink)

Those familiar with the Critique of Pure Reason will not at all be surprised that Thomas C. Vinci has found it fitting to dedicate an entire book to the Transcendental Deduction of the Categories, a chapter of the CPR that is as important to Kant’s argument for Transcendental Idealism as it is difficult to decipher. The purpose of that section is to establish the objective validity of the categories—to show, that is, that the pure concepts of the understanding apply to (...) all objects of human experience. While the general goal of the TD may be easy enough to state, the argument strategy that Kant uses to establish the objective validity of the categories is hardly easy to understand.Vinci focuses on the.. (shrink)

Beyond Einstein: Perspectives on Geometry, Gravitation, and Cosmology explores the rich interplay between mathematical and physical ideas by studying the interactions of major actors and the roles of important research communities over the course of the last century.

The aim of our study is to highlight the obvious similarities that exist between the organizational structures of the biological world, particularly in terms of the number and distribution of the petals on flower and the geometric configurations used by the great masters of European painting, both in the East but also in the West, in order to elaborate the compositional framework of paintings and icons. Taking into consideration the symbolic connotations concerning the field of biology (the primary focus this (...) article), we chose as a case study the Russian icon from the 14th century entitled “The Triumphal Entry of Jesus into Jerusalem”, exhibited and venerated across the orthodox world on Palm Sunday. Starting with the circle of the aureole of Jesus, we have marked the harmonic series of concentric circles that are distributed across the nodal points of some regular polygons with seven and nine sides, which make reference to the ergonomic formulas of distribution of the petals on a flower and highlight the fact that the beauty of the natural world was and still is the main source of inspiration in the world of arts. (shrink)