All physical theories, from classical Newtonian mechanics to relativistic quantum field theory, entail propositions concerning the geometric structure of spacetime. To give an example, the general theory of relativity entails that spacetime is curved, smooth, and four-dimensional. In this dissertation, I take the structural commitments of our theories seriously and ask: how is such structure instantiated in the physical world? Mathematically, a property like 'being curved' is perfectly well-defined insofar as we know what it means for a mathematical (...) space to be curved. But what could it mean to say that the physical world is curved? Call this the problem of physical geometry. The problem of physical geometry is a plea for foundations--a request for fundamental truth conditions for physical-geometric propositions. My chief claim is that only a substantival theory of spacetime--a theory according to which spacetime is an entity in its own right, existing over and above the material content of the world--can supply the necessary truth conditions. (shrink)

I explore some ways in which one might base an account of the fundamental metaphysics of geometry on the mathematical theory of Linear Structures recently developed by Tim Maudlin (2010). Having considered some of the challenges facing this approach, Idevelop an alternative approach, according to which the fundamental ontology includes concrete entities structurally isomorphic to functions from space-time points to real numbers.

The relationship between physics and geometry is examined in classical and quantum physics based on the view that the symmetry group of physics and the automorphism group of the geometry are the same. Examination of quantum phenomena reveals that the space-time manifold is not appropriate for quantum theory. A different conception of geometry for quantum theory on the group manifold, which may be an arbitrary Lie group, is proposed. This provides a unified description of gravity and gauge (...) fields as well as generalizations of these fields. A correspondence principle which relates the geometry of quantum physics and the geometry of classical physics is formulated. (shrink)

Discussions of the metaphysical status of spacetime assume that a spacetime theory offers a causal explanation of phenomena of relative motion, and that the fundamental philosophical question is whether the inference to that explanation is warranted. I argue that those assumptions are mistaken, because they ignore the essential character of spacetime theory as a kind of physical geometry. As such, a spacetime theory does notcausally explain phenomena of motion, but uses them to construct physicaldefinitions of basic geometrical structures (...) by coordinating them with dynamical laws. I suggest that this view of spacetime theories leads to a clearer view of the philosophical foundations of general relativity and its place in the historical evolution of spacetime theory. I also argue that this view provides a much clearer and more defensible account of what is entailed by realism concerning spacetime. (shrink)

Tim Maudlin sets out a completely new method for describing the geometrical structure of spaces, and thus a better mathematical tool for describing and understanding space-time. He presents a historical review of the development of geometry and topology, and then his original Theory of Linear Structures.

The standard mathematical account of the sub-metrical geometry of a space employs topology, whose foundational concept is the open set. This proves to be an unhappy choice for discrete spaces, and offers no insight into the physical origin of geometrical structure. I outline an alternative, the Theory of Linear Structures, whose foundational concept is the line. Application to Relativistic space-time reveals that the whole geometry of space-time derives from temporal structure. In this sense, instead of spatializing time, (...) Relativity temporalizes space. (shrink)

The aim of this paper is to discuss the relation between the observation basis and the theoretical principles of General Relativity. More specifically, this relation is analyzed with respect to constructive axiomatizations of the observation basis of space-time theories, on the one hand, and in attempts to complete them, on the other. The two approaches exclude one another so that a choice between them is necessary. I argue that the completeness approach is preferable for methodological reasons.

Putnam’s proof that time flow is incompatible with Relativity is underestimated, mostly due to Stein’s interpretation of the notion of reality in it as a two-term relation. This interpretation makes it vulnerable to easy criticism and makes various ways of escaping its conclusion possible. An alternative approach is proposed, resulting in a formalization which seems closer to Putnam’s intentions where reality is interpreted as a non-relational property. Although it makes the proof immune to all standard strategies of blocking the proof, (...) it reveals its real weak point which consists in assuming an overly strong interpretation of the principle of relativity. (shrink)

ArgumentHermann von Helmholtz’s distinction between “pure intuitive” and “physical” geometry must be counted as the most influential of his many contributions to the philosophy of science. In a series of papers from the 1860s and 70s, Helmholtz argued against Kant’s claim that our knowledge of Euclidean geometry was an a priori condition for empirical knowledge. He claimed that geometrical propositions could be meaningful only if they were taken to concern the behaviors of physical bodies used in (...) measurement, from which it followed that it was posterior to our acquaintance with this behavior. This paper argues that Helmholtz’s understanding of geometry was fundamentally shaped by his work in sense-physiology, above all on the continuum of colors. For in the course of that research, Helmholtz was forced to realize that the color-space had no inherent metrical structure. The latter was a product of axiomatic definitions of color-addition and the empirical results of such additions. Helmholtz’s development of these views is explained with detailed reference to the competing work of the mathematician Hermann Grassmann and that of the young James Clerk Maxwell. It is this separation between 1) essential properties of a continuum, 2) supplementary axioms concerning distance-measurement, and 3) the behaviors of the physical apparatus used to realize the axioms, which is definitive of Helmholtz’s arguments concerning geometry. (shrink)

Earlier in this century, many philosophers of science (for example, Rudolf Carnap) drew a fairly sharp distinction between theory and observation, between theoretical terms like 'mass' and 'electron', and observation terms like 'measures three meters in length' and 'is _2° Celsius'. By simply looking at our instruments we can ascertain what numbers our measurements yield. Creatures like mass are different: we determine mass by calculation; we never directly observe a mass. Nor an electron: this term is introduced in order to (...) explain what we observe. This (once standard) distinction between theory and observation was eventually found to be wanting. First, if the distinction holds, it is difficult to see what can characterize the relationship between theory :md observation. How can theoretical terms explain that which is itself in no way theorized? The second point leads out of the first: are not the instruments that provide us with observational material themselves creatures of theory? Is it really possible to have an observation language that is entirely barren of theory? The theory-Iadenness of observation languages is now an accept ed feature of the logic of science. Many regard such dependence of observation on theory as a virtue. If our instruments of observation do not derive their meaning from theories, whence comes that meaning? Surely - in science - we have nothing else but theories to tell us what to try to observe. (shrink)

Recently the interest in Bohm realist interpretation of quantum mechanics has grown. The important advantage of this approach lies in the possibility to introduce non-locality ab initio, and not as an "unexpected host". In this book the authors give a detailed analysis of quantum potential, the non-locality term and its role in quantum cosmology and information. The different approaches to the quantum potential are analysed, starting from the original attempt to introduce a realism of particles trajectories (influenced by de Broglie's (...) pilot wave) to the recent dynamic interpretation provided by Goldstein, Durr, Tumulka and Zanghì, and the geometrodynamic picture, with suggestion about quantum gravity. Finally we focus on the algebraic reading of Hiley and Birkbeck school, that analyse the meaning of the non-local structure of the world, bringing important consequences for the space, time and information concepts. (shrink)

From the desire to find support and confirmation for our personal sensory observations, and from the human interest in sharing our experiences with others, there emerges a basic principle of scientific method: We demand the possibility of intelligible communication and agreement concerning individuals' sensory perceptions in particular and their experiences in general. This requirement is made both for the natural and social sciences. The raw material offered for logical organization must be capable of exhibiting an inter-subjective character—such material, or protocols, (...) must be acceptable in common to all concerned. And it must be acceptable for logical organization or systematization in some science. (shrink)

From the desire to find support and confirmation for our personal sensory observations, and from the human interest in sharing our experiences with others, there emerges a basic principle of scientific method: We demand the possibility of intelligible communication and agreement concerning individuals' sensory perceptions in particular and their experiences in general. This requirement is made both for the natural and social sciences. The raw material offered for logical organization must be capable of exhibiting an inter-subjective character—such material, or protocols, (...) must be acceptable in common to all concerned. And it must be acceptable for logical organization or systematization in some science. (shrink)

Brown and Pooley’s ‘dynamical approach’ to physical theories asserts, in opposition to the orthodox position on physical geometry, that facts about physical geometry are grounded in, or explained by, facts about dynamical fields, not the other way round. John Norton has claimed that the proponent of the dynamical approach is illicitly committed to spatiotemporal presumptions in ‘constructing’ space-time from facts about dynamical symmetries. In this article, I present an abstract, algebraic formulation of field theories and (...) demonstrate that the proponent of the dynamical approach is not committed, in special relativity, to the illicit presumptions to which Norton refers. (shrink)

This paper discusses Husserl’s views on physical theories in the first volume of his Logical Investigations, and compares them with those of his contemporaries Pierre Duhem and Henri Poincaré. Poincaré’s views serve as a bridge to a discussion of Husserl’s almost unknown views on physical geometry from about 1890 on, which in comparison even with Poincaré’s—not to say Frege’s—or almost any other philosopher of his time, represented a rupture with the philosophical tradition and were much more in (...) tune with the physical geometry underlying the Einstein-Hilbert general theory of relativity developed more than two decades later. (shrink)

This unique book provides a self-contained conceptual and technical introduction to the theory of differential sheaves. This serves both the newcomer and the experienced researcher in undertaking a background-independent, natural and relational approach to "physical geometry". In this manner, this book is situated at the crossroads between the foundations of mathematical analysis with a view toward differential geometry and the foundations of theoretical physics with a view toward quantum mechanics and quantum gravity. The unifying thread is provided (...) by the theory of adjoint functors in category theory and the elucidation of the concepts of sheaf theory and homological algebra in relation to the description and analysis of dynamically constituted physical geometric spectrums. (shrink)

Previous discussions of Robb’s work on space and time have offered a philosophical focus on causal interpretations of relativity theory or a historical focus on his use of non-Euclidean geometry, or else ignored altogether in discussions of relativity at Cambridge. In this paper I focus on how Robb’s work made contact with those same foundational developments in mathematics and with their applications. This contact with applications of new mathematical logic at Göttingen and Cambridge explains the transition from his electron (...) research to his treatment of relativity in 1911 and finally to the axiomatic presentation in 1914 in terms of postulates. At the heart of Robb’s physical optics was the model of the light cone. The model underwent a transition from a working mechanical model in the Maxwellian Cambridge sense of a pedagogical and research tool to the semantic model, in the logical, model-theoretic sense. Robb tracked this transition from the 19th- to the 20th-century conception with the earliest use of the term ‘model’ in the new sense. I place his cone models in a genealogy of similar models and use their evolution to track how Robb’s physical researches were informed by his interest in geometry, logic and the foundations of mathematics. (shrink)

This paper discusses Husserl’s views on physical theories in the first volume of his Logical Investigations , and compares them with those of his contemporaries Pierre Duhem and Henri Poincaré. Poincaré’s views serve as a bridge to a discussion of Husserl’s almost unknown views on physical geometry from about 1890 on, which in comparison even with Poincaré’s—not to say Frege’s—or almost any other philosopher of his time, represented a rupture with the philosophical tradition and were much more (...) in tune with the physical geometry underlying the Einstein-Hilbert general theory of relativity developed more than two decades later. (shrink)

The standard coordinate-based formulation of the space-time theory of special relativity (Minkowski geometry) is philosophically unsatisfactory for various reasons. We here present an explicit axiomatic formulation of that theory in terms of primitives with a definitive physical interpretation, prove its equivalence to the standard coordinate formulation, and draw various philosophical conclusions concerning the physical content and assumptions of the space-time theory. The prevalent causal interpretation of physical Minkowski geometry deriving from Reichenbach is criticised on the (...) basis of the present formulation. (shrink)

InCelebratingthisDIUMessayHistoriographywe describeIEMNTorricelli’s lifeLille Universityand worksUdine University in their scientific context, including the ArchimedeanArchimedean heritage in Torricelli’s works. We analyse the changes of Torricelli’s works and explain the novelties of the edition we are offering. Then, we provide a picture of the most significant results obtained by Torricelli (1608–1647), particularly in mechanicsMechanicsand geometryGeometry. Furthermore, we also focus on the Torricelli’s methodology, specifying how he provedProved two of his achievements, given their novelty and mathematical meaning: (a) the volumeVolume of the “solido acutissimo”; (...) (b) the geometryGeometryof theLogarithmic Spirallogarithmic spiralGeometric Spiral. AnContent ad hoc ad hocAd hocContentContent of the Opera geometricaOpera geometrica is proposed as well. It is based on the original chronological sequences of the contentsContent wrote by Torricelli. This is important to evaluate the birth of the scientific matters presented by Torricelli and the editorial process of the publicationPublication. (shrink)

Differential geometry, the contemporary heir of the infinitesimal calculus of the 17th century, appears today as the most appropriate language for the description of physical reality. This holds at every level: The concept of “connexion,” for instance, is used in the construction of models of the universe as well as in the description of the interior of the proton. Nothing is apparently more contrary to the wisdom of physicists; all the same, “it works.” The pages that follow show (...) the conceptual role played by this geometry in some examples—without entering into technical details. In order to achieve this, we shall often have to abandon the complete mathematical rigor and even full definitions; however, we shall be able to give a precise description of the connection of ideas thanks to some elements of group theory. (shrink)

Using as a starting point recent and apparently incompatible conclusions by Saunders and Knox, I revisit the question of the correct spacetime setting for Newtonian physics. I argue that understood correctly, these two versions of Newtonian physics make the same claims both about the background geometry required to define the theory, and about the inertial structure of the theory. In doing so I illustrate and explore in detail the view—espoused by Knox, and also by Brown —that inertial structure is (...) defined by the dynamics governing subsystems of a larger system. This clarifies some interesting features of Newtonian physics, notably the distinction between using the theory to model subsystems of a larger whole and using it to model complete universes, and the scale-relativity of spacetime structure. _1_ Introduction _2_ Newtonian Mechanics and Galilean Spacetime _3_ Vector Relationism and Maxwellian Spacetime _4_ Recovering the Galilei Group: Dynamics of Subsystems _5_ Knox on Inertial Structure _6_ Connections on Maxwellian Spacetime _7_ Knox on Newtonian Gravity _8_ Vector Relationism and Newton–Cartan Theory _9_ Inertial Structure in Newton–Cartan Gravity _10_ Reconciling Knox and Saunders _11_ Conclusions. (shrink)

In this review article we discuss some of the applications of noncommutative geometry in physics that are of recent interest, such as noncommutative many-body systems, noncommutative extension of Special Theory of Relativity kinematics, twisted gauge theories and noncommutative gravity.

The Cartan equations defining simple spinors (renamed “pure” by C. Chevalley) are interpreted as equations of motion in compact momentum spaces, in a constructive approach in which at each step the dimensions of spinor space are doubled while those of momentum space increased by two. The construction is possible only in the frame of the geometry of simple or pure spinors, which imposes contraint equations on spinors with more than four components, and then momentum spaces result compact, isomorphic to (...) invariant-mass-spheres imbedded in each other, since the signatures result steadily Lorentzian; starting from dimension four (Minkowski) up to dimension ten with Clifford algebra ℂℓ(1, 9), where the construction naturally ends. The equations of motion met in the construction are most of those traditionally postulated ad hoc: from Weyl equations for neutrinos (and Maxwell's) to Majorana ones, to those for the electroweak model and for the nucleons interacting with the pseudoscalar pion, up to those for the 3 baryon-lepton families, steadily progressing from the description of lower energy phenomena to that of higher ones. The 3 division algebras: complex numbers, quaternions and octonions appear to be strictly correlated with Clifford algebras and then with this spinor-geometrical approach, from which they appear to gradually emerge in the construction, where they play a basic role for the physical interpretation: at the third step complex numbers generate U(1), possible origin of the electric charge and of the existence of charged—neutral fermion pairs, explaining also easily the opposite charges of proton-electron. Another U(1) appears to generate the strong charge at the fourth step. Quaternions generate the signature of space-time at the first step, the SU(2) internal symmetry of isospin and, in the gauge term, the SU(2) L one, of the electroweak model at the third step; they are also at the origin of 3 families; in number equal to that of quaternion imaginary units. At the fifth and last step octonions generate the SU(3) internal symmetry of flavour, with SU(2) isospin subgroup and, in the gauge term, the one of color, correlated with SU(2) L of the electroweak model. These 3 division algebras seem then to be at the origin of charges, families and of the groups of the Standard model. In this approach there seems to be no need of higher dimensional (>4) space-time, here generated by the four Poincaré translations, and dimensional reduction from ℂℓ(1,9) to ℂℓ(1,3) is equivalent to decoupling of the equations of motion. This spinor-geometrical approach is compatible with that based on strings, since these may be expressed bilinearly (as integrals) in terms of Majorana–Weyl simple or pure spinors which are admitted by ℂℓ(1, 9) = R(32). (shrink)

Du Châtelet holds that mathematical representations play an explanatory role in natural science. Moreover, she writes that things proceed in nature as they do in geometry. How should we square these assertions with Du Châtelet’s idealism about mathematical objects, on which they are ‘fictions’ dependent on acts of abstraction? The question is especially pressing because some of her important interlocutors (Wolff, Maupertuis, and Voltaire) denied that mathematics informs us about the properties of material things. After situating Du Châtelet in (...) this debate, this chapter argues, first, that her account of the origins of mathematical objects is less subjectivist than it might seem. Mathematical objects are non-arbitrary, public entities. While mathematical objects are partly mind-dependent, so are material things. Mathematical objects can approximate the material. Second, it is argued that this moderate metaphysical position underlies Du Châtelet’s persistent claims that mathematics is required for certain kinds of general knowledge, including in natural science. The chapter concludes with an illustrative example: an analysis of Du Châtelet’s argument that matter is continuous. A key premise in the argument is that mathematical representations and material nature correspond. (shrink)

It is proposed to remove the difficulty of nonitegrability of length in the Weyl geometry by modifying the law of parallel displacement and using “standard” vectors. The field equations are derived from a variational principle slightly different from that of Dirac and involving a parameter σ. For σ=0 one has the electromagnetic field. For σ<0 there is a vector meson field. This could be the electromagnetic field with finite-mass photons, or it could be a meson field providing the “missing (...) mass” of the universe. In cosmological models the two natural gauges are the Einstein gauge and the cosmic gauge. With the latter the universe has a fixed size, but the sizes of small systems decrease with time and their masses and energies increase, thus producing the Hubble effect. The field of a particle in this gauge is investigated, and it leads to an interesting solution of the Einstein equations that raises a question about the Schwarzschild solution. (shrink)

Il est tout à fait possible de soutenir que l’espace de Newton manque d’objectivité physique (ce qui est un corollaire de la théorie einsténienne) et néanmoins prendre tout à fait au sérieux la thèse de l’espace euclidien comme condition de possibilité de l’expérience. Condition de possibilité de l’émergence d’un sujet qui configure son monde en remettant tout point de son environnement à une métrique. Cette métrique ne serait autre que celle qui donne sens à la géométrie que l’on a appris (...) à l´école, c’est à dire, la métrique euclidienne. Tout en manquant de réalité physique, tout en n`étant pas fertile à l’heure de rendre compte de la forme de l’univers, la métrique euclidienne serait la seule modalité sous laquelle il y aurait pour l’homme une prise sur le monde. Qu’on l’appelle ou pas transcendantale, la reconnaissance de la métrique euclidienne comme condition de possibilité de l’expérience nous rapprocherait de Kant sans pour autant répudier Einstein ou Poincaré. (shrink)

The 24 components of the relativistic spin tensor consist of 3 + 3 basic spin fields and 9 + 9 constitutive fields. Empirically only three basic spin fields and nine constitutive fields are known. This empirem can be expressed by two spin axioms, one of them denying purely relativistic spin fields, and the other one relating the three additional basic fields and the nine additional constitutive fields to the known (and measurable) ones. This identification by the spin axioms is material-independent (...) and does not mix basic spin fields with constitutive properties. The approaches to the Weyssenhoff fluid and the Dirac-electron fluid found in literature are discussed with regard to these spin axioms. The conjecture is formulated, that another reduction from six to three basic spin fields which does not obey the spin axioms introduces special material properties by not allowed mixing of constitutive and basic fields. (shrink)

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)

This book offers a new interpretation of Hermann von Helmholtz's work on the epistemology of geometry. A detailed analysis of the philosophical arguments of Helmholtz's Erhaltung der Kraft shows that he took physical theories to be constrained by a regulative ideal. They must render nature "completely comprehensible", which implies that all physical magnitudes must be relations among empirically given phenomena. This conviction eventually forced Helmholtz to explain how geometry itself could be so construed. Hyder shows how (...) Helmholtz answered this question by drawing on the theory of magnitudes developed in his research on the colour-space. He argues against the dominant interpretation of Helmholtz's work by suggesting that for the latter, it is less the inductive character of geometry that makes it empirical, and rather the regulative requirement that the system of natural science be empirically closed. (shrink)

This high-level study discusses Newtonian principles and 19th-century views on electrodynamics and the aether. Additional topics include Einstein's electrodynamics of moving bodies, Minkowski spacetime, gravitational geometry, time and causality, and other subjects. Highlights include a rich exposition of the elements of the special and general theories of relativity.

The history and philosophy of science are destined to play a fundamental role in an epoch marked by a major scientific revolution. This ongoing revolution, principally affecting mathematics and physics, entails a profound upheaval of our conception of space, space–time, and, consequently, of natural laws themselves. Briefly, this revolution can be summarized by the following two trends: by the search for a unified theory of the four fundamental forces of nature, which are known, as of now, as gravity, electromagnetism, and (...) strong and weak nuclear forces; by the search for new mathematical concepts capable of elucidating and therefore explaining such a relationship. In fact, the first search is essentially dependent on the second; that is to say, that in order for a new theory of physics to come to light, the development of a deeper geometric theory capable of explaining the structure of space–time on a quantum scale appears to be necessary. On careful consideration, we notice that both of these developments converge in the direction of a unitary and fundamental tendency of modern science—which is the geometrization of theoretical physics and of natural sciences. This new emergent situation carries within it a profound conceptual change, affecting the way in which relations are conceived of, first and foremost, between mathematics and physics. This new paradigm can be summed up by the intimately interdependent points: the immense variety of physical phenomena and of natural forms follows from the equally infinite variety of geometric and topological objects that can be made out in space and from which space is made up; the second point, which ensues from the former one and which is of great historical and epistemological significance, is that mathematics is involved in rather than applied to phenomena. In other words, phenomena are effects that emerge from the geometrical structure of space–time. There is no doubt that this new conception of the relationship between the universe of mathematical ideas and objects and the world of natural phenomena is the true scientific revolution of our century, of great conceptual importance, and consequently, capable of changing our view of science and of nature at one and the same time. It is all at once of a scientific, philosophical and aesthetic order. (shrink)

Gyrogroup theory and its applications is introduced and explored, exposing the fascinating interplay between Thomas precession of special relativity theory and hyperbolic geometry. The abstract Thomas precession, called Thomas gyration, gives rise to grouplike objects called gyrogroups [A, A. Ungar, Am. J. Phys.59, 824 (1991)] the underlying axions of which are presented. The prefix gyro extensively used in terms like gyrogroups, gyroassociative and gyrocommutative laws, gyroautomorphisms, and gyrosemidirect products, stems from their underlying abstract Thomas gyration. Thomas gyration is tailor (...) made for hyperbolic geometry. In a similar way that commutative groups underlie vector spaces, gyrocommutative gyrogroups underlie gyrovector spaces. Gyrovector spaces, in turn, provide a most natural setting for hyperbolic geometry in full analogy with vector spaces that provide the setting for Euclidean geometry. As such, their applicability to relativistic physics and its spacetime geometry is obvious. (shrink)