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)

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)

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 quotation I take above as motto is from the Author's Epistle to the Reader of De Corpore. Immediately after it, Hobbes elaborates the conceit likening six sciences with the six days of divine creation. These are supplemented with divine commandment and final contemplation of "subjection to command." Thus, with some poetic license, all compartments of Hobbes's reiterated ordering of several bodies of science and "Elements of Philosophy" are indicated: De Corpore, and then De Homine and De Cive. Following Hobbes, (...) as I profess, I sunder them that I may acknowledge both their distinction and order; yet I dispute their sufficient derivation, one from another. (shrink)

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)

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)

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)

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)

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)

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)

I will discuss only one of the several entwined strands of the philosophy of space and time, the question of the relation between the nature of motion and the geometrical structure of the world.1 This topic has many of the virtues of the best philosophy of science. It is of long-standing philosophical interest and has a rich history of connections to problems of physics. It has loomed large in discussions of space and time among contemporary philosophers of science. Furthermore, (...) there is, I think, widespread agreement that recent insights here have lead to a genuine deepening of our understanding. (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 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)

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.

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 application of geometry to physics has provided us with new insightful information about many physical theories such as classical mechanics, general relativity, and quantum geometry. The geometry also plays an important role in foundations of quantum mechanics and quantum information. In this work we discuss a geometric framework for mixed quantum states represented by density matrices, where the quantum phase space of density matrices is equipped with a symplectic structure, an almost complex structure, and a (...) compatible Riemannian metric. This compatible triple allow us to investigate arbitrary quantum systems. We will also discuss some applications of the geometric framework. (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)

I use recent work on Kant and diagrammatic reasoning to develop a reconsideration of central aspects of Kant’s philosophy of geometry and its relation to spatial intuition. In particular, I reconsider in this light the relations between geometrical concepts and their schemata, and the relationship between pure and empirical intuition. I argue that diagrammatic interpretations of Kant’s theory of geometrical intuition can, at best, capture only part of what Kant’s conception involves and that, for example, they cannot explain why (...) Kant takes geometrical constructions in the style of Euclid to provide us with an a priori framework for physical space. I attempt, along the way, to shed new light on the relationship between Kant’s theory of space and the debate between Newton and Leibniz to which he was reacting, and also on the role of geometry and spatial intuition in the transcendental deduction of the categories. (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)

We discuss the relationship between logic, geometry and probability theory under the light of a novel approach to quantum probabilities which generalizes the method developed by R. T. Cox to the quantum logical approach to physical theories.

A classic problem in general relativity, long studied by both physicists and philosophers of physics, concerns whether the geodesic principle may be derived from other principles of the theory, or must be posited independently. In a recent paper [Geroch & Weatherall, "The Motion of Small Bodies in Space-Time", Comm. Math. Phys. ], Bob Geroch and I have introduced a new approach to this problem, based on a notion we call "tracking". In the present paper, I situate the main results (...) of that paper with respect to two other, related approaches, and then make some preliminary remarks on the interpretational significance of the new approach. My main suggestion is that "tracking" provides the resources for eliminating "point particles"---a problematic notion in general relativity---from the geodesic principle altogether. (shrink)

Euler developed a program which aimed to transform analysis into an autonomous discipline and reorganize the whole of mathematics around it. The implementation of this program presented many difficulties, and the result was not entirely satisfactory. Many of these difficulties concerned the integral calculus. In this paper, we deal with some topics relevant to understand Euler’s conception of analysis and how he developed and implemented his program. In particular, we examine Euler’s contribution to the construction of differential equations and his (...) notion of indefinite integrals and general integrals. We also deal with two remarkable difficulties of Euler’s program. The first concerns singular integrals, which were considered as paradoxical by Euler since they seemed to violate the generality of certain results. The second regards the explicitly use of the geometric representation and meaning of definite integrals, which was gone against his program. We clarify the nature of these difficulties and show that Euler never thought that they undermined his conception of mathematics and that a different foundation was necessary for analysis. (shrink)

We present the basic concepts of space and time, the Galilean and pseudo-Euclidean geometry. We use an elementary geometric framework of affine spaces and groups of affine transformations to illustrate the natural relationship between classical mechanics and theory of relativity, which is quite often hidden, despite its fundamental importance. We have emphasized a passage from the group of Galilean motions to the group of Poincaré transformations of a plane. In particular, a 1-parametric family of natural deformations of the Poincaré (...) group is described. We also visualized the underlying groups of Galilean, Euclidean, and pseudo-Euclidean rotations within the special linear group. (shrink)

It is well known that over the eighteenth century the calculus moved away from its geometric origins; Euler, and later Lagrange, aspired to transform it into a “purely analytical” discipline. In the 1780 s, the Portuguese mathematician José Anastácio da Cunha developed an original version of the calculus whose interpretation in view of that process presents challenges. Cunha was a strong admirer of Newton (who famously favoured geometry over algebra) and criticized Euler’s faith in analysis. However, the fundamental propositions (...) of his calculus follow the analytical trend. This appears to have been possible due to a nominalistic conception of variable that allowed him to deal with expressions as names, rather than abstract quantities. Still, Cunha tried to keep the definition of fluxion directly applicable to geometrical magnitudes. According to a friend of Cunha’s, his calculus had an algebraic (analytical) branch and a geometrical branch, and it was because of this that his definition of fluxion appeared too complex to some contemporaries. (shrink)

Husserl left many unpublished drafts explaining (or trying to) his views on spatial representation and geometry, such as, particularly, those collected in the second part of Studien zur Arithmetik und Geometrie (Hua XXI), but no completely articulate work on the subject. In this paper, I put forward an interpretation of what those views might have been. Husserl, I claim, distinguished among different conceptions of space, the space of perception (constituted from sensorial data by intentionally motivated psychic functions), that of (...) physical geometry (or idealized perceptual space), the space of the mathematical science of physical nature (in which science, not only raw perception has a word) and the abstract spaces of mathematics (free creations of the mathematical mind), each of them with its peculiar geometrical structure. Perceptual space is proto-Euclidean and the space of physical geometry Euclidean, but mathematical physics, Husserl allowed, may find it convenient to represent physical space with a non-Euclidean structure. Mathematical spaces, on their turn, can be endowed, he thinks, with any geometry mathematicians may find interesting. Many other related questions are addressed here, in particular those concerning the a priori or a posteriori character of the many geometric features of perceptual space (bearing in mind that there are at least two different notions of a priori in Husserl, which we may call the conceptual and the transcendental a priori). I conclude with an overview of Weyl’s ideas on the matter, since his philosophical conceptions are often traceable back to his former master, Husserl. (shrink)

As part of his program to unify linear algebra and geometry using the language of Clifford algebra, David Hestenes has constructed a (well-known) isomorphism between the conformal group and the orthogonal group of a space two dimensions higher, thus obtaining homogeneous coordinates for conformal geometry.(1) In this paper we show that this construction is the Clifford algebra analogue of a hyperbolic model of Euclidean geometry that has actually been known since Bolyai, Lobachevsky, and Gauss, and we explore (...) its wider invariant theoretic implications. In particular, we show that the Euclidean distance function has a very simple representation in this model, as demonstrated by J. J. Seidel.(18). (shrink)