We explore the intimate connection between spacetime geometry and electrodynamics. This link is already implicit in the constitutive relations between the field strengths and excitations, which are an essential part of the axiomatic structure of electromagnetism, clearly formulated via integration theory and differential forms. We review the foundations of classical electromagnetism based on charge and magnetic flux conservation, the Lorentz force and the constitutive relations. These relations introduce the conformal part of the metric and allow the study of (...) electrodynamics for specific spacetime geometries. At the foundational level, we discuss the possibility of generalizing the vacuum constitutive relations, by relaxing the fixed conditions of homogeneity and isotropy, and by assuming that the symmetry properties of the electro-vacuum follow the spacetime isometries. The implications of this extension are briefly discussed in the context of the intimate connection between electromagnetism and the geometry of spacetime. (shrink)

Whitehead, in two basic books, considers two different approaches to point-free geometry: the inclusion-based approach , whose primitive notions are regions and inclusion relation between regions, and the connection-based approach , where the connection relation is considered instead of the inclusion. We show that the latter cannot be reduced to the first one, although this can be done in the framework of multivalued logics.

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)

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)

The city has featured as a central image in utopian thought. In planning the foundation of the new and ideal city there is a close interconnection between ideas about urban form and the vision of the moral good. The spatial structure of the ideal city in these visions is a framing device that embodies and articulates not only political philosophy but is itself an articulation of moral and cosmological systems. This paper analyses three different utopian moments in three different historical (...) epochs – Tommaso Campanella’s City of the Sun (1602), the Choson dynasty foundation of the city of Seoul (1395) and the modernist utopian urbanism of the controversial Le Corbusier (1887–1965). In this analysis attention is drawn to the cosmological and moral visions articulated in these three ‘images of the city’ (Lynch). The opposition between rationalistic/mechanistic and religious/traditional urban design can prove to be an oversimplification that obscures the complex interrelations between the moral geometry and the natural alignments of the ideal urban form. (shrink)

Peg Rawes examines a "minor tradition" of aesthetic geometries in ontological philosophy. Developed through Kant’s aesthetic subject she explores a trajectory of geometric thinking and geometric figurations--reflective subjects, folds, passages, plenums, envelopes and horizons--in ancient Greek, post-Cartesian and twentieth-century Continental philosophies, through which productive understandings of space and embodies subjectivities are constructed. Six chapters, explore the construction of these aesthetic geometric methods and figures in a series of "geometric" texts by Kant, Plato, Proclus, Spinoza, Leibniz, Bergson, Husserl and Deleuze. In (...) each text, geometry is expressed as a uniquely embodies aesthetic activity because each respective geometric method and figure is imbued with aesthetic sensibility and geometric sense (rather than as disembodies scientific methods). An ontology of aesthetic geometric methods and figures is therefore traced from Kant’s Critical writings, back to Plato and Proclus Greek philosophy, Spinoza and Leibniz’s post-Cartesian philosophies, and forwards to Bergson’s "duration" and Husserl’s "horizons" towards Deleuze’s philosophy of sense. (shrink)

This book offers a reconstruction of the debate on non-Euclidean geometry in neo-Kantianism between the second half of the nineteenth century and the first decades of the twentieth century. Kant famously characterized space and time as a priori forms of intuitions, which lie at the foundation of mathematical knowledge. The success of his philosophical account of space was due not least to the fact that Euclidean geometry was widely considered to be a model of certainty at his time. (...) However, such later scientific developments as non-Euclidean geometries and Einstein’s general theory of relativity called into question the certainty of Euclidean geometry and posed the problem of reconsidering space as an open question for empirical research. The transformation of the concept of space from a source of knowledge to an object of research can be traced back to a tradition, which includes such mathematicians as Carl Friedrich Gauss, Bernhard Riemann, Richard Dedekind, Felix Klein, and Henri Poincaré, and which finds one of its clearest expressions in Hermann von Helmholtz’s epistemological works. Although Helmholtz formulated compelling objections to Kant, the author reconsiders different strategies for a philosophical account of the same transformation from a neo-Kantian perspective, and especially Hermann Cohen’s account of the aprioricity of mathematics in terms of applicability and Ernst Cassirer’s reformulation of the a priori of space in terms of a system of hypotheses. This book is ideal for students, scholars and researchers who wish to broaden their knowledge of non-Euclidean geometry or neo-Kantianism. (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)

This book aims to provide an overview of several topics in advanced Differential Geometry and Lie Group Theory, all of them stemming from mathematical problems in supersymmetric physical theories. It presents a mathematical illustration of the main development in geometry and symmetry theory that occurred under the fertilizing influence of supersymmetry/supergravity. The contents are mainly of mathematical nature, but each topic is introduced by historical information and enriched with motivations from high energy physics, which help the reader in (...) getting a deeper comprehension of the subject. (shrink)

In this paper I present critical evaluations of Ayer and Putnam's views on the analyticity of geometry. By drawing on the historico-philosophical work of Michael Friedman on the relativized apriori; and Roberto Torretti on the foundations of geometry, I show how we can make sense of the assertion that pure geometry is analytic in Carnap's sense.

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)

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

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.

The Foundations of Geometry was first published in 1897, and is based on Russell's Cambridge dissertation as well as lectures given during a journey through the USA. This is the first reprint, complete with a new introduction by John Slater. It provides both an insight into the foundations of Russell's philosophical thinking and an introduction to the philosophy of mathematics and logic. As such it will be an invaluable resource not only for students of philosophy, but also (...) for those interested in Russell's philosophical development. Foundations of Geometry consists of four chapters which explore the various concepts of geometry and their philosophical implications, including a historical overview of the development of geometrical theory. (shrink)

Any acceptable quantum gravity theory must allow us to recover the classical spacetime in the appropriate limit. Moreover, the spacetime geometrical notions should be intrinsically tied to the behavior of the matter that probes them. We consider some difficulties that would be confronted in attempting such an enterprise. The problems we uncover seem to go beyond the technical level to the point of questioning the overall feasibility of the project. The main issue is related to the fact that, in the (...) quantum theory, it is impossible to assign a trajectory to a physical object, and, on the other hand, according to the basic tenets of the geometrization of gravity, it is precisely the trajectories of free localized objects that define the spacetime geometry. The insights gained in this analysis should be relevant to those interested in the quest for a quantum theory of gravity and might help refocus some of its goals. (shrink)

Sobre la base que aportan las notas manuscritas de David Hilbert para cursos sobre geometría, el artículo procura contextualizar y analizar una de las contribuciones más importantes y novedosas de su célebre monografía Fundamentos de la geometría, a saber: el cálculo de segmentos lineales. Se argumenta que, además de ser un resultado matemático importante, Hilbert depositó en su aritmética de segmentos un destacado significado epistemológico y metodológico. En particular, se afirma que para Hilbert este resultado representaba un claro ejemplo de (...) uno de los rasgos más fructíferos y atractivos de su nuevo método axiomático formal, o sea, la capacidad de descubrir y exhibir conexiones estructurales o internas entre diferentes teorías matemáticas. On the basis of a set of unpublished notes for lecture courses on geometry, the paper seeks to contextualize and analyze one of the most important and original contributions of David Hilbert's celebrated monograph Foundations of Geometry, namely its arithmetic of line segments. It is argued that Hilbert attributed to his arithmetic of segments an important epistemological and methodological meaning, in addition to its relevance as an original mathematical result. In particular, it is claimed that for Hilbert his arithmetic of segments represented a clear example of one of the most fruitful and attractive traits of his new formal axiomatic method, i.e., the power to discover and exhibit inner or structural connections among different mathematical theories. (shrink)

The programme of geometrical archaeology initiated by the last Husserlian texts merits being taken further with renewed emphasis on historical discontinuities. The revolutions that geometry has gone through have led to profound changes in meaning. The arch-foundations of geometry were not actually definitively set out at the time of the first factual institution of geometry, namely Greek geometry. Instead, the full plurality of the historical forms of geometry does not just consist of a movement (...) to generalise axioms. It involves the reform of the archi-foundations themselves. We will show this using the example of two important revolutions: 1) the discovery of non-Euclidean geometries and 2) the transition to a differential geometry and a dynamic metric. We will also discuss the philosophical difficulties these two moments of re-origination brought up. These studies will follow an introductory section on historical phenomenology and the architecture of the original strata of geometry, the idea being to introduce the conceptual tools and vocabulary required for the studies in the final part. (shrink)

The paper discusses the similarity between geometry, arithmetic, and logic, specifically with respect to the question of whether applied theories of each may be revised. It argues that they can - even when the revised logic is a paraconsistent one, or the revised arithmetic is an inconsistent one. Indeed, in the case of logic, it argues that logic is not only revisable, but, during its history, it has been revised. The paper also discusses Quine's well known argument against the (...) possibility of logical deviancy. (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.

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)

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)

There is a common assumption among philosophers, shared even by many Kant scholars, that Kant had a naive faith in the absolute valid­ity of Euclidean geometry, Aristotelian logic, and Newtonian physics, and that his primary goal in the Critique of Pure Reason was to pro­vide a rational foundation upon which these classical scientific theories could be based. This, it might be thought, is the essence of his attempt to solve the problem which, as he says in a footnote to (...) the second edition Preface, "still remains a scandal to philosophy and to human reason in general" -namely, "that the existence of things outside us...must be accepted merely on faith, and that if anyone thinks good to doubt their existence, we are unable to.. (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)

This paper aims to show how the mathematical content of Hilbert's Axiom of Completeness consists in an attempt to solve the more general problem of the relationship between intuition and formalization. Hilbert found the accordance between these two sides of mathematical knowledge at a logical level, clarifying the necessary and sufficient conditions for a good formalization of geometry. We will tackle the problem of what is, for Hilbert, the definition of geometry. The solution of this problem will bring (...) out how Hilbert's conception of mathematics is not as innovative as his conception of the axiomatic method. The role that the demonstrative tools play in Hilbert's foundational reflections will also drive us to deal with the problem of the purity of methods, explicitly addressed by Hilbert. In this respect Hilbert's position is very innovative and deeply linked to his modern conception of the axiomatic method. In the end we will show that the role played by the Axiom of Completeness for geometry is the same as the Axiom of Induction for arithmetic and of Church-Turing thesis for computability theory. We end this paper arguing that set theory is the right context in which applying the axiomatic method to mathematics and we postpone to a sequel of this work the attempt to offer a solution similar to Hilbert's for the completeness of set theory. (shrink)

The critique of my protophysical approaches to operational foundation of geometry by Lucas Amiras (Journal for General Philosophy of Science Vol. 34 (2003)) concerns my first publication from 1976 but not the further 30 years of work. It does not offer any argument leading from the (erroneous) judgement “lacking success” to the conclusion “impossible”. And it is, in general, based on a philosophical defect: it ignores the principle of methodical order as leading for constructivist protophysics.

We show that the first-order theory of a large class of plane geometries and the first-order theory of their groups of motions, understood both as groups with a unary predicate singling out line-reflections, and as groups acting on sets, are mutually inter-pretable.

In lieu of an abstract, here is a brief excerpt of the content:Hume Studies Volume XXIII, Number 2, November 1997, pp. 227-244 Hume on Geometry and Infinite Divisibility in the Treatise H. MARK PRESSMAN Scholars have recognized that in the Treatise "Hume seeks to find a foundation for geometry in sense-experience."1 In this essay, I examine to what extent Hume succeeds in his attempt to ground geometry visually. I argue that the geometry Hume describes in the (...) Treatise faces a serious set of problems. Geometric Lines Hume maintains that ideas "are images" (T 6) which may be called up "when I shut my eyes" (T 3). "That we may fix the meaning of a word, figure," according to Hume, "we may revolve in our mind the ideas of circles, squares, parallelograms, triangles of different sizes and proportions, and may not rest on one image or idea" (T 22). As Arthur Pap notes, "'Idea' is in Hume's usage synonymous with 'mental image'." D.C.G. MacNabb also notes that "Hume thought of ideas as images, and primarily as visual images."2 When Hume argues, then, that "we have the idea of indivisible points, lines and surfaces conformable to the [geometer's] definition" (T 44), he is arguing that we have mental images of geometry's points, lines and planes. Consider geometric lines first. Geometers define a line "to be length without breadth or depth" (T 42). It is not apparent to everyone that we have mental images of such lines. According to "L'Art de penser" (T 43), published in 1662, it is impossible "to conceive a length without any breadth" (T 43). H. Mark Pressman is at the Department of Philosophy, University of California, Davis, 1238 Social Sciences and Humanities Building, Davis CA 95616-8673 USA. email: imn78@cvin.fresno.com 228 H. Mark Pressman We can only by an abstraction "consider the one without regarding the other" (T 43). Hume, however, offers "clear proof" (T 44), in the form of two arguments, that we actually possess ideas or mental images of geometry's lines. First, we have ideas or mental images of lengths with breadth, though these are supposed never to be without breadth and thus not geometric lines. (A mental image oflength is also a mental image with length, just as "to say the idea of extension agrees to any thing, is to say it is extended" [T 240].) Hume reduces to absurdity the view that our mental images of length are always with breadth and thus not images of geometric lines. If our length-images were always with breadth, then our mental images would be endlessly divisible in breadth, since they would always have some positive breadth (T 43). But no mental image is endlessly divisible, for the mind arrives "at an end in the division of its ideas, nor are there any possible means of evading the evidence of this conclusion" (T 27). We thus have an idea or image of length without (divisible) breadth conforming to the geometer's definition (T 44). Second, we can imagine or picture the termination of a geometric plane. But a geometric plane must terminate in a geometric line (T 44). (Proof: Assume L is a line with breadth which terminates plane P. Since L has breadth, L must contain at least two parts, w and y, exactly one of which actually terminates P. But whether it is w or y, L doesn't terminate P, which was the assumption.) Since we can picture the termination of a geometric plane, we can picture a geometric line, length without divisible breadth. It is one thing to have ideas or mental images of geometric lines which may be called up "when I shut my eyes" (T 3). It is another for such objects actually to exist "in nature." The conceptualist holds that though (a) geometry 's points, lines and planes are ideas, nevertheless (b) there are no such things in nature and that (c) there can't be such things in nature: [T]he objects of geometry... are mere ideas in the mind, and not only never did, but never can exist in nature. They never did exist; for no one will pretend to draw a line or make a... (shrink)

This paper is about Poincaré’s view of the foundations of geometry. According to the established view, which has been inherited from the logical positivists, Poincaré, like Hilbert, held that axioms in geometry are schemata that provide implicit definitions of geometric terms, a view he expresses by stating that the axioms of geometry are “definitions in disguise.” I argue that this view does not accord well with Poincaré’s core commitment in the philosophy of geometry: the view (...) that geometry is the study of groups of operations. In place of the established view I offer a revised view, according to which Poincaré held that axioms in geometry are in fact assertions about invariants of groups. Groups, as forms of the understanding, are prior in conception to the objects of geometry and afford the proper definition of those objects, according to Poincaré. Poincaré’s view therefore contrasts sharply with Kant’s foundation of geometry in a unique form of sensibility. According to my interpretation, axioms are not definitions in disguise because they themselves implicitly define their terms, but rather because they disguise the definitions which imply them. (shrink)

On the productive-operative approach to the foundations of geometry in protophysics. Attempts to establish a foundation to elementary geometry as a theory of spatial figures in Protophysics are surveyed in Section 1. An idea suggested by H. Dingler was to extract the basic properties of the geometrical primitive notions from descriptions of the operations performed in the ‘first’ production of the corresponding objects. P. Janich presents this ‘productive-operative’ approach as a succesful methodical alternative to the ‘geometry (...) of forms’ of R. Inhetveen and P. Lorenzen (Section 2). However, in his work not even one fundamental task seems to be achieved in a satisfactory manner. The article evaluates Janich’s contributions (Section 3) and discusses the shortcomings of his approach (Section 4). In this process a ‘functional-operative’ alternative emerges (Section 5). (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)

While Michel Serres’ work has become relatively well-known among social theoreticians in recent years, his explicit thematization of the foundations of human collectives has gained surprisingly little attention. This article claims that Serres’ approach to the theme of foundations can be clarified by scrutinizing the way in which he poses and answers the following three questions: How are we together? What and whom do we exclude from our togetherness and how? Who are we today? Instead of starting with (...) a ready-made order, be it in the form of individuals or society, Serres pushes social research to take up the challenge of examining the point at which order is about to emerge out of noise and chaos, but where the outcome of the process remains uncertain. Especially relevant to the discussion are Serres’ books Rome, Statues and Geometry, all three of which bear the subtitle Book of Foundations. (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)

RésuméCet article est consacré à l’histoire de la théorie locale des courbes “à double courbure”. Initiée par Clairaut en 1731, cette théorie se développe en parallèle à la théorie des surfaces et trouve son achèvement avec les formules de Serret et Frenet et leur interprétation par Darboux, en 1887. Au delà de l’analyse des contributions de nombreux mathématiciens, parmi lesquels Monge bien sûr mais aussi Fourier, Lagrange et Cauchy, notre étude donne un regard particulier sur l’évolution conjointe de l’Analyse et (...) de la Géométrie, dans une longue période riche de nombreuses remises en cause théoriques. (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)

In a series of articles dating from 1903 to 1906, Frege criticizes Hilbert’s methodology of proving the independence and consistency of various fragments of Euclidean geometry in his Foundations of Geometry. In the final part of the last article, Frege makes his own proposal as to how the independence of genuine axioms should be proved. Frege contends that independence proofs require the development of a ‘new science’ with its own basic truths. This paper aims to provide a (...) reconstruction of this New Science that meets modern standards and to examine possible problems surrounding Frege’s original proposal. The paper is organized as follows: the first two sections summarize the main points of the Frege–Hilbert controversy and discuss some issues surrounding the problem of independence proofs. Section 3 contains an informal presentation of Frege’s proposal. In section 4 a more detailed reconstruction of Frege’s New Science is set out while section 5 examines what is left out. The concluding section is devoted to a discussion of Frege’s strategy and its significance from a broader perspective. (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)

Charge, like mass in Newtonian mechanics, is an irreducible element of electromagnetic theory that must be introduced ab initio. Its origin is not properly a part of the theory. Fields are then defined in terms of forces on either masses—in the case of Newtonian mechanics, or charges in the case of electromagnetism. General Relativity changed our way of thinking about the gravitational field by replacing the concept of a force field with the curvature of space-time. Mass, however, remained an irreducible (...) element. It is shown here that the Reissner-Nordström solution to the Einstein field equations tells us that charge, like mass, has a unique space-time signature. (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)

It has been shown that for the Reissner-Nordström solution to the vacuum Einstein field equations charge, like mass, has a unique space-time signature (Marsh, Found. Phys. 38:293–300, 2008). The presence of charge results in a negative curvature. This work, which includes a discussion of effective mass, is extended here to the Kerr-Newman solution.

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)