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Adventures of the Diagonal: Non-Euclidean Mathematics and Narrative

Arkady Plotnitsky

Apostolos Doxiadis and Barry Mazur (eds)

in Circles Disturbed: The Interplay of Mathematics and Narrative

This chapter explores the relationship between narrative and non-Euclidean mathematics. It considers how non-Euclidean mathematics and the narratives accompanying it are linked: first, to the ... More

This chapter explores the relationship between narrative and non-Euclidean mathematics. It considers how non-Euclidean mathematics and the narratives accompanying it are linked: first, to the question of the potentially uncircumventable limits of thought and knowledge; and second, to the question of a certain heterogeneous and yet interactive multiplicity of concepts and of different fields such as algebra and geometry. The chapter starts with the Pythagoreans' discovery of the concept of “incommensurability,” or the irrationality of certain numbers, specifically the side and the diagonal of the square, and the narrative associated with this discovery. It then examines the transition from non-Euclidean physics to non-Euclidean mathematics by focusing on quantum mechanics and its relation to non-Euclidean epistemology. It also discusses the algebraic aspects of non-Euclidean geometry and concludes with the suggestion that non-Euclidean thinking retains the essential, shaping role of the movement of thought.Less

The Longest Journey: Bertrand Russell

Bas C. van Fraassen

in Scientific Representation: Paradoxes of Perspective

The first sustained, rigorous development of a structuralist view of science appeared in the writings of Bertrand Russell, where the philosophical motivation precedes a precise formulation drawing on ... More

The first sustained, rigorous development of a structuralist view of science appeared in the writings of Bertrand Russell, where the philosophical motivation precedes a precise formulation drawing on mathematical logic. He founded theoretical physics in a mathematics constructed along logicist lines, which is also what proved his undoing at the hands of a famous review by Newman that set the pattern for later objections to structuralist views.Less

Introduction

Ciaran McMorran

in Joyce and Geometry

This chapter explores how James Joyce evokes an overarching concern with the linear in his works, both formally (in terms of the Euclidean ideal of rectilinearity) and conceptually (in terms of ... More

This chapter explores how James Joyce evokes an overarching concern with the linear in his works, both formally (in terms of the Euclidean ideal of rectilinearity) and conceptually (in terms of linear narratives, histories, arguments, modes of thought, etc.). In particular, it considers how the non-linearity of Joyce’s works reflects a wider questioning of the straight line in modernist literature which followed the development of non-Euclidean geometries in the nineteenth and twentieth centuries. This chapter also provides an overview of the geometric babble which entered into the context of Joyce’s writing following the popularization of non-Euclidean geometry in modernist art and literature, as well as the “fashionable nonsense” associated with the application of geometric concepts in contemporary literary criticism. By referring to the source texts which informed Joyce’s articulation of multiple geometric registers, it traces his engagement with non-Euclidean geometry to his early readings of Giordano Bruno’s mathematical and philosophical works, illustrating how notions associated with the curvature of the straight line inform the structural composition of Ulysses and Finnegans Wake.Less

Geometry: Math for Math’s Sake: Non-Euclidean Geometry and Aestheticism

Andrea Henderson

in Algebraic Art: Mathematical Formalism and Victorian Culture

Edwin Abbott’s Flatland dramatizes the implications of dethroning what Victorians regarded as the preeminent representational system: Euclidean geometry. The displacement of the singular Euclidean ... More

Edwin Abbott’s Flatland dramatizes the implications of dethroning what Victorians regarded as the preeminent representational system: Euclidean geometry. The displacement of the singular Euclidean account of space with a multiplicity of non-referential spatial regimes did more than introduce the possibility of varying perspectives on the world; the challenge to the “sacredness” of Euclid met with resistance partly because it suggested the ideal of a transparent representational system was inherently untenable. Flatland explores the repercussions of this problem for the novel, shifting emphasis from the revelation of the content of character to focus on the vagaries of point of view. The characters are Euclidean figures shown the limitations of their constructions of the world, and epistemic certainty is unavailable because all representational systems are contingent. Abbott finds consolation for this loss of certainty in the formalist, aesthetic character of projective geometry, insisting on the beauty of signs in and of themselves.Less

Kant’s Theory of Geometry and Transcendental Idealism II

Thomas C. Vinci

in Space, Geometry, and Kant's Transcendental Deduction of the Categories

Chapter 4 has three main parts. The first develops an account of Kant’s mathematical method, based partly on work by Philip Kitcher, called the KV account. This account allows for epistemically ... More

Chapter 4 has three main parts. The first develops an account of Kant’s mathematical method, based partly on work by Philip Kitcher, called the KV account. This account allows for epistemically independent applications of geometrical method to objects in pure space—the mind-dependent space of imagination—as well as to objects in a putatively mind-independent empirical space. The a priori synthetic necessity of geometry is understood as counterfactual necessity. Objections from Friedman and Waxman are considered. In the former case the application constitutes pure geometry, in the latter, physical geometry. In the second part we see that Kant compares the theorems of both kinds of geometry, finds them identical, and seeks to explain this coincidence in a Second Geometrical Argument. The explanation is Transcendental Idealism. In the final part, I consider how Kant’s Euclidean theory of pure geometry fares in light of modern non-Euclidean theories of physical geometry.Less

Embrace change, considering non-Euclidean geometry

Susan D'Agostino

in How to Free Your Inner Mathematician: Notes on Mathematics and Life

“Embrace change, considering non-Euclidean geometry” offers a basic introduction to the geometry of curved surfaces in which the sum of the degrees of a triangle may equal more or less than 180°. The ... More

“Embrace change, considering non-Euclidean geometry” offers a basic introduction to the geometry of curved surfaces in which the sum of the degrees of a triangle may equal more or less than 180°. The discussion is enhanced by numerous hand-drawn sketches, including triangles on spheres and saddles. Mathematics students and enthusiasts are encouraged to embrace change as a way to alleviate boredom, understand one’s strengths, and uncover new opportunities and modes of thinking. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.Less

Metric description of a curved space

Ta-Pei Cheng

in Relativity, Gravitation and Cosmology: A Basic Introduction

Einstein's new theory of gravitation is formulated in a geometric framework of curved spacetime. Here the subject of non-Euclidean geometry is introduced by way of Gauss's theory of curved surfaces. ... More

Einstein's new theory of gravitation is formulated in a geometric framework of curved spacetime. Here the subject of non-Euclidean geometry is introduced by way of Gauss's theory of curved surfaces. Generalized (Gaussian) coordinates: A systematic way to label points in space without reference to any objects outside this space. Metric function: For a given coordinate choice, the metric determines the intrinsic geometric properties of a curved space. Geodesic equation: It describes the shortest and the straightest possible curve in a warped space and is expressed in terms of the metric function. Curvature: It is a nonlinear second derivative of the metric. As the deviation from Euclidean relations is proportional to the curvature, it measures how much the space is warped.Less

Rigor and Rigorization

John P. Burgess

in Rigor and Structure

This chapter explains what heuristic methods, ubiquitous and indispensable in the context of discovery in mathematics, are prohibited in the context of justification by the requirement of rigor as ... More

This chapter explains what heuristic methods, ubiquitous and indispensable in the context of discovery in mathematics, are prohibited in the context of justification by the requirement of rigor as now understood. The chapter describes, with special attention to the role of infinities and infinitesimals and of spatiotemporal intuition in the eighteenth-century calculus, how the requirement of rigor, honored in principle but slighted in practice during the early modern period, came to be enforced as never before during the nineteenth century. The question why such new higher standards, ultimately going far beyond the ancient level of rigor represented by Euclid, came to be insisted upon is broached, and several factors discussed that provide partial answers, especially the role of non-Euclidean geometry. But it is not pretended that any answer can be given that would completely rationalize all the historical decisions of the mathematical community.Less

A Very Short Course in Projective Geometry

Tony Robbin

in Shadows of Reality: The Fourth Dimension in Relativity, Cubism, and Modern Thought

This chapter serves as a brief lesson on projective geometry. It starts by discussing two basic theorems of projective geometry, which are concerned with perspectives from a point and perspectives ... More

This chapter serves as a brief lesson on projective geometry. It starts by discussing two basic theorems of projective geometry, which are concerned with perspectives from a point and perspectives from a line. The discussion then looks at projectivities and perspectivities, which result from several perspectivities and relate a range, respectively. This is followed by the study of analytic projective geometry and the connection between projective and projection. The chapter concludes with a discussion on the complex projective line, the projective three-space, and Felix Klein's thoughts on the presence of non-Euclidean geometries in projective geometry, specifically the topology of the projective plane.Less

The Perils of Parallels!

Nicholas Mee

in Celestial Tapestry: The Warp and Weft of Art and Mathematics

In the nineteenth century, three mathematicians—Bolyai, Gauss, and Lobachevsky—almost simultaneously discovered the possibility of non-Euclidean or hyperbolic geometries. These geometries rest on ... More

In the nineteenth century, three mathematicians—Bolyai, Gauss, and Lobachevsky—almost simultaneously discovered the possibility of non-Euclidean or hyperbolic geometries. These geometries rest on axioms that do not include the parallel postulate. This means that many results of Euclidean geometry do not hold. Spherical geometry is considered as a model to illustrate why this is the case. The mathematician Donald Coxeter inspired artist M. C. Escher to produce remarkable artworks based on the hyperbolic geometry of the Poincaré disc. Gauss attempted to measure the curvature of the space around the Earth. Since Einstein, we know that gravity curves space and time.Less

Mathematics and the Unity of Sensibility

Waxman Wayne

in Kant's Anatomy of the Intelligent Mind

This chapter focuses on Kant’s account of the possibility of mathematics in relation to the transcendental expositions of space and time in the Transcendental Aesthetic, that is, the role of ... More

This chapter focuses on Kant’s account of the possibility of mathematics in relation to the transcendental expositions of space and time in the Transcendental Aesthetic, that is, the role of sensibility therein rather than understanding (which is discussed in Chapter 15). It is argued that Kant was nothing like the Euclidean dogmatist he is commonly portrayed as being by showing that and how his account of sensibility is perfectly capable of accommodating mathematical spaces of any curvature and number of dimensions. It is also shown how this account enabled him to explain the possibility not only of geometry, but even the most abstract, purely symbolic varieties of mathematics, among which post-Fregean mathematical logic should probably be included.Less

Ezra Pound and the Materiality of the Fourth Dimension

Ian F. A. Bell

in Science in Modern Poetry: New Directions

This chapter describes how Ezra Pound discovered the aesthetic potential of the concept of Vorticism and fourth-dimensional thought and integrated this into his writings. In his ‘Vorticism’ essay, ... More

This chapter describes how Ezra Pound discovered the aesthetic potential of the concept of Vorticism and fourth-dimensional thought and integrated this into his writings. In his ‘Vorticism’ essay, Pound emphasized the extent to which the Vorticist surface belongs to advances in Euclidean geometry, offering ‘Descartian’ or analytic geometry as the creative idiom. It is suggested that modernist poetics availed itself of trajectories within fourth-dimensional thought through a telling of obliquity, and that Vorticism and its allied activities provide the most generative ground.Less

Geometry

Olivier Darrigol

in Physics and Necessity: Rationalist Pursuits from the Cartesian Past to the Quantum Present

This chapter deals with the foundations of geometry, with emphasis on Helmholtz’s approach to this problem in the late 1860s. It is first recalled how the ancient belief in the apodictic truth of the ... More

This chapter deals with the foundations of geometry, with emphasis on Helmholtz’s approach to this problem in the late 1860s. It is first recalled how the ancient belief in the apodictic truth of the axioms of geometry came under attack at the turn of the eighteenth and nineteenth centuries, making room for the new geometries of Gauss, Bolyai, Lobachevski, and Riemann. This opening of possibilities raised the question of the necessary structural kernel of any physical geometry. The central section of this chapter is a detailed account of Helmholtz’s answer to this question, according to which the locally Euclidean character of space follows from its measurability by freely mobile rigid bodies. This argument has difficulties, due to a seeming circularity in the definition of rigid bodies and to the strictness of the assumed rigidity, which requires spaces of constant curvature. It is shown how these difficulties can be circumvented, leading to the necessity of the Riemannian structure of space (with any curvature) for measurable space.Less

Non-Euclidean Geometry

Jeremy Gray

in Space: A History

Non-Euclidean geometry began as an inquiry into a possible weakness in Euclid’s Elements and became the source of the ideas that there are geometries of spaces other than the one imagined in ... More

Non-Euclidean geometry began as an inquiry into a possible weakness in Euclid’s Elements and became the source of the ideas that there are geometries of spaces other than the one imagined in elementary geometry and that many mathematical theories, not only in geometry but in algebra and analysis, can be fully and profitably axiomatized.Less

Geometry on my Mind

David D. Nolte

in Galileo Unbound: A Path Across Life, the Universe and Everything

This chapter reviews the history of modern geometry with a focus on the topics that provided the foundation for the new visualization of physics. It begins with Carl Gauss and Bernhard Riemann, who ... More

This chapter reviews the history of modern geometry with a focus on the topics that provided the foundation for the new visualization of physics. It begins with Carl Gauss and Bernhard Riemann, who redefined geometry and identified the importance of curvature for physics. Vector spaces, developed by Hermann Grassmann, Giuseppe Peano and David Hilbert, are examples of the kinds of abstract new spaces that are so important for modern physics, such as Hilbert space for quantum mechanics. Fractal geometry developed by Felix Hausdorff later provided the geometric language needed to solve problems in chaos theory. Motion cannot exist without space—trajectories are the tracks of points, mathematical or physical, through it.Less

Rehabilitating Otto Liebmann

Frederick C. Beiser

in The Genesis of Neo-Kantianism, 1796-1880

Chapter 7 discusses the early philosophical works and development of Otto Liebmann. It focuses upon the contents and motives behind his early Kant und die Epigonen and his early critique of ... More

Chapter 7 discusses the early philosophical works and development of Otto Liebmann. It focuses upon the contents and motives behind his early Kant und die Epigonen and his early critique of Schopenhauer. I attempt to rescue Liebmann’s reputation from Klaus Christian Köhnke’s severe criticisms. Liebmann is shown to be a classic 19th-century German liberal rather than a proto-national-socialist. Liebmann is treated as a crucial figure in the neo-Kantian critique of positivism and in the development of a non-psychological interpretation of Kant.Less

Memo to Readers: Overview and Synopsis

Waxman Wayne

in Kant's Anatomy of the Intelligent Mind

This introductory text signals what is new and noteworthy in the book and provides guidance for comprehending and evaluating it. It opens with some general remarks regarding the book’s relation to ... More

This introductory text signals what is new and noteworthy in the book and provides guidance for comprehending and evaluating it. It opens with some general remarks regarding the book’s relation to the same author’s Kant and the Empiricists. Understanding Understanding (2005), and recapitulates some of the themes common to both books. It then states the thesis that most directly sets this interpretation of Kant apart from others—that the categories presuppose apperception (pure self-consciousness) and not vice versa—and describes how its acceptance leads to privileging questions that seldom get asked. The Memo concludes with a part-by-part, chapter-by-chapter synopsis of the whole.Less

A Mathematical Sculptor’s Perspective on Space

George Hart

in Space: A History

In this Reflection, a mathematician discusses four sculptures he created to express important aspects of various kinds of spaces, including ordinary Euclidean, hyperbolic space. The sculptures ... More

In this Reflection, a mathematician discusses four sculptures he created to express important aspects of various kinds of spaces, including ordinary Euclidean, hyperbolic space. The sculptures represent the transcription into physical objects of conceptual ideas concerning figures and the spaces they inhabit. Different sculptures exhibit various aspects of different spaces: e.g. whereas a handheld sculpture may lack orientation, thereby exhibiting an aspect of classic Euclidean space, a large sculpture fixed to the ground may have an orientation, thereby exhibiting an aspect of physical space.Less