## 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

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691149042
- eISBN:
- 9781400842681
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691149042.003.0012
- Subject:
- Mathematics, History of Mathematics

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

## Ancient Greek Mathematics: A Role for Diagrams

*José Ferreirós*

### in Mathematical Knowledge and the Interplay of Practices

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691167510
- eISBN:
- 9781400874002
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691167510.003.0005
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter focuses on the ancient Greek tradition of geometrical proof in light of recent studies by Kenneth Manders and others. It advances the view that the borderline of elementary mathematics ... More

## Space and Geometry

*Ian Hinckfuss*

### in The Existence of Space and Time

- Published in print:
- 1975
- Published Online:
- October 2011
- ISBN:
- 9780198245193
- eISBN:
- 9780191680854
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198245193.003.0004
- Subject:
- Philosophy, Metaphysics/Epistemology

This chapter first reports the geometrical properties of space. One of the assumptions of Newtonian dynamics is that space is Euclidean; that is, space satisfies the axioms, and hence the theorems, ... More

## Time, space and mass

*JESPER LÜTZEN*

### in Mechanistic Images in Geometric Form: Heinrich Hertz's 'Principles of Mechanics'

- Published in print:
- 2005
- Published Online:
- January 2010
- ISBN:
- 9780198567370
- eISBN:
- 9780191717925
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567370.003.0011
- Subject:
- Physics, History of Physics

Heinrich Hertz introduced the three fundamental concepts of his mechanics — time, space, and mass — at the beginning of his book Principles of Mechanics. This chapter analyses Hertz's concepts of ... More

## The Longest Journey: Bertrand Russell

*Bas C. van Fraassen*

### in Scientific Representation: Paradoxes of Perspective

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199278220
- eISBN:
- 9780191707926
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199278220.003.00010
- Subject:
- Philosophy, Philosophy of Mind, Philosophy of Science

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

## Kant’s Theory of Geometry and Transcendental Idealism II

*Thomas C. Vinci*

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

- Published in print:
- 2014
- Published Online:
- October 2014
- ISBN:
- 9780199381166
- eISBN:
- 9780199381234
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199381166.003.0005
- Subject:
- Philosophy, History of Philosophy, Metaphysics/Epistemology

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

## Introduction

*Ciaran McMorran*

### in Joyce and Geometry

- Published in print:
- 2020
- Published Online:
- September 2020
- ISBN:
- 9780813066288
- eISBN:
- 9780813065267
- Item type:
- chapter

- Publisher:
- University Press of Florida
- DOI:
- 10.5744/florida/9780813066288.003.0001
- Subject:
- Literature, 20th-century Literature and Modernism

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

## Metric description of a curved space

*Ta-Pei Cheng*

### in Relativity, Gravitation and Cosmology: A Basic Introduction

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199573639
- eISBN:
- 9780191722448
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199573639.003.0005
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

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

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

*Andrea Henderson*

### in Algebraic Art: Mathematical Formalism and Victorian Culture

- Published in print:
- 2018
- Published Online:
- May 2018
- ISBN:
- 9780198809982
- eISBN:
- 9780191860140
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198809982.003.0002
- Subject:
- Literature, 19th-century Literature and Romanticism

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

## Geometry

*Olivier Darrigol*

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

- Published in print:
- 2014
- Published Online:
- June 2014
- ISBN:
- 9780198712886
- eISBN:
- 9780191781360
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198712886.003.0004
- Subject:
- Physics, History of Physics

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

## Mathematics and the Unity of Sensibility

*Waxman Wayne*

### in Kant's Anatomy of the Intelligent Mind

- Published in print:
- 2013
- Published Online:
- January 2014
- ISBN:
- 9780199328314
- eISBN:
- 9780199369348
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199328314.003.0007
- Subject:
- Philosophy, History of Philosophy

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

## The Perils of Parallels!

*Nicholas Mee*

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

- Published in print:
- 2020
- Published Online:
- September 2020
- ISBN:
- 9780198851950
- eISBN:
- 9780191886690
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198851950.003.0016
- Subject:
- Physics, History of Physics

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

## Non-Euclidean Geometry

*Jeremy Gray*

### in Space: A History

- Published in print:
- 2020
- Published Online:
- April 2020
- ISBN:
- 9780199914104
- eISBN:
- 9780190090579
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780199914104.003.0013
- Subject:
- Philosophy, Metaphysics/Epistemology, History of Philosophy

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

## Mathematical Knowledge and the Interplay of Practices

*José Ferreirós*

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691167510
- eISBN:
- 9781400874002
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691167510.001.0001
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

This book presents a new approach to the epistemology of mathematics by viewing mathematics as a human activity whose knowledge is intimately linked with practice. Charting an exciting new direction ... More

## Embrace change, considering non-Euclidean geometry

*Susan D'Agostino*

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

- Published in print:
- 2020
- Published Online:
- April 2020
- ISBN:
- 9780198843597
- eISBN:
- 9780191879388
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198843597.003.0016
- Subject:
- Mathematics, Educational Mathematics, Applied Mathematics

“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

## Rigor and Rigorization

*John P. Burgess*

### in Rigor and Structure

- Published in print:
- 2015
- Published Online:
- May 2015
- ISBN:
- 9780198722229
- eISBN:
- 9780191789076
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198722229.003.0001
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

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

## A Very Short Course in Projective Geometry

*Tony Robbin*

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

- Published in print:
- 2006
- Published Online:
- October 2013
- ISBN:
- 9780300110395
- eISBN:
- 9780300129625
- Item type:
- chapter

- Publisher:
- Yale University Press
- DOI:
- 10.12987/yale/9780300110395.003.0005
- Subject:
- Society and Culture, Technology and Society

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

## Joyce and Geometry

*Ciaran McMorran*

- Published in print:
- 2020
- Published Online:
- September 2020
- ISBN:
- 9780813066288
- eISBN:
- 9780813065267
- Item type:
- book

- Publisher:
- University Press of Florida
- DOI:
- 10.5744/florida/9780813066288.001.0001
- Subject:
- Literature, 20th-century Literature and Modernism

Following the development of non-Euclidean geometries from the mid-nineteenth century onwards, Euclid’s system came to be re-conceived as a language for describing reality rather than a set of ... More

## Semantics and the Notion of Consequence

*Jody Azzouni*

### in Tracking Reason: Proof, Consequence, and Truth

- Published in print:
- 2006
- Published Online:
- October 2011
- ISBN:
- 9780195187137
- eISBN:
- 9780199850570
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195187137.003.0015
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

Contemporary mathematics is the study of algorithmic systems of all sorts. There are two alternatives to the first-order thesis as a descriptive thesis of the logic of mature mathematics. The first ... More

## Ezra Pound and the Materiality of the Fourth Dimension

*Ian F. A. Bell*

### in Science in Modern Poetry: New Directions

- Published in print:
- 2012
- Published Online:
- June 2013
- ISBN:
- 9781846318092
- eISBN:
- 9781846317743
- Item type:
- chapter

- Publisher:
- Liverpool University Press
- DOI:
- 10.5949/UPO9781846317743.009
- Subject:
- Literature, Poetry

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