Worlds Out of Nothing

A Course in the History of Geometry in the 19th Century

Jeremy J. Gray - Collection Springer Undergraduate Mathematics Series (SUMS)

376 pages, parution le 01/02/2007

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Résumé

Worlds Out of Nothing is the first book to provide a course on the history of geometry in the 19th century. Based on the latest historical research, the book is aimed primarily at undergraduate and graduate students in mathematics but will also appeal to the reader with a general interest in the history of mathematics. Emphasis is placed on understanding the historical significance of the new mathematics: Why was it done? How - if at all - was it appreciated? What new questions did it generate?

Topics covered in the first part of the book are projective geometry, especially the concept of duality, and non-Euclidean geometry. The book then moves on to the study of the singular points of algebraic curves (Plücker's equations) and their role in resolving a paradox in the theory of duality; to Riemann's work on differential geometry; and to Beltrami's role in successfully establishing non-Euclidean geometry as a rigorous mathematical subject. The final part of the book considers how projective geometry, as exemplified by Klein's Erlangen Program, rose to prominence, and looks at Poincaré's ideas about non-Euclidean geometry and their physical and philosophical significance. It then concludes with discussions on geometry and formalism, examining the Italian contribution and Hilbert's Foundations of Geometry; geometry and physics, with a look at some of Einstein's ideas; and geometry and truth.

Three chapters are devoted to writing and assessing work in the history of mathematics, with examples of sample questions in the subject, advice on how to write essays, and comments on what instructors should be looking for.

L'auteur - Jeremy J. Gray

Jeremy Gray is the author or co-author of ten books, most recently Hidden Harmony – Geometric Fantasies: the rise of complex function theory (Springer 2013) with Umberto Bottazzini (Milan), upon which this book is based. He is also the author of Henri Poincaré: a scientific biography (Princeton U.P. 2012) and Plato’s Ghost: The Modernist Transformation of Mathematics (Princeton U.P. 2008). In 2009 he was awarded the Albert Leon Whiteman Memorial Prize of the American Mathematical Society for his work in the history of mathematics, and he was elected an inaugural Fellow of the American Mathematical Society in 2012.

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Sommaire

Mathematics in the French Revolution.
Poncelet (and Pole and Polar).
Theorems in Projective Geometry.
Poncelet's Traité.
Duality and the Duality Controversy.
Poncelet and Chasles.
Lambert and Legendre.
Gauss.#Janos Bolyai.
Lobachevskii.
To 1855.
Writing.
Möbius.
The Duality Paradox.
The Plücker Formulae.
Higher Plane Curves.
Complex Curves.
Riemann.
Differential Geometry of Surfaces.
Non-Euclidean Geometry Accepted.
Writing.#Fundamental Geometry.
Hilbert.#Italian Foundations.
The Disc Model.
The Geometry of Space.
Summary: Geometry to 1900.
The Formal Side.
The Physical Side.
Is Geometry True?
Writing.
Appendix: Von Staudt and his Influence.-

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Caractéristiques techniques

	PAPIER
Éditeur(s)	Springer
Auteur(s)	Jeremy J. Gray
Collection	Springer Undergraduate Mathematics Series (SUMS)
Parution	01/02/2007
Nb. de pages	376
Format	18 x 23,5
Couverture	Broché
Poids	652g
Intérieur	Noir et Blanc
EAN13	9781846286322