Doing Mathematics: Convention, Subject, Calculation, Analogy (2nd Edition) - Hardcover

Krieger Martin H

Hardcover

ISBN 10: 9814571830 ISBN 13: 9789814571838

Publisher: World Scientific Publishing Company, 2015

Synopsis

Doing Mathematics discusses some ways mathematicians and mathematical physicists do their work and the subject matters they uncover and fashion. The conventions they adopt, the subject areas they delimit, what they can prove and calculate about the physical world, and the analogies they discover and employ, all depend on the mathematics — what will work out and what won't. The cases studied include the central limit theorem of statistics, the sound of the shape of a drum, the connections between algebra and topology, and the series of rigorous proofs of the stability of matter. The many and varied solutions to the two-dimensional Ising model of ferromagnetism make sense as a whole when they are seen in an analogy developed by Richard Dedekind in the 1880s to algebraicize Riemann's function theory; by Robert Langlands' program in number theory and representation theory; and, by the analogy between one-dimensional quantum mechanics and two-dimensional classical statistical mechanics. In effect, we begin to see "an identity in a manifold presentation of profiles," as the phenomenologists would say.

This second edition deepens the particular examples; it describe the practical role of mathematical rigor; it suggests what might be a mathematician's philosophy of mathematics; and, it shows how an "ugly" first proof or derivation embodies essential features, only to be appreciated after many subsequent proofs. Natural scientists and mathematicians trade physical models and abstract objects, remaking them to suit their needs, discovering new roles for them as in the recent case of the Painlevé transcendents, the Tracy-Widom distribution, and Toeplitz determinants. And mathematics has provided the models and analogies, the ordinary language, for describing the everyday world, the structure of cities, or God's infinitude.

About this title

From the Back Cover

Doing Mathematics discusses some ways mathematicians do their work and the subject matter that is being worked upon and created. It argues that the conventions we adopt, the subject areas we delimit, what we can prove and calculate about the physical world, and the analogies that work for mathematicians, all depend on mathematics what will work out and what won't. And how mathematics, as it is done, is shaped and supported, or not, by convention, subject matter, calculation, and analogy. The cases studied include the central limit theorem of statistics, the sound of the shape of a drum, the connection between algebra and topology, rigorous proofs of the stability of matter, solutions to the two-dimensional Ising model of ferromagnetism, and their connection to the Langlands program in number theory and representation theory and a relationship of number theory, function theory, and analysis begun by Dedekind.

This second edition deepens each chapter: mathematical rigor and the philosophy of mathematics; finance and big data in statistics; the need for perseverance and the inevitable inelegance in a first proof; the recurrent appearance of the Bethe Ansatz and Hopf algebras in these lattice models; solutions of the Kondo model as epitomizing these themes; analogies between one-dimensional quantum mechanics and two-dimensional classical statistical mechanics; Edward Frenkel's use of the Weil threefold-analogy in the geometric Langlands program; the warehouse of mathematical objects and how it is enlarged; and how recent developments in set theory are analogous with developments in systematic theology as attempts to be articulate about what others take as vague or beyond analysis.

Edition details

• Publisher: World Scientific Publishing Company
• Publication date: 2015
• Language: English
• ISBN 10: 9814571830
• ISBN 13: 9789814571838
• Binding: Hardcover
• Edition number: 2
• Number of pages: 492