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Doing Mathematics

Author : Martin H Krieger
ISBN : 9789814571869
Genre : Mathematics
File Size : 46. 37 MB
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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. Contents:IntroductionConvention: How Means and Variances are Entrenched as StatisticsSubject: The Fields of TopologyAppendix: The Two-Dimensional Ising Model of a FerromagnetCalculation: Strategy, Structure, and Tactics in Applying Classical AnalysisAnalogy: A Syzygy Between a Research Program in Mathematics and a Research Program in PhysicsIn Concreto: The City of MathematicsAppendices:The Spontaneous Magnetization of a Two-Dimensional Ising Model (C N Yang)On the Dirac and Schwinger Corrections to the Ground-State Energy of an Atom (C Fefferman and L A Seco)Sur la Forme des Espaces Topologiques et sur les Points Fixes des Représentations (J Leray)Une Lettre à Simone Weil (A Weil) Readership: Mathematicians, physicists, philosophers and historians of science. Keywords:Means and Variances;Topology;SyzygyReviews: Reviews of the First Edition: "The book Doing Mathematics, by Martin Krieger is truly a masterpiece. He has not only explained ways of doing mathematical work to aspiring mathematicians and the intelligent laymen, but has also shown how various pieces of research work are related to each other. Even experts may not have realized such inter-relations. The cases studied include, especially, the stability of matter and the Ising model, two topics of great depth. Such clear explanations cannot be found anywhere else. Furthermore, his style of writing makes the book exceptionally enjoyable to read." T T Wu Gordon McKay Professor of Applied Physics Professor of Physics, Harvard University, USA "This is the first time I have seen a mathematician deal substantively with the issue of mathematics as culturally based, and he does it superbly and mathematically … Although this book is no easy read, it is well worth the effort, and I am sure it will stimulate and inform, perhaps even surprise, the most sophisticated of mathematical readers. It is refreshing to find such a book being published." Mathematical Reviews "Both challenging and provocative reading, Doing Mathematics sheds bright light on some of the main characteristics of the mathematical quest." Library of Science "Krieger has made some effort to accommodate different levels of readers; for example, structuring his text so that lay readers are alerted to sections that can be safely skipped and paragraphs that provide nontechnical summaries." Mathematical Association of America

Doing Mathematics

Author : Martin H. Krieger
ISBN : 9814571849
Genre : Mathematics
File Size : 66. 72 MB
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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.

Doing Physics

Author : Martin H. Krieger
ISBN : 9780253006073
Genre : Science
File Size : 84. 22 MB
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Doing Physics makes concepts of physics easier to grasp by relating them to everyday knowledge. Addressing some of the models and metaphors that physicists use to explain the physical world, Martin H. Krieger describes the conceptual world of physics by means of analogies to economics, anthropology, theater, carpentry, mechanisms such as clockworks, and machine tool design. The interaction of elementary particles or chemical species, for example, can be related to the theory of kinship—who can marry whom is like what can interact with what. Likewise, the description of physical situations in terms of interdependent particles and fields is analogous to the design of a factory with its division of labor among specialists. For the new edition, Krieger has revised the text and added a chapter on the role of mathematics and formal models in physics. Doing Physics will be of special interest to economists, political theorists, anthropologists, and sociologists as well as philosophers of science.

The Year Of Our Lord 1943

Author : Alan Jacobs
ISBN : 9780190864668
Genre : Religion
File Size : 56. 75 MB
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By early 1943, it had become increasingly clear that the Allies would win the Second World War. Around the same time, it also became increasingly clear to many Christian intellectuals on both sides of the Atlantic that the soon-to-be-victorious nations were not culturally or morally prepared for their success. A war won by technological superiority merely laid the groundwork for a post-war society governed by technocrats. These Christian intellectuals-Jacques Maritain, T. S. Eliot, C. S. Lewis, W. H. Auden, and Simone Weil, among others-sought both to articulate a sober and reflective critique of their own culture and to outline a plan for the moral and spiritual regeneration of their countries in the post-war world. In this book, Alan Jacobs explores the poems, novels, essays, reviews, and lectures of these five central figures, in which they presented, with great imaginative energy and force, pictures of the very different paths now set before the Western democracies. Working mostly separately and in ignorance of one another's ideas, the five developed a strikingly consistent argument that the only means by which democratic societies could be prepared for their world-wide economic and political dominance was through a renewal of education that was grounded in a Christian understanding of the power and limitations of human beings. The Year of Our Lord 1943 is the first book to weave together the ideas of these five intellectuals and shows why, in a time of unprecedented total war, they all thought it vital to restore Christianity to a leading role in the renewal of the Western democracies.

Mathematical Reviews

Author :
ISBN : UOM:39015067268279
Genre : Mathematics
File Size : 35. 70 MB
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Proofs And Confirmations

Author : David M. Bressoud
ISBN : 0521666465
Genre : Mathematics
File Size : 33. 90 MB
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This introduction to recent developments in algebraic combinatorics illustrates how research in mathematics actually progresses. The author recounts the dramatic search for and discovery of a proof of a counting formula conjectured in the late 1970s: the number of n x n alternating sign matrices, objects that generalize permutation matrices. While it was apparent that the conjecture must be true, the proof was elusive. As a result, researchers became drawn to this problem and made connections to aspects of the invariant theory of Jacobi, Sylvester, Cayley, MacMahon, Schur, and Young; to partitions and plane partitions; to symmetric functions; to hypergeometric and basic hypergeometric series; and, finally, to the six-vertex model of statistical mechanics. This volume is accessible to anyone with a knowledge of linear algebra, and it includes extensive exercises and Mathematica programs to help facilitate personal exploration. Students will learn what mathematicians actually do in an interesting and new area of mathematics, and even researchers in combinatorics will find something unique within Proofs and Confirmations.

Books In Print Supplement

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ISBN : STANFORD:36105025417838
Genre : American literature
File Size : 22. 45 MB
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Metaphor In Educational Discourse

Author : Lynne Cameron
ISBN : UCSC:32106015890491
Genre : Education
File Size : 71. 49 MB
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This book reports research into metaphor in use with school students. It is one of the first studies of metaphor to investigate the phenomenon in contextualised discourse, to adopt a socio-cultural approach to metaphor and to introduce the relatively new science of complex dynamic systems which offers new and potentially fruitful analogies for the analysis of educational discourse.

Proceedings Of The Annual Convention Held At

Author : American Concrete Institute. Convention
ISBN : UCAL:B2920078
Genre : Building
File Size : 35. 39 MB
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English Mechanics And The World Of Science

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ISBN : UCAL:$C208747
Genre : Technology
File Size : 66. 36 MB
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