DOING MATHEMATICS: CONVENTION, SUBJECT, CALCULATION, ANALOGY(2ND EDITION)

Book ID/图书代码： 14600015B79566

English Summary/英文概要： 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.

Chinese Summary/中文概要： 《做数学》讨论了数学家们和数学物理学家们做数学的一些方式和那些他们尚未发现和使用的课题。他们采用的惯例，他们界定的课题领域，他们能证明和计算的物理世界，以及他们发现和采用的等比，这些都依靠数学—哪些能实现，哪些不能。本书研究了包括统计学的中心极限定理，鼓状物的声音，代数和拓扑学之间的联系和一系列关于物质稳定性的严密证明。解出了铁磁体二维伊辛模型的许多不同的答案，它们作为一个整体，可以在一系列等比中看到，如19世纪80年代由理查德•戴德金到黎曼的函数论，罗伯特•朗兰兹的数字论和表示论，以及一维量子力学和二维经典统计力学的类比。事实上，现象学家会说我们开始看到了“方面资料的认证”。

第二版更深入地举了具体例子；本书写了数学严密性所扮演的实际角色；它说明了第一级证明或体现基本特征的导数有多“简陋”。自然科学家们和数学家们交换物理模型和抽象个体，重整它们让它们符合他们的需要，像在近期的潘勒韦的超验理论，特雷西-威顿的广义函数，常对角矩阵的案例里发现的它们的新角色。数学为描述我们生活的世界，城市的构成或上帝的无限提供了模型，类比和日常用语。（YYW）

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