Doing Mathematics: Convention, Subject, Calculation, Analogy 2nd edition (PDF) discusses some ways mathematical physicists and mathematicians do their work and the subject matters they uncover and fashion. The conventions they adopt, what they can prove and calculate about the physical world, the subject areas they delimit, 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 connections between topology and algebra, 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 1-dimensional quantum mechanics and 2-dimensional classical statistical mechanics. In effect, we begin to see “an identity in a manifold presentation of profiles,” as the phenomenologists would say.
This 2nd 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” derivation or first proof 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 Tracy-Widom distribution, the Painlevé transcendents, and Toeplitz determinants. And math has provided the analogies and models, the ordinary language, for describing the everyday world, the structure of cities, or God’s infinitude.
NOTE: Doing Mathematics: Convention, Subject, Calculation, Analogy 2e PDF does not come with any online codes.