Doing Mathematics ─ Convention, Subject, Calculation, Analogy

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.