This book discusses some ways of doing mathematical 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 the 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, the stability of matter, the Ising model, and the Langlands Program in number theory and representation theory.
Contents:
- Convention: How Means and Variances are Entrenched as Statistics
- Subject: The Fields of Topology
- Appendix: The Two-Dimensional Ising Model of a Ferromagnet
- Calculation: Strategy, Structure, and Tactics in Applying Classical Analysis
- Analogy: A Syzygy Between a Research Program in Mathematics and a Research Program in Physics
- In Concreto: The City of Mathematics
Readership: Mathematicians, physicists, philosophers and historians of science.
“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
“The book Doing Mathematics by Martin Krieger is truly a masterpiece. He has not only explained ways of doing mathematical work to the aspiring mathematicians and the intelligent laymen, but also shown how the 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
“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
Martin Krieger has taught at the University of California (Berkeley), the University of Minnesota (Twin Cities), MIT, and the University of Michigan (Ann Arbor). He has been a fellow at the Center for Advanced Study in the Behavioral Sciences and at the National Humanities Center. He is professor of planning at the University of Southern California. Professor Krieger was trained as physicist.Professor Krieger's earlier books include Marginalism and Discontinuity, Tools for the Crafts of Knowledge and Decision (1989), Doing Physics, How Physicists Take Hold of the World (1992), and Constitutions of Matter, Mathematically Modeling the Most Everyday of Physical Phenomena (1996).
