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Doing Mathematics: Convention, Subject, Calculation, Analogy

Martin H. Krieger
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
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See our review of the first edition. The author says in the preface

In this second edition, I have tried to deepen and clarify the text. I have come to understand more of some of the mathematics and the examples, and so I have been able to better discern my themes. In the last decade there have been remarkable advances, and some of them are relevant to the discussion. The Prolog epitomizes these themes.

The themes highlighted in the Prolog are

  • What we want when we ask "what is really going on in the mathematics?".
  • How ideas and calculations play off each other.
  • Fields of mathematics are dynamic.
  • Ugly proofs have an inner beauty.
  • Analogy is destiny.
  • Details and fine points matter.

The mathematical themes through which Krieger tries to build his argument are hearing the shape of a drum, periodic regularity and self-similarity, the connection between local and global, and general notions of "adding up".

The book is intended to be accessible to non-mathematicians "with suitable skipping."

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.

  • Introduction
  • 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
  • Appendices:
    • 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)