The Princeton Companion to Mathematics edited by Timothy Gowers, June Barrow-Green and Imre Leader - ISBN 9780691118802 - PUP 2008



Getting an overview of mathematics as a whole, not as a fragmented set of fields. Also highly recommended by Wagner. Also, wanting to be able to leverage Shun-ichi Amari's work (including Methods of Information Geometry) and thus requiring a better understanding of several key mathematical notions.

Pre-reading model

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  • I Introduction
    • 1 What Is Mathematics About?
      • 1 Algebra, Geometry, and Analysis
      • 2 The Main Branches of Mathematics
    • 2 The Language and Grammar of Mathematics
    • 3 Some Fundamental Mathematical Definitions
      • 4 Functions between Algebraic Structures
        • 3 Eigenvalues and Eigenvectors
      • 6 What Is Geometry?
        • 9 Manifolds and Differential Geometry
          • 1 Smooth Manifolds
        • 10 Riemannian Metrics
    • 4 The General Goals of Mathematical Research
  • III Mathematical Concepts
    • 17 Dimension
    • 18 Distributions
    • 21 Elliptic Curves
    • 56 Matroids
      • "matroids form a natural home for many optimization problems" (p248)
  • IV Branches of Mathematics
    • 2 Analytic Number Theory
      • 1 Introduction
    • 3 Computational Number Theory
      • 4 The Riemann Hypothesis and the Distribution of the Primes
    • 6 Algebraic Topology
    • 7 Differential Topology
    • 20 Computational Complexity by Oded Goldreich and Avi Wigderson
  • VI Mathematicians
    • 33 Niels Henrik Abel
    • 35 Carl Gustav Jacob Jacobi
    • 36 Peter Gustav Lejeune Dirichlet
    • 37 William Rowan Hamilton
    • 49 Georg Friedrich Bernhard Riemann
    • 53 Sophus Lie
    • 61 Jules Henri Poincaré
    • 63 David Hilbert
    • 65 Jacques Hadamard
    • 88 Andrei Nikolaevich Kolmogorov
      • (to find) Information Theory and Algorithms (seems inexistant!)
    • 89 Alonzo Church
    • 91 John von Neumann
    • 92 Kurt Gödel
    • 94 Alan Turing
    • 96 Nicolas Bourbaki
  • VII The Influence of Mathematics
    • 2 Mathematical Biology
      • 9 Phylogenetics and Graph Theory
    • 4 The Mathematics of Traffic in Networks
      • 5 Flow Control in the Internet
    • 5 The Mathematics of Algorithm Design
    • 7 Mathematics and Cryptography
  • VIII Final Perspectives
    • 6 Advice to a Young Mathematician
      • III Alain Connes
        • "Each generation builds a mental picture that reflects their own understanding of this world. They construct mental tools that penetrate more and more deeply into it, so that they can explore aspects of it that were previously hidden." (p1012)

See also


  • Mathematics Examples of Wolfram|Alpha
  • Sage Notebook create, collaborate on, and publish interactive worksheets. In a worksheet, one can write code using Sage, Python, and other software included in Sage.
  • PLUME, Thème : mathématiques logiciels utilisés dans les domaines de la recherche, de l’enseignement en mathématiques (pures comme appliquées), mais aussi ceux relevant de cette science (calcul scientifique, statistiques …) et pratiqués dans d’autres disciplines (physique, chimie, finance …).
  • Tricki store of useful mathematical problem-solving techniques.

Overall remarks and questions

  • main editor Timothy Gowers started the polymath project "using the comment functionality of his blog to produce mathematics collaboratively".
  • seems there is no reference to Ronald Fisher


So in the end, it was about X and was based on Y.


Point A, B and C are debatable because of e, f and j.


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Post-reading model

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Other read books linking to the The Princeton Companion to Mathematics page :

The Mathematician's Mind

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