PREFACE

LECTURE TWENTY-ONE. Order within disorder

  1. The birth of mathematical probability (1654)

LECTURE TWENTY-wo. Moving pictures versus still pictures

  1. The invention of the differential calculus (1629-1680s)

LECTURE TWENTY-THREE. Like opening and closing a door

  1. The fundamental theorem of the calculus (1669-1690s)

LECTURE TWENTY-FOUR. Power(ful) series

  1. Taylor and Maclaurin series (1715, 1742)

LECTURE TWENTY-FIVE. Yea + Yea + Yea + Yea

  1. Fourier series (1807)

LECTURE TWENTY-six. The liberation of geometry, I

  1. The discovery of non-Euclidean geometry (1829)

LECTURE TWENTY-SEVEN. The liberation of geometry, II

  1. The discovery of non-Euclidean geometry (1829)-continued

LECTURE TWENTY-EIGHT. The liberation of algebra, I

  1. The discovery of a noncommutative algebra (1843)

LECTURE TWENTY-NINE. The liberation of algebra, II

  1. The discovery of a noncommutative algebra (1843)-continued

LECTURE THIRTY. An important atomic structure

  1. The group structure (1830-1860)

LECTURE THIRTY-ONE. A remarkable codification

  1. The Erlanger Programm (1872)

LECTURE THIRTY-TWO. Pythagoras justified

  1. The arithmetization of analysis; the natural number system as a foundation of mathematics (end of the nineteenth century)

LECTURE THIRTY-THREE. Digging deeper

  1. Set theory as a foundation of mathematics (end of the nineteenth century)
  2. Abstract spaces (1906)
  3. The function concept refined by set theory (beginning of the twentieth century)

LECTURE THIRTY-FOUR. Beyond the finite

  1. Transfinite numbers (1874-1895)

LECTURE THIRTY-FIVE. Some remarkable definitions

  1. Formal axiomatics (early twentieth century)
  2. A definition of mathematics (early twentieth century)

LECTURE THIRTY-SIX. Some clarifying examples

  1. A definition of mathematics (early twentieth century)continued

LECTURE THIRTY-SEVEN. The third level

  1. Metamathematics (1899-1920)

LECTURE THIRTY-EIGHT. Mathematics as a branch of theology

  1. Godel's incompleteness theorem (1931)

LECTURE THIRTY-NINE. The dream that came true

  1. The modern electronic computing machines (1944)
  2. Resolution of the four-color conjecture (1976)

LECTURE FORTY. Apology and regrets

HINTS FOR THE SOLUTION OF SOMIE OF THE EXERCISES

INDEX

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