Principia Mathematica

book on the foundations of mathematics

The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1927, it appeared in a second edition with an important Introduction To the Second Edition, an Appendix A that replaced ✸9 and an all-new Appendix C.

Title page of the Principia Mathematica


"From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2." (✸54·43)
  • The present work was originally intended by us to be comprised in a second volume of The Principles of Mathematics. With that object in view, the writing of it was begun in 1900. But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have now arrived at what we believe to be satisfactory solutions. It therefore became necessary to make our book independent of The Principles of Mathematics.
    • Preface, vol. I, p. iii
  • We have found it necessary to give very full proofs, because otherwise it is scarcely possible to see what hypotheses are really required, or whether our results follow from our explicit premisses.
    • Preface, vol.I, p. vi
  • From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2.
    • ✸54·43. In vol. I, part II, "Prolegomena to Cardinal Arithmetic", section A, "Unit Classes and Couples", ✸54, "Cardinal Couples", p. 362
  • The above proposition [1 + 1 = 2] is occasionally useful.
    • ✸110·643. In vol. II, part III, "Cardinal Arithmetic", section B, "Addition, Multiplication and Exponentiation", ✸110, "The arithmetical sum of two classes and of two cardinals", p. 83

Quotes about Principia Mathematica

  • "But why all these pages to prove that 1 + 1 = 2?"

    "Hm … how shall I put it? It's the price you pay for being truly certain."

  • I can remember Bertrand Russell telling me of a horrible dream. He was in the top floor of the University Library, about A.D. 2100. A library assistant was going round the shelves carrying an enormous bucket, taking down book after book, glancing at them, restoring them to the shelves or dumping them into the bucket. At last he came to three large volumes which Russell could recognize as the last surviving copy of Principia Mathematica. He took down one of the volumes, turned over a few pages, seemed puzzled for a moment by the curious symbolism, closed the volume, balanced it in his hand and hesitated …
  • As a work in mathematics, Principia Mathematica soon became obsolete. Symbolic logic is also a field of philosophy, and so the study of major works from the past has a special role unlike that of the study of the history of mathematics.
    • Bernard Linsky: The Evolution of Principia Mathematica (2011)
  • It takes a huge chunk of this volume just to prove that 1 + 1 = 2. And a large part of that proof revolves around the problems of the finite and the infinite, and the paradoxes that Cantor's work had thrown up.
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