Principia Mathematica

For Isaac Newton's book containing basic laws of physics, see Philosophiæ Naturalis Principia Mathematica

The Principia Mathematica is a three-volume work on the foundations of mathematics by Alfred North Whitehead and Bertrand Russell. It was published in 1910, 1912, and 1913. In 1927, it appeared in a second edition with an important Introduction to the second edition, and different notes at the end. It is often known as PM.

The title page of the shortened version of the Principia Mathematica to *56

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 books, 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....

Hardy, G. H. (2004) [1940]. A Mathematician's Apology. Cambridge: University Press. p. 83. ISBN 978-0-521-42706-7.

The book was an attempt to describe a set of axioms, inference rules and law of noncontradiction in symbolic logic from which all mathematical truths could in principle be proved. This ambitious project is of great importance in the history of mathematics and philosophy. The authors believed that such a project could be done. However, in 1931, Gödel's incompleteness theorem proved that PM, and any other attempt, could never reach this goal. For any set of axioms and inference rules proposed, either the system must be inconsistent, or there must in fact be some truths of mathematics which could not be deduced from them.

One of the main inspirations and motivations for PM was the earlier work of Gottlob Frege on logic.

PM is not to be confused with Russell's 1903 Principles of Mathematics. PM states: "The present work was originally intended by us to be ... a second volume of Principles of Mathematics... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed..."

The Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentieth century.