The Principles of Mathematics
The book presents a view of the foundations of mathematics and has become a classic reference. It reported on developments by Giuseppe Peano, Mario Pieri, Richard Dedekind, Georg Cantor, and others. In 1937 Russell prepared a new introduction saying, "Such interest as the book now possesses is historical, and consists in the fact that it represents a certain stage in the development of its subject." Further editions were printed in 1938, 1951, 1996, and 2009.
Early reviews 
Reviews were prepared by G. E. Moore and Charles Sanders Peirce, but Moore's was never published and that of Peirce was brief and somewhat dismissive. He indicated that he thought it unoriginal, saying that the book "can hardly be called literature" and "Whoever wishes a convenient introduction to the remarkable researches into the logic of mathematics that have been made during the last sixty years [...] will do well to take up this book." However, a long and generally favorable review was written by G. H. Hardy and appeared in Times Literary Supplement (Issue #88, 18 September 1903). Hardy titles his review "The Philosophy of Mathematics" and expects the book to appeal more to philosophers than mathematicians. But he says
- [I]n spite of its five hundred pages the book is much too short. Many chapters dealing with important questions are compressed into five or six pages, and in some places, especially in the most avowedly controversial parts, the argument is almost too condensed to follow. And the philosopher who attempts to read the book will be especially puzzled by the constant presupposition of a whole philosophical system utterly unlike any of those usually accepted.
In 1904 another review appeared in Bulletin of the American Mathematical Society (11(2):74–93) written by Edwin Bidwell Wilson.He says "The delicacy of the question is such that even the greatest mathematicians and philosophers of to-day have made what seem to be substantial slips of judgement and have shown on occasions an astounding ignorance of the essence of the problem which they were discussing. ... all too frequently it has been the result of a wholly unpardonable disregard of the work already accomplished by others." Wilson recounts the developments of Peano that Russell reports, and takes the occasion to correct Henri Poincaré who had ascribed them to David Hilbert. In praise of Russell, Wilson says "Surely the present work is a monument to patience, perseverance, and thoroughness."(page 88)
Later reviews 
In 1959 Russell wrote My Philosophical Development, in which he recalled the impetus to write the Principles:
- It was at the International Congress of Philosophy in Paris in the year 1900 that I became aware of the importance of logical reform for the philosophy of mathematics. ... I was impressed by the fact that, in every discussion, [Peano] showed more precision and more logical rigour than was shown by anybody else. ... It was [Peano's works] that gave the impetus to my own views on the principles of mathematics.
Recalling the book after his later work, he provides this evaluation:
- The Principles of Mathematics, which I finished on 23rd May, 1902, turned out to be a crude and rather immature draft of the subsequent work [Principia Mathematica], from which, however, it differed in containing controversy with other philosophies of mathematics.
Such self-deprecation from the author after half a century of philosophical growth is understandable. On the other hand, Jules Vuillemin wrote in 1968
- The Principles inaugurated contemporary philosophy. Other works have won and lost the title. Such is not the case with this one. It is serious, and its wealth perseveres. Furthermore, in relation to it, in a deliberate fashion or not, it locates itself again today in the eyes of all those that believe that contemporary science has modified our representation of the universe and through this representation, our relation to ourselves and to others.
Russell's Principles of 1903 also looms large in Ivor Grattan-Guinness' study of the roots of modern logic.
In 2006, Philip Ehrlich challenged the validity of Russell's analysis of infinitesimals in the Leibniz tradition. A recent study documents the non-sequiturs in Russell's critique of the infinitesimals of Gottfried Leibniz and Hermann Cohen.
The Principles of Mathematics consists of 59 chapters divided into seven parts: indefinables in mathematics, number, quantity, order, infinity and continuity, space, matter and motion.
In chapter one, "Definition of Pure Mathematics," Russell asserts that:
The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists in the analysis of Symbolic Logic itself.
There is an anticipation of relativity physics in the final part as the last three chapters consider Newton's laws of motion, absolute and relative motion, and Hertz's dynamics. However, Russell rejects what he calls "the relational theory", and says on page 489
- For us, since absolute space and time have been admitted, there is no need to avoid absolute motion, and indeed no possibility of doing so.
In his review, Hardy (1903) says "Mr. Russell is a firm believer in absolute position in space and time, a view as much out of fashion nowadays that Chapter [58: Absolute and Relative Motion] will be read with peculiar interest."
- Russell, Bertrand (1903 (1st ed), 1938 (2nd ed)). Principles of Mathematics (2nd edition). W.W. Norton. ISBN 0-393-00249-7. "The fundamental thesis of the following pages, that mathematics and logic are identical, is one which I have never since seen any reason to modify." The quotation is from the first page of Russell's introduction to the second (1938) edition.
- Quin, Arthur (1977). The Confidence of British Philosophers. p. 221. ISBN 90-04-05397-2.
- See the first paragraph of his review of What is Meaning? and The Principles of Mathematics (1903), The Nation, v. 77, n. 1998, p. 308, Google Books Eprint, reprinted in Collected Papers of Charles Sanders Peirce v. 8 (1958), paragraph 171 footnote. The review was publicly anonymous like the other reviews (totaling over 300) that Peirce wrote for The Nation on a regular basis. Murray Murphy called the review "so brief and cursory that I am convinced that he never read the book." in Murphy, Murray (1993). The Development of Peirce's Philosophy. Hackett Pub. Co. p. 241. ISBN 0-87220-231-3. Others such as Norbert Wiener and Christine Ladd-Franklin shared Peirce's view of Russell's work. See Anellis, Irving (1995), "Peirce Rustled, Russell Pierced", Modern Logic 5, 270-328.
- Russell, My Philosophical Development, p. 65.
- Russell, My Philosophical Development, p. 74.
- J. Vuillemin (1968), p. 333.
- *Ehrlich, Philip (2006), "The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence of non-Archimedean systems of magnitudes", Archive for History of Exact Sciences 60 (1): 1–121, doi:10.1007/s00407-005-0102-4
- Katz, Mikhail; Sherry, David (2012), "Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond", Erkenntnis, arXiv:1205.0174, doi:10.1007/s10670-012-9370-y.
- Bertrand Russell, Principles of Mathematics (1903), p. 5
- Louis Couturat (1905) Les Principes des Mathematiques: avec un appendice sur la philosophie des mathématiques de Kant. Republished 1965, Georg Olms.
- Ivor Grattan-Guinness (2000) The Search for Mathematical Roots 1870-1940: Logics, Set Theories, and the Foundations of Mathematics from Cantor through Russell to Gödel. Princeton Univ. Press. ISBN 0-691-05858-X. See pages 292–302 and 310–326.
- Jules Vuillemin (1968) Leçons sur la primière philosophie de Russell, Paris: Colin.