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."

[3]Russell deconstructs pure mathematics with relations, by positing them, their converses and complements as primitive notions.

Russell had written previously on foundations of geometry, denoting, and relativism of space and time, so those topics are recounted.

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.

Wilson recounts the developments of Peano that Russell reports, and takes the occasion to correct Henri Poincaré who had ascribed them to David Hilbert.

James Feibleman, an admirer of the book, thought Russell's new preface went too far into nominalism so he wrote a rebuttal to this introduction.

Then in 2000 Grattan-Guinness published The Search for Mathematical Roots 1870 – 1940, which considered the author's circumstances, the book's composition and its shortcomings.

[15] A recent study documents the non-sequiturs in Russell's critique of the infinitesimals of Gottfried Leibniz and Hermann Cohen.