My problem with this book is less its content (the discussion on "classes" is fantastic) than the fact that Russell falls into the same hole as so many other philosopher-mathematicians: he doesn't actually know how to write. This is not as regards the symbolic aspect or definition of terms; it's that his prose is so overly loaded in "therefore we must as to proceed"-style of verbosity that it obscures half the points he tries to make. Part of this is the evolution of pedagogy over the last century of mathematics, part of it is that Russell's introduction simultaneously assumes unfamiliarity and familiarity with the concepts that he constantly oversteps into greater rigor. (Ironically, the last chapter anticipates and discusses this.)

It's an interesting primer, but one that's pretty much been overshadowed in concept (that all maths can come from philosophical axioms) by Godel's work decades later... in addition to their simply being much, much better-written works out there on maths and philosophy. Tobias Dantzig's "Number" is recommended for actual laypersons.