One significant difference in our experiences was that analysis at Berkeley, such as the definition of an integral alluded to in the book, was given via epsilon delta arguments that I think would have pleased Russell. These sorts of arguments began to develop in the 1820’s with Cauchy who began to make definitions and theorems that were in easy sight of being rigorous. Perhaps Berkeley was unusual in presenting a rigorous calculus course. Most of the other courses were also were a stone’s throw from rigor. The art of making these arguments really rigorous and yet short, is still underdeveloped. With background from earlier Berkeley courses on the development of the reals, based on Zermelo-Fraenkel set theory, one could see without undue effort the rigorous versions of the proofs for most courses.
I think that Russell assumed that the search for rigor would proceed by proposing axioms and rules for deduction and then going thru a process of pretending to believe only what you had proved—of course being guided by what you knew but had not yet proven.
It was some years later that I realized that meta-mathematics did not take this track. The new discipline did not forego any previous mathematical knowledge and applied all available mathematical tools in deducing what could be deduced from foundational axioms. This is the essence of model theory which takes a ‘God’s eye view’ of sets and wonders which sets can be captured by various axiom sets. While I was at Berkeley I was unaware of these two levels and was thus confused when I sat in on graduate logic courses by Tarski and others. (There were other impediments too.) Careful reading of the book shows this too.