The title of this work is modeled on Bertrand Russell's 1903 book The Principles of Mathematics. What is the connection? As I see it, Russell's book was an important step in his struggle to liberate himself from traditional approaches to logic and the foundations of mathematics and to replace them by an approach using as its main tool, and deriving its inspiration from, the new logic created by Frege and Peano. In the Principles, Russell is not yet actually constructing the new foundation for mathematics which he later built with A. N. Whitehead. The Principles is not the Principia. What Russell is doing in the 1903 book is to examine the conceptual problems that arise in the foundations of logic and mathematics, expose the difficulties in the earlier views and by so doing try to find guidelines for the right approach.

In this book, I am hoping in the same spirit to prepare the ground for the next revolution (in Jefferson's sense rather than Lenin's) in the foundations of mathematics. As in Russell, this involves both a critical and a constructive task, even though they cannot be separated from each other. Indeed, if there had not been the danger of confusing bibliographers, I would have given this book the double entendre title The Principles of Mathematics Revis(it)ed.

The critical part of my agenda is the easier one to describe.