This book grew out of two conferences held in August 2004 at Uppsala University: “Logicism, Intuitionism, and Formalism” and “A Symposium on Constructive Mathematics”. Twenty-four mathematicians made contributions to the book in three broad sections, namely: Logicism and Neo-Logicism; Intuitionalism and Constructive Mathematics; and Formalism. Generally, the book is about what has happened as a result of the various attempts, since the publications of Gödel disturbed the thinking of mathematicians about basic mathematical concepts, to put mathematics on a more solid foundation.

I was pleased that almost all the authors gave interesting historical insights into the foundational approach of each school of discussion and then followed up with modern results and research programs currently underway in those areas. The impression from the book is that various schools of philosophy of mathematics are alive and well, perhaps fragmenting into various sub-branches in a kind of a revival of foundational mathematics. Intuitionism and Constructive mathematics is proving extremely useful in theoretical computer science problems and other domains of applications for the other schools seems to be emerging.

At the core, the different mathematical schools differ mostly by what they consider constitutes a proof and what would best be taken as a foundation of mathematics. The schools also differ in the fundamental properties that statements can have. Statement “properties” discussed in the book included not only truth values, but also provability, decidability, computability, constructibility and allowability of statements. Thus, it is meaningful to talk about statements that are “true” but not “provable”, or statements that are “provable” but not “computable”, and so on.

Some topics covered in the book include abstraction principles, choice principles, FAN and BAR theorems, axiomatic set theory, reverse mathematics and category theory. Other topics of interest to foundational mathematics include meta-mathematical statements; Gödel’s theorems, the Liar’s paradox and other statements that reference fundamental statement properties, alternatives to axiomatic set theory for foundations; role of recursive definitions; role of recursive proofs; and the role of mathematical induction. One interesting section dealt with principle of totality in context of using the operation of pairing as a foundation for mathematics something like set theory is often used.

The book comes nicely typeset, with a good solid binding, and a solid index. While the sections are somewhat independent the book flows nicely. Overall, I found the book highly readable and enjoyable. My only caveat is the steep price; hopefully the publisher will create an e-book version to make it more affordable to a wider audience.

Collin Carbno is a specialist in process improvement and methodology. He holds a Master’s of Science Degree in theoretical physics and completed course work for Ph.D. in theoretical physics (relativistic rotating stars) in 1979 at the University of Regina. He has been employed for nearly 30 years in various IT and process work at Saskatchewan Telecommunications and currently holds a Professional Physics Designation from the Canadian Association of Physicists, and the Information System Professional designation from the Canadian Information Process Society.