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.
Preface.- Notes on the Contributors.- Introduction; Sten Lindström, Erik Palmgren.-
I. LOGICISM AND NEO-LOGICISM.-
Protocol Sentences for Lite Logicism; John Burgess.- Frege's Context Principle and Reference to Natural Numbers; Øystein Linnebo.- The Measure of Scottish Neo-Logicism; Stewart Shapiro.- Natural Logicism via the Logic of Orderly Pairing; Neil Tennant.-
II. INTUITIONISM AND CONSTRUCTIVE MATHEMATICS.-
A Constructive Version of the Lusin Separation Theorem; Peter Aczel.- Dini's Theorem in the Light of Reverse Mathematics; Josef Berger, Peter Schuster.- Journey in Apartness Space; Douglas Bridges, Luminita Vita.- Relativisation of Real Numbers to a Universe; Hajime Ishihara.- 100 years of Zermelo's Axiom of Choice: What Was the Problem With It?; Per Martin-Löf.- Intuitionism and the Anti-Justification of Bivalence; Peter Pagin.- From Intuitionistic to Point-Free Topology; Erik Palmgren.- Program Extraction in Constructive Mathematics; Helmut Schwichtenberg.- Brouwer's Approximate Fixed-Point Theorem is Equivalent to Brouwer's Fan Theorem; Wim Veldman.-
"Gödel's Modernism: On Set-Theoretic Incompleteness," Revisited; Mark van Atten, Juliette Kennedy.- Tarski's Practice and Philosophy: Between Formalism and Pragmatism; Hourya Benis Sinaceur.- The Constructive Hilbert-Program and the Limits of Martin-Löf Type Theory; Michael Rathjen.- Categories, Structures, and the Frege-Hilbert Controversy: the Status of Meta-Mathematics; Stewart Shapiro.- Beyond Hilbert's Reach?; Wilfried Sieg.- Hilbert and the Problem of Clarifying the Infinite; Sören Stenlund.-