Hintikka's Revolution: The Principles of Mathematics Revisited

The title of this book refers to Bertrand Russell's The Principles of Mathematics, but it does not deal with Russell's ideas. Jaakko Hintikka sees his book in a similar role to that enjoyed by Russell's, almost a century ago. It is 'crisis science', aimed at establishing a new paradigm in the foundations of mathematics. Hintikka presents a new formalism and a new logic, together with a new semantics. Whether this grandiose claim is borne out or not, the new 'paradigm' is provocative and insightful, and a dedicated reader will learn a lot from a careful study of this work. However, it is not an easy read. The formalisms and some of the ideas are not developed fully, and so the reader is left to supply many of the details, often via guesswork. The model is Principles of Mathematics and not Principia Mathematica. Unfortunately, there are also a number of substantial errors. The process of working through these was most educational for both of us, but we were not able to resolve every case. In addition, the book is marred by many typos, some of which destroy the intended sense. Chapter 1 concerns the purposes of logic in foundational studies, and sets the stage for the technical development to follow. Hintikka focuses on two distinct but related goals of logical studies. One is to codify the process of inference within a branch of mathematics, and the other is descriptive, where one develops a formal language to describe mathematical structures. In contemporary logic, the first goal is served by proof theory and the second by model theory. Hintikka argues that logicians have placed too much emphasis on the proof-theoretic aspects of logic at the expense of the model-theoretic. He does his part to reverse this imbalance by almost ignoring the deductive aspects of the language he develops in this book. Chapter 2 contains a brief account of Hintikka's game-theoretic semantics. Begin with a standard, first-order language whose logical terminology is