The key word of the title, foundationalism, can be defined as “the view that it is possible and desirable to reconstruct each branch of mathematics on a completely secure basis, one that is maximally immune to rational doubt”. However, the subtitle of the book determines its contents more precisely, the central thesis being that second-order (or, more generally, higher-order) logic plays an important role in the foundations of mathematics. The strongest support of such a thesis is the fact that second-order logic provides more convenient models for basic mathematical notions and theories than first-order logic.
The book consists of three parts, entitled as follows: “Orientation” (I), “Logic and Mathematics” (II), and “History and Philosophy” (III).
In the first part, consisting of two chapters: “Terms and questions” (Chapter 1), “Foundationalism and foundations of mathematics” (Chapter 2), the general purposes of logic, the problems of formal modeling of a natural language, and the essence of epistemological foundationalism in mathematics are treated. The author analyzes the relationship between two basic conceptions of logic: the foundational and the semantic one.
The second part may be considered as a technical development of second- and higher-order logic. Through the four chapters: “Theory” (Chapter 3), “Metatheory” (Chapter 4), “Second-order logic and mathematics” (Chapter 5), and “Advanced metatheory” (Chapter 6), the fundamental notions of first-order and higher-order logic are presented, including standard and Henkin semantics, followed by a corresponding treatment of completeness and compactness results and the Löwenheim-Skolem theorems.
The last part of the book consists of three chapters: “The historical ‘triumph’ of first-order languages” (Chapter 7), “Second-order logic and rule-following” (Chapter 8), and “The competition” (Chapter 9). In this part, various philosophical arguments in favor of the acceptance of second-order logic are presented.
Throughout the book, different mathematical, logical and philosophical notions, problems and viewpoints are illustrated with corresponding examples. And finally, the merit of the author’s marvelous style, making this book interesting, exciting, polemic and informative to specialists in logic, foundations, and philosophy of mathematics, but, at the same time, accessible to a large circle of readers, is to be praised.