The principles of mathematics revisited. (English) Zbl 0869.03003

Cambridge: Cambridge Univ. Press. xii, 288 p. (1996).

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This is a book written in order to present the strength and the advantages of a new logical tool in the philosophy of mathematics by recalling the main problems and topics in this field of research. This tool is Independence Friendly (IF) first-order logic, developed by the author in close connection with Gabriel Sandu. Sandu contributed a paper of his own to this volume entitled “IF first-order logic, Kripke, and 3-valued logic” (pp. 254-270), which will be reviewed separately. IF first-order logic allows the verifiers of quantifiers in a formula to be independent of each other. E.g. in the formula \((\forall x)(\exists y/\forall x) \text{S} [x,y]\) in which the specific slash notation is used to express that the choice by the verifier of a value of \(y\) must be made in ignorance of the value of \(x\). IF first-order logic is as basic as ordinary first-order logic, but much stronger since it allows to avoid most second-order considerations.
The author has organized his material in 11 chapters. Chapter 1 on “The Function of Logic and the Problem of Truth Definitions” gives a very readable account of the role of logic in mathematical theorizing, considering the status of truth definitions, model theory, and the limitations of set theory as the basis of mathematics. Chapter 2, “The Game of Logic,” gives a game-theoretical semantics for first-order logic. In chapter 3 Independence Friendly first-order logic is introduced. Under the heading “The Joy of Independence” the next chapter gives some uses of IF logic, e.g. in quantum theory, Ramsey Theory in combinatorics, computer architecture, matters of principle in the foundations of mathematics, and epistemic logic. Chapter 5, “The Complexities of Completeness” tries to make the best of the fact that IF first-order logic does not admit a complete axiomatization. Starting from the distinction between several senses of completeness, i.e., descriptive completeness, semantical completeness, deductive completeness, and the so-called “Hilbertian completeness”, the author shows that Gödel’s incompleteness theorem does not touch directly “the most important sense of completeness and incompleteness” (p. 95), i.e., descriptive completeness and incompleteness. In chapter 6 the author asks “Who’s Afraid of Alfred Tarski?”, arguing for game-theoretical truth definitions rather than for adopting Tarski’s theory of truth. He gives a first-order truth predicate of the form \((\exists X)(\text{TR}[X] \& X(y))\) where \(\text{TR}[X]\) guarantees that there is a winning strategy for the initial verifier in a semantical game (pp. 114-115). He stresses that his predicate shows that “one can develop a model theory for the powerful IF first-order languages on the first-order level” (p. 129). In chapter 7 (“The Liar Belied”) negation in IF logic is discussed. In chapter 8 on “Axiomatic Set Theory” the author argues that the privileged role of axiomatic first-order set theory as a medium of choice for model-theoretical theorizing “becomes extremely dubious in the light of the insights that the game-theoretical approach has yielded, and will yield” (p. 163). Chapter 9 gives some examples for the achievements of “IF Logic as a Framework for Mathematical Theorizing.” The author gives, e.g., some definitions of mathematical concepts in IF first-order logic or its extended version such as the notion of well-ordering, the principle of mathematical induction, the notion of power set, the Bolzano-Weierstrass theorem, the topological notions of open set and of continuity. In the last two chapters the relation of IF logic to constructivism (“Constructivism Reconstructed”) is discussed and “The Epistemology of Mathematical Objects” is re-considered.
In sum, the book is very stimulating, provoking a non-standard view on the relation between logic and mathematics. Its programme resembles at several places Paul Lorenzen’s operative mathematics and the constructivism of his Erlangen school which is, however, not mentioned at all.


03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
00A30 Philosophy of mathematics
03A05 Philosophical and critical aspects of logic and foundations
03B20 Subsystems of classical logic (including intuitionistic logic)