1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Congress of Mathematicians (ICM) in Paris has tremendous importance for all mathematicians. Moreover, a substantial part of

Examples are discussed of natural statements about irrational numbers that are equivalent, provably in ZFC, to strong set-theoretical hypotheses, and of apparently classical statements provable in ZFC of which the only known proofs use strong set-theoretical concepts. 1.

The term structuralism occurred in several branches of the humanities and the sciences in the period 1929 – 1970: in Linguistics (Ferdinand de Saussure, Roman Jakobson), Anthropology (Claude Lévi-Strauss), Developmental psychology (Jean Piaget), Literature (Workshop for potential literature, Raymond Queneau) and in Mathematics (Nicolas Bourbaki). To the layman the structuralist movement in mathematics was perhaps most visible the form of New Math, which was strongly influenced by the Bourbaki school. It has been argued in (Aubin 1997) that there were cultural connections between these movements. (See also A. Aczel 2007.) Some of these interactions may be regarded as rather superficial. The epistemologist Piaget however was very much influenced by Bourbaki, and seems to have suggested that those patterns of thought used to explain cognitive development were closely related to the mathematical “mother structures ” found by Bourbaki. On a very general level, structuralism refers to a mode of thinking involving abstraction from specifics and systematic identification and naming of common patterns. It is the relation of objects under study to each other that is of importance rather than their specific appearance, or “nature”. In mathematics, Richard Dedekind may be said to be the first structuralist. He described the positive integers (1, 2, 3,...) as positions in an infinite progression of elements (a so-called simply infinite system) 1 � � 2

We begin with a short (obviously incomplete) historical summary related to the book being reviewed. A complex number α is said to be “algebraic ” if there is a polynomial 0 ̸ = p(x) with integer coefficients such that p(α) = 0; otherwise α is said to be “transcendental”. The first recorded instance of mathematicians encountering an “irrational ” algebraic number appears to be the ancient Greeks. Indeed, if one considers a square with unit sides, then, of course, by the Pythagorean Theorem a diagonal must have length √ 2. To Pythagoras (and his followers) is also due the result that √ 2 cannot be expressed as the quotient of integers. One story has it that Hippasus, the person who revealed this irrationality, died at sea in a shipwreck, “struck by the wrath of the gods.” As history evolved, so did mathematicians ’ views about numbers. During the sixteenth century cubic equations were discussed by the Italian algebraists G. Cardano and R. Bombelli ([v1], Part C §2). In particular a very curious phenomenon appeared with equations like x 3 =15x +4. Here one readily finds that x =4isa root and, after division by x−4, that the other two roots are also real. However, the

Abstract. This is a critical reexamination of several pieces in van Heijenoort’s Selected Essays that are directly or indirectly concerned with the philosophy of logic or the relation of logic to natural language. Among the topics discussed are absolutism and relativism in logic, mass terms, the idea of a rational dictionary, and sense and identity of sense in Frege. Mathematics Subject Classification. 03-03, 03A05.