From Dedekind to Gödel. Essays on the development of the foundations of mathematics
[The articles of this volume will not be indexed individually.] The volume under review originated from a conference held at Boston University in April 1992 as a part of the 1991-92 program of the Boston Colloquium in the Philosophy of Science. One of the aims of the conference was to raise the consciousness of philosophers of the fact that still not much is known about the foundations of mathematics at the turn of the century, that though there are many books and papers on Frege and Russell, little attention was paid to many ideas in the foundational studies of various mathematicians like Weierstrass, Dedekind, Kronecker, Borel, Weyl, etc. The goal of the conference was to explore those important but relatively neglected episodes or lines in the development of the foundations of mathematics. Since the papers included in the book will not be reviewed individually, it is reasonable to present them shortly in this review. \textit{J. Webb}'s paper ``Tracking contradictions in geometry: The idea of a model from Kant to Hilbert'' discusses the problem of who actually discovered non-euclidean geometry, who actually believed in its consistency and why, and who can be said to have proved it to be free of contradiction. The author stresses the significance of the idea of a model in this domain. The paper ``Standard vs. nonstandard distinction'', by \textit{J. Hintikka}, is devoted to the discussion of a conceptual distinction, or a contrast, between the standard and the nonstandard interpretation of higher-order logic. A natural sequel to this paper is the paper ``Wittgenstein and Ramsey on infinity'' (not presented at the conference) by \textit{M. Marion}. It is shown in it that the standard/nonstandard distinction provides the proper context in which differences between Wittgenstein and Ramsey on identity are to be understood. \textit{H. M. Edwards} analyses in his paper Kronecker's views on the foundations of mathematics. \textit{D. Ch. McCart}'s paper ``The mysteries of Richard Dedekind'' is devoted to three mysteries in Dedekind's work: (1) his ``proof'' that there exists an infinite set, (2) the mystery around his famous essay on continuity where he claimed to have captured the essence of the continuum, (3) his ``free creation'' (Freie Schöpfung). \textit{C. O. Hill} in his paper ``Frege's letters'' attempts to ``engage in the (\dots) task of systematically trying to piece together what we actually can know about the philosophical content of (Frege's) letters that have been lost (it is known that Frege exchanged letters with many leading people in his field). Another paper devoted to Frege is \textit{R. G. Heck}'s (\textit{jun.}) paper on Frege's principle where Frege's Axiom V is analysed and discussed. A second paper by \textit{C. O. Hill} inquires into the origins of Husserl's ideas on completeness. The author discusses also Husserl's opinions on the role of the ideal elements. In the paper by \textit{Ph. Ehrlich}, the introduction of the idea of an Archimedean complete ordered field by H. Hahn and its influence on the development of the modern theory of magnitudes is discussed. \textit{G. H. Moore} is concerned in his paper with the light that the correspondence of Russell and Couturat in the years 1897-1903 sheds on the origins of the paradox of the largest cardinal and on Russell's Paradox. The emergence of descriptive set theory is presented in the paper by \textit{A. Kanamori}, where contributions of Cantor, Baire, Lebesgue, Hausdorff, Łuzin, Sierpiński, Suslin, von Neumann, Mirimanoff, Gödel and others are analysed. The paper by \textit{J. von Plato} ``Change against constructibility'' is devoted to the origins of modern probability theory. Connections with the change of the sense of existence in mathematics (construction vs. consistency) is discussed. \textit{W. Boss} discusses in his paper Skolem's plea for metalogical relativism and Weyl's tendencies toward a ``predicative'' variant of ``transcendental idealism'' trying to mediate between them. \textit{J. Woleński} in his paper ``On Tarski's background'' attempts to ``throw light on the cognitive conflict or dissonance of Tarski between his nominalistic and empiricistic sympathies and his ``Platonic'' mathematical practice as well as why he was so parsimonious in expressing his philosophical views''. The paper by \textit{J. Floyd} (not presented at the conference) is devoted to the discussion of the problem of reactions of Wittgenstein on Gödel's incompleteness theorem. This is done in the context of the place of the trisection example in the ``Philosophical Investigations'' and the general background of Wittgenstein's interest in Gödel. The paper by \textit{D. Føllesdal}, which closes the volume, is devoted to the discussion of connections between philosophical views of Gödel and Husserl. Summing up one should stress that the volume under review is a valuable and interesting source of information for the discussions on the foundations of mathematics at the turn of the century. Several rather less known facts and events are presented in it. The volume will certainly stimulate further research in this field.
scientific article; zbMATH DE number 808187