Some critical remarks on definitions and on philosophical and logical ideals. (English) Zbl 0897.03006

Odifreddi, Piergiorgio (ed.), Kreiseliana: about and around Georg Kreisel. Wellesley, MA: A K Peters. 417-438 (1996).
The author discusses two subjects to which Georg Kreisel offered challenges. He considers {“[...]} Truth and Usefulness especially applied to definitions in sciences. The second is the area of heroic ideals (here) exemplified with the idea of a comprehensive language for the sciences and with the properties of the language used in logic and mathematics” (p. 417). The first topic is treated by a discussion of the truth (validity) or the usefulness of definitions. In particular the author discusses the conceptions of Russell and Quine and their application to empirical sciences. Concerning definitions in logic and mathematics he considers Kreisel’s opinion that the usefulness of a definition can never be clear at the very beginning but only be realized on the basis of specific experiences with applications of this definition.
The heroic ideal of a universal language as a maximum precise instrument for all sciences is treated by discussing the programs of Leibniz and Carnap. In particular the author describes the steps towards Leibniz’s universal language as based on his arithmetical calculus. From this he distinguishes Carnap’s conception of a logical syntax. He discusses the problems of Carnap’s notion of analyticity in connection with logicism, coming to the conclusion that “nowadays no mature philosopher or logician would believe anymore in these two traditional ideals” (p. 433). Weingartner finally unmasks, following Kreisel, logical closure as a contemporary heroic ideal. He takes as an example usual systems of epistemic logic where maintaining logical closure leads to the properties of deductive infallibility and logical omniscience, incompatible with human rationality.
For the entire collection see [Zbl 0894.03002].


03A05 Philosophical and critical aspects of logic and foundations
00A30 Philosophy of mathematics
01A60 History of mathematics in the 20th century
01A45 History of mathematics in the 17th century