Traditional logic and the early history of sets, 1854-1908
This long article attempts a fresh look at well-known developments in the foundations of mathematics and set theory in the mid and late 19th century. The main attention is cast upon Riemann and Dedekind. Some nice touches include the various ways in which `Mannigfaltigkeit' was used, and the set theory implicit in Dedekind's early work in the general theory of equations; but some other novelties are unwelcome. It should have been made clear that `traditional logic' (actually there were several different ones) deployed a part-whole theory of collections whereas the mathematical logic drew upon the new theory of sets (Cantor's and to some extent Dedekind's). The algebraic logic growing out of Boole, traditional from this point of view, is strongly contrasted with mathematical logic from this but also other points of view [see the reviewer's `Living together and living apart: on the interactions between mathematics and logics from the French Revolution to the First World War', South African journal of philosophy 7, No. 2, 73-82 (1988)]. Similarly, Dedekind, Frege, Schröder and Russell are all put forwarded as advocates of `the' logistic philosophy of mathematics, whereas all four differ considerably. Schröder's envisioned logicism as an exhaustive cataloguing of mathematical theories using versions of various algebraic theories, of which Dedekind was an important contributor cast into Schröder's logic of relations. By contrast, the other three saw it as some sort of construction from allegedly basic concepts (fn. 89 is to be rejected). Among them set-theorist Dedekind seems to have followed a largely Kantian view of logic (see the rather better recent treatment by \textit{D. C. McCarty} [`The mysteries of Richard Dedekind', in J. Hintikka (ed.), From Dedekind to Gödel. Essays in the development of the foundations of mathematics. (Dordrecht: Kluwer, 1995), pp. 53-96] avoided by Frege and Russell. The absence of any discussion of Peano and his school, very influential mathematical logicians but not logistics, is painful.
scientific article; zbMATH DE number 881718
scientific article