Charles S. Peirce `On the logic of number' (Q2862668)

\textit{G. Mannoury} [Methodologisches und Philosophisches zur Elementar-Mathematik. Haarlem: P. Visser Azn. (1909; JFM 40.0084.03)] had pointed out that C. S. Peirce had provided the first rigorous foundation for the theory of finite numbers, one that was discovered independently by Dedekind, Frege, and Peano, and lamented that \textit{C. S. Peirce}'s [Sylv., Am. J. IV, 85--96 (1881; JFM 13.0055.02)] had been unjustly ignored.NEWLINENEWLINEMotivated by this remark, the author of this 1981 Fordham University Ph.D. dissertation in philosophy, sets out to present the details of the case and to highlight the ``firsts'' that the above-cited paper introduced. The first chapter presents the foundations of arithmetic and finite sets, including the notion of cardinality, introduced by Dedekind, Peano, and Peirce. In Chapter 2, the axiom systems of Peirce and Dedekind are shown to be equivalent, when phrased in formal logic, on the basis of informal set theory (``consistent with the lower reaches of ZF set theory'' (p.\ 63)). The firsts contained in Peirce's paper include: (i) use of recursive definitions for arithmetic operations, (ii) definition of Dedekind-infinite sets, (iii) the first ordinal construction of cardinals, (iv) the first abstract description of partially and totally ordered sets. Chapter 3 re-evaluates Peirce's contribution and places it in the context of his philosophy and in its historical context.