Edited By Elize Bisanz
The volume focuses on the application of Peirce’s semeiotic as a methodological tool to establish a common field for interdisciplinary research. Contributors from the fields of biology, architecture, logic, esthetics and neuroscience, among others, work on diverse research problems, unified by the idea of transcending the dyadic limitations of disciplinary restrictions and applying Peirce’s triadic method, and the structure and process of sign relations of the particular problem that has to be solved. The result is an invigorating example of methodological plasticity wherein the reader acquires an understanding of scientific observation within the complex universe of semeiosis relations.
Peirce’s “Logic of Number,” Firstorderwise (Thomas G. McLaughlin)
Thomas G. McLaughlin
Peirce’s “Logic of Number,” Firstorderwise1
In his elegant paper (Shields, Paul: “Peirce’s Axiomatization of Arithmetic”. In Houser, Nathan / Roberts, Don D. / Van Evra, James (eds.): Studies in the Logic of Charles Sanders Peirce. Indiana University Press 1997, pp. 43–52), Paul Shields has given a clear and persuasive account of the claim that can be made for Charles Peirce as one of the founders of the formal, or axiomatic, approach to the theory of natural numbers. (That he has not been widely acclaimed as such will be clear to anyone making a casual survey of the literature in the area of “Logic and Foundations,” or taking a casual poll among practitioners in that area.) Naturally enough, Shields’ discussion of Peirce’s axioms is focused on what all Nineteenth Century mathematicians who were interested in axiomatics wanted from their systems: categoricity. That is, the axioms should not only determine a class of structures including the “intended” one that originally prompted their formulation: they should literally define the motivating structure, specifying it and it alone. To do this for arithmetic; it turns out, one needs at least as much set theory as is found in second order logic, i.e., one needs the ability to talk about and manipulate not only the natural numbers themselves, but also arbitrary collections of them; and it is to such collections that the Induction Axiom must be made to apply. None of Peirce’s predecessors or late-19th-century contemporaries would – so...
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