Tuesday, 15 February 2011
The Absolute Arithmetic Continuum and its Peircean Counterpart
Ohio University, Department of Philosophy
12:05 pm, 817R Cathedral of Learning
Abstract: In a number of earlier works, we have suggested that whereas the real number system should be regarded as constituting an arithmetic continuum modulo the Archimedean axiom—the axiom that precludes the existence of infinitesimals—the ordered field No of surreal numbers may be regarded as a sort of absolute arithmetic continuum (modulo standard set theory). In the present paper, we introduce a formal replacement for the intriguing extension of the classical linear continuum sketched by Charles Sanders Peirce at the turn of the twentieth century, and point out that by limiting No to its substructure NoP consisting of its finite and infinitesimal members, one obtains a model of this Peircean linear continuum, as we call it, whose properties mimic the remarkable properties of No. In the course of so doing, we will also clarify some of the senses in which a Peircean linear continuum, so defined, realizes some of the key characteristics envisioned by Peirce for his purported linear continuum as well as draw attention to some of the differences that exist between Peirce’s intuitive conception and our proposed formal replacement thereof. Some of the differences point to limitations in Peirce’s intuitive conception while others are byproducts of the underlying set-theoretic framework which, in accordance with standard geometrical practice and in marked contrast with Peirce, treats the collection of points on a line as an actual as opposed to a potential collection. Moreover, unlike Peirce, we will go beyond a largely order-theoretic exploration of Peircean linear continua by shedding light on the ordered algebraic structure NoP inherits from No. Finally, we will suggest how modifications can be made in the theory to better capture the modal nature of Peirce’s original vision.