Lettered selves and beyond -- The alphabetic body -- Gesture and non-alphabetic writing -- Technologized mathematics -- Parallel selves -- Ghost effects.

Starting from Poincaré’s assignment of an algebraic object to a topological manifold, namely the fundamental group, this article introduces the concept of categories and their language of arrows that has, since their mid-20th-century inception, altered how large areas of mathematics, from algebra to abstract logic and computer programming, are conceptualized. The assignment of the fundamental group is an example of a functor, an arrow construction central to the notion of a category. The exposition of category theory’s arrows, which operate at (...) three distinct but deeply interconnected levels, is framed by a comparison with the language and outlook of set theory founded on the concept of membership; sets and their theorization having provided, famously through the Bourbaki initiative, the basic ontological and epistemological vocabulary for defining and handling all mathematical entities. The comparison with sets emphasizes how categories offer a form of diagrammatic argument and thought against set theory’s fidelity to syntax-based proofs; how categories invert set theory’s priority of objects and their attributes over relations by making the relations of an object to others of its kin primary; and how categories replace identity, that is, equality, between objects, by the weaker notion of isomorphism, restricting equality to identity between arrows. The article concludes with a return to topology and some remarks about the question of its possible use in articulating/characterizing cultural dynamics. (shrink)

This ambitious work puts forward a new account of mathematics-as-language that challenges the coherence of the accepted idea of infinity and suggests a startlingly new conception of counting. The author questions the familiar, classical, interpretation of whole numbers held by mathematicians and scientists, and replaces it with an original and radical alternative--what the author calls non-Euclidean arithmetic. The author's entry point is an attack on the notion of the mathematical infinite in both its potential and actual forms, an attack organized (...) around his claim that any interpretation of "endless" or "unlimited" iteration is ineradicably theological. Going further than critique of the overt metaphysics enshrined in the prevailing Platonist description of mathematics, he uncovers a covert theism, an appeal to a disembodied ghost, deep inside the mathematical community's understanding of counting. (shrink)

Existing work on the ultimate limits of computation has urged that the apparatus of real numbers should be eschewed as an investigative tool and replaced by discrete mathematics. The present paper argues for a radical extension of this viewpoint: not only the continuum but all infinitary constructs including the rationals and the potential infinite sequence of whole numbers need to be eliminated if a self-consistent investigative framework is to be achieved.