The concept of computability

I explore the conceptual foundations of Alan Turing’s analysis of computability, which still dominates thinking about computability today. I argue that Turing’s account represents a last vestige of a famous but unsuccessful program in pure mathematics, viz., Hilbert’s formalist program. It is my contention that the plausibility of Turing’s account as an analysis of the computational capacities of physical machines rests upon a number of highly problematic assumptions whose plausibility in turn is grounded in the formalist stance towards mathematics. More speci5cally, the Turing account con6ates concepts that are crucial for understanding the computational capacities of physical machines. These concepts include the idea of an “operation” or “action” that is “formal,” “mechanical,” “well-de5ned,” and “precisely described,” and the idea of a “symbol” that is “formal,” “uninterpreted,” and “shaped”. When these concepts are disentangled, the intuitive appeal of Turing’s account is signi5cantly undermined. This opens the way for exploring models of hypercomputability that are fundamentally di:erent from those currently entertained in the literature. c © 2003 Elsevier B.V. All rights reserved.