Copyright © 2010–2015 All rights reserved. Jonathan Bennett [Brackets] enclose editorial explanations. Small ·dots· enclose material that has been added, but can be read as though it were part of the original text. Occasional •bullets, and also indenting of passages that are not quotations, are meant as aids to grasping the structure of a sentence or a thought. Every four-point ellipsis . . . . indicates the omission of a brief passage that seems to present more difficulty than it is (...) worth. (shrink)

Church's and Turing's theses dogmatically assert that an informal notion of effective calculability is adequately captured by a particular mathematical concept of computability. I present an analysis of calculability that is embedded in a rich historical and philosophical context, leads to precise concepts, but dispenses with theses. To investigate effective calculability is to analyze symbolic processes that can in principle be carried out by calculators. This is a philosophical lesson we owe to Turing. Drawing on that lesson and recasting work (...) of Gandy, I formulate boundedness and locality conditions for two types of calculators, namely, human computing agents and mechanical computing devices (discrete machines). The distinctive feature of the latter is that they can carry out parallel computations. The analysis leads to axioms for discrete dynamical systems (representing human and machine computations) and allows the reduction of models of these axioms to Turing machines. Cellular automata and a variety of artificial neural nets can be shown to satisfy the axioms for machine computations. (shrink)

Wilfred Sieg. Mechanical Procedures and Mathematical Experience.

Artifacts are objects intentionally made to serve a given purpose; natural objects come into being without human intervention. I shall argue that this difference does not signal any ontological deficiency in artifacts qua artifacts. After sketching my view of artifacts as ordinary objects, I’ll argue that ways of demarcating genuine substances do not draw a line with artifacts on one side and natural objects on the other. Finally, I’ll suggest that philosophers have downgraded artifacts because they think of metaphysics as (...) resting on a distinction between what is “mind­independent” and what is “mind­ dependent.” I’ll challenge the use of any such distinction as a foundation for metaphysics. (shrink)

Accelerated Turing machines are Turing machines that perform tasks commonly regarded as impossible, such as computing the halting function. The existence of these notional machines has obvious implications concerning the theoretical limits of computability.