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Summary: CHURCH WITHOUT DOGMA:
Axioms for computability
Wilfried Sieg
Carnegie Mellon University
Pittsburgh
2
Abstract: Church's and Turing's theses dogmatically assert that an informal notion of effective
calculability is adequately captured by a particular mathematical concept of computabilty. 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.*
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