Two Dogmas of Computationalism ORON SHAGRIR Sidney M. Edelstein Center for History and Philosophy of Science, The Hebrew University, Jerusalem, Israel Abstract. This paper challenges two orthodox theses: (a) that computational processes must be algorithmic; and (b) that all computed functions must be Turing-computable. Section 2 advances the claim that the works in computability theory, including Turing’s analysis of the effective computable functions, do not substantiate the two theses. It is then shown (Section 3) that we can describe a system that computes a number-theoretic function which is not Turing-computable. The argument against the first thesis proceeds in two stages. It is first shown (Section 4) that whether a process is algorithmic depends on the way we describe the process. It is then argued (Section 5) that systems compute even if their processes are not described as algorithmic. The paper concludes with a suggestion for a semantic approach to computation. Key words: algorithm, computability, recursive function, Turing-machine, step-satisfaction, analog and digital, attractor neural nets. According to Orthodox Computationalism, computing systems differ from non- computing dynamical physical systems chiefly in two ways: in the causal structure of their processes, and in the type of functions they compute. Specifically, the ortho- dox theses are that (C1) computations are disciplined algorithmic processes, and that (C2) all computed functions must be Turing-computable. Given the Church– Turing thesis, (C2) is entailed by (C1) in the context of computability theory. It will be shown, however, that the relations between the two theses in the context of physical systems are more complex 1 . Both theses, I claim, are articles of faith. They are rooted in a misconception of our pre-theoretic notion of computation. The argument proceeds in four stages. In Section 2, I claim that there is a gulf between the concepts used in computability theory and the concepts needed to define physical computing systems. This observation is often underestimat- ed. Friends of Orthodox Computationalism are tempted to think that the notion of a computing system must be grounded in the rigorous analyses of ‘effective computability’, as analyzed in 1936 by Church, Turing, and others. However, I argue that the concepts analyzed by Church, Turing, and others, do not, in fact, substantiate the orthodox theses. In Section 3, I argue against (C2). I describe an abstract machine that computes a number-theoretic function that is not Turing-computable. The argument against (C1) proceeds in two stages. In Section 4, I argue that the distinction between algorithmic processes and other physical dynamics depends on the way we describe these processes. In Section 5, I argue that describing a process as algorithmic is not necessary for its being computational. I present an attractor neural net that Minds and Machines 7: 321–344, 1997. c 1997 Kluwer Academic Publishers. Printed in the Netherlands.