Effective procedures and computable functions
- 46 Downloads
Horsten and Roelants have raised a number of important questions about my analysis of effective procedures and my evaluation of the Church-Turing thesis. They suggest that, on my account, effective procedures cannot enter the mathematical world because they have a built-in component of causality, and, hence, that my arguments against the Church-Turing thesis miss the mark. Unfortunately, however, their reasoning is based upon a number of misunderstandings. Effective mundane procedures do not, on my view, provide an analysis of ourgeneral concept of an effective procedure; mundane procedures and Turing machine procedures are different kinds of procedure. Moreover, the same sequence ofparticular physical action can realize both a mundane procedure and a Turing machine procedure; it is sequences of particular physical actions, not mundane procedures, which “enter the world of mathematics.” I conclude by discussing whether genuinely continuous physical processes can “enter” the world of real numbers and compute real-valued functions. I argue that the same kind of correspondence assumptions that are made between non-numerical structures and the natural numbers, in the case of Turing machines and personal computers, can be made in the case of genuinely continuous, physical processes and the real numbers.
Key wordsChurch-Turing thesis effective procedure mundane procedure Turing machine procedure isomorphism compute mirror follow
Unable to display preview. Download preview PDF.
- Cleland, Carol E. (1993), ‘Is the Church-Thesis True?’,Minds and Machines 3, pp. 283–312.Google Scholar
- Fetzer, James H. (1994), ‘Mental Algorithms: Are Minds Computational Systems?’,Pragmatics & Cognition 21(1), pp. 1–29.Google Scholar
- Horsten, Leon & Roelants, Herman (1995), ‘The Church-Turing Thesis and Effective Mundane Procedures’,Minds and Machines 5, pp. 1–8 (this issue).Google Scholar
- Lewis, Harry R. & Papadimitrious, Christos H. (1981),Elements of the Theory of Computation, Englewood Cliffs: Prentice-Hall.Google Scholar
- Minsky, Marvin (1967),Computation: Finite and Infinite Machines, Englewood Cliffs: Prentice-Hall.Google Scholar
- Turing, Alan (1964), ‘Computing Machinery and Intelligence’, in A. Anderson, ed.,Minds and Machines, New Jersey: Prentice Hall, pp. 4–30.Google Scholar
- Turning, Alan (1965), ‘Systems of Logic Based on Ordinals’, in M. Davis, ed.,The Undecidable, New York: Raven Press, pp. 155–222.Google Scholar