From Logic to Physics: How the Meaning of Computation Changed over Time

Authors:
Itamar Pitowsky
Affiliations:
The Edelstein Center, Levi Building, The Hebrew University, Jerusalem, Israel
Venue:
CiE '07 Proceedings of the 3rd conference on Computability in Europe: Computation and Logic in the Real World
Year:
2007

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Abstract

The common formulation of the Church-Turing thesis runs as follows:Every computable partial function is computable by a Turing machineWhere by partial function I mean a function from a subset of natural numbers to natural numbers. As most textbooks relate, the thesis makes a connection between an intuitive notion (computable function) and a formal one (Turing machine). The claim is that the definition of a Turing machine captures the pre-analytic intuition that underlies the concept computation. Formulated in this way the Church-Turing thesis cannot be proved in the same sense that a mathematical proposition is provable. However, it can be refuted by an example of a function which is not Turing computable, but is nevertheless calculable by some procedure that is intuitively acceptable.