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What Is the Concept of Computation?

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Sailing Routes in the World of Computation (CiE 2018)

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Abstract

This essay examines the two central and equivalent ways of introducing rigorous notions of computation as starting-points. The first way explicates effective calculability of number theoretic functions as the (uniform) calculability of their values in formal calculi; Gödel, Church and Kleene initially pursued this way. The other way views mechanical procedures as transforming finite configurations via production rules, a path Turing and Post took. For both explications, one can prove important mathematical theorems, namely, absoluteness and reducibility results. These results are of great interest, but problematic when viewed as justifying the adequacy of the rigorous notions. However, the theorems and their proofs reveal features that motivate the abstract concept of a computable dynamical system. That is, so I argue, the appropriate structural axiomatization of computation.

This essay is dedicated to Martin Davis—mentor, friend, collaborator—on the occasion of his 90\(^\text {th}\) birthday.

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Notes

  1. 1.

    Gödel states then that an appeal to mechanical procedures “is required by the concept of formal system, whose essence it is that reasoning is completely replaced by mechanical operations on formulas.”

  2. 2.

    The most ambitious undertaking along these lines is the work of Dershowitz and Gurevich in their (2008) paper. My perspective on their mathematical work is detailed in (Sieg 2013a, 116–123).

  3. 3.

    \(\{1\}_0\), denotes the chain of the system \(\{1\}\) with respect to f, i.e., the intersection of all systems that contain 1 as an element and are closed under f.

  4. 4.

    In his letter to Gödel written on 30 October 1938, Post pointed out that “the whole force” of his argument for his analysis of “all finite processes of the human mind” depended on “identifying my ‘normal systems’ with any symbolic logic and my sole basis for that were the reductions I mentioned.” This letter was published in volume V of Gödel’s Collected Works.

  5. 5.

    Kolmogorov introduced operations on special kinds of graphs in his paper with Uspensky, (1963). Byrnes and I took up their work in our (1996) and generalized it to K-graphs. The production rules for those graphs are viewed as formalizations of generalized Turing machines operating on K-graphs.

  6. 6.

    For Post, there is a genuine philosophical difficulty, as he programmatically appeals to (subtle) properties of the human mind; see (Sieg et al. 2016, Sects. 7.6 and 7.7).

  7. 7.

    (Turing 1954, 15). I have analyzed this extremely interesting article, which was intended for a general audience, in (Sieg 2012). One straightforward observation should be mentioned. Turing discusses all the methodological issues surrounding “Turing’s Thesis”, but does not at all appeal to computing machines.

  8. 8.

    The mathematical details are found in my (2002) and (2009), where the earlier paper is Sect. 5.

  9. 9.

    E.g., artificial neural nets with a variety of learning algorithms, but also with back propagation.

  10. 10.

    These concerns, with respect to Gödel and Turing, are exposed in my (2013b).

  11. 11.

    In quite different ways, this goal has been pursued by (Ganesalingam and Gowers 2017) and (Sieg and Walsh 2017).

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Authors and Affiliations

  1. Carnegie Mellon University, Pittsburgh, USA

    Wilfried Sieg

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  1. Wilfried Sieg
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Correspondence to Wilfried Sieg .

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Editors and Affiliations

  1. Kiel University, Kiel, Germany

    Florin Manea

  2. Queens College, CUNY, Queens, New York, USA

    Russell G. Miller

  3. Kiel University, Kiel, Germany

    Dirk Nowotka

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Sieg, W. (2018). What Is the Concept of Computation?. In: Manea, F., Miller, R., Nowotka, D. (eds) Sailing Routes in the World of Computation. CiE 2018. Lecture Notes in Computer Science(), vol 10936. Springer, Cham. https://doi.org/10.1007/978-3-319-94418-0_39

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