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.
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.
- 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.
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.
- 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.
(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.
- 9.
E.g., artificial neural nets with a variety of learning algorithms, but also with back propagation.
- 10.
These concerns, with respect to Gödel and Turing, are exposed in my (2013b).
- 11.
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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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