Abstract
In this paper we analyze methodological and philosophical implications of algorithmic aspects of unconventional computation. At first, we describe how the classical algorithmic universe developed and analyze why it became closed in the conventional approach to computation. Then we explain how new models of algorithms turned the classical closed algorithmic universe into the open world of algorithmic constellations, allowing higher flexibility and expressive power, supporting constructivism and creativity in mathematical modeling. As Gödel’s undecidability theorems demonstrate, the closed algorithmic universe restricts essential forms of mathematical cognition. In contrast, the open algorithmic universe, and even more the open world of algorithmic constellations, remove such restrictions and enable new, richer understanding of computation.
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References
Frege, G.: Grundgesetze der Arithmetik, Begriffschriftlich Abgeleitet, Viena (1893/1903)
Nelson, R.J.: Church’s thesis and cognitive science. Notre Dame J. of Formal Logic 28(4), 581–614 (1987)
Kalmar, L.: An argument against the plausibility of Church’s thesis. In: Constructivity in Mathematics, pp. 72–80. North-Holland Publishing Co., Amsterdam (1959)
Gödel, K.: Some Remarks on the Undecidability Results (1986–1995); Gödel, K.: Collected Works, vol. II, pp. 305–306. Oxford University Press, Oxford (1972)
Burgin, M.: Super-recursive Algorithms. Springer, Heidelberg (2005)
Sloman, A.: The Irrelevance of Turing machines to AI Aaron Sloman. In: Scheutz, M. (ed.) Computationalism: New Directions. MIT Press (2002), http://www.cs.bham.ac.uk/~axs/
Shannon, C.: Mathematical Theory of the Differential Analyzer. J. Math. Physics 20, 337–354 (1941)
Abramsky, S., Jung, A.: Domain theory. In: Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science, vol. III. Oxford University Press (1994)
Edalat, A.: Domains for computation in mathematics, physics and exact real arithmetic. Bulletin of Symbolic Logic 3, 401–452 (1997)
Ko, K.: Computational Complexity of Real Functions. Birkhauser, Boston (1991)
Weihrauch, K.: Computable Analysis. An Introduction. Springer, Heidelberg (2000)
Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. In: Perspectives in Mathematical Logic, vol. 1, Springer, Berlin (1989)
Burgin, M.: The Notion of Algorithm and the Turing-Church Thesis. In: Proceedings of the VIII International Congress on Logic, Methodology and Philosophy of Science, Moscow, vol. 5, part 1, pp. 138–140 (1987)
Burgin, M.: Nonlinear Phenomena in Spaces of Algorithms. International Journal of Computer Mathematics 80(12), 1449–1476 (2003)
Gödel, K.: Über formal unentscheidbare Sätze der Principia Mathematics und ver-wandter System I. Monatshefte für Mathematik und Physik 38, 173–198 (1931)
Smullian, R.M.: Theory of Formal Systems. Princeton University Press (1962)
Mathias, A.R.D.: The Ignorance of Bourbaki. Physis Riv. Internaz. Storia Sci (N.S.) 28, 887–904 (1991)
Rogers, H.: Theory of Recursive Functions and Effective Computability. MIT Press, Cambridge Massachusetts (1987)
Burgin, M.: Arithmetic Hierarchy and Inductive Turing Machines. Notices of the Russian Academy of Sciences 299(3), 390–393 (1988)
Penrose, R.: The Emperor’s New Mind. Oxford University Press, Oxford (1989)
Burgin, M.: Topological Algorithms. In: Proceedings of the ISCA 16th International Conference “Computers and their Applications”, pp. 61–64. ISCA, Seattle (2001)
Rossini, C.: Access to Knowledge as a Foundation for an Open World. EDUCAUSE Review 45(4), 60–68 (2010)
Budapest Open Access Initiative, http://www.soros.org/openaccess/read.shtml
Nielsen, M.: Reinventing Discovery: The New Era of Networked Science. Princeton University Press, Princeton and Oxford (2012)
Dodig-Crnkovic, G., Muller, V.C.: A Dialogue Concerning Two World Systems. In: Information and Computation, pp. 107–148. World Scientific, New York (2011)
Rozenberg, G., Bäck, T.H.W., Kok, J.N. (eds.): Handbook of Natural Computing. Springer, Heidelberg (2012)
Dodig Crnkovic, G., Burgin, M.: Unconventional Algorithms: Complementarity of Axiomatics and Construction. Entropy, Special issue “Selected Papers from the Symposium on Natural/Unconventional Computing and its Philosophical Significance” (forthcoming) (2012), http://www.mdpi.com/journal/entropy/special_issues/unconvent_computing
Dodig-Crnkovic, G.: Significance of Models of Computation from Turing Model to Natural Computation. Minds and Machines 21(2), 301 (2011)
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Burgin, M., Dodig-Crnkovic, G. (2013). From the Closed Classical Algorithmic Universe to an Open World of Algorithmic Constellations. In: Dodig-Crnkovic, G., Giovagnoli, R. (eds) Computing Nature. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 7. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37225-4_16
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DOI: https://doi.org/10.1007/978-3-642-37225-4_16
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