The progress of science from a computational point of view: the drive towards ever higher solvability

Marciszewski, Witold

doi:10.2478/fcds-2019-0002

Abstract

This essay’s content is rendered by the titles of the successive sections. 1. Effective solvability versus intuitive solvability. — 2. Decidability, i.e. effective solvability, in predicate logic. The speedup phenomenon — 3. Contributions of the second-order logic to the problems of solvability — 4. The infinite progress of science in the light of Turing’s idea of the oracle. The term “oracle” is a technical counterpart of the notion of mathematical intuition.

A more detailed summary can be obtained through juxtaposing the textboxes labelled with letters A...F. Conclusion: in the progress of science an essential role is played by the feedback between intellectual intuitions (intuitive solvability) and algorithmic procedures (effective solvability).

References

[1] Benzmüller Ch., Kerber M., A Challenge for Mechanized Deduction (to find the full text in Web, ask Google for the title and select the relevant PDF), 2001.
Search in Google Scholar Back to article
[2] Benzmueller Ch., Brown Ch., The Curious Inference of Boolos in Mizar and OMEGA, in: Studies in Logic, Grammar and Rhetoric (http://logika.uwb.edu.pl/studies/index.php?page=search&vol=23) 23, 2007.
Search in Google Scholar Back to article
[3] Boolos G., A Curious Inference?, Journal of Philosophical Logic, 16, 1987, 1-12.10.1007/BF00250612
Search in Google Scholar Back to article
[4] Buss S.R., On Godel’s Theorems on Lengths of Proofs I: Number of Lines and Speedup for Arithmetics, J. Symbolic Logic, 59, 3, 1994, 737-756,.10.2307/2275906
Search in Google Scholar Back to article
[5] Fischer M.J., Rabin M.O., Super-Exponential Complexity of Presburger Arithmetic, Proceedings of the SIAM-AMS Symposium in Applied Mathematics, 7, 1974, 27-41.
Search in Google Scholar Back to article
[6] Fraenkel A.A., Abstract Set Theory, North Holland, 1976.
Search in Google Scholar Back to article
[7] Gödel K., Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik, 38, 1931, 173-198.10.1007/BF01700692
Search in Google Scholar Back to article
[8] Gödel K., Über die Lange von Beweisen, Ergeb. Math. Kolloquiums, 7, 1936, 23-24.
Search in Google Scholar Back to article
[9] Gödel K., Kurt Gödel Collected Works, vol. 1. Oxford Univ. Press, Oxford, 1986.
Search in Google Scholar Back to article
[10] Hartmanis J., Stearns R., On the computational complexity of algorithms, Transactions of the AMS, 117, 1965, 285-306.10.1090/S0002-9947-1965-0170805-7
Search in Google Scholar Back to article
[11] Hilbert D., Ackermann W., Grundzüge der theoretischen Logik, Springer, 1928.
Search in Google Scholar Back to article
[12] Hilbert D., Naturerkennen und Logik, Naturwissenschaften, Heft 47/48/49, 28.II.1930, 959-963.10.1007/BF01492194
Search in Google Scholar Back to article
[13] Hilbert D., Bernays P., Grundlagen der Mathematik, Springer, (vol. 1) 1934, (vol. 2) 1939.
Search in Google Scholar Back to article
[14] Kneale W., Kneale M., The Development of Logic, Clarendon Press, 1962.
Search in Google Scholar Back to article
[15] Kuhn T., The Structure of Scientific Revolutions, University of Chicago Press, 1962, rev. ed. 1970.
Search in Google Scholar Back to article
[16] Marciszewski W. (ed.), Dictionary of Logic as Applied in the Study of Language. Concepts, Methods, Theories, Nijhoff, 1981.10.1007/978-94-017-1253-8
Search in Google Scholar Back to article
[17] Marciszewski W., Murawski R., Mechanization of Reasoning in a Historical Perspective, Rodopi, 1995.10.1163/9789004457423
Search in Google Scholar Back to article
[18] Marciszewski W., Hypercomputational vs. computational complexity. A challenge for methodology of the social sciences, in: Free Market and Computational Complexity. Essays in Commemoration of Friedrich Hayek (1899-1992) of the series Studies in Logic, Grammar and Rhetoric (http://logika.uwb.edu.pl/studies/index.php?page=search&vol=18), 5(18), 2002.
Search in Google Scholar Back to article
[19] Marciszewski W., The Gödelian Speed-up and Other Strategies to Address Decidability and Tractability, Studies in Logic, Grammar and Rhetoric, 9(22), 2006.
Search in Google Scholar Back to article
[20] Newman M. H. A., Alan Mathison Turing, Biographical memoirs of the Royal Society, 1955, 253-263.10.1098/rsbm.1955.0019
Search in Google Scholar Back to article
[21] Placek T., Mathematical Intuitionism and Intersubjectivity: A Critical Exposition of Arguments for Intuitionism, Springer Science & Business Media, 1999.10.1007/978-94-015-9315-1
Search in Google Scholar Back to article
[22] Poincaré H., The Value of Science (French La Valeur de la Science, 1905), Dover Publications, 1958.
Search in Google Scholar Back to article
[23] Surma S. J., Deduction theorem, in: Marciszewski (ed.), Dictionary of Logic as Applied in the Study of Language. Concepts, Methods, Theories, Nijhoff, 1981, 77-81.10.1007/978-94-017-1253-8_18
Search in Google Scholar Back to article
[24] Tarski A., Logic, Semantics, Metamathematics, translated by J. H. Woodger, Clarendon Press, 1956.
Search in Google Scholar Back to article
[25] Tarski A., On some fundamental concepts of metamathematics, in: Tarski A., Logic, Semantics, Metamathematics, 1956, German original published in the proceedings of the Scientific Society of Warsaw, 1930.
Search in Google Scholar Back to article
[26] Turing A., On computable numbers with an application to the Entscheidungsproblem, Proc. of the London Math. Society, Series 2, 1936, 230-265.10.1112/plms/s2-42.1.230
Search in Google Scholar Back to article
[27] Turing A., Systems of logic defined by ordinals, Proc. Lond. Math. Soc., Ser. 2, 45, 1939, 161-228.10.1112/plms/s2-45.1.161
Search in Google Scholar Back to article
[28] Webb J.C., Mechanism, Mentalism, and Metamathematics, Reidel, 1980.10.1007/978-94-015-7653-6
Search in Google Scholar Back to article
[29] Wittgensten L., Tractatus Logico-Philosophicus, Routledge & Kegan Paul, 1921.
Search in Google Scholar Back to article

The progress of science from a computational point of view: the drive towards ever higher solvability

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