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://logika.uwb.edu.pl/studies/index.php?page=search&vol=18 ), 5(18), 2002. [19] Marciszewski W., The Gödelian Speed-up and Other Strategies to Address Decidability and Tractability, Studies in Logic, Grammar and Rhetoric , 9(22), 2006. [20] Newman M. H. A., Alan Mathison Turing, Biographical memoirs of the Royal Society , 1955, 253-263. [21] Placek T., Mathematical Intuitionism and Intersubjectivity: A Critical Exposition of Arguments for Intuitionism , Springer Science & Business Media, 1999. [22] Poincaré H., The Value of Science (French La Valeur de la Science , 1905), Dover Publications, 1958. [23] Surma S. J

Abstract

In this paper I dispute the current view that intuitionistic logic is the common basis for the three main trends of constructivism in the philosophy of mathematics: intuitionism, Russian constructivism and Bishop’s constructivism. The point is that the so-called ‘Markov’s principle’, which is accepted by Russian constructivists and rejected by the other two, is expressible in intuitionistic first-order logic, and so it appears to have the status of a logical principle. The result of appending this principle to a complete intuitionistic axiom system for first-order predicate logic constitutes a new logic, which could well be called ‘Markov’s logic’, and which should be regarded as the true logical system underlying Russian constructivism.

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Petros, Vandoulakis Ioannis, Martínez Maricarmen and Foundalis Harry. 2015. “Collective Discovery Events: Web-based Mathematical Problem-solving with Codelets”. In: Tarek R. Besold, Marco Schorlemmer, Alan Smaill (Eds) Computational Creativity Research: Towards Creative Machines . Atlantis Thinking Machines (Book 7) Atlantis/Springer, pp. 371-392. Tieszen, R. L. 1989. Mathematical Intuition: Phenomenology and Mathematical Knowledge , Dordrecht: Kluwer. Tieszen, R. L. 1992. “What is a Proof?” In: Detlefsen, M. (ed.) Proof, Logic and Formalization , London: Routledge

Schuster, New York. Reprinted in 2003, New York: Dover. Newstead, A. (2009), ‘Cantor of infinity in nature, numbers, and the divine mind’, American Catholic Philosophical Querterly 83(4), 533-553. Pesch, T. (1883), Institutiones philosophiae naturalis: Secundum principia S. Thomas Aquinatis ad usum scholasticum, Herder, Freiburg. Placek, T. (1999), Mathematical Intuitionism and Intersubjectivity. A Critical Exposition of Arguments for Intuitionism, Kluwer Academic Publishers. Prasad, G. (1993), Some great mathematicians of the nineteenth century, Dynamic informative

that such an algorithm will be found, constituted the core of his ratio- nalistic optimism. Gödel was an optimist as well, yet in the sense more moderate: that for particular problems, unsolvable in the present state of science, as the time will go, new ontological algorithms will be de- tected – owing to the great power of mathematical intuition – to base on them relevant syntactic algorithms, and thus to solve the problem at stake. 3. Formal grammar and the corresponding to it ontology at the bottom of logic and informatics The subject matter of this Section is a