Abstract
The paper is divided into three parts. First I shall talk about reduc-tionism and reduction in general. In the second chapter a historical example of reductionism will be analyzed: the principles of rationalism of the 17th century. In the third chapter different kinds of reductionisms and reductions in modern logic and mathematics will be discussed.
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References
Bernays, P. (MEW) “Mathematische Existenz und Widerspruchsfreiheit”, in: Bernays, P.: Abhandlungen zur Philosophie der Mathematik. Darmstadt 1976, pp. 92–106.
Cigler J./-Reichel H.-C.: Topologie. Eine Grundvorlesung. BI Hochschultaschenbücher, Bd. 121. Mannheim-Wien-Zürich 21987.
Cleave (AEB): “An Accont of Entailment Based on Classical Semantics”, Analysis 34, pp. 118–122.
Descartes (RD): Rules for the Direction of the Mind Indianapolis 1961.
Frege, G. (GGA): Grundgesetze der Arithmetik, Bd. I Jena 1893 (Darmstadt 1962), Bd. II Jena 1903 (Darmstadt 1962).
Gödel, K. (RML): “Russell’s Mathematical Logic”, in: Russell, B.: The Philosophy of Bertrand Russel, ed. A. Schilpp, New York 1944, pp. 125–153.
Hermes, H. (ILG): “Ideen von Leibniz zur Grundlagenforschung: Die ars inveniendi und die ars iudicandi”, in: Studia Leibniziana Supplementa, Vol. 15–4, K. Müller, H. Schepers, and W. Totok (eds.), Wiesbaden 1975.
Hillbert (GEZ): “Die Grundlegung der elementaren Zahlenlehre”, in: Mathem. Annalen 104 (1931), pp. 483–494.
Hillbert (GSA): Gesammelte Abhandlungen. Berlin 1970.
Kant, I. (KRV): Kritik der Reinen Vernunft. Riga 1781 (1787).
Körner, S. (CTh): Conceptual Thinking New York 1959.
Kreisel, G. (ACM): “On Analogies in Contemporary Mathematics”, to appear.
Kreisel, G. (HPG): “Hilbert’s Programme”, in: Benecerraf, P. and Putnam, H. (eds.): Philosophy of Mathematics. Englewood Cliffs 1964, pp. 157–180
Leibniz (GP): Grundgesetze der Philosophie. Frankfurt 1962.
Leibniz (NE): Nouveau Essais sur l’entendement humain. Frankfurt 1961. Leibniz (OF): Opuscules et fragmetns inédits de Leibniz. Paris 1903.
Lukasiewicz, J. (ASy): Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic. London 257.
Marshall, D. Jr. (LLA): “Lukasiewicz, Leibniz and the Arithmetization of the Syllogism”, in: Notre Dame Journal of Formal Logic 18(1977), pp. 235–242.
Popper, K.R. (OUN): The Open Universe. Totowa 1982.
Rescher, N. (LIP): Leibniz. An Introduction to His Philosophy. Oxford 1979.
Russell, B. (IMP): Introduction to Mathematical Philosophy. London 1919 (1960).
Russell, B. (PM):The Principles of Mathematics. London 1937 (1903).
Schurz, G. (DRS): “Das deduktive Relevanzkriterium von Stephan Körner und seine wissenschaftstheoretischen Anwendungen”, Grazer Philosophische Studien 20 (1983), pp. 149–177.
Schurz, G. (RDS): “Relevant Deduction. From Solving Paradoxes Towards a General Theory”, Erkenntnis 35,1991.
Schurz, G., Weingartner, P. (VDR): “Verisimilitude Defined by Relevant Consequence Elements. A New Reconstruction of Popper’s Original Idea”, in: Kuipers, Th. (ed.): What is Closer-to-the-Truth Amsterdam 1987, pp. 47–77.
Skolem, T. (NCZ): “Über die Nicht-Charakterisierbarkeit der Zahlenreihe mittels endlich oder abzahlbar unendlich vieler Aussagen mit ausschließlich Zahlenvariablen”, in: Fund. Math. 23 (1934), pp. 150–161.
Skolem, T. (UVC): “Über die Unmöglichkeit einer Charakterisierung der Zahlenreihe mittels eines endlichen Axiomensystems”, in: Skolem, T.: Selected Works in Logic (ed. J.E. Fenstad), Oslo 1970, pp. 345–354. (Originally published in 1933)
Wang, H. (FMP): “The Formalization of Mathematics”, Journal of Symbolic Logic 19 (1954), pp. 241–266.
Weingartner, P. (APS): “Antinomies and Paradoxes and their Solutions”, in: Studies in Soviet Thought 39 (1990), pp. 313–331.
Weingartner, P. (CEM): “On the Characterization of Entities by Means of Individuals and Properties”, Journal of Symbolic Logic 39 (1974), pp. 323–336.
Weingartner, P. (DLM): “On the Demarcation between Logic and Mathematics”, The Monist 65 (1982), pp. 38–51.
Weingartner, P. (IMS): “The Ideal of Mathematization of All Sciences and of ’More Geometrico’ in Descartes and Leibniz”, in: Shea, W. R. (ed.): Nature Mathematized Proceedings of the 3rd International Congress of History and Philosophy of Science (Montreal 1980). Dordrecht 1983, pp. 151–195.
Weingartner, P. (PDS): “A Proposal to Define a Special Type of Proposition”, in: Weingartner,, P./Schurz, G. (eds.): Philosophy of the Natural Sciences. Proceedings of the 13th International Wittgenstein-Symposium, Kirchberg/Wechsel, Austria 1988. Wien 1989, pp. 348–353.
Weingartner, P. (RCT): “Remarks on the Consequence-Class of Theories”, in: Scheibe, E. (ed.): The Role of Experience in Science. Proceedings of the 1986 Conference of the Académie Internationale de Philosophie des Science (Bruxelles) Held at the University of Heidelberg. Berlin 1988, pp. 161–180.
Weingartner, P., Schurz, G. (PSS): “Paradoxes Solved by Two Simple Relevance Criteria”, in: Logique et Analyse 3 (1986), pp. 3–40.
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Weingartner, P. (1991). Reductionism and Reduction in Logic and in Mathematics. In: Agazzi, E. (eds) The Problem of Reductionism in Science. Episteme, vol 18. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3492-7_7
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DOI: https://doi.org/10.1007/978-94-011-3492-7_7
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