Abstract
Among the distinctive features of mathematical structures or theories is their actual or potential applicability to empirical phenomena. It is the purpose of this essay to compare mathematical structures, especially those of arithmetic and the real number system, on the one hand, with empirical structures, especially those of discrete and continuous phenomena, on the other, to examine what is involved in applying mathematical to empirical structures, and to exhibit some metaphysical assumptions about their relations to each other. The essay begins with some remarks on what might be called the “empirical arithmetic of countable aggregates” (Section 1). Next an indication is given, how this empirical arithmetic is idealized into a pure arithmetic of natural numbers and integers, and, beyond, into a pure mathematical theory of rational and real numbers (Section 2). There follows a brief characterization of empirical continua (Section 3); a discussion of the application of pure numerical mathematics to empirically discrete and empirically continuous phenomena; and some remarks on the application of mathematics in general (Section 4). The paper ends by drawing attention to some relations between pure and applied mathematics on the one hand and metaphysics on the other (Section 5).
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Notes
The quoted words form part of the first sentence of the first chapter of a book by E. Landau explaining the calculation with whole, rational, irrational and complex numbers. See Grundlagen der Analysis (Leipzig, 1930 ).
See Aristotle, Physics book VI; and Brentano, Raum, Zeit und Continuum (Hamburg, 1976 ).
See Grundgesetze der Arithmetik vol. 2, p. 69 (Jena, 1903).
For a more detailed discussion of empirical continuity, see chapter IV of my Experience and Theory (London, 1966); and for a formal analysis, see J.P. Cleave, “Quasi-Boolean Algebras, Empirical Continuity and Three-Valued Logic” Zeitschr. fur Math. Logik und Grundlagen d. Mathematik 22 (1976): 481 – 500.
From Hilbert’s formalism one should distinguish the formalism of A. Robinson, whose view on the application of mathematics is similar to the one which has been taken here. See his “Formalism 64” and “Concerning Progress in the Philosophy of Mathematics” in: Selected Papers, vol. 2 ( New Haven, Conn.: Yale University Press, 1979 ).
For a more detailed account, see e.g. “Science and the Organization of Belief” in: Science, Belief and Behaviour, ed. D.H. Mellor (Cambridge, Mass.: Cambridge University Press, 1980 ), pp. 43 – 61.
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© 1986 D. Reidel Publishing Company
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Körner, S. (1986). On the Empirical Application of Mathematics and Some of its Philosophical Aspects. In: Ullmann-Margalit, E. (eds) The Kaleidoscope of Science. Boston Studies in the Philosophy of Science, vol 94. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-5496-0_1
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DOI: https://doi.org/10.1007/978-94-009-5496-0_1
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