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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).

DOI: 10.1007/978-94-009-5496-0_1

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