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MEASURING THE SIZE OF INFINITE COLLECTIONS OF NATURAL NUMBERS: WAS CANTOR’S THEORY OF INFINITE NUMBER INEVITABLE?

  • PAOLO MANCOSU (a1)
Abstract

Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collection A is properly included in a collection B then the ‘size’ of A should be less than the ‘size’ of B (part–whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part–whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, envisaged by Bolzano (Kitcher).

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*UNIVERSITY OF CALIFORNIA, BERKELEY, DEPARTMENT OF PHILOSOPHY, 314 MOSES HALL, BERKELEY, CA 94720-2390. E-mail:mancosu@socrates.berkeley.edu
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R. Arthur (1999). Infinite number and the world soul; in defense of Carlin and Leibniz. The Leibniz Review, 9, 105116.

R. Arthur (2001). Leibniz on infinite number, infinite wholes, and the whole world: A reply to Gregory Brown. The Leibniz Review, 11, 103116.

V. Benci , & M. Di Nasso (2003). Numerosities of labeled sets: A new way of counting. Advances in Mathematics, 173, 5067.

V. Benci , M. Di Nasso , & M. Forti (2006). An Aristotelean notion of size. Annals of Pure and Applied Logic, 143, 4353.

B. Bolzano (1973). Theory of Science. Dordrecht, The Netherlands: Reidel.

H. Bos (1974). Differentials, higher-order differentials and the derivative in the Leibnizian calculus. Archive for History of Exact Sciences, 14, 190.

G. Brown (2000). Leibniz on wholes, unities, and infinite number. The Leibniz Review, 10, 2151.

R. Bunn (1977). Quantitative relations between infinite sets. Annals of Science, 34, 177191.

R. Cross (1998). Infinity, continuity, and composition: The contribution of Gregory of Rimini. Medieval Philosophy and Theology, 7, 89110.

R. C. Dales (1984). Henry of Harclay on the infinite. Journal of the History of Ideas, 45, 295301.

T. Dewender (2002). Das Problem des Unendlichen im ausgehenden 14. Jahrhundert. Eine Studie mit Textedition zum Physikkommentar des Lorenz von Lindores. Amsterdam, The Netherlands: B.R. Grüner Publishing Co.

J. Duggan (1999). A general extension theorem for binary relations. Journal of Economic Theory, 86, 116.

B. Dushnik , & E. W. Miller (1941). Partially ordered sets. American Journal of Mathematics, 63, 600610.

P. Mancosu (2008a). Mathematical Explanation: Why it matters. In Paolo Mancosu , editor. The Philosophy of Mathematical Practice. Oxford, UK: Oxford University Press, pp. 134149.

P. Mancosu , editor. (2008b). The Philosophy of Mathematical Practice. Oxford, UK: Oxford University Press.

P. Mancosu , & E. Vailati (1991). Torricelli’s infinitely long solid and its philosophical reception in the XVIIth century. Isis, 82, 5070.

A. W. Moore (1990). The Infinite. London: Routledge.

J. Murdoch (1981b). Mathematics and infinity in the later middle ages. In D. O. Dahlstrom , D. T. Ozar and L. Sweeney , editors. Infinity, Proceedings of the American Catholic Philosophical Association, Vol. 55. Washington, DC, pp. 4058.

N. Rabinovitch (1970). Rabbai Hasdai Crescas (1340–1410) on numerical infinities. Isis, 61, 224230.

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The Review of Symbolic Logic
  • ISSN: 1755-0203
  • EISSN: 1755-0211
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