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

Published online by Cambridge University Press:  01 December 2009

PAOLO MANCOSU*
Affiliation:
Department of Philosophy, University of California, Berkeley
*
*UNIVERSITY OF CALIFORNIA, BERKELEY, DEPARTMENT OF PHILOSOPHY, 314 MOSES HALL, BERKELEY, CA 94720-2390. E-mail:mancosu@socrates.berkeley.edu
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Abstract

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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).

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2009

References

BIBLIOGRAPHY

Albertus, de Saxonia, (1492). Questiones Subtilissime in Libros Aristotelis de Celo et Mundo. Venetiis. Reprint Georg Olms, Hildesheim, 1986.Google Scholar
Andrikopoulos, A. (2009). General Extension Theorems for Binary Relations. Forthcoming. Available from: http://scholar.google.com/scholar?hl=fr&lr=&cites=13315085436213194747&start=20&sa=N.Google Scholar
Arthur, R. (1999). Infinite number and the world soul; in defense of Carlin and Leibniz. The Leibniz Review, 9, 105116.Google Scholar
Arthur, R. (2001). Leibniz on infinite number, infinite wholes, and the whole world: A reply to Gregory Brown. The Leibniz Review, 11, 103116.Google Scholar
Benci, V. (1995). I numeri e gli insiemi etichettati. Conferenze del seminario di matematica dell’Universita’ di Bari, Vol. 261. Bari, Italy: Laterza, pp. 29.Google Scholar
Benci, V., & Di Nasso, M. (2003). Numerosities of labeled sets: A new way of counting. Advances in Mathematics, 173, 5067.Google Scholar
Benci, V., Di Nasso, M., & Forti, M. (2006). An Aristotelean notion of size. Annals of Pure and Applied Logic, 143, 4353.Google Scholar
Benci, V., Di Nasso, M., & Forti, M. (2007). An Euclidean measure of size for mathematical universes. Logique et Analyse, 50, 4362.Google Scholar
Bianchi, L. (1984). L’Errore di Aristotele. La Polemica Contro l’Eternità del Mondo nel XIII Secolo. Firenze, Italy: La Nuova Italia.Google Scholar
Biard, J., & Celeyrette, J. (2005). De la Théologie aux Mathématiques. L’Infini au XIVeme Siécle. Paris, France: Les Belles Lettres.Google Scholar
Bolzano, B. (1837). Wissenschaftslehre. Sulzbach, Germany.Google Scholar
Bolzano, B. (1973). Theory of Science. Dordrecht, The Netherlands: Reidel.Google Scholar
Bolzano, B. (1975a). Einleitung zur Grössenlehre. Erste Begriffe der allgemeinen Grössenlehre. In Berg, Jan, editor. Gesamtausgabe, II A 7. Stuttgart-Bad Cannstatt, Germany: Friedrich Fromann Verlag.Google Scholar
Bolzano, B. (1975b). Paradoxien des Unendlichen. Hamburg, Germany: Felix Meiner Verlag. Translated as Paradoxes of the Infinite, edited by Donald A. Steele, London: Routledge and Kegan Paul, and New Haven: Yale University Press, 1950.Google Scholar
Bos, H. (1974). Differentials, higher-order differentials and the derivative in the Leibnizian calculus. Archive for History of Exact Sciences, 14, 190.Google Scholar
Bradwardine, T. (1979). Geometria Speculativa. Wiesbaden, Germany: F. Steiner Verlag.Google Scholar
Breger, H. (2008). Natural numbers and infinite cardinal number. In Hecht, H., Mikosch, R., Schwarz, I., Siebert, H., and Werthers, R., editors. Kosmos und Zahl. Stuttgart, Germany: Steiner, pp. 309318.Google Scholar
Brown, G. (2000). Leibniz on wholes, unities, and infinite number. The Leibniz Review, 10, 2151.Google Scholar
Bunn, R. (1977). Quantitative relations between infinite sets. Annals of Science, 34, 177191.Google Scholar
Burbage, F., & Chouchan, N. (1993). Leibniz et l’infini. Paris, France: PUF.Google Scholar
Buzaglo, M. (2002). The Logic of Concept Expansion. Cambridge, UK: Cambridge University Press.Google Scholar
Cantor, G. (1962). Gesammelte Abhandlungen. Hildesheim, Germany: Georg Olms.Google Scholar
Cross, R. (1998). Infinity, continuity, and composition: The contribution of Gregory of Rimini. Medieval Philosophy and Theology, 7, 89110.Google Scholar
Dales, R. C. (1984). Henry of Harclay on the infinite. Journal of the History of Ideas, 45, 295301.Google Scholar
Dauben, J. (1990). Georg Cantor. His mathematics and Philosophy of the Infinite. Princeton: Princeton University Press.Google Scholar
Dewender, T. (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.Google Scholar
Di Nasso, M., & Forti, M. (2009). Numerosities of point sets over the real line. Transactions of the American Mathematical Society. Forthcoming. Available from: http://www.dm.unipi.it/~dinasso/papers.html.Google Scholar
Duggan, J. (1999). A general extension theorem for binary relations. Journal of Economic Theory, 86, 116.Google Scholar
Duhem, P. (1955). Léonard de Vinci et les deux infinis. In Études sur Léonard de Vinci, Seconde Serie. Paris, France: De Nobele, pp. 353, 368407.Google Scholar
Dushnik, B., & Miller, E. W. (1941). Partially ordered sets. American Journal of Mathematics, 63, 600610.Google Scholar
Ferreirós, J. (1999). Labyrinth of Thought. A History of Set Theory and its Role in Modern Mathematics. Basel, Switzerland: Birkhäuser.Google Scholar
Fine, B., & Rosenberger, G. (2007). Number Theory. An Introduction Through the Distrubution of Primes. Boston, MA: Birkhäuser.Google Scholar
Galileo, (1939). Dialogues Concerning Two New Sciences. Evanston, IL: Northwestern University. Reprinted by Dover 1954.Google Scholar
Galileo, (1958). Discorsi e Dimostrazioni Intorno a Due Nuove Scienze. Torino, Italy: Boringhieri.Google Scholar
Gardies, J.-L. (1984). Pascal entre Eudoxe et Cantor. Paris, France: Vrin.Google Scholar
Gericke, H. (1977). Wie vergleicht man unendliche Mengen? Sudhoffs Archiv, 61, 5465.Google Scholar
Gilbert, T., & Rouche, N. (1996). Y-at-il vraiment autant de nombres pairs que des naturels? In Pétry, A., editor. Méthodes et Analyse Non Standard, Cahiers du Centre de Logique, Vol. 9. Louvain-la-Neuve, Belgium: Bruylant-Academia, pp. 99139.Google Scholar
Gödel, K. (1990). Collected Works. In Feferman, S., Dawson, J. W., Kleene, S. C., Moore, G. H., Solovay, R., and van Heijenoort, J., editors. Vol. II. New York: Oxford University Press.Google Scholar
Hallett, M. (1984). Cantorian Set Theory and Limitation of Size. Oxford, UK: Clarendon Press. Foreword by Michael Dummett. Reprinted in paperback, with revisions, 1986, 1988.Google Scholar
Harclay, H. (2008). Ordinary Questions. In Henninger, M., editor. Two volumes. New York: Oxford University Press.Google Scholar
Israel, J. (2002). Radical Enlightenment: Philosophy and the Making of Modernity. Oxford, UK: Oxford University Press.Google Scholar
Katz, F. M. (1981). Sets and Their Sizes. Ph.D. Dissertation, MIT. Now newly typeset (2001). Available from: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.28.7026.Google Scholar
Kirschner, S. (1997). Nicolaus Oresmes Kommentar zue Physik des Aristoteles. Kommenatr mit Edition der Quaestiones zu Buch 3 und 4 der aristotelischen Physik sowie von vier Quaestiones zu Buch 5. Wiesbaden, Germany: Franz Steiner Verlag (Sudhoff Archiv, Beihefte 39).Google Scholar
Kitcher, P. (1984). The Nature of Mathematical Knowledge. Oxford, UK: Oxford University Press.Google Scholar
Leibniz, G. W. (1875–1890). Gerhardt, C. I., editor. Die philosophischen Schriften. Berlin: Weidmann. Reprint by G. Olms.Google Scholar
Leibniz, G. W. (2001). The Labyrinth of the Continuum. New Haven, CT: Yale University Press.Google Scholar
Lévy, T. (1987). Figures de l’infini. Paris, France: Seuil.Google Scholar
Lewis, N. (2007). Robert Grosseteste. Stanford Encyclopedia of Philosophy. Available from: http://plato.stanford.edu/entries/grosseteste/.Google Scholar
Maier, A. (1949). Die Vorläufer Galileis im 14. Jahrhundert; Studien zur Naturphilosophie der Spätskolastik. Rome: Edizioni di Storia e Letteratura.Google Scholar
Maignan, E. (1673). Cursus philosophicus, Lyons, France: Johannis Grégoire. (second edition; [1st ed., Toulouse, 1652])Google Scholar
Mancosu, P. (1996). Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century. Oxford, UK: Oxford University Press.Google Scholar
Mancosu, P. (2008a). Mathematical Explanation: Why it matters. In Mancosu, Paolo, editor. The Philosophy of Mathematical Practice. Oxford, UK: Oxford University Press, pp. 134149.Google Scholar
Mancosu, P., editor. (2008b). The Philosophy of Mathematical Practice. Oxford, UK: Oxford University Press.Google Scholar
Mancosu, P., & Vailati, E. (1991). Torricelli’s infinitely long solid and its philosophical reception in the XVIIth century. Isis, 82, 5070.Google Scholar
Moore, A. W. (1990). The Infinite. London: Routledge.Google Scholar
Murdoch, J. (1981a). Henry of Harclay and the infinite. In Maierù, A., and Paravicini Bagliani, A., editors. Studi sul XIV secolo in memoria di Anneliese Maier. Roma, Italy: Edizioni di storia e letteratura, pp. 219261.Google Scholar
Murdoch, J. (1981b). Mathematics and infinity in the later middle ages. In Dahlstrom, D. O., Ozar, D. T. and Sweeney, L., editors. Infinity, Proceedings of the American Catholic Philosophical Association, Vol. 55. Washington, DC, pp. 4058.Google Scholar
Murdoch, J. (1982). Infinity and continuity. In Kretzmann, N., Kenny, A., and Pinborg, J., editors. The Cambridge History of Later Medieval Philosophy. Cambridge, UK: Cambridge University Press, pp. 564592.Google Scholar
Parker, M. (2009). Philosophical method and Galileo’s paradox of infinity. In van Kerchove, B., editor. New Perspectives on Mathematical Practices. Hackensack, NJ: World Scientific, pp. 76113.Google Scholar
Pascal, J. (2005). Anamorphoses et visions miraculeuses du père Maignan (1602–1676). MEFRIM: Mélanges de l’École française de Rome: Italie et mediterranée, 117(1), 4571.Google Scholar
Petruzzellis, N. (1968). L’infinito nel pensiero di S. Tommaso e di G. Duns Scoto. In De Doctrina Ioannis Duns Scoti, Acta Congr. Scotistici; Studia Scotistica 2, Vol. 2. Rome, Italy, 435445.Google Scholar
Pines, S. (1968). Thabit B. Qurra’s conception of number and theory of the mathematical infinite. In Actes du Onzième Congrès International d’Histoire des Sciences Sect. III: Histoire des Sciences Exactes (Astronomie, Mathématiques, Physique) (Wroclaw, 1963), pp. 160166.Google Scholar
Proclus, , (1992). A Commentary on the First book of Euclid’s Elements. In Morrow, G., editor. Princeton: Princeton University Press.Google Scholar
Purkert, W. (1987). Georg Cantor, 1845–1918. Basel, Switzerland: Birkhäuser.Google Scholar
Rabinovitch, N. (1970). Rabbai Hasdai Crescas (1340–1410) on numerical infinities. Isis, 61, 224230.Google Scholar
Riedl, C. C. (1942). Robert Grosseteste On Light. Milwaukee, WI: Marquette University Press.Google Scholar
Russell, B. (1903). Principles of Mathematics. Cambridge, UK: Cambridge University Press.Google Scholar
Sabra, A. (1997). Thabit ibn Qurra on the infinite and other puzzles; edition and translation of his discussions with ibn Usayyid. Zeitschrift für Geschichte der Arabisch-Islamischen Wissenschaften, 11, 133.Google Scholar
Saguens, J. (1703). Ioannes Philosophia Maignani scholastica sive in formam concinniorem et auctiorem scholasticam digesta, distributa in tomos quatuor. Tolosae, France: Antonium Pech.Google Scholar
Sarnowsky, J. (1989). Die aristotelisch-scholastische Theorie der Bewegung. Studien zum Kommentar Alberts von Sachsen zur Physik des Aristoteles. Münster, Germany: Achendorff.Google Scholar
Scotus, D. (1639). In VIII. Libros Physicorum Aristotelis Quaestiones. Lugdunii, France: Laurentii Durand. Reprinted by Georg Olms.Google Scholar
Sebestik, J. (1992). La paradoxe de la réflexivité des ensembles infinis: Leibniz, Gold-bach, Bolzano. In Monnoyeur, Françoise, editor. Infini des Mathématiciens, Infini des Philosophes. Paris, France: Belin, pp. 175191.Google Scholar
Sebestik, J. (2002). Logique et Mathématique chez Bernard Bolzano. Paris, France: Vrin.Google Scholar
Sesiano, J. (1988). On an algorithm for the approximation of surds from a Provençal treatise. In Hay, C., editor. Mathematics from Manuscript to Print, 1300–1600. Oxford, UK: Clarendon Press, pp. 3056.Google Scholar
Sesiano, J. (1996). Vergleiche zwischen unendlichen Mengen bei Nicolas Oresme. In Folkerts, M., editor. Mathematische Probleme im Mittelater—der lateinische und arabische Sprachbereich. Wiesbaden, Germany: Harrassowitz Verlag, pp. 361378.Google Scholar
Sorabij, R. (1983). Time Creation and the Continuum. Cornell, WI: Cornell University Press.Google Scholar
Spalt, D. (1990). Die Unendlichkeit bei Bernard Bolzano. In König, G., editor. Konzepte des mathematischen Unendlichen im 19. Jahrhundert. Göttingen, Germany: Vandenhoeck & Ruprecht, pp. 189218.Google Scholar
Szpilrajn, E. (1930). Sur l’extension de l’ordre partiel. Fundamenta Mathematicae, 16, 386389.Google Scholar
Tapp, C. (2005). Kardinalität und Kardinale. Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit. Wiesbaden, Germany: Franz Steiner Verlag.Google Scholar
Terski, F. (2006). L’anamorphose murale de la Trinité des Monts à Rome: ou l’invisible intelligible. Montpellier, France: Editions de l’Esperou.Google Scholar
Van Atten, M. (2009). A note on Leibniz’ argument against infinite wholes. Forthcoming.Google Scholar
Zellini, P. (2005). A Brief History of Infinity. London: Penguin Global.Google Scholar