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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
References
Hide All
Albertus de Saxonia, (1492). Questiones Subtilissime in Libros Aristotelis de Celo et Mundo. Venetiis. Reprint Georg Olms, Hildesheim, 1986.
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.
Arthur R. (1999). Infinite number and the world soul; in defense of Carlin and Leibniz. The Leibniz Review, 9, 105116.
Arthur R. (2001). Leibniz on infinite number, infinite wholes, and the whole world: A reply to Gregory Brown. The Leibniz Review, 11, 103116.
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.
Benci V., & Di Nasso M. (2003). Numerosities of labeled sets: A new way of counting. Advances in Mathematics, 173, 5067.
Benci V., Di Nasso M., & Forti M. (2006). An Aristotelean notion of size. Annals of Pure and Applied Logic, 143, 4353.
Benci V., Di Nasso M., & Forti M. (2007). An Euclidean measure of size for mathematical universes. Logique et Analyse, 50, 4362.
Bianchi L. (1984). L’Errore di Aristotele. La Polemica Contro l’Eternità del Mondo nel XIII Secolo. Firenze, Italy: La Nuova Italia.
Biard J., & Celeyrette J. (2005). De la Théologie aux Mathématiques. L’Infini au XIVeme Siécle. Paris, France: Les Belles Lettres.
Bolzano B. (1837). Wissenschaftslehre. Sulzbach, Germany.
Bolzano B. (1973). Theory of Science. Dordrecht, The Netherlands: Reidel.
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.
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.
Bos H. (1974). Differentials, higher-order differentials and the derivative in the Leibnizian calculus. Archive for History of Exact Sciences, 14, 190.
Bradwardine T. (1979). Geometria Speculativa. Wiesbaden, Germany: F. Steiner Verlag.
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.
Brown G. (2000). Leibniz on wholes, unities, and infinite number. The Leibniz Review, 10, 2151.
Bunn R. (1977). Quantitative relations between infinite sets. Annals of Science, 34, 177191.
Burbage F., & Chouchan N. (1993). Leibniz et l’infini. Paris, France: PUF.
Buzaglo M. (2002). The Logic of Concept Expansion. Cambridge, UK: Cambridge University Press.
Cantor G. (1962). Gesammelte Abhandlungen. Hildesheim, Germany: Georg Olms.
Cross R. (1998). Infinity, continuity, and composition: The contribution of Gregory of Rimini. Medieval Philosophy and Theology, 7, 89110.
Dales R. C. (1984). Henry of Harclay on the infinite. Journal of the History of Ideas, 45, 295301.
Dauben J. (1990). Georg Cantor. His mathematics and Philosophy of the Infinite. Princeton: Princeton University Press.
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.
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.
Duggan J. (1999). A general extension theorem for binary relations. Journal of Economic Theory, 86, 116.
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.
Dushnik B., & Miller E. W. (1941). Partially ordered sets. American Journal of Mathematics, 63, 600610.
Ferreirós J. (1999). Labyrinth of Thought. A History of Set Theory and its Role in Modern Mathematics. Basel, Switzerland: Birkhäuser.
Fine B., & Rosenberger G. (2007). Number Theory. An Introduction Through the Distrubution of Primes. Boston, MA: Birkhäuser.
Galileo (1939). Dialogues Concerning Two New Sciences. Evanston, IL: Northwestern University. Reprinted by Dover 1954.
Galileo (1958). Discorsi e Dimostrazioni Intorno a Due Nuove Scienze. Torino, Italy: Boringhieri.
Gardies J.-L. (1984). Pascal entre Eudoxe et Cantor. Paris, France: Vrin.
Gericke H. (1977). Wie vergleicht man unendliche Mengen? Sudhoffs Archiv, 61, 5465.
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.
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.
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.
Harclay H. (2008). Ordinary Questions. In Henninger M., editor. Two volumes. New York: Oxford University Press.
Israel J. (2002). Radical Enlightenment: Philosophy and the Making of Modernity. Oxford, UK: Oxford University Press.
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.
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).
Kitcher P. (1984). The Nature of Mathematical Knowledge. Oxford, UK: Oxford University Press.
Leibniz G. W. (1875–1890). Gerhardt C. I., editor. Die philosophischen Schriften. Berlin: Weidmann. Reprint by G. Olms.
Leibniz G. W. (2001). The Labyrinth of the Continuum. New Haven, CT: Yale University Press.
Lévy T. (1987). Figures de l’infini. Paris, France: Seuil.
Lewis N. (2007). Robert Grosseteste. Stanford Encyclopedia of Philosophy. Available from: http://plato.stanford.edu/entries/grosseteste/.
Maier A. (1949). Die Vorläufer Galileis im 14. Jahrhundert; Studien zur Naturphilosophie der Spätskolastik. Rome: Edizioni di Storia e Letteratura.
Maignan E. (1673). Cursus philosophicus, Lyons, France: Johannis Grégoire. (second edition; [1st ed., Toulouse, 1652])
Mancosu P. (1996). Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century. Oxford, UK: Oxford University Press.
Mancosu P. (2008a). Mathematical Explanation: Why it matters. In Mancosu Paolo, editor. The Philosophy of Mathematical Practice. Oxford, UK: Oxford University Press, pp. 134149.
Mancosu P., editor. (2008b). The Philosophy of Mathematical Practice. Oxford, UK: Oxford University Press.
Mancosu P., & Vailati E. (1991). Torricelli’s infinitely long solid and its philosophical reception in the XVIIth century. Isis, 82, 5070.
Moore A. W. (1990). The Infinite. London: Routledge.
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.
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.
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.
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.
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.
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.
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.
Proclus , (1992). A Commentary on the First book of Euclid’s Elements. In Morrow G., editor. Princeton: Princeton University Press.
Purkert W. (1987). Georg Cantor, 1845–1918. Basel, Switzerland: Birkhäuser.
Rabinovitch N. (1970). Rabbai Hasdai Crescas (1340–1410) on numerical infinities. Isis, 61, 224230.
Riedl C. C. (1942). Robert Grosseteste On Light. Milwaukee, WI: Marquette University Press.
Russell B. (1903). Principles of Mathematics. Cambridge, UK: Cambridge University Press.
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.
Saguens J. (1703). Ioannes Philosophia Maignani scholastica sive in formam concinniorem et auctiorem scholasticam digesta, distributa in tomos quatuor. Tolosae, France: Antonium Pech.
Sarnowsky J. (1989). Die aristotelisch-scholastische Theorie der Bewegung. Studien zum Kommentar Alberts von Sachsen zur Physik des Aristoteles. Münster, Germany: Achendorff.
Scotus D. (1639). In VIII. Libros Physicorum Aristotelis Quaestiones. Lugdunii, France: Laurentii Durand. Reprinted by Georg Olms.
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.
Sebestik J. (2002). Logique et Mathématique chez Bernard Bolzano. Paris, France: Vrin.
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.
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.
Sorabij R. (1983). Time Creation and the Continuum. Cornell, WI: Cornell University Press.
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.
Szpilrajn E. (1930). Sur l’extension de l’ordre partiel. Fundamenta Mathematicae, 16, 386389.
Tapp C. (2005). Kardinalität und Kardinale. Wissenschaftshistorische Aufarbeitung der Korrespondenz zwischen Georg Cantor und katholischen Theologen seiner Zeit. Wiesbaden, Germany: Franz Steiner Verlag.
Terski F. (2006). L’anamorphose murale de la Trinité des Monts à Rome: ou l’invisible intelligible. Montpellier, France: Editions de l’Esperou.
Van Atten M. (2009). A note on Leibniz’ argument against infinite wholes. Forthcoming.
Zellini P. (2005). A Brief History of Infinity. London: Penguin Global.
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