Theological Underpinnings of the Modern Philosophy of Mathematics.

Part I: Mathematics Absolutized

Vladislav Shaposhnikov 1
  • 1 Lomonosov Moscow State University

Abstract

The study is focused on the relation between theology and mathematics in the situation of increasing secularization. My main concern is nineteenth-century mathematics. Theology was present in modern mathematics not through its objects or methods, but mainly through popular philosophy, which absolutized mathematics. Moreover, modern pure mathematics was treated as a sort of quasi-theology; a long-standing alliance between theology and mathematics made it habitual to view mathematics as a divine knowledge, so when theology was discarded, mathematics naturally took its place at the top of the system of knowledge. It was that cultural expectation aimed at mathematics that was substantially responsible for a great resonance made by set-theoretic paradoxes, and, finally, the whole picture of modern mathematics.

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  • Albertson, D. (2014). Mathematical theologies: Nicholas of Cusa and the legacy of Thierry of Chartres. New York, NY: Oxford University Press.

  • Beck, L.W. (1991). Foreword. In K.C. Köhnke, The rise of Neo-Kantianism: German academic philosophy between idealism and positivism (pp. ix–xiii). Cambridge, UK: Cambridge University Press.

  • Bloor, D. (1991). Knowledge and social imagery (2nd ed.). Chicago, IL: University of Chicago Press (First published 1976).

  • Bolzano, B. (1973). Theory of Science (J. Berg, Ed., B. Terrell, Trans.). Dordrecht, Holland: D. Reidel Publishing Company. (Original work published 1837)

  • Boniface, J. (2005). Leopold Kronecker’s conception of the foundations of mathematics. Philosophia Scientiæ, cahier spécial 5, 143–156. Retrieved from http://philosophiascientiae.revues.org/384.

  • Breger, H. (2005). God and mathematics in Leibniz’s thought (pp. 485–498). In T. Koetsier & L. Bergmans (Eds.), Mathematics and the divine: A historical study. Amsterdam: Elsevier B.V.

  • Bueno, O., & Vickers, P. (Eds.). (2014). Is Science Inconsistent? Synthese, 191, 2887–3158 (A special issue).

  • Byers, W. (2007). How mathematicians think: using ambiguity, contradiction, and paradox to create mathematics. Princeton, NJ: Princeton University Press.

  • Cantor, G. (1932). Über unendliche lineare Punktmannigfaltigkeiten, Nr. 5: Grundlagen einer allgemeinen Mannigfaltigkeitslehre. In G. Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts (E. Zermelo, Ed.) (pp. 165–209). Berlin: Springer. Original work published 1883. English translation (Ewald, 1996, pp. 878–920).

  • Colyvan, M. (2008). Who’s afraid of inconsistent mathematics? ProtoSociology, 25, 24–35.

  • Colyvan, M. (2009). Applying inconsistent mathematics. In O. Bueno & Ø. Linnebo (Eds.), New waves in philosophy of mathematics (pp. 160–172). Basingstoke: Palgrave MacMillan.

  • Comte, A. (1934). Cours de philosophie positive. Tome 1: Les préliminaires généraux et la philosophie mathématique. (6 éd.). Paris: Alfred Costes. (Original work published 1830)

  • Comte, A. (1851). The philosophy of mathematics: Translated from the Cours de philosophie positive by W.M. Gillespie. New York, NY: Harper & Brothers. (Original work published 1830)

  • Dedekind, R. (1893). Was sind und was sollen die Zahlen? 2. Auflage. Braunschweig: Vieweg. First published 1888. English translation under the name “The nature and meaning of numbers” (Dedekind, 1901, pp. 29–115).

  • Dedekind, R. (1901). Essays on the theory of numbers, authorized trans. W.W. Beman. Chicago: Open Court.

  • Dedekind, R. (1932). Gesammelte mathematische Werke, Hrsg. von R. Fricke, E. Noether & Ö. Ore. (Band 3). Braunschweig: Vieweg.

  • Descartes, R. (1985). The philosophical writings of Descartes, vol. I and II (J. Cottingham, R. Stoothoff, & D. Murdoch, Trans.). Cambridge University Press.

  • Descartes, R. (1991). The philosophical writings of Descartes, vol. III: The correspondence (J. Cottingham, R. Stoothoff, D. Murdoch, & A. Kenny, Trans.). Cambridge University Press.

  • Detlefsen, M. (1998). Constructive existence claims. In M. Schirn (Ed.), The philosophy of mathematics today (pp. 307–336). New York, NY: Oxford University Press.

  • Diderot, D. (1995). Detailed explanation of the system of human knowledge. In J. le R. d’Alembert, Preliminary discourse to the Encyclopedia of Diderot (R.N. Schwab, Trans., with the collaboration of W.E. Rex) (pp. 143–157). Chicago, IL: University of Chicago Press. (Original work published 1751)

  • Eldridge, M. (2004). Naturalism. In A.T. Marsoobian & J. Ryder (Eds.), The Blackwell guide to American philosophy (pp. 52–72). Oxford, UK: Blackwell Publishing.

  • Ernest, P. (1991). The philosophy of mathematical education. London, UK: Routledge Falmer.

  • Ernest, P. (1998). Social constructivism as a philosophy of mathematics. Albany, NY: SUNY Press.

  • Ewald, W.B. (1996). From Kant to Hilbert: A source book in the foundations of mathematics. New York, NY: Oxford University Press.

  • Ferreirós, J. (2007a). Labyrinth of thought: A history of set theory and its role in modern mathematics (2nd rev. ed.). Berlin: Birkhäuser. (First published 1999)

  • Ferreirós, J. (2007b). ʹΟ Θεὸς ʹΑριϑμετίζει: The rise of pure mathematics as arithmetic with Gauss. In C. Goldstein, N. Schappacher, & J. Schwermer (Eds.), The shaping of arithmetic after C.F. Gauss’s Disquisitiones Arithmeticae (pp. 234–268). Berlin: Springer.

  • Ferreirós, J. (2008). The crisis in the foundations of mathematics. In T. Gowers (Ed.), The Princeton companion to mathematics (pp. 142–156), Princeton, NJ: Princeton University Press.

  • Frege, G. (1979). Posthumous writings (H. Hermes et al., Eds., P. Long & R. White, Trans.). Oxford, UK: Basil Blackwell.

  • Frege, G. (1984). Collected papers on mathematics, logic, and philosophy (B. McGuinness, Ed.). Oxford, UK: Basil Blackwell.

  • Frege, G. (1996). Diary: Written by Professor Dr Gottlob Frege in the time from 10 March to 9 May 1924 (G. Gabriel & W. Kienzler, Ed. and comm., R.L. Mendelsohn, Trans.). Inquiry: An Interdisciplinary Journal of Philosophy, 39, 303–342.

  • Galilei, G. (1967). Dialogue concerning the two chief world systems (2nd rev. ed.). Berkeley, CA: University of California Press.

  • Gersh, S.E. (1978). From Iamblichus to Eriugena: An investigation of the prehistory and evolution of the Pseudo-Dionysian tradition. Leiden: E.J. Brill.

  • Gersh, S.E. (1996). Concord in discourse: Harmonics and semiotics in late classical and early medieval Platonism. Berlin: Mouton de Gruyter.

  • Giaquinto, M. (2002). The search for certainty: A philosophical account of foundations of mathematics. New York, NY: Oxford University Press.

  • Heidegger, M. (2002). Nietzsche’s word: “God is dead”. In M. Heidegger, Off the beaten track (J. Young & K. Haynes, Eds. and trans.) (pp. 157–199). Cambridge, UK: Cambridge University Press. (Original work written 1943 and published 1950)

  • Hilbert, D. (1902). Mathematical problems (Lecture delivered before the International Congress of Mathematicians at Paris in 1900). Bulletin of the American Mathematical Society, 8, 437–479 (Original work published 1900)

  • Hilbert, D. (1926). Über das Unendliche. Mathematische Annalen, 95, 161–190. English translation (van Heijenoort, 1967, pp. 367–392).

  • Hilbert, D. (1992). Natur und mathematisches Erkennen: Vorlesungen, gehalten 19191920 in Göttingen (D.E. Rowe, Ed.). Basel: Birkhäuser.

  • Huttinga, W. (2014). “Marie Antoinette” or mystical depth?: Herman Bavinck on theology as queen of the sciences. In J. Eglinton & G. Harinck (Eds.), Neo-Calvinism and the French revolution (pp. 143–154). London, UK: Bloomsbury T&T Clark.

  • Kanigel, R. (1991). The man who knew infinity: A life of the genius Ramanujan. New York, NY: Charles Scribner’s Sons.

  • Kant, I. (1998). Critique of pure reason (P. Guyer & A.W. Wood, Trans. and ed.). New York, NY: Cambridge University Press. (Original work published 1781/1787)

  • Kepler, J. (1997). The harmony of the world. Philadelphia, PA: American Philosophical Society. (Original work published 1619)

  • Kline, M. (1980). Mathematics: the loss of certainty. New York, NY: Oxford University Press.

  • Kovač, S. (2008). Gödel, Kant, and the path of a science. Inquiry, 51, 147–169.

  • Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery (J. Worrall & E. Zahar, Eds.). Cambridge, UK: Cambridge University Press.

  • Lakoff, G., & Núñez, R.E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York, NY: Basic Books.

  • Leibniz, G.W. (1989). Philosophical essays (R. Ariew & D. Garber, Eds. and trans.). Indianapolis, IN: Hackett Publishing Company.

  • Lorenz, K. (1941). Kants Lehre vom Apriorischen im Lichte gegenwärtiger Biologie. Blätter für Deutsche Philosophie, 15, 94–125. English translation (Lorenz, 2009).

  • Lorenz, K. (2009). Kant’s doctrine of the a priori in the light of contemporary biology. In M. Ruse (Ed.), Philosophy after Darwin: Classic and contemporary readings (pp. 231–247). Princeton, NJ: Princeton University Press.

  • Lossky, V. (2012). Théologie dogmatique. Paris: Cerf.

  • Mancosu, P. (1998). From Brouwer to Hilbert: The debate on the foundations of mathematics in the 1920s. New York, NY: Oxford University Press.

  • Map of the system of human knowledge. In D. Goodman et al. (Project directors) The Encyclopedia of Diderot & d’Alembert: Collaborative Translation Project. University of Michigan. Retrieved from http://quod.lib.umich.edu/d/did/tree.html (Original work published 1751)

  • Martineau, H. (1896). The positive philosophy of Auguste Comte, freely translated and condensed, in three volumes. London: George Bell & Sons.

  • Menzel, C. (2001). God and mathematical Objects. In R.W. Howell &W.J. Bradley (Eds.), Mathematics in a postmodern age: A Christian perspective (pp. 65–97). Grand Rapids, MI: Eerdmans.

  • Mill, J.S. (1974). Collected works. Vol. VIII: A system of logic, ratiocinative and inductive. Books IVVI and appendices (J.M. Robson, Ed.). Toronto: University of Toronto Press. (Original work published 1843)

  • Molière (18??). Don Juan. In The dramatic works of Molière, rendered into English by H. Van Laun, a new edition, complete in six volumes (Vol. 3, pp. 69–133). Philadelphia: George Barrie.

  • Mortensen, C. (2013). Inconsistent mathematics. In E.N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy. Retrieved from http://plato.stanford.edu/entries/mathematics-inconsistent/ (First published 1996, substantive revision 2013)

  • O’Meara, D.J. (1989). Pythagoras revived: Mathematics and philosophy in late Antiquity. New York, NY: Oxford University Press.

  • Pascal, B. (1654). Le Mémorial. In Les Pensées de Blaise Pascal. Retrieved from http://www.penseesdepascal.fr/Hors/Hors1-moderne.php

  • Posy, C.J. (1998). Brouwer versus Hilbert: 1907–1928. Science in Context, 11, 291–325.

  • Pyenson, L. (1982). Relativity in Late Wilhelmian Germany: The appeal to a preestablished harmony between mathematics and physics. Archive for History of Exact Sciences, 27, 137–155.

  • Quine, W.V. (1981). Success and limits of mathematization. In W.V. Quine, Theories and things (pp. 148–155). Cambridge, MA: Belknap Press of Harvard University Press.

  • Schaeffer, J.-M. (2007). La fin de l’exception humaine. Paris: Gallimard.

  • Scott, C.A. (1900). The International Congress of Mathematicians in Paris. Bulletin of the American Mathematical Society, 7, 57–79.

  • Sepkoski, D. (2007). Nominalism and constructivism in seventeenth-century mathematical philosophy. New York, NY: Routledge.

  • Seshu Aiyar, P.V. & Ramachandra Rao, R. (1927). Srinivasa Ramanujan (1887–1920). In S. Ramanujan, Collected papers (G.H. Hardy, P.V. Seshu Aiyar, & B.M. Wilson, Eds.) (pp. xi–xix). Cambridge, UK: Cambridge University Press.

  • Shaposhnikov, V.A. (2009). Kategoriya chisla v konkretnoj metafizike Pavla Florenskogo [Number as a category in Pavel Florensky’s concrete metaphysics]. In A.N. Krichevets (Ed.), Chislo: Trudy Moskovskogo seminara po filosofii matematiki [Number: Moscow studies in the philosophy of mathematics] (pp. 341–367). Moscow: MAX Press. [In Russian]

  • Shaposhnikov, V.A. (2014). The applicability problem and a naturalistic perspective on mathematics. In G. Mints & O. Prosorov (Eds.), Philosophy, mathematics, linguistics: Aspects of interaction. Proceedings of the international scientific conference: St. Petersburg, April 21–25, 2014 (pp. 185–197). Saint Petersburg: Euler International Mathematical Institute.

  • Slaveva-Griffin, S. (2014). Number in the metaphysical landscape. In P. Remes & S. Slaveva-Griffin (Eds.), The Routledge handbook of Neoplatonism (pp. 200–215). New York, NY: Routledge.

  • Sortorius von Waltershausen, W. (1856). Gauss zum Gedächtniss, Leipzig: S. Hirzel. Système figuré des connaissances humaines. (1751). In ARTFL Encyclopédie Project (R. Morrissey & G. Roe, Eds.). University of Chicago. Retrieved from http://encyclopedie.uchicago.edu/content/système-figuré-des-connaissances-humaines-0

  • Tapp, C. (2011). Infinity in mathematics and theology. Theology and Science, 9, 91–100.

  • van Heijenoort, J. (1967). From Frege to Gödel: A source book in mathematical logic, 18791931. Cambridge, MA: Harvard University Press.

  • Vizgin, V.P. (1998). Einstein’s “cosmic religion” and Wigner’s “empirical law of epistemology”. In I.V. Filimonova & V.A. Petrov (Eds.), Fundamental problems of high energy physics and field theory: Proceedings of the XXI workshop on high energy physics and field theory, Protvino, June 2325, 1998 (pp. 90–96). Protvino: Institute for High Energy Physics. Retrieved from http://web.ihep.su/library/pubs/tconf98/ps/vizgin.pdf

  • Weber, H. (1893). Leopold Kronecker. Mathematische Annalen, 43, 1–25.

  • Weber, Z. (2009). Inconsistent mathematics. In J. Fieser & B. Dowden (Eds.), The Internet Encyclopedia of Philosophy. Retrieved from http://www.iep.utm.edu/math-inc/

  • Weyl, H. (1921). Über die neue Grundlagenkrise der Mathematik. Mathematische Zeitschrift, 10, 39–79. English translation (Mancosu, 1998, pp. 86–118).

  • White, L.A. (1947). The locus of mathematical reality: An anthropological footnote. Philosophy of Science, 14, 289–303.

  • Wigner, E. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications of Pure and Applied Mathematics, 13, 1–14.

  • Wilder, R.L. (1981). Mathematics as a cultural system. New York, NY: Pergamon Press.

  • Wilholt, T. (2006). Lost on the way from Frege to Carnap: How the philosophy of science forgot the applicability problem. Grazer Philosophische Studien, 73, 69–82.

  • Zakai, A. (2007). The rise of modern science and the decline of theology as the “queen of sciences” in the early modern era. Reformation & Renaissance Review, 9, 125–151. Reprinted in (Zakai, 2010, pp. 51–85).

  • Zakai, A. (2010). Jonathan Edwards’s philosophy of nature: The re-enchantment of the world in the age of scientific reasoning. London, UK: Continuum T&T Clark.

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