Skip to main content
SpringerLink
Log in

Mathematical Explanation: Problems and Prospects

  • Published:
Topoi Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  • Allwein, G. and Barwise, J. (eds.): 1996, Logical Reasoning With Diagrams, New York: Oxford University Press.

    Google Scholar 

  • Aspray, W. and Kitcher, P. (eds.): 1988, History and Philosophy of Modern Mathematics, Minneapolis: University of Minnesota Press.

    Google Scholar 

  • Atiyah, M. et al.: 1994, ‘Responses to “Theoretical Mathematics”: Towards a Cultural Synthesis of Mathematics and Theoretical Physics, by A. Jaffe and F. Quinn’, Bulletin of the American Mathematical Society 30(2), 178–207.

    Article  Google Scholar 

  • Barwise, J. and Etchemendy, J.: 1991, Visual Information and Valid Reasoning, in Allwein, Barwise 1996, pp. 3–26.

  • Bottazzini, U.: 1986, The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass, New-York, Berlin, Heidelberg: Springer.

    Book  Google Scholar 

  • Bottazzini, U.: 1994, ‘Three Traditions in Complex Analysis: Cauchy, Riemann, and Weierstrass’, in I. Grattan-Guinness (Ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, vol.I, London and New York: Routledge, pp. 419–431.

    Google Scholar 

  • Bouligand, G.: 1932, Premières leçons sur la théorie générale des groupes, Paris: Vuibert.

    Google Scholar 

  • Bouligand, G.: 1933, ‘L'idée de causalité en mathématiques et dans quelques théories physiques’, Revue Scientifique 71(9), 257–267.

    Google Scholar 

  • Bouligand, G.: 1934, La Causalité des Théories Mathématiques, Paris: Hermann.

    Google Scholar 

  • Bouligand, G.: 1935, ‘Géométrie et causalité’, in AA.VV., L'évolution des sciences physiques et mathématiques, Paris: Flammarion, pp. 139–176.

    Google Scholar 

  • Bouligand, G.: 1937, Structure des Théories, Paris: Hermann.

    Google Scholar 

  • Brown, J. R.: 1997, ‘Proofs and Pictures’, The British Journal for the Philosophy of Science 48, 161–186.

    Article  Google Scholar 

  • Brown, J. R.: 1999, Philosophy of Mathematics. An Introduction to a World of Proofs and Pictures, London and New York: Routledge.

    Google Scholar 

  • Burgess, J. and Ernest, P.: ‘Review of “The Mathematical Experience. Study Guide”’, Philosophia Mathematica 5, 175–188.

  • Chemla, K.: 1997, ‘What is at Stake in Mathematical Proofs from Third-Century China?’, Science in Context 10(2), 227–251.

    Article  Google Scholar 

  • Chaitin, G. J.: 1994, Response to “Theoretical Mathematics”, in Atiyah et al. 1994, pp. 4–5.

  • Couturat, L.: 1905, Les Principes des Mathématiques, Paris: Alcan.

    Google Scholar 

  • Davis, P. J. and Hersh, R.: 1981, The Mathematical Experience, Boston: Birkhäuser.

    Google Scholar 

  • Davis, P. J., Hersh, R. and Marchisotto, E. A.: 1995, The Mathematical Experience. Study Guide, Boston: Birkhäuser.

    Book  Google Scholar 

  • Desanti, J. T., 1973, ‘L'Explication en Mathématiques’, in L. Apostel et al. (Eds.), L'Explication dans les Sciences, Paris: Flammarion.

    Google Scholar 

  • Detlefsen, M. and Luker, M.: 1980, ‘The Four-colour Theorem and Mathematical Proof’, Journal of Philosophy, 803–820.

  • Duhem, P., 1906, La Théorie Physique, son object et sa structure, Paris. Second edition 1914. Second edition translated as The Aim and Structure of Physical Theory, Princeton: Princeton University Press, 1954.

    Google Scholar 

  • Friedman, M.: 1974, ‘Explanation and Scientific Understanding’, Journal of Philosophy 71, 5–19.

    Article  Google Scholar 

  • Giaquinto, M.: 1992, ‘Visualizing as a Means of Geometrical Discovery’, Mind and Language 7, 381–401.

    Google Scholar 

  • Giaquinto, M.: 1993, ‘Visualizing in Arithmetic’, Philosophy and Phenomenological Research LIII, 385–396.

    Article  Google Scholar 

  • Giaquinto, M.: 1994, ‘Epistemology of Visual Thinking in Elementary Real Analysis’, British Journal for the Philosophy of Science 44, 769–813.

    Google Scholar 

  • Gödel, K.: 1944, ‘Russell's Mathematical Logic’, in P. A. Schilpp (Ed.), The Philosophy of Bertrand Russell, 3rd edition, New York: Tudor, pp. 123–153. Reprinted in Kurt Gödel, Collected Works, 1990, vol. II, ed. Solomon Feferman et al., New York, Oxford: Oxford University Press, pp. 119–141.

    Google Scholar 

  • Gödel, K.: 1947, ‘What is Cantor's Continuum Problem?’, American Mathematical Monthly 54, 515–525. Reprinted in Kurt Gödel, Collected Works, 1990, vol. II, ed. Solomon Feferman et al., New York, Oxford: Oxford University Press, pp. 176–187 and 254–270.

    Article  Google Scholar 

  • Grosholz, E. and Breger, H. (eds.): 2000, Growth of Mathematical Knowledge, Dordrecht: Kluwer.

    Google Scholar 

  • Hanna, G., 1990, ‘Some Pedagogical Aspects of Proof’, Interchange 21(1), 6–13.

    Article  Google Scholar 

  • Hanna, G.: 1997, ‘The Ongoing Value of Proof’, Journal Mathematik Didaktik 18, 171–185.

    Google Scholar 

  • Hanna, G. and Jannke, H. N.: 1999, ‘Using Arguments from Physics to Promote Understanding of Mathematical Proofs’, Proceedings of PME 23; available on the web at http://fcis.oise.utoronto.ca/~ghanna.

  • Hersh, R.: 1993, ‘Proving is Convincing and Explaining’, Educational Studies in Mathematics 24, 389–399.

    Article  Google Scholar 

  • Hersh, R.: 1997, ‘Prove-Once More and Again’, Philosophia Mathematica 5, 153–165.

    Google Scholar 

  • Hersh, R.: 1997a, What is Mathematics, Really?, New York: Oxford University Press.

    Google Scholar 

  • Irvine, I.: 1989, ‘Epistemic Logicism and Russell's Regressive Method’, Philosophical Studies 55, 303–327.

    Article  Google Scholar 

  • Jaffe, A. and Quinn, F., 1993, ‘“Theoretical Mathematics”: Towards a Cultural Synthesis of Mathematics and Theoretical Physics’, Bulletin of the American Mathematical Society 29(2), 1–13.

    Article  Google Scholar 

  • Jaffe, A. and Quinn, F.: 1994, ‘Response to Comments on “Theoretical Mathematics”’, Bulletin of the American Mathematical Society 30(2), 208–211.

    Article  Google Scholar 

  • Jaffe, A.: 1997, ‘Proof and Evolution in Mathematics’, in “Proof and Progress in Mathematics”, ed. by A. Kanamori, Synthèse, pp. 133–146.

  • Kelly, T. R.: 1937, Explanation and Reality in the Philosophy of Émile Meyerson, Princeton: Princeton University Press.

    Google Scholar 

  • Kitcher. P.: 1975, ‘Bolzano's Ideal of Algebraic Analysis’, Studies in History and Philosophy of Science 6, 229–269.

    Article  Google Scholar 

  • Kitcher, P.: 1981, ‘Explanatory Unification’, Philosophy of Science 48, 507–531

    Article  Google Scholar 

  • Kitcher, P.: 1984, The Nature of Mathematical Knowledge, New York: Oxford University Press.

    Google Scholar 

  • Kitcher, P. and Aspray, W.: 1988, An Opinionated Introduction, in Aspray, Kitcher, 1988, pp. 3–57.

  • Kitcher, P.: 1989, ‘Explanatory Unification and the Causal Structure of the World’, in Kitcher and Salmon (Eds.), Scientific Explanation, Minnesota Studies in the Philosophy of Science, vol. XIII, Minneapolis: University of Minnesota Press.

    Google Scholar 

  • Koetsier, T.: 1991, Lakatos' Philosophy of Mathematics. A Historical Approach, Amsterdam: North Holland.

    Google Scholar 

  • Kolmogorov, A. N. and Yushkevich, A. P. (eds.): 1996, Mathematics in the 19th Century. Geometry, Analytic Function Theory, Basel-Boston-Berlin: Birkhäuser.

    Google Scholar 

  • Lakatos, I.: 1962, ‘Infinite Regress and Foundations of Mathematics’, Aristotelian Society Supplementary Volume 36, pp. 155–184. Reprinted in Lakatos 1978, pp. 3–23.

    Google Scholar 

  • Lakatos, I.: 1967, ‘A Renaissance of Empiricism in the Recent Philosophy of Mathematics?’, British Journal for the Philosophy of Science 27, 201–223. Reprinted in Lakatos 1978, pp. 24–42.

    Article  Google Scholar 

  • Lakatos, I., 1976, Proofs and Refutations, Cambridge: Cambridge University Press.

    Google Scholar 

  • Lakatos, I.: 1978, Mathematics, Science and Epistemology (Philosophical Papers, vol. 2), J. Worrall and G. Currie, eds., Cambridge: Cambridge University Press.

    Chapter  Google Scholar 

  • Lakatos, I.: 1978a, Cauchy and the Continuum: The Significance of Non-standard Analysis for the History and Philosophy of Mathematics, in Lakatos 1978, pp. 43–60.

  • MacLane, S., 1997, Despite Physicists, Proof is Essential in Mathematics, in ‘Proof and Progress in Mathematics’, ed. by A. Kanamori, Synthese 111, 147–154.

  • Mancosu, P.: 1996, Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century, New York: Oxford University Press.

    Google Scholar 

  • Mancosu, P.: 1998, ‘Hilbert and Bernays on Metamathematics’, in P. Mancosu (Ed.), From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s, New York: Oxford University Press, pp. 149–188.

    Google Scholar 

  • Mancosu, P.: 1999, ‘Bolzano and Cournot on Mathematical Explanation’, Revue d'Histoire des Sciences 52, 429–455.

    Article  Google Scholar 

  • Mancosu, P.: 2000, On Mathematical Explanation, in E. Grosholz and H. Breger eds., 2000, pp. 103–119.

  • Manders, K.: 1989, ‘Domain Extension and the Philosophy of Mathematics’, Journal of Philosophy 86, 553–562.

    Google Scholar 

  • Mazur, B.: 1997, Conjecture, in ‘Proof and Progress in Mathematics’, ed. by A. Kanamori, Synthese 111, 197–210.

  • Mehlberg, H.: 1960, ‘The Present Situation in the Philosophy of Mathematics’, Synthese 12, 380–412. Reprinted in B. H. Kazemier and D. Vuysie, Logic and Language, Dordrecht: Reidel, 1962, pp. 69–103. Quotes from the 1962 reprint.

    Article  Google Scholar 

  • Menger, K.: 1928, ‘Bemerkungen zur Grundlagenfragen’, J.d.D.M.V. 37, 213–226. Translated as ‘An Intuitionistic-formalistic Dictionary of Set-theory’, in K. Menger, 1979, Selected Papers in Logic and Foundations, Didactics, Economics, Dordrecht: Reidel, pp. 79–87.

    Google Scholar 

  • Meyerson, E.: 1921, De l'Explication dans les Sciences, Paris: Payot. Second edition 1927. English translation of the second edition by M. A. Sipfle and D. A. Sipfle, Explanation in the Sciences, Dordrecht, Boston: Kluwer, 1991.

    Google Scholar 

  • Mill, J. S.: 1843, A System of Logic, London: John P. Parker.

    Google Scholar 

  • Nehrlich, G.: 1979, ‘What Can Geometry Explain?’, British Journal for the Philosophy of Science 30, 69–83.

    Article  Google Scholar 

  • Neuenschwander, E.: 1981, ‘Studies in the History of Complex Function Theory II: Interactions Among the French School, Riemann, and Weierstrass’, Bulletin of the American Mathematical Society 5, 87–105.

    Article  Google Scholar 

  • Parsons, C.: 1990, ‘Introductory Note to Gödel 1944’, in Kurt Gödel (Ed.), Collected Works, vol. II, ed. Solomon Feferman et al., New York, Oxford: Oxford University Press, pp. 102–118.

    Google Scholar 

  • Pitt, J. C. (ed.): 1988, Theories of Explanation, Oxford: Oxford University Press.

    Google Scholar 

  • Poincaré, H.: 1895, ‘L'Espace et la Géométrie’, Revue de Métaphysique et de Morale 3, 631–646.

    Google Scholar 

  • Poincaré, H.: 1898, ‘L'oeuvre mathématique de Weierstrass’, Acta Mathematica 22, 1–18.

    Article  Google Scholar 

  • Pringsheim, A.: 1895, ‘Ueber die Entwickelung eindeutiger analytischer Funktionen in Potenzreihen’, Sitzungsberichte der mathematisch-physikalischen Classe der k. b. Akademie der Wissenschaften zu München 25, 75–92.

    Google Scholar 

  • Pringsheim, A.: 1896, ‘Ueber Vereinfachungen in der elementaren Theorie der analytischen Funktionen’, Mathematische Annalen 47, 121–154.

    Article  Google Scholar 

  • Pringsheim, A.: 1900, ‘Zur Geschichte des Taylorschen Lehrsatzes’, Bibliotheca Mathematica, Dritte Folge 1, 433–479.

    Google Scholar 

  • Pringsheim, A.: 1916, Vorlesungen über Zahlenlehre, Berlin: Teubner, Leipzig (2nd edition 1923).

    Google Scholar 

  • Pringsheim, A.: 1920, Elementare Funktionenlehre und komplexe Integration, Sitzungsberichte der mathematisch-physikalischen Classe der k. b. Akademie der Wissenschaften zu München, 145–182.

  • Pringsheim, A.: 1925, Vorlesungen über Zahlen-und Funktionenlehre, Zweiter Band, Erste Abteilung: Grundlagen der Theorie der analytischen Funktionen einer komplexen Veränderlichen, Leipzig-Berlin: B.G. Teubuer.

    Google Scholar 

  • Quine, W. V.: 1958 (1980), Elementary Logic, Cambridge, Mass: Harvard University Press.

    Google Scholar 

  • Resnik, M. and Kushner, D.: 1987, ‘Explanation, Independence and Realism in Mathematics’, British Journal for the Philosophy of Science 38, 141–158.

    Article  Google Scholar 

  • Rota, G. C.: 1997, The Phenomenology of Mathematical Proof, in “Proof and Progress in Mathematics”, ed. by A. Kanamori, Synthese 111, 183–196.

  • Ruben, D. H.: 1990, Explaining Explanation, London and New York: Routledge.

    Book  Google Scholar 

  • Ruben, D.-H. (ed.): 1993, Explanation, Oxford: Oxford University Press.

    Google Scholar 

  • Ruhla, C.: 1992, The Physics of Chance, Oxford: Oxford University Press.

    Google Scholar 

  • Russell, B.: 1906, ‘Les paradoxes de la logique’, Revue de Métaphysique et de Morale 14, 627–650. English version ‘On “insolubilia” and their solution by symbolic logic’, in Russell, 1973, pp. 190–214.

    Google Scholar 

  • Russell, B.: 1907, The Regressive Method of Discovering the Premises of Mathematics, in Russell, 1973, pp. 272–283.

  • Russell: 1911, ‘La théories des types logiques’, Revue de Métaphysique et de Morale 18, 263–301; English version in Russell, 1973, pp. 215–255.

    Google Scholar 

  • Russell, B.: 1924, ‘Logical Atomism’, in J. M. Muirhead (Ed.), Contemporary British Philosophy, first series, London: George Allen & Unwin, pp. 357–383. Reprinted in Logic and Knowledge, edited by R. C. Marsh, London: George Allen & Unwin, pp.323–343. Page numbers in text are from the latter edition.

    Google Scholar 

  • Russell, B.: 1973, Essays in Analysis, ed. by D. Lackey, London: George Allen & Unwin.

    Google Scholar 

  • Salmon, W.: 1990, Four Decades of Scientific Explanation, Minneapolis: University of Minnesota Press.

    Google Scholar 

  • Sandborg, D.: 1997, Explanation in Mathematical Practice, Ph.D. Dissertation, University of Pittsburgh.

  • Sandborg, D.: 1998, ‘Mathematical Explanation and the Theory of Why-questions’, British Journal for the Philosophy of Science 49, 603–624.

    Article  Google Scholar 

  • Sierpinska, A.: 1994, Understanding in Mathematics, London: The Falmer Press.

    Google Scholar 

  • Steiner, M.: 1978, ‘Mathematical Explanation’, Philosophical Studies 34, 135–151.

    Article  Google Scholar 

  • Steiner, M.: 1983, ‘The Philosophy of Mathematics of Imre Lakatos’, The Journal of Philosophy 80, 502–521.

    Article  Google Scholar 

  • Teller, P.: 1980,’ Computer Proof’, The Journal of Philosophy 77, 797–803.

    Article  Google Scholar 

  • Tymoczko, T.: 1979, ‘The Four-color Problem and its Philosophical Significance’, The Journal of Philosophy 76, 57–83.

    Article  Google Scholar 

  • Tymoczko, T. (ed.): 1986, New Directions in the Philosophy of Mathematics, Boston: Birkhäuser.

    Google Scholar 

  • Thurston, W.: 1994, ‘On Proof and Progress in Mathematics’, Bulletin of the American Mathematical Society 30(2), 161–177.

    Article  Google Scholar 

  • Van Fraassen, B.: 1980, The Scientific Image, Oxford: The Clarendon Press.

    Google Scholar 

  • Whitehead, A. N. and Russell, B.: 1910, Principia Mathematica, vol. 1, Cambridge: Cambridge University Press. Second edition 1927. Quotes from the 1978 reprint of the second edition.

    Google Scholar 

  • Wittgenstein, L.: 1976, Wittgenstein's Lectures on the Foundations of Mathematics, Cambridge 1939, Ithaca, NY: Cornell University Press.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. U.C, Berkeley, U.S.A

    Paolo Mancosu

Authors
  1. Paolo Mancosu
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mancosu, P. Mathematical Explanation: Problems and Prospects. Topoi 20, 97–117 (2001). https://doi.org/10.1023/A:1010621314372

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1010621314372

Keywords

Search

Navigation