Journal for General Philosophy of Science

, Volume 44, Issue 1, pp 41–61

Defending the Indispensability Argument: Atoms, Infinity and the Continuum

Article

Abstract

This paper defends the Quine-Putnam mathematical indispensability argument against two objections raised by Penelope Maddy. The objections concern scientific practices regarding the development of the atomic theory and the role of applied mathematics in the continuum and infinity. I present two alternative accounts by Stephen Brush and Alan Chalmers on the atomic theory. I argue that these two theories are consistent with Quine’s theory of scientific confirmation. I advance some novel versions of the indispensability argument. I argue that these new versions accommodate Maddy’s history of the atomic theory. Counter-examples are provided regarding the role of the mathematical continuum and mathematical infinity in science.

Keywords

Atomic theory Infinity Maddy Mathematical indispensability Quine-Putnam The continuum 

References

  1. Azzouni, J. (2004). Deflating existential consequence: A case for nominalism. New York: Oxford University Press.CrossRefGoogle Scholar
  2. Baker, A. (2009). Mathematical explanation in science. British Journal for the Philosophy of Science, 60, 611–633.CrossRefGoogle Scholar
  3. Bangu, S. (2012). The applicability of mathematics: Indispensability and ontology. London: Palgrave-Macmillan.Google Scholar
  4. Berzelius, J. (1813). Experiments on the nature of azote, hydrogen, and of ammonia and upon the degree of oxidation of which azote is susceptible. Annals of Philosophy, 2, 276–284, 357–368.Google Scholar
  5. Boltzmann, L. (1974). Reply to a lecture on happiness given by Professor Ostwald. In B. MacGuiness (Ed.), Ludwig Boltzmann. Theoretical physics and philosophical problems (pp. 173–184). Dordrecht: Reidel Publishing Company.Google Scholar
  6. Brush, S. (1968). A history of random processes, I. Brownian Movement from Brown to Perrin. Archive for History of Exact Sciences, 5, 1–36.CrossRefGoogle Scholar
  7. Brush, S. (1974). Should the history of science be rated X? Science, 183, 1164–1172.CrossRefGoogle Scholar
  8. Busch, J. (2011). Is the indispensability argument dispensable? Theoria, 77, 139–158.CrossRefGoogle Scholar
  9. Busch, J. (2012). The indispensability argument for mathematical realism and scientific realism. Journal for General Philosophy of Science, 43(1), 3–9.CrossRefGoogle Scholar
  10. Butterfield, J., & Isham, C. (2004). Spacetime and the philosophical challenge of quantum gravity. In C. Callender & N. Huggett (Eds.), Physics meets philosophy at the Planck scale—contemporary theories in quantum gravity (pp. 33–89). Cambridge: Cambridge University Press.Google Scholar
  11. Chalmers, A. (2008). Atom and aether in nineteenth-century physical science. Foundations of Chemistry, 10, 157–166.CrossRefGoogle Scholar
  12. Chalmers, A. (2009). The scientist’s atom and philosopher’s stone—how science succeeded and philosophy failed to gain knowledge of atoms. New York: Springer.Google Scholar
  13. Chalmers, A. (2011). Drawing philosophical lessons from Perrin’s Experiments on Brownian motion: A response to van Fraassen. British Journal for the Philosophy of Science, 62(4), 711–732.CrossRefGoogle Scholar
  14. Cheyne, C. (2001). Knowledge, cause, and abstract objects. Dordrecht: Kluwer.CrossRefGoogle Scholar
  15. Clark, P. (1976). Atomism versus thermodynamics. In C. Howson (Ed.), Method and appraisal in the physical sciences (pp. 41–105). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  16. Colyvan, M. (1998). In defence of indispensability. Philosophia Mathematica, 6(3), 39–62.CrossRefGoogle Scholar
  17. Colyvan, M. (2001). The indispensability of mathematics. New York: Oxford University Press.CrossRefGoogle Scholar
  18. Colyvan, M. (2006). Scientific realism and mathematical nominalism: A marriage made in hell. In C. Cheyne & J. Worrall (Eds.), Rationality and reality: Conversations with Alan Musgrave. Netherlands: Springer.Google Scholar
  19. Colyvan, M., & Easwaran, K. (2008). Mathematical and physical continuity. Australasian Journal of Logic, 6, 87–93.Google Scholar
  20. Davies, P. (1989). The new physics: A synthesis. In The new physics (pp. 1–6). Cambridge: Cambridge University Press.Google Scholar
  21. Decock, L. (2002). Quine’s weak and strong indispensability argument. Journal for General Philosophy of Science, 33, 231–250.CrossRefGoogle Scholar
  22. Devitt, M. (1984). Realism and truth. Oxford: Basil Blackwell.Google Scholar
  23. Dieveney, P. (2007). Dispensability in the indispensability argument. Synthese, 157, 105–128.CrossRefGoogle Scholar
  24. Duhem, P. (2007). La Théorie physique: son object, sa structure. Paris: Vrin.Google Scholar
  25. Einstein, A. (1949). Reply to criticism. In P. Schilpp (Ed.), Albert Einstein: Philosopher-scientist (pp. 663–688). La Salle Ill.: Open Court.Google Scholar
  26. Ellis, G. (2007). Issues in the philosophy of cosmology. In J. Butterfield & J. Earman (Eds.), Philosophy of physics (pp. 1183–1287). The Netherlands: North Holland.CrossRefGoogle Scholar
  27. Feynman, R. (1967). The character of physical law. Cambridge, MA: MIT Press.Google Scholar
  28. Feynman, R. (1985). QED: The strange theory of light and matter. Princeton, NJ: Princeton University Press.Google Scholar
  29. Feynman, R., Leighton, R., & Sands, M. (1964). The Feynman lectures on physics (Vol. ii). Reading, MA: Addison-Wesley.Google Scholar
  30. Field, H. (1980). Science without numbers. Princeton, NJ: Princeton University Press.Google Scholar
  31. Gardner, M. (1979). Realism and instrumentalism in 19th-century atomism. Philosophy of Science, 46, 1–34.CrossRefGoogle Scholar
  32. Gasperini, M. (2008). The universe before the big bang—cosmology and string theory. Berlin: Springer.Google Scholar
  33. Hawking, S., & Ellis, G. (1973). The large structure of space-time. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  34. Heller, M. (2011). Infinities in cosmology. In M. Heller & H. Woodin (Eds.), Infinity: New research frontiers (pp. 218–229). New York: Cambridge University Press.CrossRefGoogle Scholar
  35. Isham, C. (1989). Quantum gravity. In P. Davies (Ed.), The new physics (pp. 70–93). Cambridge: Cambridge University Press.Google Scholar
  36. Jacques, J. (1987). Berthelot: 1827–1907, Autopsie d’un Mythe. Paris: Belin.Google Scholar
  37. Jozsa, R. (1986). An approach to the modelling of the physical continuum. The British Journal for the Philosophy of Science, 37, 395–404.CrossRefGoogle Scholar
  38. Katz, J. (1998). Realistic rationalism. Cambridge, MA: MIT Press.Google Scholar
  39. Maddy, P. (1990). Realism in mathematics. Oxford: Oxford Clarendon Press.Google Scholar
  40. Maddy, P. (1992). Indispensability and practice. Journal of Philosophy, 89, 275–289.CrossRefGoogle Scholar
  41. Maddy, P. (1997). Naturalism in mathematics. New York: Oxford University Press.Google Scholar
  42. Maddy, P. (2000). Naturalism and the a priori. In P. Boghossian & C. Peacocke (Eds.), New essays on the a priori (pp. 92–116). Oxford: Oxford University Press.CrossRefGoogle Scholar
  43. Maddy, P. (2005). Three forms of naturalism. In S. Shapiro (Ed.), The Oxford handbook of philosophy of mathematics and logic (pp. 437–459). New York: Oxford University Press.CrossRefGoogle Scholar
  44. Maddy, P. (2007). Second philosophy—a naturalistic method. Oxford: Oxford University Press.CrossRefGoogle Scholar
  45. Majid, S. (2008). Preface. In S. Majid (Ed.), On space and time (pp. xi–xx). New York: Cambridge University Press.CrossRefGoogle Scholar
  46. Nyhof, J. (1988). Philosophical objections to the kinetic theory. The British Journal for the Philosophy of Science, 39, 81–109.CrossRefGoogle Scholar
  47. Parsons, C. (1980). Mathematical Intuition. In W. Hart (Ed.), The philosophy of mathematics (pp. 95–113). Oxford: Oxford University Press.Google Scholar
  48. Penrose, R. (2004). The road to reality—a complete guide to the laws of the universe. London: Vintage.Google Scholar
  49. Perrin, J. (1909). Brownian movement and molecular reality (F. Soddy, Trans.). London: Taylor and Francis, 1910.Google Scholar
  50. Perrin, J. (1913). Atoms (D. Hammick, Trans.). New York: Van Nostrand, 1923.Google Scholar
  51. Poincaré, H. (1913). Dernières Pensées. Paris: Flammarion.Google Scholar
  52. Poincaré, H. (1968). La Science et l’Hypothèse. Paris: Flammarion.Google Scholar
  53. Psillos, S. (2011). Moving molecules above the scientific horizon: On Perrin’s case for realism. Journal for General Philosophy of Science, 42(2), 339–363.CrossRefGoogle Scholar
  54. Putnam, H. (1971). Philosophy of logic. In S. Laurence & L. Macdonald (Eds.), Contemporary readings in foundations of metaphysics (pp. 404–434). Oxford: Blackwell.Google Scholar
  55. Putnam, H. (1979). What is mathematical truth? In Mathematics, matter and method: Philosophical papers vol. I (pp. 60–78). Cambridge: Cambridge University Press.Google Scholar
  56. Quine, W. V. (1953). From a logical point of view. Cambridge, MA: Harvard University Press.Google Scholar
  57. Quine, W. V. (1955). Posits and reality. In The ways of paradox and other essays (pp. 233–241). New York: Random House.Google Scholar
  58. Quine, W. V. (1963). Carnap and logical truth. In P. Benacerraf & H. Putnam (Eds.), Philosophy of mathematics selected readings (pp. 355–376). Cambridge: Cambridge University Press.Google Scholar
  59. Quine, W. V. (1981a). Success and limits of mathematization. In Theories and things (pp. 148–155). Cambridge, MA: Harvard University Press.Google Scholar
  60. Quine, W. V. (1981b). Things and their place in theories. In Theories and things (pp. 1–23). Cambridge, MA: Harvard University Press.Google Scholar
  61. Quine, W. V., & Ullian, J. (1970). The web of belief. New York: Random House.Google Scholar
  62. Resnik, M. (1997). Mathematics as a science of patterns. Oxford: Oxford University Press.Google Scholar
  63. Rickles, D., & French, S. (2006). Quantum gravity meets structuralism: Interweaving relations in the foundations of physics. In D. Rickles, S. French, & J. Saatsi (Eds.), The structural foundations of quantum gravity (pp. 1–39). New York: Oxford University Press.CrossRefGoogle Scholar
  64. van Fraassen, B. (2009). The perils of Perrin, in the hands of philosophers. Philosophical Studies, 143, 5–24.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade da Beira InteriorCovilhãPortugal
  2. 2.LanCog, Centro de Filosofia da Universidade de Lisboa, Alameda da UniversidadeLisboaPortugal

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