Skip to main content

Defending the Indispensability Argument: Atoms, Infinity and the Continuum


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.

This is a preview of subscription content, access via your institution.


  1. 1.

    According to Maddy (1997, 134), Quine’s theory of confirmation includes the doctrine of confirmational holism, the criterion of ontological commitment and the theoretical benefits of scientific confirmation.

  2. 2.

    See also Bangu (2012, chapter 9) for a discussion of these objections.

  3. 3.

    See Colyvan (2001, 102) and Colyvan and Easwaran (2008).

  4. 4.

    For example, Quine (1963, 367; 1981a, 149–150) and Putnam (1971, 425; 1979, 74).

  5. 5.

    As in the case of argument Q-P1, I assume that there is a hidden naturalist premise in argument Q-P2.

  6. 6.

    The atomic debates had two sides: the physical side essentially based on the kinetic theory and the chemical side essentially based on the theory of thermodynamics.

  7. 7.

    Nyhof (1988) argues in the same direction as Brush.

  8. 8.

    For Mach see Nyhof (1988, 86–91); for Berthelot see Jacques (1987).

  9. 9.

    Maddy (2005) seems to subscribe to this same naturalistic view, but in other passages (e.g., Maddy 1997, 182) she seems to endorse other formulations of Quine’s naturalism.

  10. 10.

    In this vein, Clark (1976) argues that there were two competing research programmes during the nineteenth century—the atomic-kinetic research programme and the thermodynamics research programme. Clark denies external explanations concerning scientific evidence and makes the opposite claim: ‘it was the degeneration of the kinetic programme compared with the empirical progress of thermodynamics which accounts for the rise of scientific positivism’ (Clark 1976, 44). Clark’s arguments are shaped by the philosophy of science developed by Lakatos. Here, I will not discuss Clark’s view.

  11. 11.

    Berzelius (1813, 359).

  12. 12.

    Nyhof (1988, 90).

  13. 13.

    Clark (1976, 74).

  14. 14.

    As quoted in Clark (1976, 42).

  15. 15.

    For a full discussion of this point, see Chalmers (2009, 227–246). Here, I am ignoring Chalmers’ epistemic views on how scientific theories are confirmed: ‘a theory is supported only if the evidence conforms to predictions following in a natural rather than a contrived way from the theory in conjunction with subsidiary assumptions that themselves have independent support’ Chalmers (2009, 238). For a detailed analysis of this aspect, see also Chalmers (2011).

  16. 16.

    Other tensions in Maddy’s history are noted by van Fraassen (2009, footnote 2).

  17. 17.

    See also Quine (1955, 234) and Quine (1953, 40–41).

  18. 18.

    See Psillos (2011) for a detailed analysis of five key participants (Duhem, Stallo, Ostwald, Poincaré and Boltzmann) in the philosophical controversy concerning atomic theory. Psillos identifies two lines of dispute concerning the atomic hypothesis. By 1900, it was either perceived to be instrumentally indispensable (Stallo-Duhem-Mach) or was perceived to be epistemically precarious (Ostwald-Poincaré-Boltzmann).

  19. 19.

    This is a dispute that I cannot address in this paper. Thus, I will consider MC’s hypothesis, as other historians of science can address this dispute.

  20. 20.

    For example: ‘a good portion of scepticism was based on considerations even the supporters of atoms considered reasonable’ (Maddy 1997, 138); ‘there was still scepticism among respectable scientists for respectable reasons about the existence of atoms’ (Maddy 2000, 109).

  21. 21.

    Maddy (1997, 139). Here, the words ‘virtues’ and benefits’ are interchangeable.

  22. 22.

    ‘Epistemic values’ is the contemporary jargon for ‘theoretical benefits’.

  23. 23.

    For example, mild belief is the attitude that we should take ‘to the entities of a theory that is doing well but is not thoroughly established’; agnosticism is the attitude that we should take ‘to the entities of untried speculations at the frontiers’, Devitt (1984, 121).

  24. 24.

    Other versions of the indispensability argument are advanced by Resnik (1997, chapter 3), (the pragmatic indispensability argument), Decock (2002; weak/strong indispensability arguments are detached from Quine’s works) and Baker (2009; the enhanced indispensability argument).

  25. 25.

    Prima facie, this argument is redundant, as we could have reasons to be agnostic about numerous things, even if they are not indispensable to our scientific theories (not even when these are part of our speculations). However, in this case, I should recommend outright disbelief.

  26. 26.

    It may be objected that Quine’s criterion of ontological commitment is inconsistent with the gradation suggested by the arguments. Our scientific theories are either true or false. Hence, either certain objects lie in the domain of the quantification or not. There is no room for a middle ground. To this, I reply that this objection only shifts the problem to the truth value of the scientific theories. As, in light of Quine’s theory of confirmation, truth values and ontological commitments are on par, the above refined versions may be adapted into arguments regarding our epistemic attitudes concerning the truth value of scientific theories. It may be also objected that the same scientific statements can be common to different types of scientific theories. Therefore, it is possible that we should simultaneously be strongly committed, mildly committed and ontologically agnostic regarding entities that are indispensable to different types of scientific theories. To this objection, I reply that if an entity is indispensable to different types of scientific theories, then our type of commitment is determined by the stronger commitment of all available information on the entity in question. Here, there is a sort of separatism, but it is not the ‘current’ separatist objection to confirmational holism between mathematical entities versus empirical entities. See Busch (2011, 2012), Colyvan (2006) and Dieveney (2007) for a discussion of the separatist objection.

  27. 27.

    See Gardner (1979) for this analysis. Similar to my arguments, he argues ‘that there was a gradual transition from an instrumentalist to a realistic acceptance [of the atomic theory], because of gradual increases in its predictive power’ (Gardner 1979, 1).

  28. 28.

    The Hubble constant is proportional to the space–time curvature. ‘The most recent measurements seem to indicate that the spatial curvature is almost zero (…) and that the total energy density is very close to the critical value [which depends on the current value of the Hubble constant, the speed of light and Newton’s gravitational constant]’ (Gasperini 2008, 29).

  29. 29.

    This technical scientific account was based on Gasperini (2008).

  30. 30.

    These theorems do not take quantum gravity effects into account. Nonetheless, these theorems imply that quantum gravity must, in the limit, recover the classical geometry of space–time provided by general relativity. That is, singularities are also reproduced as limiting cases of quantum gravity. See Heller (2011, 223–224).

  31. 31.

    Heller (2011, 223).

  32. 32.

    See Ellis (2007, 1268) for arguments in the direction that claims about infinity are unverifiable.

  33. 33.

    ‘[M]athematical methods have been elaborated to deal with infinities. Although we are unable to work directly with infinities, we apply to them various versions of the ‘strategy of limits.’ The standard ‘going to the limit’ in a convergent series is the simplest and best-known method of this kind.’ Heller (2011, 224). Note, however, that this is not an idealisation, as we are attempting to address quantities that are ‘known’ to be infinite. Moreover, to the best of my knowledge, cosmological models do not involve any prior explicit idealisation concerning these quantities. Singularities are a scientific consequence of our best cosmological theories.

  34. 34.

    Here, I will not address the contemporary discussion regarding the relationalist or substantivalist views of space–time. The philosophical issues regarding the continuum go back to Zeno’s paradoxes.

  35. 35.

    A space is compact if it is closed and bounded.

  36. 36.

    See Penrose (2004, 62, 574).

  37. 37.

    ‘In the quantum theory of a particle, the attribute of position (and most other physical properties), when they exist, still take values in the classical model R’ (Jozsa 1986, 395).

  38. 38.

    See Butterfield and Isham (2004, 84–85).

  39. 39.

    Butterfield and Isham (2004, 34).

  40. 40.

    Rickles and French (2006, 7–14).


  1. Azzouni, J. (2004). Deflating existential consequence: A case for nominalism. New York: Oxford University Press.

    Book  Google Scholar 

  2. Baker, A. (2009). Mathematical explanation in science. British Journal for the Philosophy of Science, 60, 611–633.

    Article  Google 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.

  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.

    Article  Google Scholar 

  7. Brush, S. (1974). Should the history of science be rated X? Science, 183, 1164–1172.

    Article  Google Scholar 

  8. Busch, J. (2011). Is the indispensability argument dispensable? Theoria, 77, 139–158.

    Article  Google Scholar 

  9. Busch, J. (2012). The indispensability argument for mathematical realism and scientific realism. Journal for General Philosophy of Science, 43(1), 3–9.

    Article  Google 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.

    Article  Google 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.

    Article  Google Scholar 

  14. Cheyne, C. (2001). Knowledge, cause, and abstract objects. Dordrecht: Kluwer.

    Book  Google 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.

    Chapter  Google Scholar 

  16. Colyvan, M. (1998). In defence of indispensability. Philosophia Mathematica, 6(3), 39–62.

    Article  Google Scholar 

  17. Colyvan, M. (2001). The indispensability of mathematics. New York: Oxford University Press.

    Book  Google 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.

  21. Decock, L. (2002). Quine’s weak and strong indispensability argument. Journal for General Philosophy of Science, 33, 231–250.

    Article  Google 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.

    Article  Google 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.

    Chapter  Google Scholar 

  27. Feynman, R. (1967). The character of physical law. Cambridge, MA: MIT Press.

  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.

  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.

    Article  Google 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.

    Book  Google 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.

    Chapter  Google Scholar 

  35. Isham, C. (1989). Quantum gravity. In P. Davies (Ed.), The new physics (pp. 70–93). Cambridge: Cambridge University Press.

  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.

    Article  Google Scholar 

  38. Katz, J. (1998). Realistic rationalism. Cambridge, MA: MIT Press.

  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.

    Article  Google 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.

    Chapter  Google 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.

    Chapter  Google Scholar 

  44. Maddy, P. (2007). Second philosophy—a naturalistic method. Oxford: Oxford University Press.

    Book  Google Scholar 

  45. Majid, S. (2008). Preface. In S. Majid (Ed.), On space and time (pp. xi–xx). New York: Cambridge University Press.

    Chapter  Google Scholar 

  46. Nyhof, J. (1988). Philosophical objections to the kinetic theory. The British Journal for the Philosophy of Science, 39, 81–109.

    Article  Google 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.

  50. Perrin, J. (1913). Atoms (D. Hammick, Trans.). New York: Van Nostrand, 1923.

  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.

    Article  Google 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.

  56. Quine, W. V. (1953). From a logical point of view. Cambridge, MA: Harvard University Press.

  57. Quine, W. V. (1955). Posits and reality. In The ways of paradox and other essays (pp. 233–241). New York: Random House.

  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.

  60. Quine, W. V. (1981b). Things and their place in theories. In Theories and things (pp. 1–23). Cambridge, MA: Harvard University Press.

  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.

    Chapter  Google Scholar 

  64. van Fraassen, B. (2009). The perils of Perrin, in the hands of philosophers. Philosophical Studies, 143, 5–24.

    Article  Google Scholar 

Download references


Thanks to the audiences and individuals who commented on previous versions of this paper, especially, Alexander Bird, James Ladyman, Øystein Linnebo, John Mayberry, Leon Horsten and José Díez. I would also like to thank two anonymous referees for this journal for their comments and suggestions. This work was supported by grant SFRH/BPD/46847/2008, the project PTDC/FIL-FCI/109991/2009, “Hilbert’s Legacy in the Philosophy of Mathematics”, and the project PTDC/FIL–FIL/121209/2010, “Online Companion to Problems of Analytical Philosophy”, from Fundação para a Ciência e Tecnologia.

Author information



Corresponding author

Correspondence to Eduardo Castro.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Castro, E. Defending the Indispensability Argument: Atoms, Infinity and the Continuum. J Gen Philos Sci 44, 41–61 (2013).

Download citation


  • Atomic theory
  • Infinity
  • Maddy
  • Mathematical indispensability
  • Quine-Putnam
  • The continuum