Skip to main content
SpringerLink
Log in

The Epistemic Indispensability Argument

  • Article
  • Published:
Journal for General Philosophy of Science Aims and scope Submit manuscript

Abstract

This article elaborates the epistemic indispensability argument, which fully embraces the epistemic contribution of mathematics to science, but rejects the contention that such a contribution is a reason for granting reality to mathematicalia. Section 1 introduces the distinction between ontological and epistemic readings of the indispensability argument. Section 2 outlines some of the main flaws of the first premise of the ontological reading. Section 3 advances the epistemic indispensability argument in view of both applied and pure mathematics. And Sect. 4 makes a case for the epistemic approach, which firstly calls into question the appeal to inference to the best explanation in the defense of the indispensability claim; secondly, distinguishes between mathematical and physical posits; and thirdly, argues that even though some may think that inference to the best explanation works in the postulation of physical posits, no similar considerations are available for postulating mathematicalia.

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

Similar content being viewed by others

Notes

  1. For an elaboration of unrestricted mathematical platonism, see Balaguer (1998).

  2. Note that my argument focuses on distinguishing those versions of the indispensability argument that draw ontological consequences concerning mathematicalia, and those that propose to read the indispensability claim in an epistemic key. I am aware of other versions of the indispensability argument. For a recent survey of the matter, see Panza and Sereni (2016).

  3. For critical appraisals of the argument, see Baker (2005, 2009), Saatsi (2011), Bangu (2012), Pincock (2012) and Bueno and French (2018), among others.

  4. Alternative constructions of the argument focus on the ontological reading as well. For instance, Resnik (1997) pays attention to the pragmatic component of the use of mathematics in scientific theorizing, whereas Baker (2009) endorses the reality of those mathematical entities that indispensably partake in scientific explanations. As in the standard OIA, however, both take their approaches to lead us to the conclusion that at least some mathematicalia exist.

  5. Note, however, that this is not always the case. Liggins (2016, 533 and ff.) argues for a form of indispensability that avoids both conformational holism and naturalism. See also Baker (2005, 224).

  6. Our argument may seem to fall prey of what Putnam calls the “intellectual dishonesty” (1971, 347) of not believing in the existence of what one daily presupposes. In our case, mathematics partakes in scientific theorizing routinely. Hence, why should we not believe in the reality of mathematicalia? (Or so the platonist would challenge us.) Against Putnam’s view, I am inclined to endorse the distinction between quantifier commitment and ontological commitment advocated by Azzouni (2004) and Bueno (2005), i.e., scientific theories routinely quantify over mathematical posits without entailing ontological commitment. Beyond the mathematical setting, we likewise quantify over fictional entities in literature, folklore, or else, without entailing existence of any sort.

  7. For discussion in this direction, but that concentrates on the explanatory indispensability of mathematics specifically, see Bangu (2008) and Saatsi (2011, 2016).

  8. For the sake of brevity, I shall look into these two criteria only in terms of experimental access and causal role. I am aware that other strategies are available in recent literature. Azzouni (1997) advances the criterion of thick epistemic access for separating the mathematical from the physical. Another criterion is discussed by Colyvan (1999) and Melia (2002), namely, unification in theory and in ontology. Concerning the latter, Melia (2002, 75) observes that the postulation of both mathematical and physical posits may contribute to the attractiveness and utility of theories, but points out that they do so differently. He suggests a case for demonstrating that mathematical unification does not entail physical unification. The use of complex numbers rather than real algebra helps us solve different equations in a unified way. This does not imply physical unification, he argues, nor does it prove that mathematical entities are on an ontological par with physical posits. By contrast, physical unification may give us a hint about the reality of new layers of phenomena, as in the case of Einstein’s postulation of a four-dimensional space-time, which unified the Newtonian gravitational theory and new phenomena related to gravitational lensing and redshift; or as in the case of Maxwell’s unification of electromagnetic phenomena and optical phenomena in his equations describing the behaviour of the electromagnetic field.

  9. This criterion has also been contested within the philosophy of mathematics. Bigelow’s (1988) physicalist Pythagoreanism and Franklin’s (2014) Aristotelian structural realism grant mathematical entities causal powers by placing them in the physical world. At present, for reasons of space, I leave these strategies aside. From a different perspective, Colyvan (2006) and Baker (2009) defend the view that mathematical entities play a genuine explanatory role in science, which, if granted, not only would justify their appearance as variables in current best scientific theories, but would also give us reason to believe that they are part and parcel of the fabric of reality. I briefly return to this issue in Sect. 5.

  10. We can also add now that the postulation of dark matter contributes to the unificatory power of explanations of physical phenomena, bringing together aspects as diverse as the total mass of the universe, the curvature observed in the orbits of galaxies, light lensing, and so forth.

References

  • Arabatzis, T. (2006). Representing electrons. A biographical approach to theoretical entities. Chicago and London: The University of Chicago Press.

    Google Scholar 

  • Azzouni, J. (1997). Thick epistemic access: Distinguishing the mathematical from the empirical. The Journal of Philosophy, 94(9), 472–484.

    Google Scholar 

  • Azzouni, J. (2004). Deflating existential consequence. New York: Oxford University Press.

    Book  Google Scholar 

  • Baker, A. (2005). Are there genuine mathematical explanations of physical phenomena? Mind, 114(454), 223–238.

    Article  Google Scholar 

  • Baker, A. (2009). Mathematical explanation in science. British Journal for the Philosophy of Science, 60(3), 611–633.

    Article  Google Scholar 

  • Baker, A., & Colyvan, M. (2011). Indexing and mathematical explanation. Philosophia Mathematica, 19(3), 323–334.

    Article  Google Scholar 

  • Balaguer, M. (1998). Platonism and anti-platonism in mathematics. Oxford: Oxford University Press.

    Google Scholar 

  • Bangu, S. (2008). Inference to the best explanation and mathematical realism. Synthese, 160(1), 13–20.

    Article  Google Scholar 

  • Bangu, S. (2012). The applicability of mathematics in science: Indispensability and ontology. London: Palgrave Macmillan.

    Google Scholar 

  • Bigelow, J. (1988). The reality of numbers. A physicalist’s philosophy of mathematics. Oxford: Clarendon Press.

    Google Scholar 

  • Bueno, O. (2005). Dirac and the dispensability of mathematics. Studies in History and Philosophy of Modern Physics, 36(3), 465–490.

    Article  Google Scholar 

  • Bueno, O. (2009). Mathematical fictionalism. In O. Bueno, & O. Linnebo (Eds.), New waves in the philosophy of mathematics (pp. 59–79). New York: Palgrave Macmillan.

    Chapter  Google Scholar 

  • Bueno, O. (2016). An anti-realist application of the application of mathematics. Philosophical Studies, 173(10), 2591–2604. https://doi.org/10.1007/s11098-016-0670-y.

    Article  Google Scholar 

  • Bueno, O., & Colyvan, M. (2011). An inferential conception of the application of mathematics. Nous, 45(2), 345–374.

    Article  Google Scholar 

  • Bueno, O., & French, S. (2018). Applying mathematics: Immersion, inference, interpretation. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Clowe, D., Bradac, M., Gonzalez, A. H., Marketevich, M., Randall, S. W., & Zaritsky, D. (2006). A direct empirical proof of the existence of dark matter. The Astrophysical Journal Letters, 648(2), L109–L113.

    Article  Google Scholar 

  • Colyvan, M. (1999). Confirmation theory and indispensability. Philosophical Studies, 96(1), 1–19.

    Article  Google Scholar 

  • Colyvan, M. (2001). The indispensability of mathematics. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Colyvan, M. (2002). Mathematics and aesthetics considerations in science. Mind, 111(441), 69–74.

    Article  Google Scholar 

  • 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 (pp. 225–237). Dordrecht: Springer.

    Chapter  Google Scholar 

  • Colyvan, M. (2012). An introduction to the philosophy of mathematics. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Daly, C., & Langford, S. (2009). Mathematical explanation and indispensability arguments. The Philosophical Quarterly, 59(237), 641–658.

    Article  Google Scholar 

  • Dyson, F. (1962). Mathematics in the physical sciences. Scientific American, 211(3), 128–146.

    Article  Google Scholar 

  • Field, H. (1980). Science without numbers. A defence of nominalism. Princeton: Princeton University Press.

    Google Scholar 

  • Field, H. (1989). Realism, mathematics and modality. New York: Basil Blackwell.

    Google Scholar 

  • Franklin, J. (2014). An aristotelian realist philosophy of mathematics. Mathematics as the science of quantity and structure. London: Palgrave Macmillan.

    Book  Google Scholar 

  • Freese, K. (2006). The dark side of the universe. Nuclear Instruments and Methods in Physics Research, A, 559(2), 337–340.

    Article  Google Scholar 

  • Lange, M. (2002). An introduction to the philosophy of physics: Locality, fields, energy, and mass. New York: Blackwell.

    Google Scholar 

  • Leng, M. (2002). What’s wrong with indispensability? (Or, the case for recreational mathematics). Synthese, 131(3), 395–417.

    Article  Google Scholar 

  • Leng, M. (2010). Mathematics and reality. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Liggins, D. (2016). Grounding and the indispensability argument. Synthese, 193(2), 531–548. https://doi.org/10.1007/s11229-014-0478-2.

    Article  Google Scholar 

  • Maddy, P. (1990). Realism in mathematics. Oxford: Oxford University Press.

    Google Scholar 

  • Maddy, P. (1992). Indispensability and practice. The Journal of Philosophy, 89(6), 275–289.

    Article  Google Scholar 

  • Maddy, P. (1995). Naturalism and ontology. Philosophia Mathematica, 3(3), 248–270.

    Article  Google Scholar 

  • Maddy, P. (1997). Naturalism in mathematics. Oxford: Oxford University Press.

    Google Scholar 

  • Melia, J. (2000). Weaseling away the indispensability argument. Mind, 109(435), 435–479.

    Article  Google Scholar 

  • Melia, J. (2002). Response to Colyvan. Mind, 111(441), 75–79.

    Article  Google Scholar 

  • Morrison, M. (2015). Reconstructing reality. Models, mathematics, and simulations. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Musgrave, A. (1986). Arithmetical platonism: Is Wright wrong or must Field yield? In M. Fricke (Ed.), Essays in honour of Bob Durrant (pp. 90–110). Dunedin: Otago University Philosophy Department.

    Google Scholar 

  • Panza, M., & Sereni, A. (2016). The varieties of indispensability arguments. Synthese, 193(2), 469–516. https://doi.org/10.1007/s11229-015-0977-9.

    Article  Google Scholar 

  • Pincock, C. (2012). Mathematics and scientific representation. Oxford: Oxford University Press.

    Book  Google Scholar 

  • Psillos, S. (2012). Anti-nominalistic scientific realism: A defence. In A. Bird, B. Ellis, & H. Sankey (Eds.), Properties, powers, and structures. Issues in the metaphysics of realism (pp. 63–80). New York and London: Routledge.

    Google Scholar 

  • Putnam, H. (1971). Philosophy of logic. In H. Putnam (Ed.), Mathematics, matter and method: Philosophical papers (Vol. 1, pp. 323–357). Cambridge: Cambridge University Press.

    Google Scholar 

  • Quine, W. V. O. (1948). On what there is. The Review of Metaphysics, 2(5), 21–38.

    Google Scholar 

  • Quine, W. V. O. (1951). Two dogmas of empiricism. The Philosophical Review, 60(1), 20–43.

    Article  Google Scholar 

  • Quine, W. V. O. (1981). Theories and things. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Quine, W. V. O. (2004). Quintessence. In R. F. Gibson Jr. (Ed.), Basic readings from the philosophy of W. V. Quine. Cambridge, MA: The Belknap Press of Harvard University Press.

    Google Scholar 

  • Resnik, M. D. (1997). Mathematics as a science of patterns. Oxford: Clarendon Press.

    Google Scholar 

  • Saatsi, J. (2011). The enhanced indispensability argument: Representational versus explanatory role of mathematics in science. British Journal for the Philosophy of Science, 62(1), 143–154.

    Article  Google Scholar 

  • Saatsi, J. (2016). On the ‘indiepensable explanatory role’ of mathematics. Mind, 125(500), 1045–1070.

    Article  Google Scholar 

  • Sober, E. (1993). Mathematics and indispensability. The Philosophical Review, 102(1), 35–57.

    Article  Google Scholar 

  • Steiner, M. (1995). The applicabilities of mathematics. Philosophia Mathematica, 3(2), 129–156.

    Article  Google Scholar 

  • Steiner, M. (1998). The applicability of mathematics as a philosophical problem. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Tegmark, M. (2014). Our mathematical universe. My quest for the ultimate nature of reality. London: Penguin Books.

    Google Scholar 

  • Weinberg, S. (1993). Dreams of a final theory. London: Vintage.

    Book  Google Scholar 

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

    Article  Google Scholar 

Download references

Acknowledgements

This article is a result of the governmental funded research Grant FONDECYT de Iniciación, No. 11160324, “The Physico-Mathematical Structure of Scientific Laws: On the Roles of Mathematics, Models, Measurements, and Metaphysics in the Construction of Laws in Physics,” CONICYT, Chile.

Author information

Authors and Affiliations

  1. Departamento de Filosofía, Universidad de Chile, Av. Capitán Ignacio Carrera Pinto, 1025, Ñuñoa, RM, Chile

    Cristian Soto

Authors
  1. Cristian Soto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cristian Soto.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Soto, C. The Epistemic Indispensability Argument. J Gen Philos Sci 50, 145–161 (2019). https://doi.org/10.1007/s10838-018-9437-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10838-018-9437-9

Keywords

Search

Navigation