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 to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
For an elaboration of unrestricted mathematical platonism, see Balaguer (1998).
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).
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.
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.
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.
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.
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.
Arabatzis, T. (2006). Representing electrons. A biographical approach to theoretical entities. Chicago and London: The University of Chicago Press.
Azzouni, J. (1997). Thick epistemic access: Distinguishing the mathematical from the empirical. The Journal of Philosophy, 94(9), 472–484.
Azzouni, J. (2004). Deflating existential consequence. New York: Oxford University Press.
Baker, A. (2005). Are there genuine mathematical explanations of physical phenomena? Mind, 114(454), 223–238.
Baker, A. (2009). Mathematical explanation in science. British Journal for the Philosophy of Science, 60(3), 611–633.
Baker, A., & Colyvan, M. (2011). Indexing and mathematical explanation. Philosophia Mathematica, 19(3), 323–334.
Balaguer, M. (1998). Platonism and anti-platonism in mathematics. Oxford: Oxford University Press.
Bangu, S. (2008). Inference to the best explanation and mathematical realism. Synthese, 160(1), 13–20.
Bangu, S. (2012). The applicability of mathematics in science: Indispensability and ontology. London: Palgrave Macmillan.
Bigelow, J. (1988). The reality of numbers. A physicalist’s philosophy of mathematics. Oxford: Clarendon Press.
Bueno, O. (2005). Dirac and the dispensability of mathematics. Studies in History and Philosophy of Modern Physics, 36(3), 465–490.
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.
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.
Bueno, O., & Colyvan, M. (2011). An inferential conception of the application of mathematics. Nous, 45(2), 345–374.
Bueno, O., & French, S. (2018). Applying mathematics: Immersion, inference, interpretation. Oxford: Oxford University Press.
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.
Colyvan, M. (1999). Confirmation theory and indispensability. Philosophical Studies, 96(1), 1–19.
Colyvan, M. (2001). The indispensability of mathematics. Oxford: Oxford University Press.
Colyvan, M. (2002). Mathematics and aesthetics considerations in science. Mind, 111(441), 69–74.
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.
Colyvan, M. (2012). An introduction to the philosophy of mathematics. Cambridge: Cambridge University Press.
Daly, C., & Langford, S. (2009). Mathematical explanation and indispensability arguments. The Philosophical Quarterly, 59(237), 641–658.
Dyson, F. (1962). Mathematics in the physical sciences. Scientific American, 211(3), 128–146.
Field, H. (1980). Science without numbers. A defence of nominalism. Princeton: Princeton University Press.
Field, H. (1989). Realism, mathematics and modality. New York: Basil Blackwell.
Franklin, J. (2014). An aristotelian realist philosophy of mathematics. Mathematics as the science of quantity and structure. London: Palgrave Macmillan.
Freese, K. (2006). The dark side of the universe. Nuclear Instruments and Methods in Physics Research, A, 559(2), 337–340.
Lange, M. (2002). An introduction to the philosophy of physics: Locality, fields, energy, and mass. New York: Blackwell.
Leng, M. (2002). What’s wrong with indispensability? (Or, the case for recreational mathematics). Synthese, 131(3), 395–417.
Leng, M. (2010). Mathematics and reality. Oxford: Oxford University Press.
Liggins, D. (2016). Grounding and the indispensability argument. Synthese, 193(2), 531–548. https://doi.org/10.1007/s11229-014-0478-2.
Maddy, P. (1990). Realism in mathematics. Oxford: Oxford University Press.
Maddy, P. (1992). Indispensability and practice. The Journal of Philosophy, 89(6), 275–289.
Maddy, P. (1995). Naturalism and ontology. Philosophia Mathematica, 3(3), 248–270.
Maddy, P. (1997). Naturalism in mathematics. Oxford: Oxford University Press.
Melia, J. (2000). Weaseling away the indispensability argument. Mind, 109(435), 435–479.
Melia, J. (2002). Response to Colyvan. Mind, 111(441), 75–79.
Morrison, M. (2015). Reconstructing reality. Models, mathematics, and simulations. Oxford: Oxford University Press.
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.
Panza, M., & Sereni, A. (2016). The varieties of indispensability arguments. Synthese, 193(2), 469–516. https://doi.org/10.1007/s11229-015-0977-9.
Pincock, C. (2012). Mathematics and scientific representation. Oxford: Oxford University Press.
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.
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.
Quine, W. V. O. (1948). On what there is. The Review of Metaphysics, 2(5), 21–38.
Quine, W. V. O. (1951). Two dogmas of empiricism. The Philosophical Review, 60(1), 20–43.
Quine, W. V. O. (1981). Theories and things. Cambridge, MA: Harvard University Press.
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.
Resnik, M. D. (1997). Mathematics as a science of patterns. Oxford: Clarendon Press.
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.
Saatsi, J. (2016). On the ‘indiepensable explanatory role’ of mathematics. Mind, 125(500), 1045–1070.
Sober, E. (1993). Mathematics and indispensability. The Philosophical Review, 102(1), 35–57.
Steiner, M. (1995). The applicabilities of mathematics. Philosophia Mathematica, 3(2), 129–156.
Steiner, M. (1998). The applicability of mathematics as a philosophical problem. Cambridge, MA: Harvard University Press.
Tegmark, M. (2014). Our mathematical universe. My quest for the ultimate nature of reality. London: Penguin Books.
Weinberg, S. (1993). Dreams of a final theory. London: Vintage.
Wigner, E. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications on Pure and Applied Mathematics, 13, 1–14.
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.
About this article
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
- Indispensability argument
- Epistemic approach
- Inference to the best explanation