About this topic
Summary One of the lines of reasoning in support of mathematical platonism employs the fact that mathematical theories find applications in sciences which, at least prima facie, concern themselves with the physical world. From the indispensability of mathematics in science the argument moves to the indispensability of reference to mathematical objects in science. Further on, since we, supposedly, have good reasons to accept the existence of objects our best scientific theories have to refer to, we should accept the existence of such mathematical objects, on a par with the existence of electrons and other invisible entities postulated by such scientific theories. Accordingly, the argument has been attacked on different grounds. Some deny the indispensability of mathematics in science, some claim that indispensability of mathematical theories is not the same as the indispensability of reference to mathematical objects, some insist that this approach doesn't make justice to the difference between a priori mathematical knowledge and a posteriori scientific knowledge, some worry that applied mathematics is only a part of theoretical mathematics and some suggest that best scientific theories don't have to be our guide to metaphysics.
Key works Loci classici are Quine 1961Quine 1981Putnam 1975 and Putnam 1971. Further considerations can be found for instance in Parsons 1979Chihara 1973 and   Maddy 1992. Field 1980 is directed at showing the dispensability of mathematics in science. An extensive defence of the indispensability argument have been mounted by Colyvan 2003.
Introductions Start with Colyvan 2008 (and references therein).
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  1. The Indispensability of Moral Principles in Governance.M. E. Abam - 2011 - Sophia: An African Journal of Philosophy 10 (2).
  2. El argumento de indispensabilidad en matemáticas.Anastasio Aleman - 1999 - Teorema: International Journal of Philosophy 18 (2):49-61.
  3. Convenient Myths: Reconciling Indispensability And Ontological Relativity.N. Alphonse - 2011 - Florida Philosophical Review 11 (1):36-53.
    Metaphysical naturalism centers on the claim that any answer to the question "what exists?" must be framed in agreement with our overall best scientific theory of the world. Naturalists hold that objects which play a central role in facilitating the overall simplicity and elegance of our scientific theory are accorded a special status—in short they have attained "indispensability." As advanced by Penelope Maddy, the Argument from Scientific Practice is designed to show that indispensability is fundamentally incompatible with another core naturalistic (...)
  4. Troubles with Indispensability: Applying Pure Mathematics in Physical Theory.Anthony Peressini - 1997 - Philosophia Mathematica 5 (3):210-227.
    Much of the current thought concerning mathematical ontology in volves in some way the Quine/Putnam indispensability argument. The indispensability approach needs to be more thoroughly specified, however, before substantive progress can be made in assessing it. To this end I examine in some detail the ways in which pure mathematics is applied to physical theory; such considerations give rise to three specific issues with which the indispensability approach must come to grips.
  5. Mark Colyvan, The Indispensability of Mathematics.Matija Arko - 2007 - Croatian Journal of Philosophy 19:118-121.
  6. The Indispensability of Mathematics.Matija Arko - 2007 - Croatian Journal of Philosophy 7 (1):118-121.
  7. Calculus as Geometry.Frank Arntenius & Cian Dorr - 2012 - In Frank Arntzenius (ed.), Space, Time and Stuff. Oxford University Press.
    We attempt to extend the nominalistic project initiated in Hartry Field's Science Without Numbers to modern physical theories based in differential geometry.
  8. On "on What There Is".Jody Azzouni - 1998 - Pacific Philosophical Quarterly 79 (1):1–18.
    All sides in the recent debates over the Quine‐Putnam Indispensability thesis presuppose Quine's criterion for determining what a discourse is ontologically committed to. I subject the criterion to scrutiny, especially in regard to the available competitor‐criteria, asking what means of evaluation there are for comparing alternative criteria against each other. Finding none, the paper concludes that ontological questions, in a certain sense, are philosophically indeterminate.
  9. Applied Mathematics, Existential Commitment and the Quine-Putnam Indispensability Thesis.Jody Azzouni - 1997 - Philosophia Mathematica 5 (3):193-209.
    The ramifications are explored of taking physical theories to commit their advocates only to ‘physically real’ entities, where ‘physically real’ means ‘causally efficacious’ (e.g., actual particles moving through space, such as dust motes), the ‘physically significant’ (e.g., centers of mass), and the merely mathematical—despite the fact that, in ordinary physical theory, all three sorts of posits are quantified over. It's argued that when such theories are regimented, existential quantification, even when interpreted ‘objectually’ (that is, in terms of satisfaction via variables, (...)
  10. Indispensibility and the Multiple Reducibility of Mathematical Objects.Alan Baker - manuscript
  11. Parsimony and Inference to the Best Mathematical Explanation.Alan Baker - forthcoming - Synthese:1-18.
    Indispensability-based arguments for mathematical platonism are typically motivated by drawing an analogy between abstract mathematical objects and concrete scientific posits. In this paper, I argue that mathematics can sometimes help to reduce our concrete ontological, ideological, and structural commitments. My focus is on optimization explanations, and in particular the case study involving periodical cicadas. I argue that in this case, stronger mathematical apparatus yields explanations that have fewer concrete commitments. The nominalist cannot accept these more parsimonious explanations without embracing the (...)
  12. Mathematics and Explanatory Generality.Alan Baker - forthcoming - Philosophia Mathematica:nkw021.
    According to one popular nominalist picture, even when mathematics features indispensably in scientific explanations, this mathematics plays only a purely representational role: physical facts are represented, and these exclusively carry the explanatory load. I think that this view is mistaken, and that there are cases where mathematics itself plays an explanatory role. I distinguish two kinds of explanatory generality: scope generality and topic generality. Using the well-known periodical-cicada example, and also a new case study involving bicycle gears, I argue that (...)
  13. Russell Marcus. Autonomy Platonism and the Indispensability Argument.Alan Baker - 2016 - Philosophia Mathematica 24 (3):422-424.
  14. Mathematical Explanation in Science.Alan Baker - 2009 - British Journal for the Philosophy of Science 60 (3):611-633.
    Does mathematics ever play an explanatory role in science? If so then this opens the way for scientific realists to argue for the existence of mathematical entities using inference to the best explanation. Elsewhere I have argued, using a case study involving the prime-numbered life cycles of periodical cicadas, that there are examples of indispensable mathematical explanations of purely physical phenomena. In this paper I respond to objections to this claim that have been made by various philosophers, and I discuss (...)
  15. Are There Genuine Mathematical Explanations of Physical Phenomena?Alan Baker - 2005 - Mind 114 (454):223-238.
    Many explanations in science make use of mathematics. But are there cases where the mathematical component of a scientific explanation is explanatory in its own right? This issue of mathematical explanations in science has been for the most part neglected. I argue that there are genuine mathematical explanations in science, and present in some detail an example of such an explanation, taken from evolutionary biology, involving periodical cicadas. I also indicate how the answer to my title question impacts on broader (...)
  16. The Indispensability Argument and Multiple Foundations for Mathematics.Alan Baker - 2003 - Philosophical Quarterly 53 (210):49–67.
    One recent trend in the philosophy of mathematics has been to approach the central epistemological and metaphysical issues concerning mathematics from the perspective of the applications of mathematics to describing the world, especially within the context of empirical science. A second area of activity is where philosophy of mathematics intersects with foundational issues in mathematics, including debates over the choice of set-theoretic axioms, and over whether category theory, for example, may provide an alternative foundation for mathematics. My central claim is (...)
  17. Mathematics, Indispensability and Scientific Progress.Alan Baker - 2001 - Erkenntnis 55 (1):85-116.
  18. Indispensability and the Existence of Mathematical Objects.Alan Richard Baker - 1999 - Dissertation, Princeton University
    According to the so-called "Indispensability Argument", the central role played by mathematics in science gives us sufficient reason to believe in the existence of abstract mathematical objects such as numbers, sets, and functions. The Indispensability Argument may be formulated as follows: We ought rationally to believe our best available scientific theories. Mathematics is indispensable for science. we ought to believe in the existence of mathematical objects. Platonism is the view that there exist enough abstract mathematical objects to make the bulk (...)
  19. A Fictionalist Account of the Indispensable Applications of Mathematics.Mark Balaguer - 1996 - Philosophical Studies 83 (3):291 - 314.
  20. Indispensability and Explanation.Sorin Bangu - 2013 - British Journal for the Philosophy of Science 64 (2):255-277.
    The question as to whether there are mathematical explanations of physical phenomena has recently received a great deal of attention in the literature. The answer is potentially relevant for the ontology of mathematics; if affirmative, it would support a new version of the indispensability argument for mathematical realism. In this article, I first review critically a few examples of such explanations and advance a general analysis of the desiderata to be satisfied by them. Second, in an attempt to strengthen the (...)
  21. The Applicability of Mathematics in Science: Indispensability and Ontology.Sorin Bangu - 2012 - Palgrave-Macmillan.
  22. Wigner's Puzzle for Mathematical Naturalism.Sorin Bangu - 2009 - International Studies in the Philosophy of Science 23 (3):245-263.
    I argue that a recent version of the doctrine of mathematical naturalism faces difficulties arising in connection with Wigner's old puzzle about the applicability of mathematics to natural science. I discuss the strategies to solve the puzzle and I show that they may not be available to the naturalist.
  23. Inference to the Best Explanation and Mathematical Realism.Sorin Bangu - 2007 - Synthese 160 (1):13-20.
    Arguing for mathematical realism on the basis of Field’s explanationist version of the Quine–Putnam Indispensability argument, Alan Baker has recently claimed to have found an instance of a genuine mathematical explanation of a physical phenomenon. While I agree that Baker presents a very interesting example in which mathematics plays an essential explanatory role, I show that this example, and the argument built upon it, begs the question against the mathematical nominalist.
  24. Inference to the Best Explanation and Mathematical Realism.Sorin Ioan Bangu - 2008 - Synthese 160 (1):13-20.
    Arguing for mathematical realism on the basis of Field’s explanationist version of the Quine–Putnam Indispensability argument, Alan Baker has recently claimed to have found an instance of a genuine mathematical explanation of a physical phenomenon. While I agree that Baker presents a very interesting example in which mathematics plays an essential explanatory role, I show that this example, and the argument built upon it, begs the question against the mathematical nominalist.
  25. Discipline and Experience: The Mathematical Way in the Scientific Revolution. Peter Dear.Peter Barker - 1997 - Isis 88 (1):122-124.
  26. Mathematical Explanation and Epistemology: Please Mind the Gap.Sam Baron - 2016 - Ratio 29 (2):149-167.
    This paper draws together two strands in the debate over the existence of mathematical objects. The first strand concerns the notion of extra-mathematical explanation: the explanation of physical facts, in part, by facts about mathematical objects. The second strand concerns the access problem for platonism: the problem of how to account for knowledge of mathematical objects. I argue for the following conditional: if there are extra-mathematical explanations, then the core thesis of the access problem is false. This has implications for (...)
  27. A Truthmaker Indispensability Argument.Sam Baron - 2013 - Synthese 190 (12):2413-2427.
    Recently, nominalists have made a case against the Quine–Putnam indispensability argument for mathematical Platonism by taking issue with Quine’s criterion of ontological commitment. In this paper I propose and defend an indispensability argument founded on an alternative criterion of ontological commitment: that advocated by David Armstrong. By defending such an argument I place the burden back onto the nominalist to defend her favourite criterion of ontological commitment and, furthermore, show that criterion cannot be used to formulate a plausible form of (...)
  28. Can Indispensability‐Driven Platonists Be (Serious) Presentists?Sam Baron - 2013 - Theoria 79 (3):153-173.
    In this article I consider what it would take to combine a certain kind of mathematical Platonism with serious presentism. I argue that a Platonist moved to accept the existence of mathematical objects on the basis of an indispensability argument faces a significant challenge if she wishes to accept presentism. This is because, on the one hand, the indispensability argument can be reformulated as a new argument for the existence of past entities and, on the other hand, if one accepts (...)
  29. Optimisation and Mathematical Explanation: Doing the Lévy Walk.Sam Baron - 2013 - Synthese 3 (3):1-21.
    The indispensability argument seeks to establish the existence of mathematical objects. The success of the indispensability argument turns on finding cases of genuine extra- mathematical explanation. In this paper, I identify a new case of extra- mathematical explanation, involving the search patterns of fully-aquatic marine predators. I go on to use this case to predict the prevalence of extra- mathematical explanation in science.
  30. Time Enough for Explanation.Sam Baron & Mark Colyvan - 2016 - Journal of Philosophy 113 (2):61-88.
    The present paper advances an analogy between cases of extra-mathematical explanation and cases of what might be termed ‘extra-logical explanation’: the explanation of a physical fact by a logical fact. A particular case of extra-logical explanation is identified that arises in the philosophical literature on time travel. This instance of extra-logical explanation is subsequently shown to be of a piece with cases of extra-mathematical explanation. Using this analogy, we argue extra-mathematical explanation is part of a broader class of non-causal explanation. (...)
  31. On the Explanatory Role of Mathematics in Empirical Science.Robert Batterman - 2010 - British Journal for the Philosophy of Science 61 (1):1-25.
    This paper examines contemporary attempts to explicate the explanatory role of mathematics in the physical sciences. Most such approaches involve developing so-called mapping accounts of the relationships between the physical world and mathematical structures. The paper argues that the use of idealizations in physical theorizing poses serious difficulties for such mapping accounts. A new approach to the applicability of mathematics is proposed.
  32. The Applicability of Mathematics to Physical Modality.Nora Berenstain - 2016 - Synthese:1-17.
    This paper argues that scientific realism commits us to a metaphysical determination relation between the mathematical entities that are indispensible to scientific explanation and the modal structure of the empirical phenomena those entities explain. The argument presupposes that scientific realism commits us to the indispensability argument. The viewpresented here is that the indispensability of mathematics commits us not only to the existence of mathematical structures and entities but to a metaphysical determination relation between those entities and the modal structure of (...)
  33. The Quinean Quandary and the Indispensability of Nonnaturalized Epistemology.Alan Berger - 2003 - Philosophical Forum 34 (3-4):367–382.
  34. The Indispensability Argument for Induction.Lukáš Bielik - 2015 - Balkan Journal of Philosophy 7 (1):45-54.
    Developing the ideas presented in Jacquette, the paper presents an indispensability argument aimed at justification of induction. First, Hume’s problem of induction is introduced via slightly different reconstructions. Second, several traditional attempts to solve Hume’s problem are presented. Finally, Jacquette’s proposal to justify induction by an indispensability argument is developed. I conclude with presenting a kind of indispensability argument for induction.
  35. The Indispensability Argument – a New Chance for Empiricism in Mathematics?Tomasz Bigaj - 2003 - Foundations of Science 8 (2):173-200.
    In recent years, the so-calledindispensability argument has been given a lotof attention by philosophers of mathematics.This argument for the existence of mathematicalobjects makes use of the fact, neglected inclassical schools of philosophy of mathematics,that mathematics is part of our best scientifictheories, and therefore should receive similarsupport to these theories. However, thisobservation raises the question about the exactnature of the alleged connection betweenexperience and mathematics (for example: is itpossible to falsify empirically anymathematical theorems?). In my paper I wouldlike to address this (...)
  36. The Indispensability of Internalism.Laurence BonJour - 2001 - Philosophical Topics 29 (1/2):47-65.
  37. Marco Panza and Andrea Sereni. Plato's Problem: An Introduction to Mathematical Platonism. London and New York: Palgrave Macmillan, 2013. ISBN 978-0-230-36548-3 (Hbk); 978-0-230-36549-0 (Pbk); 978-1-13726147-2 (E-Book); 978-1-13729813-3 (Pdf). Pp. Xi + 306. [REVIEW]James Robert Brown - 2013 - Philosophia Mathematica (1):nkt031.
  38. Platonism, Naturalism, and Mathematical Knowledge.James Robert Brown - 2011 - Routledge.
    This study addresses a central theme in current philosophy: Platonism vs Naturalism and provides accounts of both approaches to mathematics, crucially discussing Quine, Maddy, Kitcher, Lakoff, Colyvan, and many others. Beginning with accounts of both approaches, Brown defends Platonism by arguing that only a Platonistic approach can account for concept acquisition in a number of special cases in the sciences. He also argues for a particular view of applied mathematics, a view that supports Platonism against Naturalist alternatives. Not only does (...)
  39. Chemical Atomism: A Case Study in Confirmation and Ontology.Joshua D. K. Brown - 2015 - Synthese 192 (2):453-485.
    Quine, taking the molecular constitution of matter as a paradigmatic example, offers an account of the relation between theory confirmation and ontology. Elsewhere, he deploys a similar ontological methodology to argue for the existence of mathematical objects. Penelope Maddy considers the atomic/molecular theory in more historical detail. She argues that the actual ontological practices of science display a positivistic demand for “direct observation,” and that fulfillment of this demand allows us to distinguish molecules and other physical objects from mathematical abstracta. (...)
  40. Symposium: The Relation Between the Mathematical and the Physical.Léon Brunschvicg - 1923 - Aristotelian Society Supplementary Volume 3 (1):42 - 55.
  41. The Relation Between the Mathematical and the Physical.Léon Brunschvicg - 1923 - Aristotelian Society Supplementary Volume 3 (1):42-55.
  42. Putnam and the Indispensability of Mathematics.Otávio Bueno - 2013 - Principia: An International Journal of Epistemology 17 (2):217.
  43. Dirac and the Dispensability of Mathematics.Otavio Bueno - 2005 - Studies in History and Philosophy of Science Part B 36 (3):465-490.
    In this paper, 1 examine the role of the delta function in Dirac’s formulation of quantum mechanics (QM), and I discuss, more generally, the role of mathematics in theory construction. It has been argued that mathematical theories play an indispensable role in physics, particularly in QM [Colyvan, M. (2001). The inrlispensability of mathematics. Oxford University Press: Oxford]. As I argue here, at least in the case of the delta function, Dirac was very clear about its rlispensability. I first discuss the (...)
  44. Quine's Double Standard: Undermining the Indispensability Argument Via the Indeterminacy of Reference.Otávio Bueno - 2003 - Principia 7 (1-2):17-39.
    Quine has famously put forward the indispensability argument to force belief in the existence of mathematical objects (such as classes) due to their indispensability to our best theories of the world (Quine 1960). Quine has also advocated the indeterminacy of reference argument, according to which reference is dramatically indeterminate: given a language, there’s no unique reference relation for that language (see Quine 1969a). In this paper, I argue that these two arguments are in conflict with each other. Whereas the indispensability (...)
  45. On Representing the Relationship Between the Mathematical and the Empirical.Otávio Bueno, Steven French & James Ladyman - 2002 - Philosophy of Science 69 (3):497-518.
    We examine, from the partial structures perspective, two forms of applicability of mathematics: at the “bottom” level, the applicability of theoretical structures to the “appearances”, and at the “top” level, the applicability of mathematical to physical theories. We argue that, to accommodate these two forms of applicability, the partial structures approach needs to be extended to include a notion of “partial homomorphism”. As a case study, we present London's analysis of the superfluid behavior of liquid helium in terms of Bose‐Einstein (...)
  46. Quine, Analyticity and Philosophy of Mathematics.John P. Burgess - 2004 - Philosophical Quarterly 54 (214):38–55.
    Quine correctly argues that Carnap's distinction between internal and external questions rests on a distinction between analytic and synthetic, which Quine rejects. I argue that Quine needs something like Carnap's distinction to enable him to explain the obviousness of elementary mathematics, while at the same time continuing to maintain as he does that the ultimate ground for holding mathematics to be a body of truths lies in the contribution that mathematics makes to our overall scientific theory of the world. Quine's (...)
  47. Sets and Point-Sets: Five Grades of Set-Theoretic Involvement in Geometry.John P. Burgess - 1988 - PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1988:456 - 463.
    The consequences for the theory of sets of points of the assumption of sets of sets of points, sets of sets of sets of points, and so on, are surveyed, as more generally are the differences among the geometric theories of points, of finite point-sets, of point-sets, of point-set-sets, and of sets of all ranks.
  48. Can the New Indispensability Argument Be Saved From Euclidean Rescues?Jacob Busch - 2012 - Synthese 187 (2):489-508.
    The traditional formulation of the indispensability argument for the existence of mathematical entities (IA) has been criticised due to its reliance on confirmational holism. Recently a formulation of IA that works without appeal to confirmational holism has been defended. This recent formulation is meant to be superior to the traditional formulation in virtue of it not being subject to the kind of criticism that pertains to confirmational holism. I shall argue that a proponent of the version of IA that works (...)
  49. Is the Indispensability Argument Dispensable?Jacob Busch - 2011 - Theoria 77 (2):139-158.
    When the indispensability argument for mathematical entities (IA) is spelled out, it would appear confirmational holism is needed for the argument to work. It has been argued that confirmational holism is a dispensable premise in the argument if a construal of naturalism, according to which it is denied that we can take different epistemic attitudes towards different parts of our scientific theories, is adopted. I argue that the suggested variety of naturalism will only appeal to a limited number of philosophers. (...)
  50. Scientific Realism and the Indispensability Argument for Mathematical Realism: A Marriage Made in Hell.Jacob Busch - 2011 - International Studies in the Philosophy of Science 25 (4):307-325.
    An emphasis on explanatory contribution is central to a recent formulation of the indispensability argument for mathematical realism. Because scientific realism is argued for by means of inference to the best explanation, it has been further argued that being a scientific realist entails a commitment to IA and thus to mathematical realism. It has, however, gone largely unnoticed that the way that IBE is argued to be truth conducive involves citing successful applications of IBE and tracing this success over time. (...)
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