Quine–Putnam indispensability argument

The second premise of the indispensability argument states mathematical objects are indispensable to our best scientific theories. In this context, indispensability is not the same as ineliminability because any entity can be eliminated from a theoretical system given appropriate adjustments to the other parts of the system.[23] Therefore, dispensability requires an entity is eliminable without sacrificing the attractiveness of the theory. For example, to be dispensable, an entity must be eliminable without causing the theory to become less simple, less explanatorily successful, or less theoretically virtuous.[24] Furthermore, if an entity is dispensable to a theory, an equivalent theory can be formulated without it.[25] This is the case, for example, if each sentence in one theory is a paraphrase of a sentence in another or if the two theories predict the same empirical observations.[26]

According to the Stanford Encyclopedia of Philosophy, the most-influential argument against the indispensability argument comes from Field.[27] It rejects the claim mathematical objects are indispensable to science;[28] Field tries to show this by reformulating or "nominalising" scientific theories so they do not refer to mathematical objects.[29] In support of this project, Field has offered a reformulation of Newtonian physics in terms of the relationships between space-time points. Instead of referring to numerical distances, Field's reformulation uses relationships such as "between" and "congruent" to recover the theory without implying the existence of numbers.[30] John Burgess and Mark Balaguer have taken steps to extend this nominalising project to areas of modern physics including quantum mechanics.[31] Philosophers such as David Malament and Otávio Bueno dispute whether such reformulations are successful or even possible, particularly in the case of quantum mechanics.[32]

Field's alternative to platonism is mathematical fictionalism, according to which mathematical theories are false because they make claims about abstract mathematical objects even though abstract objects do not exist.[33] As part of his argument against the indispensability argument, Field has tried to explain how and why mathematics is used in science despite being false and ultimately dispensable.[28] His argument is based on the idea mathematics is conservative. A mathematical theory is conservative if, when combined with a scientific theory, it does not produce any additional non-mathematical consequences the scientific theory would not have already had. This implies mathematics can be used by scientific theories without making the predictions of science false, even if the mathematics itself is false,[34] and therefore explains how it is possible for such mathematics to be useful for science.[35] Field thinks mathematics is useful for science because mathematical language provides a useful shorthand for talking about complex physical systems.[31]

Another approach to denying that mathematical entities are indispensable to science is to reformulate mathematical theories themselves so they do not imply the existence of mathematical objects. Charles Chihara, Geoffrey Hellman, and Putnam have offered modal reformulations of mathematics that replace all references to mathematical objects with claims about possibilities.[31]

Confirmational holism is the view scientific theories cannot be confirmed in isolation and must be confirmed as wholes. According to Michael Resnik, one example of this is the hypothesis an observer will see oil and water separate out if they are added together because they do not mix. Confirmation of this hypothesis relies on assumptions that must be confirmed alongside it such as the absence of any chemical that will interfere with their separation and that the eyes of the observer are functioning well enough to observe the separation.[36] Because mathematical theories are assumed by scientific theories, confirmational holism implies the empirical confirmations of scientific theories also support these mathematical theories.[37]

According to a counterargument by Maddy, the theses of naturalism and confirmational holism that make up the first premise of the indispensability argument are in tension with one another. Maddy said naturalism tells us we should respect the methods used by scientists as the best method for uncovering the truth, but scientists do not seem to act as though we should believe in all of the entities that are indispensable to science.[27] To illustrate this point, Maddy uses the example of atomic theory; she said despite the atom being indispensable to scientists' best theories by 1860, their reality was not universally accepted until 1913 when they were put to a direct experimental test.[38] Maddy also appeals to the fact scientists use mathematical idealizations such as assuming bodies of water to be infinitely deep without regard for the trueness of such applications of mathematics. According to Maddy, this indicates scientists do not view the indispensable use of mathematics for science as justification for the belief in mathematics or mathematical entities. Overall, Maddy said we should side with naturalism and reject confirmational holism, meaning we do not need to believe in all of the entities that are indispensable to science.[27]

Elliott Sober claims that mathematical theories are not tested in the same way as scientific theories. Whilst scientific theories compete with alternatives to find which theory has the most empirical support, there are no alternatives for mathematical theory to compete with because all scientific theories share the same mathematical core. As a result, according to Sober, mathematical theories do not share the empirical support of our best scientific theories so we should reject confirmational holism.[39]

Since these counterarguments have been raised, a number of philosophers—including Resnik, Alan Baker, Patrick Dieveney, David Liggins, Jacob Busch, and Andrea Sereni— have argued confirmational holism can be eliminated from the argument.[40] For example, Resnik has offered a pragmatic indispensability argument that "claims that the justification for doing science ... also justifies our accepting as true such mathematics as science uses".[41]

Another key part of the argument is the concept of ontological commitment. To say we should have an ontological commitment to an entity means we should believe that entity exists. Quine believed we should have ontological commitment to all the entities to which our best scientific theories are themselves committed.[42] According to Quine's "criterion of ontological commitment", the commitments of a theory can be found by translating or "regimenting" the theory from ordinary language into first-order logic. This criterion says the ontological commitments of the theory are all of the objects over which the regimented theory quantifies; the existential quantifier for Quine was the natural equivalent of the ordinary language term "there is", which he believed obviously carries ontological commitment.[43] Quine thought it is important to translate our best scientific theories into first-order logic because ordinary language is ambiguous whereas logic can make the commitments of a theory more precise. Translating theories to first-order logic also has advantages over translating them to higher-order logics such as second-order logic. Whilst second-order logic has the same expressive power as first-order logic, it lacks some of the technical strengths of first-order logic such as completeness and compactness. Second-order logic also allows quantification over properties like "redness" but whether we have ontological commitment to properties is controversial.[15] According to Quine, such quantification is simply ungrammatical.[44]

Jody Azzouni has objected to Quine's criterion of ontological commitment, saying the existential quantifier in first-order logic need not be interpreted as always carrying ontological commitment.[45] According to Azzouni, the ordinary language equivalent of existential quantification "there is" is often used in sentences without implying ontological commitment. In particular, Azzouni points to the use of "there is" when referring to fictional objects in sentences such as "there are fictional detectives who are admired by some real detectives".[46] According to Azzouni, for us to have ontological commitment to an entity, we must have the right level of epistemic access to it. This means, for example, that it must overcome some epistemic burdens for us to be able to postulate it. But according to Azzouni, mathematical entities are "mere posits" that can be postulated by anyone at any time by "simply writing down a set of axioms" so we do not need to treat them as real.[47]

More-modern presentations of the argument do not necessarily accept Quine's criterion of ontological commitment and may allow for ontological commitments to be directly determined from ordinary language.[48][lower-alpha 4]

In his counterargument, Joseph Melia appeals to a practice he calls weaseling, which occurs when a person makes a statement and then later withdraws something implied by that statement. An example of weaseling is the statement: "Everyone who came to the seminar had a handout. Except the person who came in late." Whilst this statement can be interpreted as being self-contradictory, it is more charitable to interpret it as coherently making the claim "Except for the person who came in late, everyone who came to the seminar had a handout". Melia said a similar situation occurs in scientists' use of statements that imply the existence of mathematical objects. According to Melia, whilst scientists use statements that imply the existence of mathematics in their theories, "almost all scientists ... deny that there are such things as mathematical objects".[50] As in the seminar-handout example, Melia said it is most charitable to interpret the scientists not as contradicting themselves but rather as weaseling away their commitment to mathematical objects. According to Melia, weaseling is acceptable in this case because the role of mathematics in science is not genuinely explanatory and is solely used to "make more things sayable about concrete objects". As a result, he said it is acceptable to not believe in the mathematical objects that scientists weasel away.[51]

Inspired by Maddy's and Sober's arguments against confirmational holism,[52] as well as Melia's argument we can suspend belief in mathematics if it does not play a genuinely explanatory role in science,[53] Colyvan and Baker have defended an explanatory version of the argument.[54][lower-alpha 5] This version of the argument attempts to remove the reliance on confirmational holism by replacing it with an inference to the best explanation. It states we are justified in believing in mathematical objects because they appear in our best scientific explanations, not because they inherit the empirical support of our best theories.[57] It is presented by the Internet Encyclopedia of Philosophy in the following form:[54]

• There are genuinely mathematical explanations of empirical phenomena.
• We ought to be committed to the theoretical posits in such explanations.
• Therefore, we ought to be committed to the entities postulated by the mathematics in question.

Number line visualizing why prime-numbered life cycles are advantageous compared to non-prime life cycles. If predators have life cycles of 3 or 4 years, they quickly synchronise with a non-prime life cycle such as a life cycle of 12 years. But they will not synchronise with a 13-year periodical cicada's life cycle until 39 and 52 years have passed, respectively.

An example of mathematics' explanatory indispensability presented by Baker is the periodic cicada, a type of insect that has life cycles of 13 or 17 years. It is hypothesised this is an evolutionary advantage because 13 and 17 are prime numbers. Because prime numbers have no non-trivial factors, this means it is less likely predators can synchronize with the cicadas' life cycles. Baker said this is an explanation in which mathematics, specifically number theory, plays a key role in explaining an empirical phenomenon.[52] Other important examples are explanations of the hexagonal structure of bee honeycomb, the existence of antipodes on the Earth's surface that have identical temperature and pressure, the connection between Minkowski space and Lorentz contraction, and the impossibility of crossing all seven bridges of Königsberg only once in a walk across the city.[58] The main response to this form of the argument, which philosophers such as Melia, Chris Daly, Simon Langford, and Juha Saatsi adopted, is to deny there are genuinely mathematical explanations of empirical phenomena, instead framing the role of mathematics as representational or indexical.[59]

An early indispensability argument came from Gottlob Frege

The argument is historically associated with Willard Quine and Hilary Putnam but it can be traced to earlier thinkers such as Gottlob Frege and Kurt Gödel. In his arguments against mathematical formalism—a view that argues mathematics is akin to a game like chess with rules about how mathematical symbols such as "2" can be manipulated—Frege said in 1903 "it is applicability alone which elevates arithmetic from a game to the rank of a science". Gödel, concerned about the axioms of set theory, said in a 1947 paper if a new axiom were to have enough verifiable consequences, it "would have to be accepted at least in the same sense as any well‐established physical theory".[60] Frege's and Gödel's indispensability arguments differ from later versions of the argument because they lack features such as naturalism and subordination of practice, which were introduced by Quine, leading some philosophers including Pieranna Garavaso to say they are not genuine examples of the indispensability argument.[61]

Whilst developing his philosophical view of confirmational holism, Quine was influenced by Pierre Duhem.[62] At the beginning of the twentieth century, Duhem defended the law of inertia from critics who said it is devoid of empirical content and unfalsifiable.[36] These critics based this claim on the fact the law does not make any observable predictions without positing some observational frame of reference and that falsifying instances can always be avoided by changing the choice of reference frame. Duhem responded by saying the law produces predictions when paired with auxiliary hypotheses fixing the frame of reference and is therefore no different from any other physical theory.[63] Duhem said although individual hypotheses may make no observable predictions alone, they can be confirmed as parts of systems of hypotheses. Quine extended this idea to mathematical hypotheses. Analogous to the law of inertia, mathematical hypotheses hold no empirical content on their own but according to Quine, they can share in the empirical confirmations of the systems of hypotheses in which they are contained.[64] This thesis later came to be known as the Duhem–Quine thesis.[65]

Quine described his naturalism as the "abandonment of the goal of a first philosophy. It sees natural science as an inquiry into reality, fallible and corrigible but not answerable to any supra-scientific tribunal, and not in need of any justification beyond observation and the hypothetico-deductive method."[66] The term "first philosophy" is used in reference to Descartes' Meditations on First Philosophy, in which Descartes used his method of doubt in an attempt to secure the foundations of science. Quine said Descartes' attempts to provide the foundations for science had failed and said the project of finding a foundational justification for science should be rejected because he believed philosophy could never provide a method of justification more convincing than the scientific method.[67] Quine was also influenced by the logical positivists such as his teacher Rudolf Carnap; his naturalism was formulated in response to many of their ideas.[68] For the logical positivists, all justified beliefs were reducible to sense data, including our knowledge of ordinary objects such as trees.[69] Quine criticized sense data as self-defeating, saying we must believe in ordinary objects to organize our experiences of the world. He also said because science is our best theory of how sense-experience gives us beliefs about ordinary objects, we should believe in it as well.[70] Whilst the logical positivists said individual claims must be supported by sense data, Quine's confirmational holism means scientific theory is inherently tied up with mathematical theory and so evidence for scientific theories can justify belief in mathematical objects despite them not being directly perceived.[69]

Whilst eventually becoming a self-described "reluctant platonist" due to his formulation of the indispensability argument,[71] Quine was sympathetic to nominalism from the early stages of his career.[72] In a 1946 lecture, he said; "I will put my cards on the table now and avow my prejudices: I should like to be able to accept nominalism"; in 1947 he released a paper with Nelson Goodman titled "Steps toward a Constructive Nominalism" as part of a joint project to "set up a nominalistic language in which all of natural science can be expressed".[73] In a letter to Joseph Henry Woodger the following year, however, Quine said he was becoming more convinced "the assumption of abstract entities and the assumptions of the external world are assumptions of the same sort".[74] He subsequently released the 1948 paper "On What There Is", in which he said "[t]he analogy between the myth of mathematics and the myth of physics is ... strikingly close", marking a shift towards his eventual acceptance of a "reluctant platonism".[75]

Throughout the 1950s, Quine regularly mentioned platonism, nominalism, and constructivism as plausible views, and he had not yet reached a definitive conclusion about which is correct.[76] It is unclear exactly when Quine accepted platonism; in 1953, he distanced himself from the claims of nominalism in his 1947 paper with Goodman but by 1956, Goodman was still describing Quine's "defection" from nominalism as "still somewhat tentative".[77] According to Lieven Decock, Quine had accepted the need for abstract mathematical entities by the publication of his 1960 book Word and Object, in which he wrote "a thoroughgoing nominalist doctrine is too much to live up to".[78] However, whilst he released suggestions of the indispensability argument in a number of papers, he never gave it a detailed formulation.[79]

Putnam gave the argument its first explicit presentation in his 1971 book Philosophy of Logic in which he attributed it to Quine.[80] He stated the argument as "quantification over mathematical entities is indispensable for science, both formal and physical; therefore we should accept such quantification; but this commits us to accepting the existence of the mathematical entities in question." He also wrote Quine had "for years stressed both the indispensability of quantification over mathematical entities and the intellectual dishonesty of denying the existence of what one daily presupposes."[81] Putnam's endorsement of Quine's version of the argument is disputed. The Internet Encyclopedia of Philosophy states; "In his early work, Hilary Putnam accepted Quine's version of the indispensability argument"[82] and Liggins said the argument has been attributed to Putnam by many philosophers of mathematics. Liggins and Bueno, however, said Putnam never endorsed the argument and only presented it as an argument from Quine.[83] Putnam has said he differed with Quine in his attitude to the argument from at least 1975.[84] Features of the argument that Putnam came to disagree with include its reliance on a single, regimented, best theory.[82]

In 1975, Putnam formulated his own indispensability argument based on the no miracles argument in the philosophy of science, which argues the success of science can only be explained by scientific realism without being rendered miraculous. He wrote that year: "I believe that the positive argument for realism [in science] has an analogue in the case of mathematical realism. Here too, I believe, realism is the only philosophy that doesn't make the success of the science a miracle".[85] The Internet Encyclopedia of Philosophy terms this version of the argument as "Putnam's success argument" and presents it in the form:[82]

• Mathematics succeeds as the language of science.
• There must be a reason for the success of mathematics as the language of science.
• No positions other than realism in mathematics provide a reason.
• Therefore, realism in mathematics must be correct.[lower-alpha 6]

The first and second premises of the argument have been seen as uncontroversial so discussion of this argument has been focused on the third premise. Other positions that have attempted to provide a reason for the success of mathematics includes Field's reformulations of science, which explain the usefulness of mathematics as a conservative shorthand.[82] Putnam has criticized Field's reformulations as only applying to classical physics and for being unlikely to be able to be extended to future fundamental physics.[88]

Chihara, in his 1973 nominalist book Ontology and the Vicious Circle Principle, was one of the earliest philosophers to attempt to reformulate mathematics in response to Quine's arguments.[89] Field's Science without Numbers followed in 1980 and dominated discussion about the indispensability argument throughout the 1980s and 1990s.[90] According to Russell Marcus, Science without Numbers marked the beginning of the scholarship developing the debate over the argument.[91]

Colyvan's formulation in his 1998 paper "In Defence of Indispensability" and his 2001 book The Indispensability of Mathematics is often seen as the standard or "canonical" formulation of the argument within more-recent philosophical work.[92] Colyvan's version of the argument has been influential in debates in contemporary philosophy of mathematics.[93] It differs in key ways from the arguments presented by Quine and Putnam. Quine's version of the argument relies on translating scientific theories from ordinary language into first-order logic to determine its ontological commitments whereas the modern version allows ontological commitments to be directly determined from ordinary language. Putnam's arguments were for the objectivity of mathematics but not necessarily for mathematical objects.[94] Colyvan has said "the attribution to Quine and Putnam [is] an acknowledgement of intellectual debts rather than an indication that the argument, as presented, would be endorsed in every detail by either Quine or Putnam".[95] Putnam has distanced himself from this version of the argument, saying; "From my point of view, Colyvan's description of my argument(s) is far from right" and has contrasted his indispensability argument with "the fictitious 'Quine–Putnam indispensability argument' ".[96]

• Antunes, Henrique (2018). "On Existence, Inconsistency, and Indispensability". Principia. 22 (1): 07–34. doi:10.5007/1808-1711.2018v22n1p7. ISSN 1808-1711.
• Asay, Jamin (2020). "Mathematics". A Theory of Truthmaking: Metaphysics, Ontology, and Reality. Cambridge University Press. pp. 223–246. ISBN 978-1-108-75946-5.
• Balaguer, Mark (2018). "Fictionalism in the Philosophy of Mathematics". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy (Fall 2018 ed.). Metaphysics Research Lab, Stanford University.
• Bangu, Sorin (2012). The Applicability of Mathematics in Science: Indispensability and Ontology. New Directions in the Philosophy of Science. Palgrave Macmillan. ISBN 978-0-230-28520-0.
• Bangu, Sorin (2013). "Indispensability and Explanation". British Journal for the Philosophy of Science. 64 (2): 255–277. doi:10.1093/bjps/axs026. ISSN 0007-0882.
• Blackburn, Simon (2008). "Duhem thesis". The Oxford Dictionary of Philosophy (2nd ed.). Oxford University Press. p. 106. ISBN 978-0-19-954143-0.
• Bostock, David (2009). Philosophy of Mathematics: An Introduction. Wiley-Blackwell. ISBN 978-1-4051-8991-0. OCLC 232002229.
• Bueno, Otávio (2013). "Putnam and the Indispensability of Mathematics". Principia. 17 (2): 217–234. doi:10.5007/1808-1711.2013v17n2p217. ISSN 1808-1711.
• Bueno, Otávio (2018). "Putnam's indispensability argument revisited, reassessed, revived" (PDF). Theoria. 33 (2): 201–218. doi:10.1387/theoria.18473.
• Bueno, Otávio (2020). "Nominalism in the Philosophy of Mathematics". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy (Fall 2020 ed.). Metaphysics Research Lab, Stanford University.
• Burgess, John P. (2013). "Quine's Philosophy of Logic and Mathematics". In Harman, Gilbert; Lepore, Ernie (eds.). A Companion to W.V.O. Quine. Blackwell Companions to Philosophy. Wiley-Blackwell. pp. 281–295. doi:10.1002/9781118607992.ch14. ISBN 978-0-47-067210-5.
• Burgess, John P.; Rosen, Gideon A. (1997). A Subject With No Object: Strategies for Nominalistic Interpretation of Mathematics. Oxford University Press. ISBN 0-19-825012-6.
• Busch, Jacob; Sereni, Andrea (2012). "Indispensability Arguments and Their Quinean Heritage". Disputatio. 4 (32): 343–360. doi:10.2478/disp-2012-0003. ISSN 0873-626X.
• Colyvan, Mark (2001). The Indispensability of Mathematics. Oxford University Press. ISBN 978-0-19-516661-3.
• Colyvan, Mark (2012). An Introduction to the Philosophy of Mathematics. Cambridge University Press. ISBN 978-0-521-82602-0.
• Colyvan, Mark (2019). "Indispensability Arguments in the Philosophy of Mathematics". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy (Spring 2019 ed.). Metaphysics Research Lab, Stanford University.
• Daly, Chris; Langford, Simon (2009). "Mathematical Explanation and Indispensability Arguments". The Philosophical Quarterly. 59 (237): 641–658. doi:10.1111/j.1467-9213.2008.601.x. ISSN 0031-8094.
• Decock, Lieven (2002). "Quine's Weak and Strong Indispensability Argument". Journal for General Philosophy of Science. 33 (2): 231–250. doi:10.1023/A:1022471707916. ISSN 0925-4560. JSTOR 25171232. S2CID 117002868.
• Franklin, James (2009). "Aristotelian Realism". In Irvine, Andrew D. (ed.). Philosophy of Mathematics. Elsevier. pp. 103–155. doi:10.1016/b978-0-444-51555-1.50007-9. ISBN 978-0-444-51555-1.
• Ginammi, Michele (2016). "The Applicability of Mathematics and the Indispensability Arguments". Revue de la Société de philosophie des sciences. École normale supérieure. 3 (1): 59–68. doi:10.20416/lsrsps.v3i1.313.
• Horsten, Leon (2019). "Philosophy of Mathematics". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy (Spring 2019 ed.). Metaphysics Research Lab, Stanford University.
• Knowles, Robert; Liggins, David (2015). "Good weasel hunting" (PDF). Synthese. 192 (10): 3397–3412. doi:10.1007/s11229-015-0711-7. ISSN 0039-7857. S2CID 31490461.
• Leng, Mary (2005). "Mathematical Explanation". In Cellucci, Carlo; Gillies, Donald A. (eds.). Mathematical Reasoning and Heuristics. King's College Publications. pp. 167–189.
• Leng, Mary (2010). Mathematics and Reality. Oxford University Press. ISBN 978-0-19-928079-7.
• Liggins, David (2008). "Quine, Putnam, and the 'Quine–Putnam' Indispensability Argument" (PDF). Erkenntnis. 68 (1): 113–127. doi:10.1007/s10670-007-9081-y. S2CID 170649798.
• Liggins, David (2012). "Weaseling and the Content of Science". Mind. 121 (484): 997–1005. doi:10.1093/mind/fzs112. ISSN 0026-4423.
• Linnebo, Øystein (2017). Philosophy of Mathematics. Princeton University Press. ISBN 978-1-4008-8524-4.
• Maddy, Penelope (2005). "Three Forms of Naturalism". In Shapiro, Stewart (ed.). The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press. pp. 437–459. doi:10.1093/0195148770.003.0013. ISBN 978-0-19-514877-0.
• Maddy, Penelope (2007). Second Philosophy: A Naturalistic Method. Oxford University Press. ISBN 978-0-199-27366-9.
• Mancosu, Paolo (2010). "Quine and Tarski on Nominalism". The Adventure of Reason. Oxford University Press. pp. 387–410. ISBN 978-0-19-954653-4.
• Mancosu, Paolo (2018). "Explanation in Mathematics". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy (Summer 2018 ed.). Metaphysics Research Lab, Stanford University.
• Marcus, Russell. "The Indispensability Argument in the Philosophy of Mathematics". Internet Encyclopedia of Philosophy. Retrieved 23 August 2022.
• Marcus, Russell (2014). "The holistic presumptions of the indispensability argument". Synthese. 191 (15): 3575–3594. doi:10.1007/s11229-014-0481-7. ISSN 0039-7857. S2CID 8245787.
• Marcus, Russell (2015). Autonomy Platonism and the Indispensability Argument. Lexington Books. ISBN 978-0-7391-7313-8.
• McPherson, Tristram; Plunkett, David (2015). "Deliberative Indispensability and Epistemic Justification". In Shafer-Landau, Russ (ed.). Oxford Studies in Metaethics. Vol. 10. Oxford University Press. pp. 104–133. ISBN 978-0-198-73869-5.
• Melia, Joseph (2017). "Does the World Contain States of Affairs? No". In Barnes, Elizabeth (ed.). Current Controversies in Metaphysics. Routledge. pp. 92–102. ISBN 978-0-203-73560-2.
• Molinini, Daniele (2016). "Evidence, explanation and enhanced indispensability". Synthese. 193 (2): 403–422. doi:10.1007/s11229-014-0494-2. ISSN 0039-7857. S2CID 7657901.
• Molinini, Daniele; Pataut, Fabrice; Sereni, Andrea (2016). "Indispensability and explanation: an overview and introduction". Synthese. 193 (2): 317–332. doi:10.1007/s11229-015-0998-4. ISSN 0039-7857. S2CID 38346150.
• Nolan, Daniel (2005). David Lewis. Acumen Publishing. ISBN 978-1-84465-307-2.
• Panza, Marco; Sereni, Andrea (2013). Plato's Problem: An Introduction to Mathematical Platonism. Palgrave Macmillan. ISBN 978-0-230-36549-0.
• Panza, Marco; Sereni, Andrea (2015). "On the Indispensable Premises of the Indispensability Argument". In Lolli, Gabriele; Panza, Marco; Venturi, Giorgio (eds.). From Logic to Practice: Italian Studies in the Philosophy of Mathematics. Boston Studies in the Philosophy and History of Science. Vol. 308. Springer. ISBN 978-3-319-10433-1.
• Panza, Marco; Sereni, Andrea (2016). "The varieties of indispensability arguments" (PDF). Synthese. 193 (2): 469–516. doi:10.1007/s11229-015-0977-9. ISSN 1573-0964. S2CID 255060875.
• Putnam, Hilary (2012). "Indispensability Arguments in the Philosophy of Mathematics". Philosophy in an Age of Science: Physics, Mathematics and Skepticism. Harvard University Press. pp. 181–201. doi:10.2307/j.ctv1nzfgrb.13. hdl:1885/145370. ISBN 978-0-674-26915-6.
• Resnik, Michael (2005). "Quine and the Web of Belief". In Shapiro, Stewart (ed.). The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press. pp. 412–436. doi:10.1093/0195148770.003.0012. ISBN 978-0-195-14877-0.
• Sereni, Andrea (2015). "Frege, Indispensability, and the Compatibilist Heresy". Philosophia Mathematica. 23 (1): 11–30. doi:10.1093/philmat/nkt046. ISSN 0031-8019.
• Shapiro, Stewart (2000). Thinking about Mathematics. Oxford University Press. ISBN 978-0-192-89306-2.
• Sinclair, Neil; Leibowitz, Uri D. (2016). "Introduction". In Leibowitz, Uri D.; Sinclair, Neil (eds.). Explanation in Ethics and Mathematics: Debunking and Dispensability. Oxford University Press. pp. 1–20. ISBN 978-0-19-182432-6.
• Verhaegh, Sander (2018). Working from Within: The Nature and Development of Quine's Naturalism. Oxford University Press. ISBN 978-0-19-091316-8.
• Weatherson, Brian (2021). "David Lewis". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy (Winter 2021 ed.). Metaphysics Research Lab, Stanford University.