Quine–Putnam indispensability argument

The Quine–Putnam indispensability argument, also known simply as the indispensability argument, is an argument in the philosophy of mathematics for the existence of abstract mathematical objects such as numbers and sets, a position known as mathematical platonism. Named after the philosophers Willard Quine and Hilary Putnam, the argument is one of the most important arguments in the philosophy of mathematics and is widely considered to be one of the best arguments for platonism.

Willard Quine
Hilary Putnam

Although indispensability arguments in philosophy of mathematics date back to thinkers such as Gottlob Frege and Kurt Gödel, Quine's version of the argument was unique for introducing to the argument various of his philosophical positions such as naturalism, confirmational holism, and his criteria of ontological commitment. Putnam gave Quine's argument its first detailed formulation in his 1971 book Philosophy of Logic. However, he later came to disagree with various aspects of Quine's thought, formulating his own indispensability argument based on the successful application of mathematics within science. The modern form of the Quine–Putnam indispensability argument is influenced by both Quine and Putnam but also differs in important ways from their formulations. The modern form of the argument is presented by the Stanford Encyclopedia of Philosophy as:[1]

  1. We ought to have ontological commitment[lower-alpha 1] to all and only the entities that are indispensable to our best scientific theories.
  2. Mathematical entities are indispensable to our best scientific theories.
  3. Therefore, we ought to have ontological commitment to mathematical entities.

Philosophers that reject the existence of abstract objects (nominalists) have argued against both the premises of this argument. The most influential argument against the indispensability argument, primarily advanced by Hartry Field, denies the indispensability of mathematical entities to science. This argument is often supported by attempts to reformulate scientific theories without reference to mathematical entities. The premise that we should believe in all the entities of science has also been subject to criticism, most influentially by Penelope Maddy and Elliott Sober. The arguments of Maddy and Sober inspired a new explanatory version of the argument, supported by Alan Baker and Mark Colyvan, that argues that mathematics is indispensable to scientific explanations.

Background

In his 1973 paper "Mathematical Truth", Paul Benacerraf presented a dilemma for the philosophy of mathematics. According to Benacerraf, mathematical sentences seem to imply the existence of mathematical objects such as numbers, but if such objects were to exist, then they would be unknowable to us.[4] That mathematical sentences seem to imply the existence of mathematical objects is supported by appeal to the idea that mathematics should not have its own special semantics. According to this line of thought, if the sentence "Mars is a planet" implies the existence of Mars and ascribes to it the property of being a planet, then "2 is a prime number" should similarly imply the existence of the number 2 and ascribe to it the property of being prime.[5] On the other hand, such mathematical objects would be abstract objects; objects that do not have causal powers (i.e. that cannot cause things to happen) and that have no spatio-temporal location.[6] Benacerraf argued, on the basis of the causal theory of knowledge, that we could not know about mathematical objects because they cannot come into causal contact with us. Since Benacerraf's initial formulation, this epistemological problem has been generalized beyond the causal theory of knowledge. For example, Hartry Field frames the problem more generally as issuing a challenge to provide a mechanism by which our mathematical beliefs could reflect accurately the properties of abstract mathematical objects.[7]

The philosophy of mathematics is split into two main views: platonism and nominalism. Platonism argues for the existence of abstract mathematical objects such as numbers and sets whilst nominalism argues against the existence of such objects.[8] Each of these views can easily overcome one part of Benacerraf's dilemma but has problems overcoming the other. As nominalism rejects the existence of mathematical objects, it faces no epistemological problem, but it does face problems concerning the semantic half of the dilemma. Platonism maintains a continuity between the semantics of ordinary sentences and mathematical sentences by claiming that mathematical sentences indeed do refer to abstract mathematical objects, but it has difficulty explaining how we can know about such objects.[9] The indispensability argument aims to overcome the epistemological problem posed against platonism by providing a justification for belief in abstract mathematical objects.[4]

Overview of the argument

Two important components of the indispensability argument are naturalism and confirmational holism.[10] Naturalism rejects the notion of a first philosophy which could provide a justification for science that is more convincing than the methods of science themselves.[11] Instead, naturalism views philosophy not as preceding science but as continuous with science and views science as providing a full characterization of the world.[10] Quine summarized naturalism as "the recognition that it is within science itself, and not in some prior philosophy, that reality is to be identified and described."[12]

Confirmational holism is the view that scientific theories cannot be confirmed in isolation and must be confirmed as wholes. An example given by Michael Resnik is of the hypothesis that an observer will observe a mixture of oil and water separate because oil and water do not mix. Any confirmation of this hypothesis relies on certain assumptions which must be confirmed alongside it such as that there is no chemical which will interfere with their separation and that the eyes of the observer are working properly to observe the separation.[13] Similarly, because mathematical theories are assumed by scientific theories, confirmational holism implies that the empirical confirmations of scientific theories also support these mathematical theories.[14] Naturalism and confirmational holism together justify that we should believe in science and specifically that we should believe in the entirety of science and nothing other than science.[10]

Another major part of the indispensability argument is mathematization or the idea that there are some mathematical objects which are indispensable to our best scientific theories.[15] Indispensability in the context of the indispensability argument does not mean ineliminability. This is because any entity can be eliminated from a theoretical system given appropriate adjustments to the other parts of the system.[16] Therefore, dispensability requires that an entity be 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 in any way.[17]

The Stanford Encyclopedia of Philosophy presents the argument in the following form with naturalism and confirmational holism making up the first premise and the indispensability of mathematics making up the second premise:[1]

  1. We ought to have ontological commitment[lower-alpha 1] to all and only the entities that are indispensable to our best scientific theories.
  2. Mathematical entities are indispensable to our best scientific theories.
  3. Therefore, we ought to have ontological commitment to mathematical entities.

The indispensability argument differs from other arguments for platonism because it only argues for belief in the parts of mathematics that are indispensable to science, meaning that it does not necessarily justify belief in the most abstract parts of set theory which Quine called "mathematical recreation … without ontological rights".[18] The argument has also been interpreted as making mathematical knowledge a posteriori, because mathematical truths can only be established via the empirical confirmation of scientific theories to which they are indispensable, and as making mathematical truths contingent, because empirically known truths are generally contingent. These features contradict a traditional view of mathematics as a priori and necessary and so are controversial features of the argument, although they have found some acceptance amongst proponents of the argument, including Mark Colyvan.[19]

Some other general features of indispensability arguments in philosophy of mathematics are theory construction and subordination of practice. Theory construction refers to the idea that we require the construction of theories about the world to understand our sensible experiences. Subordination of practice refers to the feature of the argument that mathematics as a discipline depends on the natural sciences for its legitimacy because it is mathematics' indispensability to science which acts as a justification for belief in mathematics.[15]

Counterarguments

The most influential argument against the indispensability argument comes from Hartry Field.[20] He argues that mathematical entities are dispensable to science by advancing two separate positions. The first is the position that it is possible for scientific theories to be "nominalized" so that they do not refer to mathematical objects.[21] In support of this idea, Field has offered a reformulation of Newtonian physics in terms of the relationships between spacetime 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.[22] Steps to extend this nominalizing project to areas of modern physics including quantum mechanics have been taken by John Burgess and Mark Balaguer.[23] The second position explains how mathematics can be effectively used by science even if it is false.[24] In this explanation, Field appeals to the concept of conservativeness. A mathematical theory is conservative if, when combined with a scientific theory, it does not produce any additional non-mathematical consequences that the scientific theory would not have already had. This implies that mathematics can be used by scientific theories without making the predictions of science false, even if the mathematics itself is false.[25] Therefore, conservativeness explains how it is possible for mathematics to be useful for science despite being false.[26] The specific way in which Field thinks mathematics is useful for science is that mathematical language provides a useful shorthand for talking about complex physical systems.[23] Another nominalizing approach to undermining the indispensability argument is to reformulate mathematical theories themselves so that they do not imply the existence of mathematical objects. Charles Chihara, Geoffrey Hellman, and Putnam himself have all offered modal reformulations of mathematics which replace all references to mathematical objects with claims about possibilities.[23]

Penelope Maddy has argued, against the first premise of the argument, that we do not need to have an ontological commitment to all of the entities indispensable to science. Specifically, Maddy has argued that the theses of naturalism and confirmational holism that make up the first premise are in tension with one another. According to Maddy, naturalism tells us that we should respect the methods used by scientists as the best method for uncovering the truth, but scientists do not seem to act as if we should believe in all the entities indispensable to science.[20] To illustrate this point, Maddy uses the example of atomic theory; she argues that despite the atom being indispensable to scientists' best theories by 1860, there was not universal acceptance of their reality until 1913 when they were put to a direct experimental test.[27] Maddy also appeals to the fact that scientists utilize mathematical idealizations such as assuming bodies of water to be infinitely deep without regard for whether or not such applications of mathematics are true. According to Maddy, this indicates that scientists do not view the indispensable use of mathematics for science as justification for the belief in mathematics or mathematical entities. Overall, Maddy thinks that we should side with naturalism and reject confirmational holism, meaning that we do not need to believe in all of the entities indispensable to science.[20] Another criticism of confirmational holism comes from Elliott Sober who argues 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, Sober believes that mathematical theories do not share the empirical support of our best scientific theories, and so we should reject confirmational holism.[28]

Jody Azzouni and Joseph Melia argue that the indispensability of mathematics to science does not imply that we ought to have ontological commitment to it.[29] Melia appeals to a practice which he calls weaseling. Weaseling occurs when a person communicates an idea by making a statement and then taking back a thing implied by that statement. An everyday life example of weaseling is "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 can also be more charitably interpreted as coherently making the claim that "Except for the person who came in late, everyone who came to the seminar had a handout." Melia argues that 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." As in the seminar handout example, Melia claims that it is most charitable to interpret the scientists not as contradicting themselves but rather as weaseling away their commitment to mathematical objects. Melia thinks that scientists only use weaseling in order to "make more things sayable about concrete objects" and as a result, we need not believe in the mathematical objects which they weasel away.[30]

The subordination of mathematical practice to the natural sciences by this argument has also faced criticism. Charles Parsons has argued contrary to the indispensability argument that mathematical truths seem obvious and immediate without the need for empirical support. Similarly, Maddy has argued that mathematicians do not seem to believe that their practice is restricted in any way by the activity of the natural sciences. For example, mathematicians' arguments over the axioms of Zermelo–Fraenkel set theory do not appeal to their applications to the natural sciences. As a result, Maddy believes that mathematics should be viewed as its own science with its own methods and ontological commitments, entirely separate from the natural sciences.[31]

Historical development

Early statements and influences on Quine
An early indispensability argument came from Gottlob Frege

The argument is historically associated with Willard Quine and Hilary Putnam but it can be traced back to earlier thinkers such as Gottlob Frege and Kurt Gödel. In his arguments against mathematical formalism—a view that argues that mathematics is akin to a game like chess with rules about how mathematical symbols such as "2" can be manipulated—Frege argued "it is applicability alone which elevates arithmetic from a game to the rank of a science." Gödel, concerned about the axioms of set theory, argued in a 1947 paper that if a new axiom were to have enough verifiable consequences, then it "would have to be accepted at least in the same sense as any well‐established physical theory."[32] Frege and Gödel's indispensability arguments differ from later versions of the argument in that they lack features such as naturalism and subordination of practice which were introduced by Quine, leading philosophers such as Pieranna Garavaso to argue that they are not genuine examples of the indispensability argument.[33]

In developing his philosophical view of confirmational holism, Quine was influenced by Pierre Duhem.[34] In 1906, Duhem argued against the idea of crucial experiments in physics by arguing that it is impossible to know if such experiments have falsified the target theory or other auxiliary hypotheses and assumptions.[35] In his 1951 essay "Two Dogmas of Empiricism", Quine proposed a stronger version of Duhem's thesis which extended the idea to the rest of science and even the laws of logic. Quine's version also differed from Duhem's in that whilst Duhem believed that it is impossible to know if there is a saving set of auxiliary hypotheses which have been falsified instead of the target theory, Quine thought that in principle saving hypotheses always exist. This thesis later came to be known as the Duhem–Quine thesis.[36]

Quine has 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."[37] 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 felt that Descartes' attempts to provide the foundations for science had failed and that the scientific method itself was a more convincing justification for belief in science.[11] Quine was also influenced by the logical positivists such as his teacher Rudolf Carnap, his naturalism being formulated in response to many of their ideas.[38] For the logical positivists, all justified beliefs were reducible to sense data, including our knowledge of ordinary objects such as trees.[39] Quine criticized sense data as self-defeating, instead arguing that we must believe in ordinary objects in order to organize our experiences of the world and that as science is our best theory of how sense experience gives us beliefs about ordinary objects, we should believe in it as well.[40] Whilst the logical positivists believed that individual claims must be supported by sense data, Quine's confirmational holism meant that scientific theory was inherently tied up with mathematical theory and so evidence for scientific theories could justify belief in mathematical objects despite them not being directly perceived.[39]

Quine's version of the argument

Although Quine never gave a detailed formulation of the argument, it was later presented explicitly for the first time by Putnam in his 1971 book Philosophy of Logic in which he attributed the argument to Quine.[41] It is given in the Internet Encyclopedia of Philosophy as:[2]

  1. We should believe the theory which best accounts for our sense experience.
  2. If we believe a theory, we must believe in its ontological commitments.[lower-alpha 1]
  3. The ontological commitments of any theory are the objects over which that theory first-order quantifies.
  4. The theory which best accounts for our sense experience first-order quantifies over mathematical objects.
  5. Therefore, we should believe that mathematical objects exist.

This argument assumes that science provides the best theory to account for sense experiences, which is supported by Quine's naturalism. This version of the argument also relies on Quine's theory of how to determine the ontological commitments of a theory. For Quine, to determine the ontological commitments of a theory requires translating (or "regimenting") the theory from ordinary language into first-order logic.[42]

Quine argues for the use of first-order logic instead of ordinary language or higher-order logics such as second-order logic for various reasons. Translating to logic is preferable to using ordinary language because ordinary language is ambiguous and it can sometimes be unclear what it implies exists. For example, "The tooth fairy does not exist" seemingly ascribes to "the tooth fairy" the property of non-existence despite referring to it. Translating to first-order logic allows for the use of words in sentences without assuming that they refer to something, a process called semantic ascent. This ability of first-order logic means that it can be used to find the existence claims of the theory. Whilst second-order logic has the same expressive power as first-order logic, it also lacks some of the technical strengths of first-order logic such as completeness and compactness. Furthermore, second-order logic means that the existence claims of a theory will include controversial entities such as properties like "redness".[42]

The final step of Quine's argument is to show that regimented scientific theories do in fact make existence claims about mathematical objects. The equations used in science when regimented into first-order logic seem to imply the existence of functions and numbers. For example, Coulomb's law, which describes the force between charged particles, implies the existence of a function which maps the numbers representing the magnitudes of the charges of the particles onto the number representing the strength of the force between them. Throughout the sciences various other mathematical entities are also used which are quantified over by the regimented theory, including Hilbert spaces, hyperbolic geometry, and many aspects of statistics. So that there are enough sets to construct all these numbers and functions, this approach also requires the theory to posit the axioms of set theory.[43]

Putnam's success argument

Whilst Putnam gave the first explicit presentation of Quine's version of the argument, he later came to formulate his own version of the argument, disagreeing with the reliance of Quine's argument on a single, regimented, best theory.[lower-alpha 2] Putnam's argument instead focused on the success of mathematics to argue for realism against mathematical fictionalists who believe that mathematical statements are useful fictions.[44] Putnam's mathematical realism can be split into sentence realism and object realism. Sentence realism claims that mathematical sentences can be true or false. Object realism claims that mathematical objects such as numbers exist. Putnam's argument can be used to argue for sentence realism or object realism,[44] although Putnam himself used the argument to argue for sentence realism and not object realism; Putnam's own view was a reformulation of mathematics in terms of modal logic that maintained mathematical objectivity without being committed to mathematical objects.[46] The argument is written in the Internet Encyclopedia of Philosophy in the following form:[44]

  1. Mathematics succeeds as the language of science.
  2. There must be a reason for the success of mathematics as the language of science.
  3. No positions other than realism in mathematics provides a reason.
  4. Therefore, realism in mathematics must be correct.

This argument is analogous to the no miracles argument in the philosophy of science which argues that the success of science can only be explained by scientific realism without it being miraculous. This was one of Putnam's motivations for formulating the argument, writing in 1975 "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".[47] 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 being a useful and conservative shorthand.[44] 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.[48]

Continued development of the argument

According to philosopher Otávio Bueno "the argument's canonical formulation" (which is given in §Overview of the argument) was developed by Mark Colyvan.[49] This modern version of the argument has been influential in contemporary arguments in philosophy of mathematics.[50] However, it differs in key ways from the arguments presented by Quine and Putnam. Quine's version of the argument relied on translating scientific theories from ordinary language into first-order logic in order to determine its ontological commitments whereas the modern version allows for ontological commitments to be determined directly from ordinary language. Putnam's arguments were for the objectivity of mathematics but not necessarily for mathematical objects.[51] Colyvan has said that "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."[52] 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"".[53]

A periodical cicada

In response to Maddy and Sober's arguments against confirmational holism, an explanatory version of the argument has been defended by Colyvan and Alan Baker.[54] This version of the argument attempts to remove the reliance on confirmational holism by replacing it with an inference to the best explanation. Specifically, it says that 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.[55] It is presented by the Internet Encyclopedia of Philosophy in the following form:[56]

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

An example of mathematics' explanatory indispensability presented by Baker is the periodic cicada case. Periodical cicadas are a type of insect that have life cycles of 13 or 17 years. It is hypothesized that this acts as an evolutionary advantage because 13 and 17 are prime numbers and so have no non-trivial factors. This means that it is less likely that predators can synchronize with the cicadas' life cycles. Baker argues that this is an explanation in which mathematics, specifically number theory, is playing a key role in explaining an empirical phenomenon.[57] Some other examples given by Stewart Shapiro are that the fact that 191 tiles will not fit a rectangular area is explained by 191 being a prime number and that raindrops being spherical requires both surface tension and the mathematical properties of spheres to be explained.[58]

Influence

According to philosopher James Franklin, the indispensability argument is widely considered to be the best argument for platonism in the philosophy of mathematics.[59] The Stanford Encyclopedia of Philosophy identifies the argument as one of the major arguments in the debate between mathematical realism and mathematical anti-realism alongside Benacerraf's epistemological problem for platonism, Benacerraf's identification problem, and Benacerraf's argument for platonism that there should be uniformity between mathematical and non-mathematical semantics. According to the Stanford Encyclopedia of Philosophy, some within the field see it as the only good argument for platonism.[60]

Notes
  1. An ontological commitment to an entity is a commitment to believing in the existence of that entity.[2] In some places in this article, the term ontic commitment has been replaced with ontological commitment when quoting the form of arguments. This is because they are synonymous terms with the latter being far more common in the philosophical literature.[3]
  2. It is disputed whether or not Putnam ever endorsed Quine's version of the argument; the Internet Encyclopedia of Philosophy states that "In his early work, Hilary Putnam accepted Quine's version of the indispensability argument"[44] and Liggins 2008 notes that the argument has been attributed to Putnam by many philosophers of mathematics, but Liggins and Bueno 2018 claim that Putnam never endorsed the argument and only presented it as an argument from Quine. Putnam himself has said on the topic:
    In sum, my "indispensability" argument was an argument for the objectivity of mathematics in a realist sense—that is, for the idea that mathematical truth must not be identified with provability. Quine’s indispensability argument was an argument for "reluctant Platonism," which he himself characterized as accepting the existence of "intangible objects" (numbers and sets). The difference in our attitudes goes back at least to 1975.[45]

References

Citations
  1. Colyvan 2019, §1.
  2. Marcus, §2.
  3. Parent 2020, endnote 9.
  4. Marcus.
  5. Colyvan 2012, pp. 9–10.
  6. Colyvan 2012, p. 1.
  7. Colyvan 2012, pp. 10–12.
  8. Colyvan 2012, p. 9.
  9. Shapiro 2000, pp. 31–32; Colyvan 2012, pp. 9–10.
  10. Colyvan 2019, §3.
  11. Maddy 2005, p. 438.
  12. Maddy 2005, p. 437. Primary source: Quine 1981a, p. 21.
  13. Resnik 2005, p. 414.
  14. Horsten 2019, §3.2.
  15. Marcus, §6.
  16. Colyvan 2019, §2. See also footnote 3.
  17. Colyvan 2019, §2.
  18. Colyvan 2019, §2; Marcus, §7; Bostock 2009, pp. 276–277. Primary source: Quine 1998, p. 400.
  19. Marcus, §7; Colyvan 2001, Ch. 6.
  20. Colyvan 2019, §4.
  21. Linnebo 2017, pp. 105–106.
  22. Colyvan 2001, p. 72.
  23. Marcus, §7.
  24. Colyvan 2019, §4; Colyvan 2001, p. 69. (Note: Both citations refer to this as the first part of Field's argument, not the second.)
  25. Colyvan 2001, p. 71.
  26. Bueno 2020, §3.1.
  27. Colyvan 2001, p. 92.
  28. Colyvan 2019, §4; Bostock 2009, p. 278; Resnik 2005, p. 419.
  29. Bueno 2010, p. 10.
  30. Liggins 2012, pp. 998–999; Knowles & Liggins 2015, pp. 3398–3399. Primary source: Melia 2000, p. 469.
  31. Horsten 2019, §3.2; Colyvan 2019, §4; Bostock 2009, p. 278.
  32. Colyvan 2001, pp. 8–9. Primary sources: Frege 1903, §91; Gödel 1947, §3.
  33. Marcus, §6; Sereni 2015.
  34. Maddy 2007, p. 91.
  35. Harding 1975, p. ix.
  36. Harding 1975, pp. xi–xii.
  37. Marcus, §2a; Shapiro 2000, p. 212. Primary source: Quine 1981b, p. 67.
  38. Shapiro 2000, p. 212; Marcus, §2a.
  39. Marcus, §2a.
  40. Maddy 2007, p. 442; Marcus, §2a.
  41. Bueno 2018, pp. 202–203; Shapiro 2000, p. 216; Sereni 2015, footnote 2.
  42. Marcus, §2c.
  43. Marcus, §2d.
  44. Marcus, §3.
  45. Putnam 2012, p. 183.
  46. Bueno 2013, p. 227; Bueno 2018, pp. 201–202; Colyvan 2001, pp. 2–3; Putnam 2012, pp. 182–183.
  47. Marcus, §3. Primary source: Putnam 1979, p. 73..
  48. Putnam 2012, pp. 190–192.
  49. Bueno 2018, p. 203.
  50. Sereni 2015, §2.1; Marcus 2014, p. 3576.
  51. Colyvan 2019; Liggins 2008, §5.
  52. Colyvan 2019, footnote 1.
  53. Putnam 2012, pp. 182, 186.
  54. Colyvan 2019, §5; Marcus, §5.
  55. Marcus 2014, pp. 3583–3584; Leng 2005.
  56. Marcus, §5.
  57. Colyvan 2019, §5.
  58. Shapiro 2000, p. 217.
  59. Franklin 2009, p. 134.
  60. Colyvan 2019, §6.

Sources

Primary sources

This section provides the primary sources for quotes mentioned in the article prose which are cited to other secondary sources. It also provides primary sources mentioned in the article prose.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.