Quine–Putnam indispensability argument

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Willard Quine

Hilary Putnam

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, it 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.

Although elements of the indispensability argument can be dated back to thinkers such as Gottlob Frege and Kurt Gödel, Quine's development of the argument was unique for introducing to it a number of his philosophical positions such as his naturalism, confirmational holism, and criterion 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 thinking, formulating his own indispensability argument based on the no miracles argument in the philosophy of science. A standard form of the argument in contemporary philosophy is credited to Mark Colyvan; whilst being influenced by both Quine and Putnam, it also differs in important ways from their own formulations. It is presented in the Stanford Encyclopedia of Philosophy as:

1. We ought to have ontological commitment 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 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. 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. 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. 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.

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. 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. The indispensability argument aims to overcome the epistemological problem posed against platonism by providing a justification for belief in abstract mathematical objects.

Overview of the argument

Two important components of the indispensability argument are naturalism and confirmational holism. Naturalism rejects the notion that philosophy can provide a foundational justification for science as it views the methods of science themselves as the most persuasive methods of justification. Instead, naturalism views philosophy not as preceding science but as continuous with it and views science as providing a full characterization of the world. 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."

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 oil and water separate out if added together because they 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. Similarly, because mathematical theories are assumed by scientific theories, confirmational holism implies that the empirical confirmations of scientific theories also support these mathematical theories. Naturalism and confirmational holism accepted together imply that we should believe in science and specifically that we should believe in the entirety of science and nothing other than science.

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. 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. 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. Furthermore, for an entity to be truly dispensable to a theory, eliminating it from that theory must result in a theory that is equivalent in some meaningful way to the original theory according to an appropriate equivalence relation. For example, Quine takes theories to be equivalent if each sentence in one theory is a paraphrase of a sentence in the other, or else exactly the same as a sentence in the other. Another example of an equivalence relation used in the argument is observational equivalence, according to which two theories are equivalent if they predict the same observation statements.

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. We ought to have ontological commitment 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.

According to Quine's criterion of ontological commitment, the ontological commitments of a theory can be found by translating (or "regimenting") the theory from ordinary language into first-order logic. This criterion specifies that the ontological commitments of the theory are all of the objects over which the regimented theory quantifies; the existential quantifier for Quine being the natural equivalent of the ordinary language term "there is" which he believed obviously carries ontological commitment. Quine felt that it was 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 to higher-order logics such as second-order logic. 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, translation to second-order logic makes the existence claims of a theory include controversial entities such as properties like "redness". More modern presentations of the argument do not necessarily accept Quine's criterion of ontological commitment and may allow for ontological commitments to be determined directly from ordinary language.

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". 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 Colyvan.

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.

Counterarguments

According to the Stanford Encyclopedia of Philosophy, the most influential argument against the indispensability argument comes from Field. As a mathematical fictionalist, Field believes that mathematical theories make claims about abstract mathematical objects but that there are no such things because abstract objects do not exist, and so mathematical theory is untrue. His argument against the indispensability argument comes in two parts. The first part is an argument that there are no mathematical objects that are indispensable to science and the second is an explanation for why the dispensable parts are useful for science. The first part comes in the form of an attempt to "nominalize" scientific theories so that they do not refer to mathematical objects. In support of this project, 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. Steps to extend this nominalizing project to areas of modern physics including quantum mechanics have been taken by John Burgess and Mark Balaguer. In the second part of his argument, Field appeals to the concept of conservativeness to explain why mathematics can be useful to science despite being false. 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, and therefore explains how it is possible for such mathematics to be useful for science. 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. 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.

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. 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. 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. 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.

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. Azzouni objects to Quine's criterion of ontological commitment, arguing that the existential quantifier in first-order logic need not be interpreted as always carrying ontological commitment. Instead, he argues that there should be a distinction between quantifier commitment and ontological commitment. For Azzouni, whilst existential quantification is enough to entail quantifier commitment, it does not mean we have any ontological commitment. Azzouni motivates this argument by pointing out that 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 "there is" when referring to fictional objects in sentences such as "there are fictional detectives who are admired by some real detectives". Azzouni further argues that we don't need to take mathematical entities as real because our epistemic access to them is "ultrathin". Our access is "ultrathin" because mathematical entities are "mere posits" which can be postulated by anyone at any time by "simply writing down a set of axioms" without the need to overcome any epistemic burdens.

Melia appeals to a practice which he calls weaseling. Weaseling occurs when a person makes a statement and then later takes 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 is more charitable to interpret it 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.

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.

Historical development

Precursors 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 in 1903 that "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." 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 some philosophers including Pieranna Garavaso to argue that they are not genuine examples of the indispensability argument.

In developing his philosophical view of confirmational holism, Quine was influenced by Pierre Duhem. At the beginning of the twentieth century, Duhem defended the law of inertia from critics that claimed that it was devoid of empirical content and unfalsifiable. Such critics based this claim on the fact that the law does not make any observable predictions without positing some observational frame of reference and that falsifying instances could always be avoided by changing the choice of reference frame. Duhem responded by pointing out that the law produces predictions when paired with auxiliary hypotheses fixing the frame of reference and was therefore no different from any other physical theory. In other words, he claimed that although individual hypotheses may make no observable predictions alone, they can nonetheless be confirmed as parts of systems of hypotheses. Quine expanded this idea to mathematical hypotheses which, analogously to the law of inertia, hold no empirical content on their own, but, according to Quine, share in the empirical confirmations of the systems of hypotheses in which they are contained. This thesis later came to be known as the Duhem–Quine thesis.

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." 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 argued that the project of finding a foundational justification for science should be rejected as he believed that philosophy could never provide a method of justification more convincing than the scientific method itself. 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. For the logical positivists, all justified beliefs were reducible to sense data, including our knowledge of ordinary objects such as trees. 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. 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.

Quine and Putnam

Whilst eventually becoming a self-described "reluctant platonist" due to his formulation of the indispensability argument, Quine was sympathetic to nominalism from the early stages of his career. 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" and 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". However, in a letter to Joseph Henry Woodger the following year, he said that he was becoming more and more convinced that "the assumption of abstract entities and the assumptions of the external world are assumptions of the same sort". He subsequently released the 1948 paper "On What There Is" in which he claimed "[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".

Throughout the 1950s, Quine regularly mentioned platonism, nominalism and constructivism as plausible views and he was still to come to a definitive conclusion about which was right. During this time, he released a number of papers which alluded to the idea that we should accept the existence of mathematical entities due to their indispensability to science, including "Two Dogmas of Empiricism" (1951), "Posits and Reality" (1955) and "Speaking of Objects" (1958). Nonetheless, it is unclear when exactly 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". According to Lieven Decock, Quine had finally accepted the need for abstract mathematical entities by the publication of his 1960 book Word and Object, in which he wrote that "a thoroughgoing nominalist doctrine is too much to live up to". However, whilst Quine continued to release suggestions of the argument in papers such as "Carnap and Logical Truth" (1963) and "Existence and Quantification" (1969), he never gave it a detailed formulation.

The argument was given its first explicit presentation by Putnam in his 1971 book Philosophy of Logic in which he attributed it to Quine. 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 that 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." 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" and David Liggins notes that the argument has been attributed to Putnam by many philosophers of mathematics, but Liggins and Otávio Bueno claim that Putnam never endorsed the argument and only presented it as an argument from Quine. Putnam himself has said on the topic that he differed with Quine in his attitude to the argument from at least 1975. Amongst the features of the argument that Putnam came to disagree with was its reliance on a single, regimented, best theory.

It was in 1975 that Putnam formulated his own indispensability argument based on the no miracles argument in the philosophy of science which argues that 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". Putnam's argument was an argument against mathematical fictionalists, who claim that mathematical statements are merely useful fictions, and for sentence realism. Sentence realism, as opposed to platonism or object realism, only requires that mathematical sentences are objectively true or false and not that mathematical objects actually exist. Putnam's own view was a reformulation of mathematics in terms of modal logic that maintained mathematical objectivity without being committed to mathematical objects. The Internet Encyclopedia of Philosophy calls Putnam's version of the argument "Putnam's success argument" and presents it in the form:

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.

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. 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.

Continued development of the argument

Following the arguments of Quine and Putnam throughout the 1960s and 1970s, Field released his Science without Numbers in 1980 and papers from Maddy and Sober followed in 1992 and 1993. In 1995, Resnik gave a Quinean formulation of the indispensability argument explicitly depending on naturalism and holism: "if mathematics is an indispensable component of science, then, by holism, whatever evidence we have for science is also evidence for the mathematical objects and mathematical principles it presupposes. So, by naturalism, mathematics is true, and the existence of mathematical objects is as well grounded as that of the other entities posited by science." In response to Maddy and Sober's arguments against confirmational holism, he also presented a pragmatic indispensability argument that doesn't depend on holism, but on mathematics' practical utility to science. According to Resnik, the argument "claims that the justification for doing science ... also justifies our accepting as true such mathematics as science uses."

Releasing the 1998 paper "In Defence of Indispensability" and 2001 book The Indispensability of Mathematics, Mark Colyvan is often credited with giving the standard or "canonical" formulation of the argument within more recent philosophical work and his version (which is given in §Overview of the argument) has been influential in debates in contemporary philosophy of mathematics. 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. 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." 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'".

A periodical cicada

Also inspired by Maddy and Sober's arguments against confirmational holism, an explanatory version of the argument has been defended by Colyvan and Alan Baker. The connection between the indispensability argument and mathematical explanations was first raised by Field in 1989 and he was later cited by Baker as originating an explanatory version of the argument, although he didn't fully formulate such an argument. Early papers advancing this form of the argument were released by Colyvan and Baker in the late 1990s and early 2000s, with Baker giving it an explicit formulation in his 2005 paper "Are There Genuine Mathematical Explanations of Physical Phenomena?" 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. It is presented by the Internet Encyclopedia of Philosophy in the following form:

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. Other important examples are the explanations of the hexagonal structure of bee's 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 and returning to the same spot. The main response to this form of the argument, adopted by philosophers such as Melia, Chris Daly, Simon Langford and Juha Saatsi, is to deny that there are genuinely mathematical explanations of empirical phenomena, instead framing the role of mathematics as representational or indexical.

Influence

According to James Franklin, the indispensability argument is widely considered to be the best argument for platonism in the philosophy of mathematics. The Stanford Encyclopedia of Philosophy identifies it 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.

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