The Quine–Putnam indispensability argument is an argument in the
philosophy of mathematics
Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics. Central questions posed include whether or not mathem ...
for the existence of
abstract mathematical object
A mathematical object is an abstract concept arising in mathematics. Typically, a mathematical object can be a value that can be assigned to a Glossary of mathematical symbols, symbol, and therefore can be involved in formulas. Commonly encounter ...
s such as numbers and sets, a position known as
mathematical platonism
Mathematical Platonism is the form of realism that suggests that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. This is often claimed to be the view most people have of numbers.
...
. It was named after the philosophers
Willard Van Orman Quine
Willard Van Orman Quine ( ; known to his friends as "Van"; June 25, 1908 – December 25, 2000) was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century" ...
and
Hilary Putnam
Hilary Whitehall Putnam (; July 31, 1926 – March 13, 2016) was an American philosopher, mathematician, computer scientist, and figure in analytic philosophy in the second half of the 20th century. He contributed to the studies of philosophy of ...
, and is one of the most important arguments in the philosophy of mathematics.
Although elements of the indispensability argument may have originated with thinkers such as
Gottlob Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philos ...
and
Kurt Gödel
Kurt Friedrich Gödel ( ; ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly ...
, Quine's development of the argument was unique for introducing to it a number of his philosophical positions such as
naturalism,
confirmational holism, and the criterion of
ontological commitment
Ontology is the philosophical study of being. It is traditionally understood as the subdiscipline of metaphysics focused on the most general features of reality. As one of the most fundamental concepts, being encompasses all of reality and every ...
. Putnam gave Quine's argument its first detailed formulation in his 1971 book ''Philosophy of Logic''. He later came to disagree with various aspects of Quine's thinking, however, and formulated his own indispensability argument based on the
no miracles argument in the
philosophy of science
Philosophy of science is the branch of philosophy concerned with the foundations, methods, and implications of science. Amongst its central questions are the difference between science and non-science, the reliability of scientific theories, ...
. A standard form of the argument in contemporary philosophy is credited to
Mark Colyvan; whilst being influenced by both Quine and Putnam, it differs in important ways from their formulations. It is presented in the ''
Stanford Encyclopedia of Philosophy
The ''Stanford Encyclopedia of Philosophy'' (''SEP'') is a freely available online philosophy resource published and maintained by Stanford University, encompassing both an online encyclopedia of philosophy and peer-reviewed original publication ...
'':
* We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.
* Mathematical entities are indispensable to our best scientific theories.
* Therefore, we ought to have ontological commitment to mathematical entities.
Nominalists
In metaphysics, nominalism is the view that universals and abstract objects do not actually exist other than being merely names or labels. There are two main versions of nominalism. One denies the existence of universals—that which can be inst ...
, philosophers who reject the existence of abstract objects, have argued against both premises of this argument. An influential argument by
Hartry Field claims that mathematical entities are dispensable to science. This argument has been supported by attempts to demonstrate that scientific and mathematical theories can be reformulated to remove all references to mathematical entities. Other philosophers, including
Penelope Maddy
Penelope Maddy (born 4 July 1950) is an American philosopher. Maddy is Distinguished Professor Emerita of Logic and Philosophy of Science and of Mathematics at the University of California, Irvine. She is well known for her influential work in the ...
,
Elliott Sober
Elliott R. Sober (born 6 June 1948) is an American philosopher. He is noted for his work in philosophy of biology and general philosophy of science. Sober is Hans Reichenbach Professor and William F. Vilas Research Professor Emeritus in the Depar ...
, and
Joseph Melia, have argued that we do not need to believe in all of the entities that are indispensable to science. The arguments of these writers inspired a new
explanatory version of the argument, which
Alan Baker and Mark Colyvan support, that argues mathematics is indispensable to specific scientific explanations as well as whole theories.
Background
In his 1973 paper "Mathematical Truth",
Paul Benacerraf
Paul Joseph Salomon Benacerraf (; 26 March 1930 – 13 January 2025) was a French-born American philosopher working in the field of the philosophy of mathematics who taught at Princeton University his entire career, from 1960 until his retirement ...
raised a problem for the
philosophy of mathematics
Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics. Central questions posed include whether or not mathem ...
. According to Benacerraf, mathematical sentences such as "two is a prime number" imply the existence of
mathematical object
A mathematical object is an abstract concept arising in mathematics. Typically, a mathematical object can be a value that can be assigned to a Glossary of mathematical symbols, symbol, and therefore can be involved in formulas. Commonly encounter ...
s. He supported this claim with the idea that mathematics should not have its own special
semantics
Semantics is the study of linguistic Meaning (philosophy), meaning. It examines what meaning is, how words get their meaning, and how the meaning of a complex expression depends on its parts. Part of this process involves the distinction betwee ...
. In other words, the meaning of mathematical sentences should follow the same rules as any other type of sentence. For example, if the sentence "Mars is a planet" implies the existence of the planet Mars, then the sentence "two is a prime number" should also imply the existence of the number two. But according to Benacerraf, if mathematical objects existed, they would be unknowable. This is because mathematical objects, if they exist, are
abstract objects
In philosophy and the arts, a fundamental distinction exists between abstract and concrete entities. While there is no universally accepted definition, common examples illustrate the difference: numbers, sets, and ideas are typically classified ...
: objects that cannot cause things to happen and that have no location in space and time. Benacerraf argued, on the basis of the
causal theory of knowledge
"A Causal Theory of Knowing" is a philosophical essay written by Alvin Goldman in 1967, published in ''The Journal of Philosophy''. It is based on existing theories of knowledge in the realm of epistemology, the study of philosophy through the scop ...
, that it would be impossible to know about such objects because they cannot come into causal contact with us. This is called Benacerraf's epistemological problem because it concerns the
epistemology
Epistemology is the branch of philosophy that examines the nature, origin, and limits of knowledge. Also called "the theory of knowledge", it explores different types of knowledge, such as propositional knowledge about facts, practical knowle ...
of mathematics, that is, how we come to know what we do about mathematics.
The philosophy of mathematics is split into two main strands:
platonism
Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary Platonists do not necessarily accept all doctrines of Plato. Platonism has had a profound effect on Western thought. At the most fundam ...
and
nominalism
In metaphysics, nominalism is the view that universals and abstract objects do not actually exist other than being merely names or labels. There are two main versions of nominalism. One denies the existence of universals—that which can be inst ...
. Platonism holds that there exist abstract mathematical objects such as numbers and sets whilst nominalism denies their existence. Each of these views faces issues due to the problem raised by Benacerraf. Because nominalism rejects the existence of mathematical objects, it faces no epistemological problem but it does face problems concerning the idea that mathematics should not have its own special semantics. Platonism does not face problems concerning the semantic half of the dilemma but it has difficulty explaining how it is possible to know about mathematical objects.
The indispensability argument aims to overcome the epistemological problem posed against platonism by providing a justification for belief in abstract mathematical objects. It is part of a broad class of indispensability arguments most commonly applied in the philosophy of mathematics, but which also includes arguments in the
philosophy of language
Philosophy of language refers to the philosophical study of the nature of language. It investigates the relationship between language, language users, and the world. Investigations may include inquiry into the nature of Meaning (philosophy), me ...
and
ethics
Ethics is the philosophy, philosophical study of Morality, moral phenomena. Also called moral philosophy, it investigates Normativity, normative questions about what people ought to do or which behavior is morally right. Its main branches inclu ...
. In the most general sense, indispensability arguments aim to support their conclusion based on the claim that the truth of the conclusion is indispensable or necessary for a certain purpose. When applied in the field of
ontology
Ontology is the philosophical study of existence, being. It is traditionally understood as the subdiscipline of metaphysics focused on the most general features of reality. As one of the most fundamental concepts, being encompasses all of realit ...
—the study of what exists—they exemplify a
Quinean strategy for establishing the existence of controversial entities that cannot be directly investigated. According to this strategy, the indispensability of these entities for formulating a theory of other less controversial entities counts as evidence for their existence. In the case of philosophy of mathematics, the indispensability of mathematical entities for formulating scientific theories is taken as evidence for the existence of those mathematical entities.
Overview of the argument
Mark Colyvan presents the argument in the ''
Stanford Encyclopedia of Philosophy
The ''Stanford Encyclopedia of Philosophy'' (''SEP'') is a freely available online philosophy resource published and maintained by Stanford University, encompassing both an online encyclopedia of philosophy and peer-reviewed original publication ...
'' in the following form:
* We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.
* Mathematical entities are indispensable to our best scientific theories.
* Therefore, we ought to have ontological commitment to mathematical entities.
Here, an ontological commitment to an entity is a commitment to believing that that entity exists. The first premise is based on two fundamental assumptions:
naturalism and
confirmational holism. According to naturalism, we should look to our best scientific theories to determine what we have best reason to believe exists. 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. Therefore, according to confirmational holism, if we should believe in science, then we should believe in ''all'' of science, including any of the mathematics that is assumed by our best scientific theories. The argument is mainly aimed at nominalists that are
scientific realists as it attempts to justify belief in mathematical entities in a manner similar to the justification for belief in theoretical entities such as
electron
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
s or
quark
A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nucleus, atomic nuclei ...
s; Quine held that such nominalists have a "double standard" with regards to ontology.
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. It does not necessarily justify belief in the most abstract parts of set theory, which Quine called "mathematical recreation … without ontological rights". Some philosophers infer from the argument that mathematical knowledge is ''
a posteriori
('from the earlier') and ('from the later') are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on experience. knowledge is independent from any experience. Examples include ...
'' because it implies mathematical truths can only be established via the empirical confirmation of scientific theories to which they are indispensable. This also indicates mathematical truths are
contingent since empirically known truths are generally contingent. Such a position is controversial because it contradicts the traditional view of mathematical knowledge as ''
a priori
('from the earlier') and ('from the later') are Latin phrases used in philosophy to distinguish types of knowledge, Justification (epistemology), justification, or argument by their reliance on experience. knowledge is independent from any ...
'' knowledge of
necessary truths.
Whilst Quine's original argument is an argument for platonism, indispensability arguments can also be constructed to argue for the weaker claim of sentence realism—the claim that mathematical theory is objectively true. This is a weaker claim because it does not necessarily imply there are abstract mathematical objects.
Major concepts
Indispensability
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. Indispensability instead means that an entity cannot be eliminated without reducing the attractiveness of the theory. The attractiveness of the theory can be evaluated in terms of theoretical virtues such as
explanatory power
Explanatory power is the ability of a hypothesis or theory to explain the subject matter effectively to which it pertains. Its opposite is ''explanatory impotence''.
In the past, various criteria or measures for explanatory power have been prop ...
,
empirical adequacy
In philosophy of science, constructive empiricism is a form of empiricism. While it is sometimes referred to as an empiricist form of structuralism, its main proponent, Bas van Fraassen, has consistently distinguished between the two views.
Overvi ...
and
simplicity
Simplicity is the state or quality of being wikt:simple, simple. Something easy to understand or explain seems simple, in contrast to something complicated. Alternatively, as Herbert A. Simon suggests, something is simple or Complexity, complex ...
. Furthermore, if an entity is dispensable to a theory, an equivalent theory can be formulated without it. 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.
According to the ''Stanford Encyclopedia of Philosophy'', one of the most influential arguments against the indispensability argument comes from
Hartry Field. It rejects the claim that mathematical objects are indispensable to science; Field has supported this argument by reformulating or "nominalizing" scientific theories so they do not refer to mathematical objects. As part of this project, Field has offered a reformulation of
Newtonian physics
Classical mechanics is a physical theory describing the motion of objects such as projectiles, parts of machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics involved substantial change in the methods ...
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. John Burgess and Mark Balaguer have taken steps to extend this nominalizing project to areas of
modern physics
Modern physics is a branch of physics that developed in the early 20th century and onward or branches greatly influenced by early 20th century physics. Notable branches of modern physics include quantum mechanics, special relativity, and genera ...
, including
quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
. Philosophers such as
David Malament and
Otávio Bueno dispute whether such reformulations are successful or even possible, particularly in the case of quantum mechanics.
Field's alternative to platonism is
mathematical fictionalism, according to which mathematical theories are false because they refer to abstract objects which do not exist. As part of his argument against the indispensability argument, Field has tried to explain how it is possible for false mathematical statements to be used by science without making scientific predictions false. His argument is based on the idea that mathematics is
conservative
Conservatism is a cultural, social, and political philosophy and ideology that seeks to promote and preserve traditional institutions, customs, and values. The central tenets of conservatism may vary in relation to the culture and civiliza ...
. A mathematical theory is conservative if, when combined with a scientific theory, it does not imply anything about the physical world that the scientific theory alone would not have already. This explains how it is possible for mathematics to be used by scientific theories without making the predictions of science false. In addition, Field has attempted to specify how exactly mathematics is useful in application. Field thinks mathematics is useful for science because mathematical language provides a useful shorthand for talking about complex physical systems.
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.
Naturalism
The naturalism underlying the indispensability argument is a form of
methodological naturalism
In philosophy, naturalism is the idea that only natural laws and forces (as opposed to supernatural ones) operate in the universe. In its primary sense, it is also known as ontological naturalism, metaphysical naturalism, pure naturalism, phi ...
that asserts the primacy of the scientific method for determining the truth. In other words, according to Quine's naturalism, our best scientific theories are the best guide to what exists. This form of naturalism rejects the idea that philosophy precedes and ultimately justifies belief in science, instead holding that science and philosophy are continuous with one another as part of a single, unified investigation into the world. As such, this form of naturalism precludes the idea of a prior philosophy that can overturn the ontological commitments of science. This is in contrast to
metaphysical forms of naturalism, which rule out the existence of abstract objects because they are not physical. An example of such a naturalism is supported by
David Armstrong. It holds a principle called the
Eleatic principle, which states that only causal entities exist and there are no non-causal entities. Quine's naturalism claims such a principle cannot be used to overturn our best scientific theories' ontological commitment to mathematical entities because philosophical principles cannot overrule science.
Quine held his naturalism as a fundamental assumption but later philosophers have provided arguments to support it. The most common arguments in support of Quinean naturalism are track-record arguments. These are arguments that appeal to science's successful track record compared to philosophy and other disciplines.
David Lewis famously made such an argument in a passage from his 1991 book ''Parts of Classes'', deriding the track record of philosophy compared to mathematics and arguing that the idea of philosophy overriding science is absurd. Critics of the track record argument have argued that it goes too far, discrediting philosophical arguments and methods entirely, and contest the idea that philosophy can be uniformly judged to have had a bad track record.
Quine's naturalism has also been criticized by
Penelope Maddy
Penelope Maddy (born 4 July 1950) is an American philosopher. Maddy is Distinguished Professor Emerita of Logic and Philosophy of Science and of Mathematics at the University of California, Irvine. She is well known for her influential work in the ...
for contradicting mathematical practice. According to the indispensability argument, mathematics is subordinated to the natural sciences in the sense that its legitimacy depends on them. But Maddy argues mathematicians do not seem to believe their practice is restricted in any way by the activity of the natural sciences. For example, mathematicians' arguments over the
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
s of
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...
do not appeal to their applications to the natural sciences. Similarly,
Charles Parsons has argued that mathematical truths seem immediately obvious in a way that suggests they do not depend on the results of our best theories.
Confirmational holism
Confirmational holism is the view that scientific theories and hypotheses cannot be confirmed in isolation and must be confirmed together as part of a larger cluster of theories. An example of this idea provided by
Michael Resnik is of the hypothesis that an observer will see oil and water separate out if they are added together because they do not mix. This hypothesis cannot be confirmed in isolation because it relies on assumptions 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. Because mathematical theories are likewise assumed by scientific theories, confirmational holism implies the empirical confirmations of scientific theories also support these mathematical theories.
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 that we should respect the methods used by scientists as the best method for uncovering the truth, but scientists do not act as if we should believe in all of the entities that are indispensable to science. To illustrate this point, Maddy uses the example of
atomic theory
Atomic theory is the scientific theory that matter is composed of particles called atoms. The definition of the word "atom" has changed over the years in response to scientific discoveries. Initially, it referred to a hypothetical concept of ...
; she states that 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. Maddy, and others such as
Mary Leng, also appeal to the fact that scientists use mathematical
idealizations—such as assuming bodies of water to be infinitely deep—without regard for whether they 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 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.
Another counterargument due to
Elliott Sober
Elliott R. Sober (born 6 June 1948) is an American philosopher. He is noted for his work in philosophy of biology and general philosophy of science. Sober is Hans Reichenbach Professor and William F. Vilas Research Professor Emeritus in the Depar ...
claims that mathematical theories are not tested in the same way as scientific theories. Sober states that scientific theories compete with alternatives to find which theory has the most empirical support. But 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.
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 that confirmational holism can be eliminated from the argument. For example, Resnik has offered a pragmatic indispensability argument focused less on the notion of evidence and more on the practical importance of mathematics in conducting scientific enquiry.
Ontological commitment
Another key part of the argument is the concept of
ontological commitment
Ontology is the philosophical study of being. It is traditionally understood as the subdiscipline of metaphysics focused on the most general features of reality. As one of the most fundamental concepts, being encompasses all of reality and every ...
. The ontological commitments of a theory are all the things that exist according to that theory. Quine believed that people should be ontologically committed to the same entities that their best scientific theories are committed, in the sense that they should be committed to believing they exist. He formulated a "criterion of ontological commitment", which aims to uncover the commitments of scientific theories by
translating or "regimenting" them from ordinary language into
first-order logic
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...
. In ordinary language, Quine believed the term "there is" must carry ontological commitment; to say "there is" something means that that thing exists. And for Quine, the
existential quantifier
Existentialism is a family of philosophy, philosophical views and inquiry that explore the human individual's struggle to lead an Authenticity (philosophy), authentic life despite the apparent Absurdity#The Absurd, absurdity or incomprehensibili ...
in first-order logic was the natural equivalent of "there is". Therefore, Quine's criterion takes the ontological commitments of the theory to be all of the objects over which the regimented theory
quantifies.
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 logic
In mathematics and logic, a higher-order logic (abbreviated HOL) is a form of logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are m ...
s such as
second-order logic
In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory.
First-order logic quantifies on ...
. 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
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it ...
. Second-order logic also allows quantification over
properties
Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property.
Property may also refer to:
Philosophy and science
* Property (philosophy), in philosophy and logic, an abstraction characterizing an ...
like "redness", but whether we have ontological commitment to properties is controversial. According to Quine, such quantification is simply ungrammatical.
Jody Azzouni has objected to Quine's criterion of ontological commitment, saying that the existential quantifier in first-order logic does not always carry ontological commitment. 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". 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.
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.
Mathematical explanation
One potential issue with the argument, raised by
Joseph Melia, is that it does not account for the role of mathematics in science. According to Melia, we only need to believe in mathematics if it is indispensable to science in the right kind of way. In particular, it needs to be indispensable to scientific explanations. But according to Melia, mathematics plays a purely representational role in science, it merely "
akesmore things sayable about concrete objects". He argues that it is legitimate to withdraw commitment to mathematics for this reason, citing a linguistic phenomenon he calls "weaseling". This is when a person makes a statement and then later withdraws something implied by that statement. An example of weaseling used to express information in an everyday context is "Everyone who came to the seminar had a handout. But the person who came in late didn't get one." Here, seemingly contradictory information is conveyed, but read charitably it simply states that everyone apart from the person who came in late got a handout. Similarly, according to Melia, although mathematics is indispensable to science "almost all scientists ... deny that there are such things as mathematical objects", implying that commitment to mathematical objects is being weaseled away. For Melia, such weaseling is acceptable because mathematics does not play a genuinely explanatory role in science.
Inspired both by the arguments against confirmational holism and Melia's argument that we can suspend belief in mathematics if it does not play a genuinely explanatory role in science, Colyvan and Baker have defended an
explanatory version of the indispensability argument. This version of the argument attempts to remove the reliance on confirmational holism by replacing it with an
inference to the best explanation
Abductive reasoning (also called abduction,For example: abductive inference, or retroduction) is a form of logical inference that seeks the simplest and most likely conclusion from a set of observations. It was formulated and advanced by Ameri ...
. 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. It is presented by the ''Internet Encyclopedia of Philosophy'' in the following form:
* 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.

An example of mathematics' explanatory indispensability presented by Baker is the
periodic cicada, a type of insect that usually has life cycles of 13 or 17 years. It is hypothesized that this is an evolutionary advantage because 13 and 17 are
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s. Because prime numbers have no non-trivial factors, this means it is less likely predators can synchronize with the cicadas' life cycles. Baker said that this is an explanation in which mathematics, specifically
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, plays a key role in explaining an empirical phenomenon.
Other important examples are explanations of the
hexagonal structure of bee honeycombs and the impossibility of crossing all
seven bridges of Königsberg
The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler, in 1736, laid the foundations of graph theory and prefigured the idea of topology.
The city of Königsberg in Prussia ...
only once in a walk across the city. The main response to this form of the argument, which philosophers such as Melia, Chris Daly, Simon Langford, and Juha Saatsi have adopted, is to deny there are genuinely mathematical explanations of empirical phenomena, instead framing the role of mathematics as representational or
indexical
In semiotics, linguistics, anthropology, and philosophy of language, indexicality is the phenomenon of a '' sign'' pointing to (or ''indexing'') some element in the context in which it occurs. A sign that signifies indexically is called an index o ...
.
Historical development
Precursors and influences on Quine
The argument is historically associated with
Willard Van Orman Quine
Willard Van Orman Quine ( ; known to his friends as "Van"; June 25, 1908 – December 25, 2000) was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century" ...
and
Hilary Putnam
Hilary Whitehall Putnam (; July 31, 1926 – March 13, 2016) was an American philosopher, mathematician, computer scientist, and figure in analytic philosophy in the second half of the 20th century. He contributed to the studies of philosophy of ...
but it can be traced to earlier thinkers such as
Gottlob Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philos ...
and
Kurt Gödel
Kurt Friedrich Gödel ( ; ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly ...
. In his arguments against
mathematical formalism—a view that likens mathematics to a game like chess with rules about how mathematical symbols such as "2" can be manipulated—Frege said in 1893 that "it is applicability alone which elevates arithmetic from a game to the rank of a science". Gödel, in a 1947 paper on the axioms of
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
, said that 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". Frege's and Gödel's arguments differ from the later Quinean indispensability argument because they lack features such as naturalism and subordination of practice, leading some philosophers, including
Pieranna Garavaso, to say that they are not genuine examples of the indispensability argument.
Whilst developing his philosophical view of confirmational holism, Quine was influenced by
Pierre Duhem
Pierre Maurice Marie Duhem (; 9 June 1861 – 14 September 1916) was a French theoretical physicist who made significant contributions to thermodynamics, hydrodynamics, and the theory of Elasticity (physics), elasticity. Duhem was also a prolif ...
. At the beginning of the twentieth century, Duhem defended the
law of inertia from critics who said that it is devoid of empirical content and
unfalsifiable
Falsifiability (or refutability) is a deductive standard of evaluation of scientific theories and hypotheses, introduced by the philosopher of science Karl Popper in his book '' The Logic of Scientific Discovery'' (1934). A theory or hypothesi ...
. These critics based this claim on the fact that the law does not make any observable predictions without positing some observational
frame of reference
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system, whose origin (mathematics), origin, orientation (geometry), orientation, and scale (geometry), scale have been specified in physical space. It ...
and that falsifying instances can always be avoided by changing the choice of reference frame. Duhem responded by saying that the law produces predictions when paired with auxiliary hypotheses fixing the frame of reference and is therefore no different from any other physical theory. Duhem said that 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, claiming that although mathematical hypotheses hold no empirical content on their own, they can 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
In philosophy of science, the Duhem–Quine thesis, also called the Duhem–Quine problem, says that unambiguous falsifications of a scientific hypothesis are impossible, because an empirical test of the hypothesis requires one or more back ...
.
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
The hypothetico-deductive model or method is a proposed description of the scientific method. According to it, scientific inquiry proceeds by formulating a hypothesis in a form that can be falsifiable, using a test on observable data where the ou ...
." 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 that Descartes' attempts to provide the foundations for science had failed and that 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.
Quine was also influenced by the
logical positivists
Logical positivism, also known as logical empiricism or neo-positivism, was a philosophical movement, in the empiricist tradition, that sought to formulate a scientific philosophy in which philosophical discourse would be, in the perception of ...
, such as his teacher
Rudolf Carnap
Rudolf Carnap (; ; 18 May 1891 – 14 September 1970) was a German-language philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle and an advocate of logical positivism.
...
; his naturalism was 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, saying that we must believe in ordinary objects to organize our experiences of the world. He also said that because 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 said that 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 their not being directly perceived.
Quine and Putnam
Whilst he eventually became a 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". He and
Nelson Goodman
Henry Nelson Goodman (7 August 1906 – 25 November 1998) was an American philosopher, known for his work on counterfactuals, mereology, the problem of induction, irrealism, and aesthetics.
Life and career
Goodman was born in Somerville, Ma ...
subsequently released a joint 1947 paper titled "Steps toward a Constructive Nominalism" as part of an ongoing project of Quine's to "set up a nominalistic language in which all of natural science can be expressed". In a letter to
Joseph Henry Woodger the following year, however, Quine said that he was becoming more convinced "the assumption of abstract entities and the assumptions of the external world are assumptions of the same sort". He later released the 1948 paper "On What There Is", in which he said that "
e 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 had not yet reached a definitive conclusion about which was correct. 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". According to Lieven Decock, Quine had accepted the need for abstract mathematical entities by the publication of his 1960 book ''
Word and Object
''Word and Object'', philosopher Willard Van Orman Quine's most famous work, expands on ideas in ''From a Logical Point of View'' (1953) and reformulates earlier arguments like his attack on the analytic–synthetic distinction from " Two Dogmas ...
'', in which he wrote "a thoroughgoing nominalist doctrine is too much to live up to". However, whilst he released suggestions of the indispensability argument in a number of papers, he never gave it a detailed formulation.
Putnam gave the argument its first explicit presentation 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 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". Putnam's endorsement of Quine's version of the argument is disputed. The ''
Internet Encyclopedia of Philosophy
The ''Internet Encyclopedia of Philosophy'' (''IEP'') is a scholarly online encyclopedia with around 900 articles about philosophy, philosophers, and related topics. The IEP publishes only peer review, peer-reviewed and blind-refereed original p ...
'' states: "In his early work, Hilary Putnam accepted Quine's version of the indispensability argument." Liggins and Bueno, however, argue that Putnam never endorsed the argument and only presented it as an argument from Quine. In a 1990 lecture, Putnam said that he had shared Quine's views on the indispensability argument since 1948 when he was a student at Harvard, but that he had since come to disagree with them. He later said that he differed with Quine in his attitude to the argument from at least 1975. Features of the argument that Putnam came to disagree with include its reliance on a single, regimented, best theory.
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
Scientific realism is the philosophical view that the universe described by science (including both observable and unobservable aspects) exists independently of our perceptions, and that verified scientific theories are at least approximately true ...
without being rendered miraculous. He wrote that year: "I believe that the positive argument for realism
n sciencehas 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." The ''Internet Encyclopedia of Philosophy'' terms this version of the argument "Putnam's success argument" and presents it in the following form:
* 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.
According to the ''Internet Encyclopedia of Philosophy'', 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 include Field's reformulations of science, which explain the usefulness of mathematics as a conservative shorthand. Putnam has criticized Field's reformulations for only applying to
classical physics
Classical physics refers to physics theories that are non-quantum or both non-quantum and non-relativistic, depending on the context. In historical discussions, ''classical physics'' refers to pre-1900 physics, while '' modern physics'' refers to ...
and for being unlikely to be able to be extended to future fundamental physics.
Continued development of the argument
According to
Ian Hacking
Ian MacDougall Hacking (February 18, 1936 – May 10, 2023) was a Canadian philosopher specializing in the philosophy of science. Throughout his career, he won numerous awards, such as the Killam Prize for the Humanities and the Balzan Prize, ...
, there was no "concerted challenge" to the indispensability argument for a number of decades after Quine first raised it. Chihara, in his 1973 book ''Ontology and the Vicious Circle Principle'', was one of the earliest philosophers to attempt to reformulate mathematics in response to Quine's arguments. Field followed with ''
Science Without Numbers'' in 1980 and dominated discussion about the indispensability argument throughout the 1980s and 1990s. With the introduction of arguments against the first premise of the argument, initially by Maddy in the 1990s and continued by Melia and others in the 2000s, Field's approach has come to be known as "Hard Road Nominalism" due to the difficulty of creating technical reconstructions of science that it requires. Approaches attacking the first premise, in contrast, have come to be known as "Easy Road Nominalism".
Colyvan is often seen as presenting the standard or "canonical" formulation of the argument within more recent philosophical work, and
his version of the argument has been influential within
contemporary philosophy
Contemporary philosophy is the present period in the history of Western philosophy beginning at the early 20th century with the increasing professionalization of the discipline and the rise of analytic and continental philosophy.
The phrase "con ...
of mathematics. 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, which is not explicitly required by Colyvan's formulation. Putnam's arguments were for the objectivity of mathematics but not necessarily for mathematical objects. Putnam has explicitly 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. Colyvan has said "the attribution to Quine and Putnam
san acknowledgement of intellectual debts rather than an indication that the argument, as presented, would be endorsed in every detail by either Quine or Putnam".
Influence
The indispensability argument is widely, though not universally, considered to be the best argument for platonism in the philosophy of mathematics.
According to the ''Stanford Encyclopedia of Philosophy'', some within the field see it as the only good argument for platonism. It is one of just a few arguments that have come to dominate the debate between mathematical realism and mathematical anti-realism. In contemporary philosophy, many types of nominalism define themselves in opposition to the indispensability argument, and it is generally seen as the most important argument to overcome for nominalist views such as fictionalism.
Quine's and Putnam's arguments have also been influential outside philosophy of mathematics, inspiring indispensability arguments in other areas of philosophy. For example,
David Lewis, who was a student of Quine, used an indispensability argument to argue for
modal realism
Modal realism is the view propounded by the philosopher David Lewis that all possible worlds are real in the same way as is the actual world: they are "of a kind with this world of ours." It states that possible worlds exist, possible worlds are ...
in his 1986 book ''
On the Plurality of Worlds
''On the Plurality of Worlds'' (1986) is a book by the philosopher David Lewis that defends the thesis of modal realism. "The thesis states that the world we are part of is but one of a plurality of worlds," as he writes in the preface, "and ...
''. According to his argument, quantification over
possible world
A possible world is a complete and consistent way the world is or could have been. Possible worlds are widely used as a formal device in logic, philosophy, and linguistics in order to provide a semantics for intensional and modal logic. Their met ...
s is indispensable to our best philosophical theories, so we should believe in their
concrete
Concrete is a composite material composed of aggregate bound together with a fluid cement that cures to a solid over time. It is the second-most-used substance (after water), the most–widely used building material, and the most-manufactur ...
existence. Other indispensability arguments in
metaphysics
Metaphysics is the branch of philosophy that examines the basic structure of reality. It is traditionally seen as the study of mind-independent features of the world, but some theorists view it as an inquiry into the conceptual framework of ...
are defended by philosophers such as
David Armstrong,
Graeme Forbes, and
Alvin Plantinga
Alvin Carl Plantinga (born November 15, 1932) is an American analytic philosophy, analytic philosopher who works primarily in the fields of philosophy of religion, epistemology (particularly on issues involving theory of justification, epistemic ...
, who have argued for the existence of
states of affairs
In philosophy, a state of affairs (), also known as a situation, is a way the actual world must be in order to make some given ''proposition'' about the actual world true; in other words, a state of affairs is a ''truth-maker'', whereas a proposit ...
due to the indispensable theoretical role they play in our best philosophical theories of
truthmakers,
modality
Modality may refer to:
Humanities
* Modality (theology), the organization and structure of the church, as distinct from sodality or parachurch organizations
* Modality (music), in music, the subject concerning certain diatonic scales
* Modalit ...
, and possible worlds. In the field of ethics,
David Enoch
David Enoch (; 1901–1949) was an Israeli chess player.
Biography
David Enoch was born in Oświęcim in 1901. He emigrated to Berlin after the First World War. He tied for 6-7th at Berlin 1927 ( Alfred Brinckmann won), and took 10th at Berlin ...
has expanded the criterion of ontological commitment used in the Quine–Putnam indispensability argument to argue for
moral realism
Moral realism (also ethical realism) is the position that ethical sentences express propositions that refer to objective features of the world (that is, features independent of subjective opinion), some of which may be true to the extent that t ...
. According to Enoch's "deliberative indispensability argument", indispensability to deliberation is just as ontologically committing as indispensability to science, and moral facts are indispensable to deliberation. Therefore, according to Enoch, we should believe in moral facts.
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{{DEFAULTSORT:Quine-Putnam indispensability argument
Philosophy of mathematics
Philosophical arguments
Willard Van Orman Quine