The Quine–Putnam 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. It was named after the philosophers Willard Van Orman Quine and Hilary Putnam, 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 and Kurt Gödel, 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. 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. 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:
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, 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, Elliott Sober, 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 raised a problem for the philosophy of mathematics. According to Benacerraf, mathematical sentences such as "two is a prime number" imply the existence of mathematical objects. He supported this claim with the idea that mathematics should not have its own special semantics. 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: 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, that it would be impossible to know about such objects because they cannot come into causal contact with anyone. This is called Benacerraf's epistemological problem because it concerns the epistemology of mathematics, that is, how people know what they do about mathematics.
The philosophy of mathematics is split into two main strands: platonism and nominalism. 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 and ethics. 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—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 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 electrons or quarks; 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 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 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, empirical adequacy and simplicity. 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 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, including quantum mechanics. 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. 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