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 The naturalism underlying the indispensability argument is a form of
methodological naturalism 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 science's ontological commitment to mathematics 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 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
axioms of
Zermelo–Fraenkel set theory 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 science.
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 entails respecting the methods used by scientists as the best method for uncovering the truth, but scientists do not act as if they are required to believe in all of the entities that are indispensable to science. To illustrate this point, Maddy uses the example of
atomic theory; 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 argues for siding with naturalism and rejecting 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 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 scientific theories, so confirmational holism should be rejected. 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. 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. 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 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 scientific theories into first-order logic because ordinary language is ambiguous, whereas logic can make the commitments of a theory more precise. Translating theories to first-order logic also has advantages over translating them to
higher-order logics such as
second-order logic. Whilst second-order logic has the same expressive power as first-order logic, it lacks some of the technical strengths of first-order logic such as
completeness and
compactness. Second-order logic also allows quantification over
properties like "redness", but whether there are ontological commitments 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, to have ontological commitment to an entity, there must be the right level of epistemic access to it. This means, for example, that it must overcome some epistemic burdens to be postulated. 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 they do not need to be treated 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 "[makes] more 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 belief in mathematics can be suspended 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. 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 numbers. Because prime numbers have no non-trivial factors, this means it is less likely predators can synchronize with the cicadas' life cycles. Baker said that this is an explanation in which mathematics, specifically
number theory, 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 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. == Historical development ==