Quine (Arguments of the Philosophers)

Indispensability Arguments in the Philosophy of Mathematics

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Indeed, see Putnam for an explicit rejection of key elements of the argument as presented here. Still, the argument presented here is influential in the contemporary literature. Think of the attribution to Quine and Putnam as 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. Most scientific realists accept inference to the best explanation. Indeed, it might be said that inference to the best explanation is the cornerstone of scientific realism. But, as we saw in note 1, inference to the best explanation may be seen as a kind of indispensability argument, so any realist who accepts the former while rejecting the latter is in a somewhat delicate position.

If the vocabulary of the theory can be partitioned in the way that Craig's theorem requires, then the theory can be reaxiomatized so that apparent reference to any given theoretical entity is eliminated. See Field , p.

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It turns out that the details of the formulation of naturalism are crucial to the argument. See Maddy for a slightly different formulation that doesn't support the conclusion of the Quine-Putnam argument. The issue of how likely modern theories such as general relativity and quantum mechanics are to yield to nominalisation is a very interesting and controversial matter. David Malament argues that quantum mechanics, for one, is likely to resist nominalisation because of the central role infinite-dimensional Hilbert spaces play in the theory.

For example, the continuum hypothesis is the assertion that the cardinality of the real numbers is the first non-denumerable cardinal. It turns out that neither this hypothesis nor its negation is provable from the ZFC axioms; the question of the size of the continuum is independent of ZFC. It is tempting to formulate Maddy's third objection as follows:. This, however, I think is to misrepresent Maddy's argument. For starters, the indispensability argument can be made to work with P1 reading: More importantly, I think that Maddy's objection is subtler than this formulation.

Once we settle on first-order logic as a canonical language, we must specify a method for determining the commitments of a theory in that language. Reading existential claims seems straightforward. For example, R2 says that there is a thing which is a hat, and which is red. But, theories do not determine their own interpretations. Quine relies on standard Tarskian model-theoretic methods to interpret first-order theories. On a Tarskian interpretation, or semantics, we ascend to a metalanguage to construct a domain of quantification for a given theory.

The objects in the domain that make the theory come out true are the commitments of the theory. For an accessible discussion of Tarskian semantics, see Tarski We construct scientific theory in the most effective and attractive way we can. We balance formal considerations, like the elegance of the mathematics involved, with an attempt to account for the broadest sensory evidence. The more comprehensive and elegant the theory, the more we are compelled to believe it, even if it tells us that the world is not the way we thought it is. If the theory yields a heliocentric model of the solar system, or the bending of rays of light, then we are committed to heliocentrism or bent light rays.

Our commitments are the byproducts of this neutral process. The final step of QI involves simply looking at the domain of the theory we have constructed.

When we write our best theory in our first-order language, we discover that the theory includes physical laws which refer to functions, sets, and numbers. In addition to the charged particles over which the universal quantifiers in front range, there is an existential quantification over a function, f. In order to ensure that there are enough sets to construct these numbers and functions, our ideal theory must include set-theoretic axioms, perhaps those of Zermelo-Fraenkel set theory, ZF. The full theory of ZF is unnecessary for scientific purposes; there will be some sets which are never needed and some numbers which are never used to measure any real quantity.

But, we take a full set theory in order to make our larger theory as elegant as possible. We can derive from the axioms of any adequate set theory a vast universe of sets. So, CL contains or entails several existential mathematical claims. According to QI, we should believe that these mathematical objects exist. Examples such as CLR abound. Real numbers are used for measurement throughout physics, and other sciences. Quantum mechanics makes essential use of Hilbert spaces and probability functions. The theory of relativity invokes the hyperbolic space of Lobachevskian geometry.

Economics is full of analytic functions. Psychology uses a wide range of statistics. Opponents of the indispensability argument have developed sophisticated strategies for re-interpreting apparently ineliminable uses of mathematics, especially in physics. Some reinterpretations use alethic modalities necessity and possibility to replace mathematical objects. Others replace numbers with space-time points or regions.

It is quite easy, but technical, to rewrite first-order theories in order to avoid quantifying over mathematical objects. For an excellent survey of dispensabilist strategies, and further references, see Burgess and Rosen ; for more recent work, see Melia and Melia Other versions of the argument attempt to avoid some of these controversial claims. The position Putnam calls realism in mathematics is ambiguous between two compatible views.

Sentence realism is the claim that sentences of mathematics can be true or false. Object realism is the claim that mathematical objects exist. Most object realists are sentence realists, though some sentence realists, including some structuralists, deny object realism. Indispensability arguments may be taken to establish either sentence realism or object realism.

Quine was an object realist. Michael Resnik presents an indispensability argument for sentence realism; see section 4. Realism contrasts most obviously with fictionalism , on which there are no mathematical objects, and many mathematical sentences considered to be true by the realist are taken to be false. To understand the contrast between realism and fictionalism, consider the following two paradigm mathematical claims, the first existential and the second conditional. The fictionalist claims that mathematical existence claims like E are false since prime numbers are numbers and there are no numbers.

The standard conditional interpretation of C is that if any two angles are consecutive in a parallelogram, then they are supplementary. If there are no mathematical objects, then standard truth-table semantics for the material conditional entail that C, having a false antecedent, is true. The fictionalist claims that conditional statements which refer to mathematical objects, such as C, are only vacuously true, if true.

The success argument emphasizes the success of science, rather than the construction and interpretation of a best theory. The scientific success argument relies on the claim that any position other than realism makes the success of science miraculous. The mathematical success argument claims that the success of mathematics can only be explained by a realist attitude toward its theorems and objects. One potential criticism of any indispensability argument is that by making the justification of our mathematical beliefs depend on our justification for believing in science, our mathematical beliefs become only as strong as our confidence in science.

It is notoriously difficult to establish the truth of scientific theory. Some philosophers, such as Nancy Cartwright and Bas van Fraassen, have argued that science, or much of it, is false, in part due to idealizations. See Cartwright and van Fraassen The success of science may be explained by its usefulness, without presuming that scientific theories are true.

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Still, even if science were only useful rather than true, PS claims that our mathematical beliefs may be justified by the uses of mathematics in science. The problems with scientific realism focus on the incompleteness and error of contemporary scientific theory. These problems need not infect our beliefs in the mathematics used in science. A tool may work fine, even on a broken machine. His argument for that premise is essentially a rejection of the argument that mathematics could be indispensable, yet not true.

Such a holist has no external perspective from which to evaluate the mathematics in scientific theory as merely useful. In contrast, the opponent of PS may abandon the claim that our most sincere commitments are found in the quantifications of our single best theory. Instead, such an opponent might claim that only objects which have causal relations to ordinary physical objects exist. It explains the applicability of mathematics to the natural world, non-miraculously, since any possible state of the natural world will be described by some mathematical theory.

Similarly, and more influentially, Hartry Field has argued that the reason that mathematics is successful as the language of science is because it is conservative over nominalist versions of scientific theories. See Field , especially the preliminary remarks and Chapter 1. In other words, Field claims that mathematics is just a convenient shorthand for a theory which includes no mathematical axioms. In the pragmatic argument, Resnik first links mathematical and scientific justification.

Arguments of the Philosophers

RP, like PS, avoids the problems that may undermine our confidence in science. Even if our best scientific theories are false, their undeniable practical utility still justifies our using them. RP states that we need to presume the truth of mathematics even if science is merely useful.

The key premises for RP, then, are the first two. If we can also take mathematics to be merely useful, then those premises are unjustified. The question for the proponent of RP, then, is how to determine whether science really presumes the existence of mathematical objects, and mathematical truth. How do we determine the commitments of scientific theory?

We could ask scientists about their beliefs, but they may work without considering the question of mathematical truth at all. Like PS, RP seems liable to the critic who claims that the same laws and derivations in science can be stated while taking mathematics to be merely useful. See Azzouni , Leng , and Melia The defender of RP needs a procedure for determining the commitments of science that blocks such a response, if not a more general procedure for determining our commitments.

Alan Baker and Mark Colyvan have defended an explanatory indispensability argument Mancosu Mathematical explanation and scientific explanation are difficult and controversial topics, beyond the scope of this article. Still, two comments are appropriate. First, it is unclear whether EI is intended as a greater demand on the indispensabilist than the standard indispensability argument. Does the platonist have to show that mathematical objects are indispensable in both our best theories and our best explanations, or just in one of them?

Conversely, must the nominalist dispense with mathematics in both theories and explanations?

EI2 refers to the theoretical posits postulated by explanations, but does not tell us how we are supposed to figure out what an explanation posits. If the commitments of a scientific explanation are found in the best scientific theory used in that explanation, then EI is no improvement on QI. If, on the other hand, EI is supposed to be a new and independent argument, its proponents must present a new and independent criterion for determining the commitments of explanations. Given the development of the explanatory indispensability argument and the current interest in mathematical explanation, it is likely that more work will be done on these questions.

Putnam and Resnik maintain that we are committed to mathematics because of the ineliminable role that mathematical objects play in scientific theory. Proponents of the explanatory argument argue that our mathematical beliefs are justified by the role that mathematics plays in scientific explanations. Indispensability arguments in the philosophy of mathematics can be quite general, and can rely on supposedly indispensable uses of mathematics in a wide variety of contexts.

For instance, in later work, Putnam defends belief in mathematical objects for their indispensability in explaining our mathematical intuitions. Since he thinks that our mathematical intuitions derive exclusively from our sense experience, this later argument may still be classified as an indispensability argument. Here are some characteristics of many indispensability arguments in the philosophy of mathematics, no matter how general:.

Although the indispensability argument is a late twentieth century development, earlier philosophers may have held versions of the argument. The most influential approach to denying the indispensability argument is to reject the claim that mathematics is essential to science.

The main strategy for this response is to introduce scientific or mathematical theories which entail all of the consequences of standard theories, but which do not refer to mathematical objects. Such nominalizing strategies break into two groups. In the first group are theories which show how to eliminate quantification over mathematical objects within scientific theories.

Hartry Field has shown how we can reformulate some physical theories to quantify over space-time regions rather than over sets. See Field and Field According to Field, mathematics is useful because it is a convenient shorthand for more complicated statements about physical quantities. See Burgess , Burgess a, and Burgess and Rosen Mark Balaguer has presented steps toward nominalizing quantum mechanics.

The second group of nominalizing strategies attempts to reformulate mathematical theories to avoid commitments to mathematical objects. Charles Chihara Chihara , Geoffrey Hellman Hellman , and Hilary Putnam Putnam b and Putnam a have all explored modal reformulations of mathematical theories. Modal reformulations replace claims about mathematical objects with claims about possibility.

Another line of criticism of the indispensability argument is that the argument is insufficient to generate a satisfying mathematical ontology. But, standard set theory entails the existence of much larger cardinalities. The indispensabilist can justify extending mathematical ontology a little bit beyond those objects explicitly required for science, for simplicity and rounding out. But few indispensabilists have shown interest in justifying beliefs in, say, inaccessible cardinals. Though, see Colyvan for such an attempt. Thus, the indispensabilist has a restricted ontology.

Indispensabilists may welcome these departures from traditional interpretations of mathematics. For example, see Colyvan , Chapter 6. Indispensability arguments need not be restricted to the philosophy of mathematics. Considered more generally, an indispensability argument is an inference to the best explanation which transfers evidence for one set of claims to another. If the transfer crosses disciplinary lines, we can call the argument an inter-theoretic indispensability argument.

If evidence is transferred within a theory, we can call the argument an intra-theoretic indispensability argument.

Willard van Orman Quine

The indispensability argument in the philosophy of mathematics transfers evidence from natural science to mathematics. Thus, this argument is an inter-theoretic indispensability argument. One might apply inter-theoretic indispensability arguments in other areas. For example, one could argue that we should believe in gravitational fields physics because they are ineliminable from our explanations of why zebras do not go flying off into space biology.

We might think that biological laws reduce , in some sense, to physical laws, or we might think that they are independent of physics, or supervenient on physics. Still, our beliefs in some basic claims of physics seem indispensable to other sciences. As an example of an intra-theoretic indispensability argument, consider the justification for our believing in the existence of atoms. Atomic theory makes accurate predictions which extend to the observable world. It has led to a deeper understanding of the world, as well as further successful research.

Despite our lacking direct perception of atoms, they play an indispensable role in atomic theory. According to atomic theory, atoms exist. Thus, according to an intra-theoretic indispensability argument, we should believe that atoms exist. But, it might be defended by using an intra-theoretic indispensability argument: There are at least three ways of arguing for empirical justification of mathematics. The first is to argue, as John Stuart Mill did, that mathematical beliefs are about ordinary, physical objects to which we have sensory access.

The second is to argue that, although mathematical beliefs are about abstract mathematical objects, we have sensory access to such objects. The currently most popular way to justify mathematics empirically is to argue:. Our experiences with physical objects justify our mathematical beliefs. This articles discusses the following version of the argument: We should believe the theory which best accounts for our sense experience.

If we believe a theory, we must believe in its ontic commitments. The ontic commitments of any theory are the objects over which that theory first-order quantifies. The theory which best accounts for our sense experience first-order quantifies over mathematical objects. We should believe that mathematical objects exist. The remainder of this section discusses each of the premises of QI in turn.

A Best Theory The first premise of QI is that we should believe the theory which best accounts for our sense experience, that is, we should believe our best scientific theory. Believing Our Best Theory The second premise of QI states that our belief in a theory naturally extends to the objects which that theory posits. We should take first-order logic as our canonical language only if: We need a single canonical language; B.

It really is adequate; and C. There is no other adequate language. The current president of the United States does not have three children. The tooth fairy does not exist. To see how second-order logic works, consider the inference R. There is a red shirt. There is a red hat. So, there is something redness that some shirt and some hat share.

If we let Sx mean that x is a shirt, and Px mean that x is a property, and so forth, then we have the following valid second-order inference RS: Mathematization The final step of QI involves simply looking at the domain of the theory we have constructed. There is a prime number greater than any you have ever thought of. The consecutive angles of any parallelogram are supplementary. Mathematics succeeds as the language of science. There must be a reason for the success of mathematics as the language of science.

Willard van Orman Quine (Stanford Encyclopedia of Philosophy)

No positions other than realism in mathematics provide a reason. So, realism in mathematics must be correct. Realism best explains the success of mathematics as the language of science. In stating its laws and conducting its derivations, science assumes the existence of many mathematical objects and the truth of much mathematics. These assumptions are indispensable to the pursuit of science; moreover, many of the important conclusions drawn from and within science could not be drawn without taking mathematical claims to be true. So, we are justified in drawing conclusions from and within science only if we are justified in taking the mathematics used in science to be true.

We are justified in using science to explain and predict. The only way we know of using science thus involves drawing conclusions from and within it.

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So, by RP3, we are justified in taking mathematics to be true Resnik There are genuinely mathematical explanations of empirical phenomena. We ought to be committed to the theoretical posits in such explanations. We ought to be committed to the entities postulated by the mathematics in question. Here are some characteristics of many indispensability arguments in the philosophy of mathematics, no matter how general: The job of the philosopher, as of the scientist, is exclusively to understand our sensible experience of the physical world.

In order to explain our sensible experience we construct a theory, or theories, of the physical world. We find our commitments exclusively in our best theory or theories. Some mathematical objects are ineliminable from our best theory or theories. Mathematical practice depends for its legitimacy on natural scientific practice.

Responses to the Indispensability Argument The most influential approach to denying the indispensability argument is to reject the claim that mathematics is essential to science. Inter-theoretic and Intra-theoretic Indispensability Arguments Indispensability arguments need not be restricted to the philosophy of mathematics. Conclusion There are at least three ways of arguing for empirical justification of mathematics.

The currently most popular way to justify mathematics empirically is to argue: Mathematical beliefs are about abstract objects; B. We have experiences only with physical objects; and yet C. This is the indispensability argument in the philosophy of mathematics. References and Further Reading Azzouni, Jody. A Case for Nominalism. Platonism and Anti-Platonism in Mathematics. Selected Readings , second edition, Cambridge: Cambridge University Press, Lawrence Erlblum Associates, Burgess, John, and Gideon Rosen. A Subject with No Object.

How the Laws of Physics Lie. Constructabiliy and Mathematical Existence. The Indispensability of Mathematics.

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