# Nominalism, Modern

# NOMINALISM, MODERN

In its main contemporary sense, nominalism is the thesis that abstract entities do not exist. Equivalently, it is the thesis that everything that does exist is a concrete object. Since there is no generally accepted account of the abstract-concrete distinction, and since it remains genuinely unclear how certain (putative) entities are to be classified, the content of modern nominalism is to some degree unsettled. Certain consequences of the view are, however, tolerably clear. For example, it is widely agreed that the objects of pure mathematics—numbers, sets, functions, abstract geometrical spaces, and so on—are to be classified as abstract. It is also widely agreed that certain objects of metaphysics and semantics—propositions, meanings, properties and relations, and so on—must be abstract if they exist at all. Modern nominalists thus commit themselves to rejecting these paradigmatic abstract entities and hence to rejecting any scientific, mathematical, or philosophical theory according to which such things exist. In this sense nominalism is standardly opposed to platonism (or, less commonly, antinominalism).

## Early History

The first significant philosophical system in the modern period to insist on the existence of abstract objects is due to Gottlob Frege. Frege (1980) held that the truths of pure mathematics concern a domain of mind-independent abstract entities. Frege (1984) further held that any adequate account of thought and language must allow that meaningful linguistic expressions are associated, not simply with concrete worldly items, but also with senses (*Sinne* ), and that for various reasons these linguistic senses must exist in a "third realm," distinct both from the realm of subjective mental items and the realm of sensible, concrete things. Frege's vigorous defense of platonism in semantics and the philosophy of mathematics forms the background for the emergence of modern nominalism in the 1920s.

The Warsaw school of logicians centered around Stanisław Leśniewski and Tadeusz Kotarbiński set itself the task of reconstructing modern logic and mathematics along nominalistic lines. Kotarbiński's reism, for example, was a methodological position according to which, wherever possible, statements that apparently concern abstract entities (e.g., "Bonds of brotherhood unite Orsetes and Electra") are to be replaced by statements that concern only concrete entities and parts thereof (e.g., "Orestes is Electra's brother") (Kotarbiński 1955). The principal motivation for the program was to prevent scientific work in these areas from becoming embroiled in ancient metaphysical and epistemological controversies. The nominalistic project was introduced into Anglophone philosophy by W. V. Quine, who first encountered it in conversations with Leśniewski and Alfred Tarski in 1933. Quine's main positive contribution to the program was the seminal 1947 manifesto "Steps Toward a Constructive Nominalism," coauthored with Nelson Goodman. Quine soon abandoned nominalism in favor of a moderate and distinctive form of platonism. It may nonetheless be said that all subsequent discussion of modern nominalism in the Anglophone tradition derives directly from this paper.

## Motivations for Nominalism

In "Steps Toward a Constructive Nominalism" Goodman and Quine defend their rejection of abstract entities by invoking "a philosophical intuition that cannot be justified by appeal to anything more ultimate" (1947: 97). In subsequent years philosophers have sought to provide a more explicit motivation for the view.

### occam's razor

According to a slogan associated with the tradition of medieval nominalism, "entities are not to be multiplied beyond necessity." Some modern nominalists appeal to this principle in motivating their position. These writers typically concede that existing scientific and mathematical theories entail the existence of abstract entities and are therefore nominalistically unacceptable. They maintain, however, that it is possible to produce nominalistically adequate versions of, or surrogates for, these theories, and so to "dispense" with abstract objects. Occam's razor is then invoked to argue that when such parsimonious surrogates are available, it is rational to reject the standard platonistic theories and to embrace the surrogates instead.

Much of the constructive work in the nominalist tradition consists in providing nominalistic surrogates for existing theories. Roughly speaking, a nominalistic surrogate T_{N} for a platonistic theory T_{P} is a theory whose quantifiers range only over concrete objects, but which is nonetheless fit to do much of the same theoretical or explanatory work as the original. For example, standard formalizations of physical theories involve quantifiers that range over both concrete physical entities (particles, fields, points, and regions of space-time, etc.) and mathematical entities (real numbers, vectors, functions, etc.) A nominalistic alternative to (say) classical electrodynamics would be a theory whose quantifiers range only over concrete objects, but whose predictive and explanatory power exactly matched that of the standard platonistic formulations.

Nominalistic surrogates for standard theories have been developed in a number of domains (Field 1980, Hodes 1984, Chihara 1990, Balaguer 1998). However, the significance of these reconstructive programs is open to doubt for several reasons. For example, while the nominalistic surrogates do indeed typically posit fewer entities than the platonistic originals, they are typically inferior to the originals in other respects. In some cases they require a substantial extension of the extensional first-order logic that suffices for platonistically formulated science. In most cases the nominalistic theory is significantly less perspicuous and flexible than its platonistic counterpart.

One may therefore concede that other things being equal, nominalistic theories are to be preferred on grounds of parsimony, while insisting that since other things are not equal, Occam's razor has no clear application. A more profound challenge is directed at the razor itself. Contemporary philosophers who cite ontological parsimony as a basis for theory choice often suppose that the principle derives its authority from its role in the sciences. But as critics have pointed out (Burgess and Rosen 1997), there is scant evidence that scientists accept the principle in its most general form. Scientists may be concerned to minimize the number of physical mechanisms or fundamental laws in the theories they accept. But working scientists and mathematicians have shown no interest in reducing the number of abstract entities posited by the mathematical theories they invoke. To the contrary, in mathematics and mathematical physics there is some concern to maximize the range of mathematical objects and structures (Maddy 1997). If this is correct, then proponents of the Occamist case for nominalism must maintain that the impulse to ontological parsimony to which they appeal is not a principle of scientific methodology, but an independently compelling philosophical principle.

### the access problem

The most widely cited ground for nominalism derives from Paul Benacerraf (1973). Benacerraf notes that since abstract mathematical objects are causally inert and therefore incapable of affecting our senses, even indirectly, there is a question as to how one might come to know that they exist. Benacerraf invokes the *causal theory of knowledge*, originally proposed by Alvin Goldman (1967) for other purposes, according to which, roughly, a person S knows that p only if S stands in some suitable causal relation to the objects with which p is concerned. This principle entails that true claims about abstract objects cannot be known to be true, even if they are true. And while this does not entail that there are no abstract entities, it does entail that platonism is unstable in the following sense: Proponents of a Platonistic theory must concede that they cannot know whether the theory they accept is true. However, as critics were quick to point out the causal theory of knowledge on which Benacerraf relies is objectionable on other grounds (Steiner 1975). In the subsequent debate nominalists rarely invoke this or any other detailed theory of knowledge. Instead, they maintain that the causal inefficacy of the abstract leaves our access to the abstract domain an utter mystery. Since it is clearly desirable to avoid such mysteries, this provides a motivation for pursuing, and perhaps also for accepting, nominalistic alternatives to standard theories.

### the dispensability argument

Hartry H. Field (1980, 1989) provides a number of motivations for nominalism that do not depend on the causal theory of knowledge. Field begins with a question for the platonist: What reason might one have for believing the claims of standard mathematics? If one has reason to believe the axioms, then one might acquire reason to believe the theorems by constructing proofs. So the question becomes: What reason might one have for believing the axioms of standard mathematics? Since the axioms involve substantial existential claims, it is hard to see how they could be known *a priori* (but see Wright 1983, Hale 1988). And since these claims concern causally inert abstract entities, it seems clear that they cannot be verified directly by observation or experiment. Field thus concludes that the only reason one can have for believing the axioms is that they play an indispensable role in one or another well-confirmed scientific theory. Earlier writers (Quine 1960, Putnam 1971) defended platonism in this way. For example, Hilary Putnam (1971) notes that since the laws of physics are standardly formulated in mathematical terms, someone who denies the existence of (say) real numbers is not in a position to formulate, much less to employ, even the most elementary laws of physics. Quine and Putnam thus offer the following indispensability argument for platonism:

(1) One is justified in believing that abstract objects exist if, but only if, theories that entail the existence of such objects are indispensable for scientific purposes.

(2) Standard mathematics entails the existence of abstract objects.

(3) Standard mathematics is indispensable for scientific purposes.

(4) Therefore, one is justified in believing that abstract objects exist.

Field rejects premise (3), thereby turning the argument on its head. He argues that in certain cases it is possible to produce reasonably attractive nominalistic versions of standard platonistic theories: versions in which the only objects posited are material bodies and space-time regions. Field maintains that to the extent that such nominalistic surrogates are available, they establish that abstract objects are dispensable for scientific purposes. The construction of such surrogates thus undercuts the only reason one might have had for believing in abstract objects, and so provides a roundabout motivation for nominalism.

Field concedes that the nominalistic alternatives he constructs are in certain respects inferior to the standard platonistic theories on which they are based. They are typically unwieldy and imperspicuous: Derivations are typically longer and harder to follow. Field concedes that it would be unreasonable for working scientists to use these nominalistic theories for most purposes and hence that platonistic theories are indispensable in practice. His central claim is that they are nonetheless dispensable in principle and that for the purposes of the Quine-Putnam challenge dispensability in principle is what matters.

One distinctive ingredient in Field's view is a demonstration that scientists who accept only the nominalistic physics that Field constructs are nonetheless entitled to use platonistic mathematics in the course of their work. This claim is supported by a formal result. Let T_{P} be a standard platonistic theory, and let T_{N} be a nominalistic surrogate for T_{P} constructed according to Field's method. It may then be shown (with certain important qualifications) that for any nominalistic statement S—that is, any statement whose quantifiers are restricted to concrete entities—S is a theorem of T_{P} if and only if S is a theorem of T_{N}. This conservative extension theorem supports the claim that a theorist who accepts T_{N} may legitimately employ the full mathematical resources of T_{P} for the purpose of deriving nominalistic claims about the concrete world (for a discussion on this, see Shapiro 1983, Burgess and Rosen 1997). Such theorists may then legitimately regard the mathematical apparatus of T_{P} as a useful fiction in which they indulge for various practical purposes. Field's version of nominalism is thus a form of fictionalism about mathematical objects.

Field's work has provoked an intense critical response (Irvine 1990). Field himself notes that his procedures for nominalizing platonistic theories are inapplicable to an important class of theories, including Albert Einstein's general theory of relativity and quantum mechanics, and hence that it remains an open question whether platonistic theories are dispensable even in principle for the purposes of contemporary physics (compare Balaguer 1996). Others wonder why Platonistic theories that are indispensable in practice should not provide one with adequate grounds for believing in the abstract objects they posit. Perhaps the most fundamental philosophical response to Field's approach calls into question premise (1) of the indispensability argument, which is also a crucial premise in Field's positive defense of nominalism. In effect, the premise asserts that abstract objects have the status of theoretical entities, in the sense that one acquires reason for believing in them only when the assumption of their existence is required for some urgent scientific purpose.

Against this, critics maintain that some propositions about abstract objects—for example, the claim that there is a number between 3 and 5, or the claim that Jane Austen wrote six novels—are perfectly ordinary claims. Anyone who has learned basic arithmetic can supply a reason for believing that there is a number between 3 and 5 (Parson 1986), and anyone who knows how to use the library can verify that Austen wrote six novels. It is a presupposition of the debate between Field and proponents of the indispensability argument that these relatively nontheoretical justifications for platonistic claims are inadequate. But this claim may be challenged. If one's ordinary reasons for believing platonistic claims are good enough, then the fact that such claims are dispensable for certain theoretical purposes has no immediate bearing on the debate over nominalism.

## Revolutionary versus Hermeneutic Nominalism

In the nominalist tradition that runs from Goodman and Quine (1947) to Field (1980), it is generally conceded that since standard mathematics entails the existence of abstract objects, the nominalist must supply an alternative to standard mathematics, both pure and applied. This alternative might take the form of a genuinely novel formulation, as with Field's nominalistic version of Newtonian gravitational theory. But it may also take the form of a reinterpretation of existing theories. On this approach the nominalist proceeds by supplying a revisionary account of the meanings of mathematical statements. For example, the nominalist may maintain that while existential arithmetical statements like "There is a number between 3 and 5" in fact affirm the existence of abstract entities, they should be reinterpreted as claims about (say) concrete numeral inscriptions. In either case the nominalist must argue for a revision in accepted science and mathematics. Nominalist programs of this sort have thus been labeled *revolutionary* (Burgess 1983).

Revolutionary nominalism is contrasted with hermeneutic nominalism. Hermeneutic nominalists maintain that it is a mistake to interpret ordinary mathematics as involving claims about abstract objects in the first place. They might maintain, for example, that as they are ordinarily understood, existential claims like "There is a number between 3 and 5" are in fact claims about concrete numeral inscriptions and hence that such claims might be true even if there were no abstract entities. On this sort of account nominalism requires no revision in settled doctrine.

The most straightforward version of hermeneutic nominalism would maintain that abstract singular terms like *3* and *the cosine function* denote particular concrete objects. Claims of this sort are rarely plausible, however, and so proposals in this domain are typically more complex. For example, Geoffrey Hellman (1989) proposes that a statement S in the language of arithmetic is true if and only if a certain modal condition holds: (a) there might have been an infinite sequence of objects satisfying the axioms of arithmetic, and (b), if there had been such a sequence, a certain structural condition derived from S would have been true of it. Hellman then argues that since this sort of modal claim might be true even if there are in fact no abstract objects, the original mathematical claim is nominalistically acceptable, appearances to the contrary notwithstanding.

There are two main objections to hermeneutic proposals of this sort. The first notes that since such claims are ultimately claims in empirical linguistics—they are claims about the meanings of ordinary mathematical statements—they require empirical support and that in the relevant cases no such support has been forthcoming (Burgess 1983). The second notes that even if hermeneutic nominalists' semantic claims were tenable, it is not clear that they would serve their purpose. Unlike their revolutionary counterparts, hermeneutic nominalists do not deny the claims of standard mathematics. But these claims include *existence theorems* : assertions of the form "There exists a number n such that …" Hermeneutic nominalists must therefore allow that these ordinary existence claims are true and hence that by their own lights, numbers and the like exist. On the face of it, however, this claim is incompatible with their nominalism (Alston 1958, Burgess and Rosen 1997; see also Stanley 2001).

## Contemporary Fictionalism

As they are usually understood, the programs of revolutionary and hermeneutic nominalism both require detailed constructive work. Theorists proceed by constructing an autonomous, independently intelligible nominalistic theory T_{N}, which is then used either to replace or to interpret the original (apparently) platonistic theory, T_{P}. The development of a suitable theory T_{N} is typically a nontrivial task, which in many cases requires a profound analysis of the original.

However, some nominalists maintain that detailed constructions of this sort are unnecessary. Easy fictionalism, as the approach is sometimes called, holds that even in the absence of an autonomous nominalistic alternative, nominalists may make free use of standard mathematics and of other platonistic theories without thereby committing themselves to the existence of abstract objects.

Consider for example the claim (S): (S) the mass (in grams) of *A* = 3.6. On its face (S) asserts that the object *A* stands in a certain relation to a number. The claim is literally true only if two conditions are satisfied: on the concrete side, the object *A* must have a certain intrinsic property—a property for which one may have no standard name that does not invoke a relation to numbers; and on the abstract side, the number 3.6 must exist. To maintain the literal truth of (S) is thus to maintain that abstract objects exist. But consider the claim that things are, in all concrete respects, as if (S) were true. The suggestion is that this claim says just what (S) says about the intrinsic configuration of the concrete world, while making no claim whatsoever about the existence of abstract entities.

Easy fictionalists propose that as a matter of convenience one routinely pretends that abstract objects of various sorts exist and that one conveys information about the concrete world by endorsing theories that purport to affirm relations between concrete things and abstract things. Their suggestion is that in "endorsing" these theories, one commits oneself only to the nominalistically acceptable claim that things are, in all concrete respects, as if one's theories are true.

Easy fictionalism comes in a number of varieties. It may be put forward as a hermeneutic proposal, describing the attitude that scientists and mathematicians normally adopt toward their own claims about abstract entities (Yablo 2001). More commonly, it is put forward as a revolutionary proposal. Here, the suggestion is that in light of the arguments in favor of nominalism, it would be rational (or at least, rationally permissible) to adopt a fictionalist attitude toward discourse about abstract objects (Balaguer 1998, Rosen 2001). The main challenge for easy fictionalism is to provide a clear account of the central idiom, "Things are, in all concrete respects, as if S were true," or perhaps, "According to the fiction of mathematical objects, S." The most natural account involves a counterfactual conditional. To say that things are in all concrete respects as if S were true is to say that if there were abstract objects (and the concrete world were just as it is in all intrinsic respects), then S would be true. But counterfactuals of this sort are problematic. It is widely held that the existence of abstract objects could not possibly be a contingent matter (Hale and Wright 1992; compare Field 1993). And if this is right, then by nominalists' own lights, such conditionals involve a necessarily false antecedent. A second challenge for the approach is to provide an account of pure mathematics, where the aim of the discourse is not simply to provide information about the configuration of the concrete world.

** See also ** Realism and Naturalism, Mathematical.

## Bibliography

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Balaguer, Mark. *Platonism and Anti-Platonism in Mathematics*. New York: Oxford University Press, 1998.

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Burgess, John P., and Gideon Rosen. *A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics*. New York: Oxford University Press, 1997.

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Field, Hartry H. *Science without Numbers: A Defence of Nominalism*. Princeton, NJ: Princeton University Press, 1980.

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Frege, Gottlob. "Thoughts." In *Collected Papers on Mathematics, Logic, and Philosophy*, translated by Max Black and edited by Brian McGuiness. New York: Blackwell, 1984. This essay was originally published in German under the title "Der Gedanke" in 1918.

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Stanley, Jason. "Hermeneutic Fictionalism." In *Midwest Studies in Philosophy. No. 25: Figurative Language*, edited by H. Wettstein and P. French, 36–71. New York: Blackwell, 2001.

Steiner, Mark. *Mathematical Knowledge*. Ithaca, NY: Cornell University Press, 1975.

Wright, Crispin. *Frege's Conception of Numbers as Objects*. Aberdeen, Scotland: Aberdeen University Press, 1983.

Yablo, Stephen. "Go Figure: A Path through Fictionalism." In *Midwest Studies in Philosophy. No. 25: Figurative Language*, edited by H. Wettstein and P. French, 77–102. New York: Blackwell, 2001.

*Gideon Rosen (2005)*

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