Nominalism is a metaphysical view in philosophy according to which general or abstract terms and predicates exist, while universals or abstract objects, which are sometimes thought to correspond to these terms, do not exist.[1] Thus, there are at least two main versions of nominalism. One version denies the existence of universals—things that can be instantiated or exemplified by many particular things (e.g. strength, humanity). The other version specifically denies the existence of abstract objects—objects that do not exist in space and time.[2]
Most nominalists have held that only physical particulars in space and time are real, and that universals exist only post res, that is, subsequent to particular things.[3] However, some versions of nominalism hold that some particulars are abstract entities (e.g. numbers), while others are concrete entities—entities that do exist in space and time (e.g. tables, chairs).
Nominalism is primarily a position on the problem of universals, which dates back at least to Plato, and is opposed to realism—the view that universals do exist over and above particulars. However, the name "nominalism" emerged out of debates in medieval philosophy with Roscellinus. As John Stuart Mill once wrote, the early versions of nominalism were that "there is nothing general except names", hence the prefix "nomin-". This, however, is a more dated use of the term that is now considered to be a specific version of what is now called "nominalism".
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Plato was perhaps the first[4] writer in Western philosophy to clearly state a non-Nominalist position, which he plainly endorsed:
...We customarily hypothesize a single form in connection with each of the many things to which we apply the same name. ... For example, there are many beds and tables. ... But there are only two forms of such furniture, one of the bed and one of the table. (Republic 596a-b, trans. Grube)
What about someone who believes in beautiful things, but doesn't believe in the beautiful itself…? Don't you think he is living in a dream rather than a wakened state? (Republic 476c)
The Platonic universals corresponding to the names "bed" and "beautiful" were the Form of the Bed and the Form of the Beautiful, or the Bed Itself and the Beautiful Itself. Platonic Forms were the first universals posited as such in philosophy.[4]
Our term "universal" is due to the English translation of Aristotle's technical term katholou which he coined specially for the purpose of discussing the problem of universals.[5] Katholou is a contraction of the phrase kata holou, meaning "on the whole".[6]
Aristotle famously rejected certain aspects of Plato's Theory of Forms, but he clearly rejected Nominalism as well:
...'Man', and indeed every general predicate, signifies not an individual, but some quality, or quantity or relation, or something of that sort. (Sophistical Refutations xxii, 178b37, trans. Pickard-Cambridge)
Nominalism arose in reaction to the problem of universals, specifically accounting for the fact that some things are of the same type. For example, Fluffy and Kitzler are both cats, or, the fact that certain properties are repeatable, such as: the grass, the shirt, and Kermit the Frog are green. One wants to know in virtue of what are Fluffy and Kitzler both cats, and what makes the grass, the shirt, and Kermit green.
The realist answer is that all the green things are green in virtue of the existence of a universal; a single abstract thing, in this case, that is a part of all the green things. With respect to the color of the grass, the shirt and Kermit, one of their parts is identical. In this respect, the three parts are literally one. Greenness is repeatable because there is one thing that manifests itself wherever there are green things.
Nominalism denies the existence of universals. The motivation for this flows from several concerns, the first one being where they might exist. Plato famously held, on one interpretation, that there is a realm of abstract forms or universals apart from the physical world (see theory of the forms). Particular physical objects merely exemplify or instantiate the universal. But this raises the question: Where is this universal realm? One possibility is that it is outside of space and time. A view sympathetic with this possibility holds that, precisely because some form is immanent in several physical objects, it must also transcend each of those physical objects; in this way, the forms are "transcendant" only insofar as they are "immanent" in many physical objects. In other words, immanence implies transcendence; they are not opposed to one another. (Nor, on this view, would there be a separate "world" or "realm" of forms that is distinct from the physical world, thus shirking much of the worry about where to locate a "universal realm".) However, naturalists assert that nothing is outside of space and time. Some Neoplatonists, such as the pagan philosopher Plotinus and the Christian philosopher St. Augustine, imply (anticipating conceptualism) that universals are contained within the mind of God. To complicate things, what is the nature of the instantiation or exemplification relation?
Conceptualists hold a position intermediate between nominalism and realism, saying that universals exist only within the mind and have no external or substantial reality.
Moderate realists hold that there is no realm in which universals exist, but rather universals are located in space and time wherever they are manifest. Now, recall that a universal, like greenness, is supposed to be a single thing. Nominalists consider it unusual that there could be a single thing that exists in multiple places simultaneously. The realist maintains that all the instances of greenness are held together by the exemplification relation, but this relation cannot be explained.
Finally, many philosophers prefer simpler ontologies populated with only the bare minimum of types of entities, or as W. V. Quine said "They have a taste for 'desert landscapes.'" They attempt to express everything that they want to explain without using universals such as "catness" or "chairness."
There are various forms of nominalism ranging from extreme to almost-realist. One extreme is predicate nominalism, which states that Fluffy and Kitzler, for example, are both cats simply because the predicate 'is a cat' applies to both of them. And this is the case for all similarity of attribute among objects. The main criticism of this view is that it does not provide a sufficient solution to the problem of universals. It fails to provide an account of what makes it the case that a group of things warrant having the same predicate applied to them[7]
Resemblance nominalists believe that 'cat' applies to both cats because Fluffy and Kitzler resemble an exemplar cat closely enough to be classed together with it as members of its kind, or that they differ from each other (and other cats) quite less than they differ from other things, and this warrants classing them together.[8] Some resemblance nominalists will concede that the resemblance relation is itself a universal, but is the only universal necessary. This betrays the spirit of nominalism. Others argue that each resemblance relation is a particular, and is a resemblance relation simply in virtue of its resemblance to other resemblance relations. This generates an infinite regress, but many argue that it is not vicious.[9]
Another form of nominalism is trope theory. A trope is a particular instance of a property, like the specific greenness of a shirt. One might argue that there is a primitive, objective resemblance relation that holds among like tropes. Another route is to argue that all apparent tropes are constructed out of more primitive tropes and that the most primitive tropes are the entities of complete physics. Primitive trope resemblance may thus be accounted for in terms of causal indiscernibility. Two tropes are exactly resembling if substituting one for the other would make no difference to the events in which they are taking part. Varying degrees of resemblance at the macro level can be explained by varying degrees of resemblance at the micro level, and micro-level resemblance is explained in terms of something no less robustly physical than causal power. David Armstrong, perhaps the most prominent contemporary realist, argues that such a trope-based variant of nominalism has promise, but holds that it is unable to account for the laws of nature in the way his theory of universals can.
Ian Hacking has also argued that much of what is called social constructionism of science in contemporary times is actually motivated by an unstated nominalist metaphysical view. For this reason, he claims, scientists and constructionists tend to "shout past each other".[10]
The notion that philosophy, especially ontology and the philosophy of mathematics should abstain from set theory owes much to the writings of Nelson Goodman (see especially Goodman 1977), who argued that concrete and abstract entities having no parts, called individuals exist. Collections of individuals likewise exist, but two collections having the same individuals are the same collection.
The principle of extensionality in set theory assures us that any matching pair of curly braces enclosing one or more instances of the same individuals denote the same set. Hence {a,b}, {b,a}, {a,b,a,b} are all the same set. For Goodman and other nominalists, {a,b} is also identical to {a,{b}}, {b,{a,b}}, and any combination of matching curly braces and one or more instances of a and b, as long as a and b are names of individuals and not of collections of individuals. Goodman, Richard Milton Martin, and Willard Quine all advocated reasoning about collectivities by means of a theory of virtual sets (see especially Quine 1969), one making possible all elementary operations on sets except that the universe of a quantified variable cannot contain any virtual sets.
In the foundation of mathematics, nominalism has come to mean doing mathematics without assuming that sets in the mathematical sense exist. In practice, this means that quantified variables may range over universes of numbers, points, primitive ordered pairs, and other abstract ontological primitives, but not over sets whose members are such individuals. To date, only a small fraction of the corpus of modern mathematics can be rederived in a nominalistic fashion.
A critique of nominalist reconstructions in mathematics was undertaken by Burgess (1983) and Burgess and Rosen (1997). Burgess distinguished two types of nominalist reconstructions. Thus, hermeneutic nominalism is the hypothesis that science, properly interpreted, already dispenses with mathematical objects (entities) such as numbers and sets. Meanwhile, revolutionary nominalism is the project of replacing current scientific theories by alternatives dispensing with mathematical objects, see (Burgess, 1983, p. 96). A recent study extends the Burgessian critique to three nominalistic reconstructions: the reconstruction of analysis by Georg Cantor, Richard Dedekind, and Karl Weierstrass that dispensed with infinitesimals; the constructivist re-reconstruction of Weiertrassian analysis by Errett Bishop that dispensed with the law of excluded middle; and the hermeneutic reconstruction, by Carl Boyer, Judith Grabiner, and others, of Cauchy's foundational contribution to analysis that dispensed with Cauchy's infinitesimals.[11]