Nominalism Totally Explained

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The American Heritage Dictionary, Fourth Edition, defines nominalism as "the doctrine holding that abstract concepts, general terms, or universals have no independent existence but exist only as names." Nominalism has also been defined as a philosophical position that various objects labeled by the same term have nothing in common but their name. Nominalism is the view that only actual physical particulars are real, and that universals exist only post res, that is, subsequent to particular things.
The term "nominalism" and its contrary, realism, emerged out of debates in medieval philosophy. Medieval realism, which is quite distinct from realism in the modern sense, holds that when humans employ descriptive terms such as "green" or "soft," they're unwittingly referring to Forms, abstract objects whose existence is wholly independent of the physical world. Medieval realism is ultimately grounded in the metaphysical doctrines of Plato. By contrast, nominalism holds that verbal abstractions employed by humans are only manners of speaking, having no existence beyond human thought and discourse.

The problem of universals

   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's a part of all the green things. With respect to the colour 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's 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 that there's 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's outside of space and time. However, naturalists assert that nothing is outside of space and time. Some Neoplatonists, such as the pagan philosopher Plotinus and the Christian philosopher 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's no realm in which universals exist, but rather universals are located in space and time wherever they're 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 can't be explained.
   Philosophers who delve deeply into the workings of the human brain, such as Daniel Dennett, reject the idea that there's some "greenness" in the real world, only circumstances that cause the brain to react with the judgment "green."
   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."

Varieties of nominalism

There are various forms of nominalism ranging from extreme to almost-realist. One extreme is "predicate" nominalism. Fluffy and Kitzler are both cats simply because the predicate 'cat' applies to both of them. However, the realist will object as to what the predicate applies. 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. 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 agree that it isn't vicious.
Another form of resemblance nominalism attempts to build upon a theory of tropes. A trope is a particular instance of a property, like the specific greenness of a shirt. One might argue that there's 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're 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 Malet Armstrong, perhaps the most prominent contemporary realist, argues that such a trope-based variant of nominalism has promise, but holds that it's 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."
Strong proponents of this school of thought include John Locke and George Berkeley.

Analytic philosophy and mathematics

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, 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 can't 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. On mathematical nominalism, see Burgess and Rosen (1997).

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