Editing Nominalism (section)

===Mathematical nominalism<!--'Mathematical nominalism' redirects here-->===
A 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 1940 and 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. Goodman was himself drawing heavily on the work of [[Stanisław Leśniewski]], especially his [[mereology]], which was itself a reaction to the paradoxes associated with Cantorian set theory. Leśniewski denied the existence of the [[empty set]] and held that any [[singleton (mathematics)|singleton]] was identical to the individual inside it. Classes corresponding to what are held to be species or genera are concrete sums of their concrete constituting individuals. For example, the class of philosophers is nothing but the sum of all concrete, individual philosophers.

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 proponents of '''mathematical nominalism'''<!--boldface per WP:R#PLA-->,<ref name=SEP-Nom>Bueno, Otávio, 2013, "[https://plato.stanford.edu/entries/nominalism-mathematics/ Nominalism in the Philosophy of Mathematics]" in the [[Stanford Encyclopedia of Philosophy]].</ref> {''a'', ''b''} is also identical to {''a'', {''b''}{{spaces}}}, {''b'', {''a'', ''b''}{{spaces}}}, 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_(mathematics)|universe]] of a quantified variable cannot contain any virtual sets.

In the [[foundations of mathematics]], nominalism has come to mean doing mathematics without assuming that [[Set (mathematics)|sets]] in the mathematical sense exist. In practice, this means that [[Quantifier (logic)|quantified variables]] may range over [[Universe_(mathematics)|universes]] of [[number]]s, [[point (geometry)|points]], primitive [[ordered pair]]s, and other abstract ontological primitives, but not over sets whose members are such individuals. Only a small fraction of the corpus of modern mathematics can be rederived in a nominalistic fashion.