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Quine proposed a distinction between ontology - the doctrine of what there is - and ideology - the complex terms and predicates expressible in one's theoretical language (though in his 1971 paper he expanded the category to include concepts that one could discern even if one does not have the verbal resources to express). I am curious how this distinction manifests itself in the context of phil of math. Quine used an example of first-order theory of real numbers - if the theory has the predicate ``x is a natural number,'' then that predicate is a part of the theory's ideology, otherwise it is not, and this does not change the theory's ontology. However, if we endorse the set-theoretic reduction of mathematical objects, does it follow that our mathematical theories are only ontologically committed to sets, while all mathematical concepts - numbers, groups, functions, spaces - are part of the ideology but not ontology of our mathematical theory?

  • Technically, Quine accepts as ontological those and only those objects that can be values of quantified variables of a first order theory after the paraphrase. Since functions, structures, relations, etc., paraphrase into sets they can be said to be "derived" ontology (so can desks and chairs, they are paraphrasable into atoms). The rest of machinery, including predicates, connectives and quantifiers, are mere ideology. See also his Ontology and Ideology Revisited (1983) – Conifold Feb 6 '18 at 4:02
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You must take into consideration that much of Quine's "technical" and philosophical work on logic and set theory can be read as a comment and refinement of W&R's Principia.

For an overview, you can see at least the introductory chapters of Set Theory and its Logic, revised ed.1969.

See page 9:

The schematic predicate letters 'F','G',... attach to variables to make dummy clauses 'Fxy', 'Gx', 'Gy', etc., as expository aids when we want to talk about the outward form of a complex sentence without specifying the component clauses.

So the letters 'F', 'G', ... are never to be thought of as variables, taking say attributes or classes as values. Abstract objects, these or others, may of course still belong to our universe of discourse, over which our genuine variables 'x', 'y',... of quantification are allowed to range. But the letters 'F', 'G',... are withheld from quantifiers, indeed withheld from sentences altogether, and used only as dummies in depicting the forms of unspecified sentences.

And see page 35:

We might therefore do well hereafter to speak of the abstraction notation '{x: Fx}' not as virtual, for us, but as noncommittal: its use merely carries no general presumption of existence of the class (nor, if it exists, of its sethood).

Whether to say in particular that there is such a class as {x: Fx} will depend on what sentence we interpret 'Fx' to represent and what axioms of existence we may eventually decide to adopt.

Thus, about e.g. natural numbers, because in set theory every object is a set, and because we can prove, from the axioms of set theory, that there is a set N that behaves like the set of natural numbers, we can conclude that in the framework of set theory, numbers are "real".

See page 81:

We have been admitting the numbers into the range of values of our variables of quantification, but we have not yet considered what sorts of things numbers are to be. We did see how to define 'N' given '0' and 'S'; the construing in turn of '0' and 'S' is what is needed, now, to fix the idea of number.

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