Full Idea
We must distinguish between 'realism in ontology' - that mathematical objects exist - and 'realism in truth-value', which is suggested by the model-theoretic framework - that each well-formed meaningful sentence is non-vacuously either true or false.
Gist of Idea
We distinguish realism 'in ontology' (for objects), and 'in truth-value' (for being either true or false)
Source
Stewart Shapiro (Philosophy of Mathematics [1997], Intro)
Book Reference
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.4
A Reaction
My inclination is fairly strongly towards realism of the second kind, but not of the first. A view about the notion of a 'truth-maker' might therefore be required. What do the truths refer to? Answer: not objects, but abstractions from objects.