I always want to say "I think there are numbers - but I understand existence in a thin logical sense". But I feel kindof dishonest saying this. It's too much like the sleazy "Of course P - but I don't mean that in any deep philosophical way" which happens when Wittgensteinians get lazy.
So here are some actual concrete ways in which I differ from other platonists (i.e. other people who believe there are mathematical objects).
1. I don't think we need to posit numbers to explain how there can be unknowable mathematical facts.
2. I think fictionalism/if-then-sim is is perfectly coherent. We could have had a mathematical practice which was completely based around mathematical properties, and studying their relaitons to one another e.g.: `Insofar as anything heirarchy-of-sets-izes, it's mathematically necessary that it satisfies the contiuum hypothesis'.
And here's an attempt to say what having only "a thin logical notion of existence" means:
When we ask what objects exist, this is equivalent to asking what sentences with a given logical form (Ex) Fx are true. So far, this is just Quinean orthodoxy.
But now the question is: what makes a given sentence (say, of English) have a certain logical form?
Now, I think having existential form is just a matter of what inferences can be made with that sentence, and what other -contrasting- sentences are in the language. We cook up various logical categories in order to best represent, and exploit, patterns in which inferences are truth preserving. Furthermore, there's noting special about objects, and object expressions. Each component of a sentence (be it concept-word, object-word, connective or opporator) makes a systematic contribution to the truth conditions of the sentences it figures in (i.e. the class of possible situations where the sentence is true).
On this view, choices about the logical form of a sentence wind up not being very deep - the question is just what's the most elegant way to capture certain inference relations.
In contrast, (I propose) having a "thick" notion of objecthood and existence, means thinking that there IS something more than elegant summary of inference relations at stake when we decide how to cut sentences up into concepts and objects. For example, you might think
1. It's easy to learn statements which don't imply that any objects exist (all bachelors are unmarried), whereas learning statements that do imply the existence of at least one object (there are some bachelors) is harder.
2. The *world* has a logical structure too! - so the most elegant way of cutting up your sentences to capture inference relations might still be wrong, because it fails to respect the logical structure of the world.
[Oh yes, they are kindof seductive. More about why they are wrong later.]