I think I generally agree with the gist of Dorr's paper. He seems to say:
1. We ordinarily talk about the existence of abstract objects, or talk in such a way as to imply their existence.
2. In a more formal manner of speaking, we find ourselves concluding that only physical objects exist.
Dorr seems to conclude that in a funny sense of 'exist', abstract objects exist. But properly understood, we see that abstract objects do not exist.
But it seems to me that abstract objects exist, and that it is only in a funny sense of 'exist' that abstract objects can be said to not exist. A restricted sense of 'exist' seems to me to be a funny sense of 'exist'.
I think that I can spell out my uneasiness in another manner. Consider his notion of 'superficial' and 'fundamental' sentences. It seems that I can superficially say:
3. This brick is red and rectangular.
But it seems that fundamentally we may have deep reasons to doubt that colors and shapes exist in the world in the same way that physical stuff like bricks exist in the world. It might be a fundamental truth that color and shape is a state of a perceiver. There may be a very real sense in which the ultimate particles or processes of theoretical physics are colorless and without a definite shape.
So far so good, it seems to me. But consider: if I am making the claim that fundamentally everything is physical, what can I mean by 'physical' here? If fundamentally there are no abstract entities, or even any middle sized entities which I observe in every day life, what is there?
It seems likely to me that the honest-to-god 'real' and 'noumenal' entities of the world are the theoretically posited unobservable entities of science. I'm going to assume that
4. Only the theoretically posited entities of science exist
has to be taken in a fundamental sense. It seems that this is false in some superficial sense, since someone could show me their hands and say: "These exist don't they? And they are not theoretically posited!"
But I think we have good reason to doubt the accuracy of the Manifest Image of the world which is painted by common-sense alone, and accept the accuracy of the Scientific Image of the world which is painted by theoretical sciences. But then: even if we accept a physicalism which holds that only the physical entities posited by theoretical science exist, don't we still have room to posit abstract entities in the same method?
Given how far modern science, even thoroughly materialistic in nature, has drifted from the simple notion of physical atoms bouncing around in the void, why cannot we at least in principle make room for abstract objects in our science next to 'pure' or 'objectless processes' which we may posit, or the fundamental particles which themselves seem rather abstract and hardly concrete.
It seems perhaps we should have a sort of narrow and broad sense of physicalism. Abstract entities may not be narrowly physical like particles, but they still may be construed as being broadly physical, like lightening is physical, or a C#ing. I hope we can resist a reductive or eliminative materialism, and an urge to reduce or eliminate abstract entities.
I guess maybe I'm hinting at this: we may posit entities superficially in a rather superficial picture of the world. Maybe this is how we come to posit abstract entities. Instead of dropping this superficial picture altogether, it seems we should look for its fundamental counterparts in a fundamental picture of the world. I don't see why even superficially posited abstract entities couldn't find counterparts in our more fundamental picture of the world.
Lastly, it still seems that even if abstract entities only have a 'nominal' or superficial existence, they still have an existence. And if we do end up pushing down fundamental existence to the fundamental particles of science, then the existence of tables and chairs seems perfectly on par with the existence of abstract entities!