Some of the more pressing arguments against materialism involve their inability to account for abstract objects such as meaning or reference. A few definitions before I continue:
Abstract object: an object that exists outside of time or space and that is causally inert (that is, it doesn't effect matter in any way except possibly through the action of a mind). Common examples are sets, numbers, propositions, facts, and meanings.
Materialism: the position that all facts, or all truths are at base truths about matter. All properties are reducible to physical properties. All things are material things. This position specifically rules out not only God and immortal souls, but also abstract objects, moral goodness, and even minds and thoughts, strictly speaking.
Naturalism: the position that all truths are natural truths. Naturalism is weaker than materialisms; it rules out supernatural entities such as God and immortal souls, but it doesn't necessarily rule out natural things like minds, moral goodness, or abstract objects.
There don't seem to be hardly any naturalists today that are not materialists (this is no doubt due in large part to the astonishing successes of scientific reductionism); however, some of the strongest arguments against materialism involve their inability to account for abstract objects. For example, what physical property could account for the fact that when I think about my car, I am thinking about my car? What physical property of my brain, or the car, or the space between my brain and the car, or anything else could constitute the fact that what I am thinking about is the car?
It can't be just a structureless token in my brain; if the token is structureless then an identical token could constitute a reference to something else, so there is nothing physical about the token that makes it about the car. It can't be a similarity or homomorphism between structures in my brain and the car, because those structures can be decomposed into structureless tokens, which, as I previously argued, can't represent anything, so they can't uniquely represent their corresponding parts of the structure of the car. It can't be a causal chain from the car to my brain, because the causal chain is gone by the time I'm thinking about the car, and the physical property that make my brain refer to the car has to be a property that exists when I'm doing the referring.
This is a serious problem for materialism, one I've never seen a good answer to, and there are other serious problems that come up if you don't acknowledge the existence of abstract objects. However, there is a response if the materialist is willing to retreat to naturalism, and that response is to acknowledge that abstract objects exist. One possible formulation would be the position that all contingent facts are facts about matter, but that there are non-contingent facts that are about abstract objects.
So, the question: I don't know of any philosophers who take this position or what the position would be called. Can anyone provide that information?