The problem of intentionality vis-à-vis physicalism is not an empirical problem; it is a categorical problem. Formal operations, for instance, are determinate in a way that physical operations cannot be. Intellection is, for instance, universally abstract in a way that physical “signs” cannot be. The contents of sensory experience are analytically non-identical with the physical correlates we infer as their causal substrate. And as for intentionality…
Intentionality and physical order are simply, categorically mutually irreducible. This is hardly a “pet claim” of Thomists. Read some J. Levine, or J. Searle, C. S. Pierce, or F. Brentano, or J. Kim, or W. Vallicella, or S. Kripke, or K. Gödel, or D. Melser. Indeed, read some D. Dennett: he is so committed to physicalism, and yet aware of the intentionality problem, that he denies the latter on behalf of the former.
The issue is simply not one that can be overcome by “more brain studies.” Intentionality, and its related immateriality, is, like purpose and action (cf. R. Taylor’s Action and Purpose), simply not reducible to behavioral categories. Intentionality, purpose, action, formal order––these are simply “their own things” and not to be trifled with. Certainly, it is true to say that intellection occurs “naturally” insofar as it is metaphysically contiguous with the operations of its natural agents; but this must be qualified by the fact that nature, thus, operates with both material and immaterial powers.
In any case, consider the following scenarios (which I borrow from R. Taylor):
a. A man attending a debate raises his hand in order to get the attention of debater Q, and thereby both attracts the attention of debater Q and scares off a fly.
b. A man attending a debate raises his hand in order to get the attention of the moderator, and thereby both attracts the attention of the moderator and scares off a fly.
c. A man attending a debate raises his hand in order to scare off a fly, and thereby both attracts the attention of debater Q and scares off the fly.
Behaviorally, and I should say neurologically, these events are indiscernible. Only on the supposition of a distinct purpose (“in order to”) can they be differentiated. Likewise with intentionality. Physically indiscernible phenomena can have different intentionality, different meaning. Physical causation is, to palm off of Walker Percy and Pierce, metaphysical dyadic, whereas language–-qua intentionality in action––is conceptually triadic, and, indeed, intersubjectively tetradic. Physical things only stand in formal, theoretical bonds with each others as we evoke those bonds by the intentional, immaterial power of referential language. An atom simply does not and cannot “refer to” something else; but language can and, incessantly, does “conscript” an atoms, and hordes or atoms, for such bonds.