If matter (wholly material beings) could think, then matter would not be matter as currently understood.
Can abstracta think? Sets count as abstracta. Can a set think? Could the set of primes contemplate itself and think the thought, I am a set, and each of my members is a prime number? Given what we know sets to be from set theory, sets cannot think. It is the same with matter. Given what we know or believe matter to be from current physics, matter cannot think. To think is to think about something, and it is this intrinsic aboutness or original intentionality that proves embarrassing for materialism. I have expatiated on this over many, many posts and I can't repeat myself here. (Here is a characteristic post.)
But couldn't matter have occult powers, powers presently hidden from our best physics, including the power to think? Well, could sets have occult powers that a more penetrating set theory would lay bare? Should we pin our hopes on future set theory? Obviously not. Why not? Because it makes no sense to think of sets as subjects of intentional states. We know a priori that the set of primes cannot lust after the set of evens. It is impossible in a very strong sense: it is broadly logically impossible.
Of course, there is a big difference between sets and brains. We know enough about sets to know a priori that sets cannot think. But perhaps we don't yet know enough about the human brain. So I don't dogmatically claim that matter could not have occult or hidden powers. Maybe the meat between my ears does have the power to think. But then that meat is not matter in any sense we currently understand. And that is my point. You can posit occult powers if you like, and pin your hopes on a future science that will lay them bare; but then you are going well beyond the empirical evidence and engaging in high-flying speculations that ought to seem unseemly to hard-headed empiricistic and scientistic types.
Such types are known to complain about spook stuff and ghosts-in-machines. But to impute occult powers, powers beyond our ken, to brain matter does not seem to be much of an improvement. For that is a sort of dualism too. There are the physical properties and powers we know about, and the physical properties and powers we know nothing about but posit to avoid the absurdities of identity materialism and eliminativism. So instead of an ontological property dualism or an ontological substance dualism we have an epistemological property dualism, a dualism as between properties and powers we know about and properties and powers we have no idea about.
There is, second, the ontological dualism as between thinking and feeling matter and ordinary hunks of matter that do not think or feel. Even the materialist must admit that there is a huge difference between Einstein and a piece of chalk. How explain that some parcels of matter think and some do not?
It is worth noting that the reverent gushing of the neuro-scientistic types over the incredible complexity (pound the lectern!) of the brain does absolutely nothing to reduce the unintelligibility of the notion that it is brains or parts of brains that are the subjects of intentional and qualitative mental states. For it is unintelligible how ramping up complexity can trigger a metabasis eis allo genos, a shift into another genus. Are you telling me that meat that means is just meat that is more complex than ordinary meat? You might as well say that the leap from unmeaning meat to meaning meat is a miracle. Will you say that consciousness emerges from certain parcels of sufficiently complex matter? But then it is not matter any more, is it? It is an emergent from matter. Emergentism is a form of ontological dualism. What's more, the word 'emergence' merely papers over the difficulty, labeling the problem without solving it. Do you materialists believe in miracle meat or mystery meat? Do you believe in magic?
There is, third, a dualism within the brain as between those parts of it that are presumably thinking and feeling and those other parts that perform more mundane functions. Why are some brain states mental and others not?
The materialist operates with a conception of matter tied to current physics. On that conception of matter, it is simply unintelligible to to say that brains feel or think. I tend to hold that this unintelligibility is a very good reason to hold that it is not my brain or any part thereof that thinks when I think, and that it is not my brain or any part thereof that feels when I feel. (I am using 'think' in the broad Cartesian sense to cover all instances of intentionality, and 'feel' to cover all non-intentional conscious states and events.)
"But from the fact that such-and-such is unintelligible to us now it does not follow that it is not the case." True. Two possibilities. It might be the case that p even though we will never understand how it is possible that p, and it might be the case that p, even though we cannot understand at present how it is possible that p. The first is a mysterian position, the second is not mysterian but a pin-hopes-on-future-science position.
My thesis is that it is reasonable to hold that when I think and feel it is not my brain or any part of it that thinks or feels. But who knows? Maybe future science will prove me wrong. It is just that I wouldn't lay any money on being wrong.