One argument against the position that the mind can act causally on the body, and the body also act causally on the mind (two-way interaction), is simply the point that the mind and the body are wholly different kinds of ‘stuff’. That one is immaterial, whereas the other is material, and it seems to be logically impossible for something immaterial to have any interaction with that which is material, and vice versa. This was perhaps the main problem to which many or most early modern philosophies were proposed in order to answer (Leibniz, Spinoza, Malebranche, etc.). It is agreed by every philosopher, or just about every philosopher, even back to the pre-socratics like Diogenes or even the atomists like Democritus, that two entirely different ‘kinds’ of stuff cannot causally interact.
The modern materialist/physicalist will often offer arguments against mind-body two-way interaction which reformulates this same objection. For example, materialists will claim that if the mind ‘isn’t anywhere‘ and is entirely immaterial, then it just cannot have causal interaction with matter, because it isn’t logically possible for something immaterial to have causal interactions with the material realm. Here, there are two steps to take in order to demonstrate that the argument is immodest. First, just because something is immaterial, does not mean it is abstract. I can accept that it is part of the very definition of abstract objects, like platonic forms, that they are causally effete (having no causal relations is part of the definition of ‘abstract’ in opposition to ‘concrete’). It may be the case that mathematical propositions are immaterial and abstract, but it does not follow from something’s being immaterial that it is abstract. One need only maintain that the mind/soul is immaterial and concrete, while material things are also concrete. It can be in virtue of their both being concrete that they can have causal interaction. This position is modest insofar as it offers no analysis of what it means for something to be material, and allows for any number of positions (from Idealism, to Leibnizian phenomenalism, to ‘metaphysical’ materialism/physicalism) to be adopted.
I am attracted to the view that matter supervenes on concrete immaterial substances, kind of like Leibniz’s phenomenalism, but this ‘metaphysical concretism’ doesn’t require any commitment that strong. It only just requires that it is logically possible that an immaterial thing can be concrete rather than abstract, and that is sufficient to allow it to have causal relations with other concrete things, including ‘material’ things.