The problem of interaction

The main argument against platonism is the problem of how we as human beings can interact with and obtain knowledge of, the platonic realm. In some ways one might say that arguments for platonism, by way of persuasiveness, stand or fall with the way in which they deal with this problem.
As I see it, the problem lies in the way in which we intuitively imagine the meaning of interactions. After all, a philosophical argument, given that it is logically correct, will be persuasive exactly if it is intuitively appealing to a majority.
Despite the modern advances in physics, most people probably still imagine interaction between two entities as two completely separate bodies, and some point in which they touch, or with some ray of energy going between them. This picture of interaction is the psychological source of the problems with dualism and platonism alike, because we can't have any direct physical touch-contact with things that are non-material, and moreover, we don't observe any "rays of energy" or anything similar going from somewhere outside the universe into our brains. So how can humans interact with the platonic realm? This question is very deep and difficult, so we can only begin to sketch an answer, which is by no means complete at this stage.

As stated above, the salient point is the ways in which we can model interaction. We have seen that the Newtonian billiard-ball model of interaction won't do, and neither will the more modern wave-mechanistic paradigm. After all, numbers do not hit us the way balls do, and neither do they induce processes in our brains after being picked up, as waves, by our ears or eyes. What other ways could there possibly be in which to model interaction? The answer may come if we consider analogies with more concrete phenomena. One example is:

How does memory interact with consciousness?

We know that memory is 'dislocated' in the brain, that is, there is no specific parts of the brain that carry memories. Still consciousness clearly interacts with at least some of our memories. We currently do not understand how this works, but whatever the mechanism, it seems very likely that it must be an interaction of a type very different from our everyday life intuitions. It is clear that consciousness and memory are partially separate entities, in the sense that we can be conscious without remembering the names of all the capitals of the world, and conversely we have memories that we are not immediately conscious of. Still these two entities must be considered as somehow intertwined, because consciousness is not physically touching a new point when it digs up old memories. The point is that it is not necessarily a question of an interaction that takes place as a transposition in space. We tend to think about interactions as points in space such that before the interaction there were two separate bodies, and during the interaction there is at least one point in space where the two entities meet. With our present understanding of consciousness and memory, this touching-point model seems to be inadequate.
I do not pretend to be able to explain how consciousness interacts with memory, but I claim that this phenomenon points to a new way of understanding interactions between entities, specifically between 'dislocated' entities.
It is naive and misleading to think about the platonic realm of mathematical entities as somehow 'outside' our physical universe. In the same way it would be wrong to think about memory as spatially located outside of consciousness. Still, not all our memories are in our consciousness, and indeed, at least some of our memories exist independently of our wanting them to be there, and independently of whether we are currently conscious of them or not.

This is an important point which I should follow up another time. Hopefully there will be more to come on this subject.