I've confused you. My apologies. What I meant with this sentence:

"we can model whatever our brains do with kinematic equations"

Was that we can model whatever our brains do _while catching a ball etc_ by means of kinematic equations. I did not mean that we can model everything our brains do, i.e. the function of the brain, in general. If we could model an entire brain just by kinematic equations we wouldn't need any AI research, and I wouldn't be arguing that we don't know what our brains do when they solve problems that we solve using kinematic equations. Our disagreement is about the solutions our brain finds to that kind of problem.

>> Not only that, but our brains can do this in two completely different ways, one of which is conscious and deliberate (what we call "doing math") and the other of which is instinctive and subconscious (developing sensory-motor skills).

That's my problem with all this - the "subconscious" part. I don't really understand what it means. When I catch a ball, I do it entirely consciously, and I know exactly what I'm doing: I'm extending my hand to catch the ball. I may not be able to articulate every little muscle movement, or describe precisely the position of my arms, my hand, my fingers, the ball, etc, but I do know with great accuracy where those objects are in space, and where they are in relation with each other. I cannot introspect into the intellectual mechanisms by which I know those things, but I do know them, so they're not "subconscious".

The difference you point out, between doing maths with pen-and-paper (or computers) and performing a task without having to do maths-with-pen-and-paper, is, I think, the difference between having a formal language that is powerful enough to describe all the objects and functions I describe above (hand position, muscle movement etc), on the one hand, and not having such a language on the other hand. Somehow humans are able to come up with formal languages with the power to describe some of the things we do, like catching balls etc, and many other things besides. As a side note, we do not have a formal language -we do not have the mathematics- to describe our ability to come up with formal languages, yet. That was be one of the original goals of AI research, although it has now fallen by the wayside, in the process of chasing benchmark performance.

I digress. When I speak of "formal languages", I mean more broadly formal systems, like mathematics (of which logic is one branch, btw). When I speak of a "model" in my earlier comment, I mean a formalism that describes various kinds of human capability, like our catching-balls example. Kinematic equations, that's one such model. But a model is not the thing it, well, models. Is my claim.

I hope this is clear and apologies if it's not. Most of our discussion is not on things of my expertise so I'm trying to find the best way to say them. Also, this is a much less technical discussion and so much less precise, than I'm used to. I hope I'm not wasting your time with needless philosophising.

On the other hand, I think this kind of conversation would be made much easier if we didn't assume human brains. Our ability to navigate, and interact with, our environment, is shared to a greater or lesser extent with many animals that aren't humans and don't have human brains, so whatever we can do with our brains thanks to that shared ability, must also share an underlying system- because we all evolved from the same, very distant, animal ancestors, ultimately, and we must have inherited the same basic firmware as it were.