In the correspondence with Princess Elisabeth, Descartes argues that human knowledge is based on a number of primitive notions. Besides the most general such notions (being, duration, number), which apply to everything, there are certainly more specific notions that apply to particular fields of knowledge, and out of which the other notions are developed. From the notion of extension we get the further notions of shape and motion, and are therefore able to do physics. From the notion of thought we get further notions like understanding and will. And we have a third in addition to these: the notion of the union of soul and body, i.e., thought and extension, from which we have notions like the the power of the soul to move the body or the power of the body to affect the soul's sensations and passions. All of these are primitive; you can't, for instance, reduce the third to the other two. He goes on to say that we can have clear and distinct ideas of the first two, but a confused and indistinct idea of the third.
In a famously obscure passage, Descartes argues that we see this third notion at work in the Aristotelian account of weight, gravitas, because we confuse the union of soul and body with the union of body and body. In Aristotelian gravitas, a physical object is taken to have a quality directing it to its natural place at the center of the earth. The way he puts it in the Sixth Replies, when he held something like the Aristotelian theory, what he held was that gravitas was distinct from the physical object itself (he analogizes it to clothing) and carried it toward the center of the earth as if it had some awareness (cognitio) of that center. In reality, this is wrong: on Descartes's account, there is nothing to gravitas that is different from the moving body itself, and you can't have anything like this directedness without cognition, and cognition belongs to thought, not extension. As Princess Elisabeth will later respond, it's odd to explain this essential primitive notion by a theory that Descartes regards as false. But Descartes's point is that, while a mistake, it is a very easy mistake to make because it seems to make sense -- and the only reason it can be so easy to make and seem so consistently to make sense is if we actually do have some kind of basic idea in which something like it happens. This is the primitive notion of union between soul and bod, in which the mind designates a goal and the body moves in accordance with it.
An implication of this that is often not remarked enough is that Descartes's account of the union of mind and body is teleological. This is why it is relevant to Princess Elisabeth's worry. She had pointed out that in Cartesian physics, all motion of bodies is explained by initiating push, determination of the way something moves, and shape and texture; none of these explain how mind can move the body in the particular way it does. (Her worry is not, as it is often put, how the mind can move the body at all, the bare interaction, but how it can move the body in cases where the body's motion is sufficiently sophisticated that the way it moves had to have been determined by the mind.) Descartes's response is that it's a different kind of motion, one that is not reducible to extension; it is a non-mechanical motion like people under the influence of Aristotle attributed to gravitas. It's a kind of motion we have to admit, because the way our bodies move is clearly goal-directed, but goal-direction is a matter of thought, not extension; so there must here be some kind of union between thought and extension, despite the fact that thought and extension are different things.
Naturally, this still leaves open the question of how this works. Descartes's account in the Sixth Replies suggests that goal-directedness is something that can only be attributed directly to a substance, and in particular the substance of the mind. Physics, based wholly on the notion of extension, has no room for final causes. But in the motion of the body, the body's motion must somehow fall within the ambit of the mind's teleology, so that any explanation on the basis of this physics will be inadequate for explaining the motion of the body. There is never any explanation for how this works, though; the closest he comes is in the Passions, when he talks about its being a special motion instituted by nature. This is puzzling, and as Princess Elisabeth goes on to point out, it seems cleaner either to reject all teleology when it comes to bodies or else assume that bodies can have the teleology that Descartes insists only belongs to minds. It's worth pointing out, however, that it would explain why Descartes is so confident that rational behavior is a sign of other minds -- the goal-directedess of rational behavior is something that he would regard as literally impossible for a mechanical system.
