The next obvious question that arises is how Descartes actually proves that extension and thought exclude each other. Nowhere do we see him explicitly proving this. In fact, steps (4) and (5) are terribly compressed. Both contain two claims: P is the essence of S; P is not connected to Q.

First, how does Descartes know that thought is the essence of mind and extension is the essence of body? For this part of the claim he is resting on the suppressed premise we saw in the last section: P is the essence of S if and only if I can conceive of S attributing only P to it. He sees that he can conceive of mind by attributing only thought to it and of body by attributing only extension.

Now, though, the big question: How does he know that extension and thought are not like squareness and shape? How does he know that they exclude each other? Though he never explicitly states his reasoning, it is not that difficult to figure out. Extension and all of its modes can be captured by mathematical language. It seems intuitive to believe, however, that the modes of thought (experience or consciousness) cannot be expressed mathematically. After all, one would be hard pressed to imagine how to mathematically express the experience of seeing red. Since it is of the nature of extension to be expressible in mathematical terms, and of the nature of thought to lack this possibility, he is able to conclude that neither is a mode of the other. Extension and thought exclude one another.

Given all this, we can now see clearly how Descartes is able to conclude the argument. He knows that mind and body have different essences that are mutually exclusive. That means that when he clearly and distinctly conceives of one, he is not secretly also conceiving of the other. He also knows that God can bring about the separation of anything we can clearly and distinctly perceive. Since we can clearly and distinctly perceive of mind without body and of body without mind, God can bring it about that mind can exist without body and body without mind. In other words, they are really distinct.

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