Omitting his definitions and axioms for the moment, we can look at his proofs. Thus we have the first:
1: A substance is prior in nature to its states. This is evident from D3 and D5.
The two definitions are of “substance” and “mode,” which latter he equates with “state of a substance.” However, neither definition explains “prior in nature,” nor is this found in any of the other definitions and axioms.
Thus his argument does not follow. But we can grant that the claim is fairly reasonable in any case, and would follow according to many reasonable definitions of “prior in nature,” and according to reasonable axioms.
He proceeds to his second proof:
2: Two substances having different attributes have nothing in common with one another. This is also evident from D3. For each ·substance· must be in itself and be conceived through itself, which is to say that the concept of the one doesn’t involve the concept of the other.
D3 and D4 (which must be used here although he does not cite it explicitly in the proof) say:
D3: By ‘substance’ I understand: what is in itself and is conceived through itself, i.e. that whose concept doesn’t have to be formed out of the concept of something else. D4: By ‘attribute’ I understand: what the intellect perceives of a substance as constituting its essence.
Thus when he speaks of “substances having different attributes,” he means ones which are intellectually perceived as being different in their essence.
Once again, however, “have nothing in common” is not found in his definitions. However, it occurs once in his axioms, namely in A5:
A5: If two things have nothing in common, they can’t be understood through one another—that is, the concept of one doesn’t involve the concept of the other.
The axiom is pretty reasonable, at least taken in a certain way. If there is no idea common to the ideas of two things, the idea of one won’t be included in the idea of the other. But Spinoza is attempting to draw the conclusion that “if two substances have different attributes, i.e. are different in essence, then they have nothing in common.” But this does not seem to follow from a reasonable understanding of D3 and D4, nor from the definitions together with the axioms. “Dog” and “cat” might be substances, and the idea of dog does not include that of cat, nor cat the idea of dog, but they have “animal” in common. So his conclusion is not evident from the definition, nor does it follow logically from his definitions and axioms, nor does it seem to be true.
And this is only the second supposed proof out of 36 in part 1 of his book.
I would suggest that there are at least two problems with his whole project. First, Spinoza knows where he wants to get, and it is not somewhere good. Among other things, he is aiming for proposition 14:
14: God is the only substance that can exist or be conceived.
This is closely related to proposition 2, since if it is true that two different things can have nothing in common, then it is impossible for more than one thing to exist, since otherwise existence would be something in common to various things.
Proposition 14 is absolutely false taken in any reasonable way. Consequently, since Spinoza is absolutely determined to arrive at a false proposition, he will necessarily employ falsehoods or logical mistakes along the way.
There is a second problem with his project. Geometry speaks about a very limited portion of reality. For this reason it is possible to come to most of its conclusions using a limited variety of definitions and axioms. But ethics and metaphysics, the latter of which is the actual topic of his first book, are much wider in scope. Consequently, if you want to say much that is relevant about them, it is impossible in principle to proceed from a small number of axioms and definitions. A small number of axioms and definitions will necessarily include only a small number of terms, and speaking about ethics and metaphysics requires a large number of terms. For example, suppose I wanted to prove everything on this blog using the method of definitions and axioms. Since I have probably used thousands of terms, hundreds or thousands of definitions and axioms would be required. There would simply be no other way to get the desired conclusions. In a similar way, we saw even in the first few proofs that Spinoza has a similar problem; he wants to speak about a very broad subject, but he wants to start with just a few definitions and axioms.
And if you do employ hundreds of axioms, of course, there is very little chance that anyone is going to grant all of them. They will at least argue that some of them might be mistaken, and thus your proofs will lose the complete certainty that you were looking for from the geometrical method.