The geometric method and the one substance

Baruch (Benedict) Spinoza’s masterwork, the Ethics (1677, published after his death), carries the full title Ethica, ordine geometrico demonstrata — “Ethics, demonstrated in geometrical order” (more geometrico). It is built exactly like a geometry textbook: a chain of numbered definitions, axioms, propositions, and proofs, each theorem deduced from what came before.

This is not a gimmick. The form is the philosophy. Spinoza is a rationalist: he holds that reality is through-and-through intelligible — fully transparent to reason. If the world really hangs together by necessity, then a correct philosophy should be able to deduce its structure the way you deduce the properties of a triangle. The geometric order is the rationalist ideal made visible on the page.

Everything turns on three defined terms. First, substance (substantia): “that which is in itself and is conceived through itself” (Ethics I, Def. 3). To be in itself is to be self-subsistent — to depend on nothing else to exist. To be conceived through itself is to be self-explanatory — you don’t have to bring in anything else to understand what it is.

Second, an attribute (attributum): “what the intellect perceives of a substance as constituting its essence” (Ethics I, Def. 4). An attribute is a fundamental way a substance is — a basic kind of reality it exhibits, grasped by the intellect as its very nature.

Third, a mode (modus): “the affections of a substance, or that which is in another through which it is also conceived” (Ethics I, Def. 5). A mode is a modification — it exists in something else and is conceived through something else. It is the exact opposite of a substance: dependent in both being and explanation.

Now the argument for substance monism — the claim that there is only one substance. It runs through several early propositions of Ethics I. Step one: a substance must be self-caused (causa sui) and cannot be produced by anything external. Why? Because if something else produced it, it would be conceived through that other thing — and then, by definition, it wouldn’t be a substance at all. So a substance owes its existence to nothing but itself; existence belongs to its very essence (Ethics I, Props. 6–7).

Step two: two distinct substances cannot share an attribute (Ethics I, Prop. 5). If two supposed substances had the very same attribute, there would be nothing to tell them apart — nothing to make them two rather than one. Since each substance is grasped entirely through its attribute(s), sharing an attribute would collapse them into the same thing.

Step three: a substance is infinite. A substance can’t be limited by another substance (there’s nothing external it depends on), and to be limited by something of the same nature would just be the case Prop. 5 already ruled out. An unlimited substance is infinite — indeed possessing infinitely many attributes, each expressing its essence (this is why Spinoza defines God as “a substance consisting of infinite attributes,” Ethics I, Def. 6).

Put the steps together. There must exist an infinite substance possessing all attributes. But if that being already has every attribute, then no second substance is possible: any candidate would have to bring an attribute of its own, yet all attributes are already taken — and by Prop. 5 it cannot share one. Therefore there can be exactly one substance (Ethics I, Prop. 14: “Except God, no substance can be or be conceived”).