The Geometrical Method
Spinoza's Ethics Demonstrated in Geometrical Order consists of:
(a) Definitions (of the meanings of certain terms; note, these are 'real' definitions, i.e. not simply made up, but true definitions that capture the essences or natures of things)
(b) Axioms (self-evident truths without need of proof)
(c) Postulates, which are similar to axioms, except that they may fall short of being self-evident
(d) Propositions (and corollaries and lemmas) (not self-evident, but truths deduced from definitions, and/or axioms, and/or other previously demonstrated propositions)
(e) Proofs (or demonstrations) (laying out the deduction from the definitions, axioms and previously demonstrated propositions)
There are also:
(g) Notes ("scholia")
The point of the geometrical mode of presentation is that every conclusion, or Proposition, is a formal deduction, that is, a truth arrived at on the basis of definitions, axioms, postulates, and previously deduced conclusions. So to every conclusion can be added "Q. E. D." (quod erat demonstrandum, "which was to be demonstrated"), as in the geometrical proofs in Euclid's Elements (Stoicheia, written c. 300 B.C.). Hence the entire work consists of reasoning a priori from definitions and axioms, etc., to conclusions about the world.
The point of the Ethics is to show what a blessed peace (acquiescentia) of mind, or man's highest happiness, is, by demonstrating knowledge of the right way of living, or ethics, from the metaphysics and physics of the universe. As one commentator has put it, "It consists in showing that our happiness and well-being lie not in a life enslaved to the passions and to the transitory goods we ordinarily pursue; nor in the related unreflective attachment to the supersitions that pass as religion, but rather in the life of reason." (Nadler) The first two books of the Ethics provide the metaphysics and physics that are the foundation for the ethics.
Here are reconstuctions of some of the proofs, or demonstrations, of the propositions, or conclusions, of Bk. I of the Ethics, "On God".
Here are the Propositions of Bk. I:
(1) "Substance is by nature prior to its [modes]" (p. 159)
(2) "Two substances having different attributes have nothing in common" (p. 159)
(3) "When things [e.g. substances] have nothing in common, one cannot be the cause of the other." (p. 159)
(4) "Two [substances]... are distinguished from one another either by... attributes... or by... [modes]" (p. 159)
(5) "there cannot be two or more substances of the same nature or attribute" (p. 159)
This simply means that if there are two substances, then to distinguish between them they must have different natures or essences, and thus different ways of conceiving of their natures or essences, i.e. different attributes. Otherwise there is only the one substance, with the one nature or essence.
(To use a non-Spinozistic example, there cannot be 'two golds', or 'two waters'. If two substances have the same nature or essence (e.g. atomic no. 79, or H20), then they are the same substance.)
(6) "One substance cannot be produced by another substance" (p. 159)
(7) "Existence belongs to the nature of substance" (p. 160)
(8) "Every substance is necessarily infinite." (p. 160)
(9) "The more reality or being a thing has, the more attributes it has." (p. 161)
(10) "Each attribute of one substance must be conceived through itself." (p. 161)
(11) "God, or substance consisting of infinite attributes, each of which expresses eternal and infinite essence, necessarily exists." (p. 162)
(12) "No attribute of substance can be truly conceived from which it would follow that substance can be divided." (p. 163)
(13) "Absolutely infinite substance is indivisible." (p. 163)
(14) "There can be, or be conceived, no other substance but God." (p. 163)
(Corollary 2: "It follows that the thing extended and the thing thinking are either attributes of God or affections [modes] of the attributes of God." (p. 163)
(15) "Whatever is, is in God, and nothing can be or be conceived without God." (p. 163)
(16) "From the necessity of the divine nature [i.e. infinite nature or essence of infinite substance, or God] there must follow infinite things in infinite ways (that is, everything that can come within the scope of infinite intellect." (p. 165)
(17) "God acts solely from the laws of his own nature, constrained by none." (p. 166)
(18) "God is the immanent, not the transitive, cause of all things." (p. 167)
(19) "God [is eternal], that is, all the attributes of God are eternal." (p. 167)
(20) "God's existence and his essence are one and the same." (p. 168)
(21) "All things that follow from the absolute nature of any attribute of God must have existed always, and as infinite; that is, through the said attribute they are eternal and infinite." (p. 168)
(22) "Whatever follows from some attribute of God, insofar as the attribute is modified by a modification that exists necessarily and as infinite through that same attribute, must also exist both necessarily and as infinite." (p. 169)
(23) "Every mode which exists necessarily and as infinite must have necessarily followed either from the absolute nature of some attribute of God or from some attribute modified by a modification which exists necessarily and as infinite." (p. 169)
If extension is an example of an attribute, or a way of conceiving of the nature or essence of a substance, then this means that there cannot be two substances that share the attribute of extension.
(c) God is infinite substance
1. God, like every substance, exists (because like every substance, God is self-caused).
2. But God is a substance with infinite attributes (i.e. God 'has' infinite attributes).
3. So, God 'has' all of the attributes that it is possible to have.
4. So, there are no attributes left for any other substance to have.
5. Substances can't share attributes.
6. Substances can't exist without attributes (i.e. without natures or essences).
---> God, or infinite substance, is the ONLY substance that exists.
This is the position of Substance Monism (i.e. there is only one substance). Since, according to Spinoza, the single substance that exists is infinite substance, or God, his position is simply the position that only God exists. There is only one substance, but it is an infinite one.
This is Spinoza's claim that there is only one, "Deus, sive Natura" (God, or Nature). (He also says "God, or substance")
Infinite substance, or God, is one. It is indivisible, or has no parts. It is eternal, or has no beginning and no end (and it has no duration).
Infinite substance or God may be thought of all in one, in which case it is called "Natura naturans" (Nature naturing). Or, infinite substance or God may be thought of in terms of all that follows from God, in which case it is called "Natura naturata" (Nature natured).