As with Hobbes and Descartes, Spinoza is impressed with mathematics as a kind of knowledge - because of the certainty of its proofs and demonstrations. More than any other thinker we have seen, however, Spinoza carries to the furthest extreme the peculiarly modern effort to establish a complete philosophy in which mathematics is not simply a part (as with Plato, for example): rather, this philosophy is one entirely subsumed under the mathematical model. (In this sense, Spinoza "out-Descartes Descartes.")

This means that Spinoza's assumptions at the outset preclude what we have seen to be a considerable problem in the modern period - especially with Hobbes and Descartes. That is, especially with Descartes, who recognizes a considerable split between the world of sense-knowledge and the mathematical dimensions of mind, - the problem we are left facing is one of explaining or justifying the relation between pure mathematics and the sensory world. (In Descartes, of course, this is a particular expression of the larger problem of explaining a relation between mind and body.) Spinoza, by contrast, simply assumes that reality is co-extensive with the world of mathematical entities and relations. That is, there simply is no other reality than reality as mathematically described: anything which presents itself as "reality" - but which is not mathematically "capturable" - is simply illusion.

This position simply draws the consequence, notice, of Descartes' original project of establishing certain knowledge - a project which begins by rejecting sense-knowledge and ultimately centers on the self as a mathematically-thinking substance. Again, in Descartes, it is an unresolved problem as to how such a mathematical self can "return" to the world of sense-knowledge as a world different from the mathematical self. Spinoza avoids this problem at the outset by simply assuming that there is no world other than the mathematical world. So Descartes' problem does not arise in the first place.

To say this in slightly fancier terms: while Descartes ends with a radical dualism (a philosophic system in which relation between diverse substances is problematic), - Spinoza develops a radical monism. (As we shall see, his problem will be just the reverse: his monism will exclude "ordinary reality" as illusory. While this saves him from the Cartesian problem of explaining relation, it is hardly more satisfactory as a philosophy. Those of you with some history of philosophy background might compare Spinoza in this way to Parmenides.)

Given this monism, however - one in which reality is directly mathematically formulable -, everything in Spinoza's system then follows with a deductive clarity that is simply breathtaking. Since such a pure, deductive system is precisely what he intended to construct - he can be said to have succeeded in his project -- however odd the results may appear.

In order to see how this works, however, we will need to examine some of the definitions and proofs - as provided by Jones - with some care.

See Jones, p. 194 -- especially the definitions of substance, attibute, mode, and God.

In the definition of substance we can see the assumption of a direct correlation between thought and reality: a substance is defined both as what is in itself, and what is conceived through itself. This is to say that we both know substance ("conceive") substance through itself (by itself, apart from anything else) - and that substance directly correlates to this conception of it, because substance is in fact what is in itself (by itself, distinct from anything else.)

Notice also that the definition of a mode - as a modification of substance - reiterates this correlation: "that which exists in, and is conceived through, something other than itself [emphasis added]."

As well, a mode or modification of substance is hence something which is both different from substance (because unlike substance, it does not exist in itself, etc.) and yet connected to substance (because it can only exist and be conceived through something other - i.e., either another mode or substance.) This definition, that is, assumes at the outself the possibility of what we will see to be Spinoza's internally differentiated monism - a monism in which a single, ultimate substance (God) is contains within it some differences and distinctions.

Finally, notice that in his "Axioms," - Spinoza again assumes this correlation between thought and being: Axiom V states that "Things which have nothing in common cannot be understood, the one by means of the other; the conception of one does not involve the conception of the other." That is, since two things in reality have nothing in common (or, in other terms, share no common point of identity) - so they have no conceptual relation either.

Again, given these definitions and axioms, a complete system follows: you logicians should just love this.

So he says, for example, as proposition II, that "Two substances, whose attributes are different, have nothing in common." This follows directly from the definition of a substance: "For each [substance] must exist in itself, and be conceived through itself; in other words, the conception of one does not imply the conception of the other."

Not only does this follow directly from the definition of substance (and, notice, Axiom V, above): this also begins to illustrate a further point. While differences may exist, as it were, underneath a substance (as, for example, a substance may have a number of different attributes), -- the difference between substances will prove to be ultimate. That is, the difference between substances will preclude any relation whatsoever. (And so, Spinoza will eventually argue that the choice is between an absolute dualism (i.e., that there exist two substances - which have absolutely no relation to each other: see proposition III) OR there can only be one substance. Absolute dualism will result in absurdity: hence there can only be one substance - i.e., God.)

THE NATURE OF SPINOZA'S ARGUMENT

As you can begin to see in the proofs for the propositions, Spinoza makes considerable use of an argument form that looks like this:

A --> B

B --> C

C -/-> n

~n

/. ~A

In addition, Spinoza's argument is further reminiscent of Descartes' tendency to establish either/or choices: this is to say that Spinoza, like Descartes, are typically modern thinkers insofar as they "work" primarily in a logic emphasizing difference to the exclusion of connection.

We can see this in the second proof for God's existence - Jones, p. 197. The proof starts with just the claim that "Of everything whatsoever a cause or reason must be assigned, either for its existence, or for its non-existence...." [That is, Spinoza's metaphysics involves a sort of "on/off" reality: either something exists - or it doesn't. There are no intermediate levels of existence or being, such as we find with Greek and Medieval thinkers.]

Likewise, "This reason or cause must either be contained in the nature of the thing in question, or be external to it."

Given these starting assumptions, the proof continues by claiming that we are hence faced with two further choices (i.e., yet a third either/or). On the one hand, reason for the non-existence of a thing "is indicated in its nature, namely, because it would involve a contradiction...." On the other hand, reason for the existence of a thing, at least in the case of a triangle, follows "from the order of universal nature in extension."

Spinoza is assuming here - as we should expect - that the "nature of extension" is directly correlative to the world of Euclidean geometry. Given such an assumption, - i.e., given that reality, as co-extensive with mathematical entities and relations, can be determined solely on the basis of mathematics - then there clearly must be the possibility of deriving in deductive fashion the existence of the triangle from the more fundamental axioms which describe "the order of universal nature in extension," - or there is not. That is, given the assumption that reality can be deduced from fundamental axioms, - a fourth either/or follows: either you can deduce the existence of a thing - or you cannot.

This leads to the fifth and final either/or. Continuing with the proof: "...it must follow, either that a triangle necessarily exists, or that it is impossible that it should exist. So much is self-evident. It follows therefrom that a thing necessarily exists, if no cause or reason be granted which prevents its existence."

This is to say:

a thing either exists (a) or it doesn't (b) (first premise);

where reality is co-extensive with a deductive, axiomatic system, it then follows that:

(a) as existing - it necessarily exists (because existence is co-extensive with a deductive, axiomatic system: its existence follows logically from those axioms)

(b) as non-existing - it necessarily does not exist (its existence must involve a logical contradiction within the axiomatic system)

And so: if we cannot show that a thing cannot exist (i.e., if we cannot show that a thing's existence involves a logical contradiction, as with the square circle), then the only other possibility is that it must exist (its existence must follow logically from the axioms of the system.)

From this it further follows that to show that God exists requires only that we show that the claim "God does not exist" involves a contradiction, - or, as Spinoza puts is, we must show some cause or reason which prevents the existence of God.

But in the face of this demand, we are again faced with an either/or. Either such a reason or cause must be drawn from the very nature of God [as with a square circle - we show that a circle by its nature cannot involve a square]: or we draw such a reason/cause from another substance.

But of course, the first option entangles us in a contradiction: "For if it [our reason/cause against the existence of God] were of the same nature, [i.e., we draw this reason/cause from God], God, by that very fact, would be admitted to exist [just as we admit the circle exists in drawing reason/cause from its nature that it cannot be a square circle]."

Our second option also will not work: "But substance of another nature could have nothing in common with God (by Prop. ii), and therefore would be unable either to cause or to destroy his existence." That is, because Spinoza has already defined substance in such a way that the difference between substances is absolute (i.e., there can be no logical/causal relation between two substances - see above, p. 4), no cause or reason drawn from a substance different from God can apply to God's existence or non-existence.

Spinoza concludes his argument this way: since our second option can't work, we can only prove the non-existence of God by the first option. But since this is contradictory, - especially given the kind of being we conceive God to be -, there is no possibility of proving the non-existence of God. And so, it neatly follows that God exists. In his words:

As, then, a reason or cause which would annul the divine existence cannot be drawn from anything external to the divine nature, such cause must perforce, if God does not exist, be drawn from God's own nature, which would involve a contradiction. To make such an affirmation about a beding absolutely infinite and supremely perfect, is absurd; therefore, neither in the nature of God, nor externally to his nature, can a cause or reason be assigned which would annul his existence. Therefore, God necessarily exists. Q.E.D.

All that is left, then, is to prove that God is the only substance that exists. This follows quite neatly, as Spinoza points out, from the definition of God (def. vi) and the necessary existence of God (Prop. XI, the second proof for which we have just examined). But if any additional substance besides God were to exist, "it would have to be explained by some attribute of God, and thus two substances with the same attribute would exist, which (by Prop. v.) is absurd; therefore, besides God no substance can be granted, or, consequently, be conceived....Q.E.D."(198)

OTHER COMMENTS

As Jones points out here, Spinoza's God qua the only substance in the universe stands as "the totality of everything that is." This totality, further, consists in a series of attributes and modes which eventually stand as a series of individual things, each of which is linked - both causally and implicatorily - to the next.

In particular, Jones is correct to point out that "the conclusion that God is the totality of everything that is follows from a rigidly logical application of the traditional definition of substance."(199) Spinoza's metaphysics, you should know, is important in the history of philosophy in part because it stands as a reductio ad absurdum argument against substance metaphysics. That is, if Spinoza is what the definition of substance leads to - there must be something wrong with the definition.

As well, I'd like you to notice here how the business of analogy - more precisely, the lack of it - plays a crucial role. Spinoza rejects the possibility of analogical speech - and insists on a (sixth) either/or: either terms are used univocally or equivocally. On the one hand, then, as Jones elaborates, if substance is to be used univocally - i.e., if there is to be one and only one meaning for the term - then the notion of "finite substances," (substances dependent on God for their existence) in fact contradicts the meaning of substance (something dependent on nothing for its existence). As we saw, it was precisely Thomas' use of analogical equivocation that allowed him to use terms such as "being," etc., in different but related senses with regard to God, on the one hand, and to the created order, on the other hand.

On the other hand, as the passage Jones quotes (201) makes clear, Spinoza entertains only the possibility of pure equivocation - i.e., use of terms in different, purely unrelated senses. It turns out that this sort of argument - which rejects one sense or another in equivocal terms as groundless - is used widely in the modern period, and constitutes part of the modern period's tendency toward the either/or. For example, Descartes uses this sort of argument to suggest that nothing can meaningfully be said or thought about God, because our language and thought are grounded in the sense-world.