Here Spinoza uses the force of ex necessitate or "necessity" to describe the operation of divine nature. In this usage, Spinoza denies teleology, that is, he specifically denies that God or the universe has a direction in mind. Yet, at the same time, Spinoza makes clear that God or the universe is not operating mindlessly as he specifically identifies intellectum infinitum or "infinite intellect" of God. This distinction highlights a key way in which Spinoza's God is different than those of traditional religions. The "infinite intellect" does not solve problems, but simply expresses solutions.
His demonstration does not bear the geometric simplicity of others. Instead, his comment si modo ad hoc attendat or "if someone would attend to this" reflects the language of his scholia or commentaries. His definitional emphasis of the first half of the demonstration reflects a continuation of the tightness of cause and effect - the effect is only known if the cause is known. Spinoza's system (as I discussed in I.P15S) is grounded in an intuition of the infallibility of clear reason. Here he is providing an example of the components of clear reason - definitions are more real insofar as they are more property laden. This is unlike our conventional scientific thought in which specifics are stripped away to discover essences. Spinoza is the thinker of the specifics, rather than generics. Within the specificity of each individual mode, there are infinite individual things. Rather than counting infinite angels on an imaginary pin, Spinoza articulates infinite properties discoverable on each "real" thing defined. He takes this framework to develop the second half of the demonstration as an analogy to the "infinitely infinite" already presented in I.P6.
Aristotle worked under a framework of four causes: efficient, material, formal and final. In this proposition, Spinoza is clearly denying that God operates with a final cause. In the corollary, Spinoza focuses on God's role. First, God is the efficient cause. Nothing else is truly causal. Second, God is the immediate, not accidental or derivative cause. God does not operate through some non-God intermediaries. Third, God is the first cause and does not share any aspect of that causal pattern with anything else.
Ex necessitate divinæ naturæ infinita infinitis modis (hoc est omnia quæ sub intellectum infinitum cadere possunt) sequi debent.
Translated as,
From the necessity of divine nature, infinite things must follow from infinite modes (that is, everything which is able to fall within infinite intellect.)
Demonstratio: Hæc propositio unicuique manifesta esse debet si modo ad hoc attendat quod ex data cujuscunque rei definitione plures proprietates intellectus concludit, quæ revera ex eadem (hoc est ipsa rei essentia) necessario sequuntur et eo plures quo plus realitatis rei definitio exprimit hoc est quo plus realitatis rei definitæ essentia involvit. Cum autem natura divina infinita absolute attributa habeat (per definitionem 6) quorum etiam unumquodque infinitam essentiam in suo genere exprimit, ex ejusdem ergo necessitate infinita infinitis modis (hoc est omnia quæ sub intellectum infinitum cadere possunt) necessario sequi debent. Q.E.D.
Translated as,
This proposition ought to be clear to every single person if only he (or she) would attend to this - that the intellect deduces more properties from the definition given of a specified thing, which (increase in the number of properties) really follows necessarily from the definition (that is, the essence itself of the thing) and the definition of the thing expresses more properties where there is more reality - that is, where the essence of the thing defined involves more reality. Moreover, since divine nature has absolutely infinite attributes (by P6) each of which still express infinite essence in their own kind, from the same necessity, infinite things must follow from infinite modes (that is, everything which is able to fall within the infinite intellect.)
Corollarium I: Hinc sequitur Deum omnium rerum quæ sub intellectum infinitum cadere possunt, esse causam efficientem.
Translated as,
From this, it follows that God is the efficient cause of everything which is able to fall within the infinite intellect.
Corollarium II: Sequitur II Deum causam esse per se, non vero per accidens.
Translated as,
It follows, second, that God is the cause through itself, truly not through an accidental cause.
Corollarium III: Sequitur III Deum esse absolute causam primam.
Translated as,
It follows, third, that God is absolutely the first cause.
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