Spinoza's geometric method of explanation tends to appear cumbersome,
tedious and redundant. This proposition seems self-evident from I.D4
where attribute was defined as that which the intellect perceived as
constituting the essence of substance. If the essences are the same,
then certainly the substances would be the same and indistinguishable.
So why the labored approach?
Since substance was not defined as
infinite or as solitary in I.D3, but only as "in itself" or "conceived
through itself," Spinoza must extrapolate all the other characteristics
of substance solely from the "in this definition and the logic of the
axioms. In this way, if someone rejects Spinoza's conclusions about
substance, then the initial definition of substance - that there is
something that is "in itself" or "conceived through itself" should serve
as the sole source of that rejection.
Of note, Spinoza
introduces equivalency between natura or "nature" and attributum or
"attribute." This add-on of natura is significant because it maintain
the duality of being (represented by natura) and knowing (represented by
attributum). Importantly to me, this duality is fundamental (rerum
natura) here.
The Demonstration of P5 just says that if there are two substances and two attributes, then there must be one substance per attribute. If there is only a single attribute, then even with a difference in the modifications, there is only one substance, because modifications don't make an essential change. Essential changes are perceived only through attributes but are still present through differences in nature.
In rerum natura non possunt dari duae aut plures substantiae ejusdem naturae sive attributi.
Translated as,
There cannot exist in the universe two or more substances having the same nature or attribute.
Also translated as,
In the nature of things, two or more substances of the same nature or attribute cannot be granted.
Demonstratio: Si darentur plures distinctae, deberent inter se distingui vel ex diversitate attributorum vel ex diversitate affectionum (per propositionem praecedentem). Si tantum ex diversitate attributorum, concedetur ergo non dari nisi unam ejusdem attributi. At si ex diversitate affectionum, cum substantia sit prior natura suis affectionibus (per propositionem 1) depositis ergo affectionibus et
in se considerata hoc est (per definitionem 3 et axioma 6) vere considerata, non poterit concipi ab alia distingui hoc est (per propositionem praecedentem) non poterunt dari plures sed tantum una. Q.E.D.
Translated as,
If several distinct substances be granted, they must be distinguished one from the other, either by the difference of their attributes, or by the difference of their modifications (by P4). If only by the difference of their attributes, it will be granted that there cannot be more than one with an identical attribute. If by the difference of their modifications—as substance is naturally prior to its modifications (by P1),—it follows that setting the modifications aside, and considering substance in itself, that is truly, (by D3 and A6), there cannot be conceived one substance different from another—that is (by P4), there cannot be granted several substances, but one substance only.
Also translated as,
If there two or more distinct substances were given, they would have to be distinguished from one another either from a difference in their attributes, or from a difference in their modifications (by preceding proposition). If only from a difference in their attributes, then it will be conceded that there cannot be but one substance with the same attribute. Yet if from a difference in their modifications, as a substance is prior in nature to its modifications (by P1), by putting modifications aside and considering substance in itself, (by D3), that is considering it truly (by A6), one substance cannot be conceived to be distinguished from another substance, so (by the preceding proposition), there cannot be many substances, but only one (of the same nature or attribute).
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