Spinoza's Ethics: I.P5: One (Attribute) to One (Substance)

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).

Spinoza's Ethics: I.P4: Distinction

This proposition builds on I.D3, I.D4 and I.D5. As I discussed in I.D5, I am translating affectionum as "modifications" rather than "affections" to avoid confusion with contemporary usage of the term. As he says in I.P2, a difference in the attributes of the substances would be a difference essentially, that is, nothing would be in common between them, unlike differences in modifications (that are conceived through another, not just through themselves). 

The demonstration raises the issue of why Spinoza included the phrase extra intellectum or "outside the intellect". At first blush, this raise the issue of appearing to being a domain within a domain and would violate the simplicity of all is substance and modifications. However, the intellect is simply a modification of substance with no violation. So what is Spinoza doing here?

First, given his sparse wording, this phrase is important, especially since he repeats it twice. I think he provides an initial clue in I.D4. That discussion highlighted the difference between concipitur or "conceive" and percipit or "perceive" where my post distinguished them as "view from nowhere" and "view from somewhere." Confirming this notion with verb forms, concipitur is rendered with the passive voice in Latin while percipit is rendered with the active voice. The subjective element or  interior space comes into view. This word implied a thinking subject or observer. Of course, both the proposition and the demonstration emphasize the word "distinguish."

To make this discussion more clear, it might be helpful to look at what might be meant if he had used the phrase "inside the intellect." In I.D3, he defines that a substance may be conceived through itself without reference to anything else. On the other hand,  in I.D5, modifications must be conceived through another. As he posited in I.P1, substances are primary, while modifications are secondary. Further, by I.D1 and by I.A7, substances must exist while modifications may not exist. This distinction is important because it means that I can conceive a true idea about modifications even if its object does not exist. But with substance, no substance can exist only "inside the intellect."  So his discussion here is not meant to address the issues that these modifications "inside the intellect" raise.

Duae aut plures res distinctae vel inter se distinguuntur ex diversitate attributorum substantiarum vel ex diversitate earundum affectionum.

Translated as,

Two or more distinct things are distinguished one from the other, either by the difference of the attributes of the substances, or by the difference of their modifications.

Also translated as,

Two or more things are made distinct by distinguishing among themselves either by a difference in the attributes of their substances or by a difference in their modifications.

Demonstratio: Omnia quae sunt vel in se vel in alio sunt (per axioma 1) hoc est (per definitiones 3 et 5) extra intellectum nihil datur praeter substantias earumque affectiones. Nihil ergo extra intellectum datur per quod plures res distingui inter se possunt praeter substantias sive quod idem est (per definitionem 4) earum attributa earumque affectiones.

Translated as,

Everything which exists, exists either in itself or in something else (I.A1),—that is (by I.D3 and I.D5), nothing is granted in addition to the understanding, except substance and its modifications. Nothing is, therefore, given besides the understanding, by which several things may be distinguished one from the other, except the substances, or, in other words (I.D4) their attributes and modifications.

Also translated as,

All things which are either exist in itself or in something else (by A1), that is (by D3 and D5), outside the intellect nothing is given except substances and their modifications. So nothing, outside the intellect, is given through which multiple things can be distinguished from one another except substances (or their attributes - which is the same thing by I.D4) and their modifications.

Spinoza's Ethics: I.P3: Causality Needs Commonality

In the Proposition, he states that if things have nothing in common with one another (invicem or "interactively" is not included here), then they must not be in at least one another. These descriptions like a set theory where two sets do not overlap or even overlap sets which overlap. If causality is immanent, as in "a circle causes roundness," then causality would require commonness. For Spinoza, to say Thing Two is caused by Thing One is the same as saying Thing Two is in Thing One. The effect is in the cause. This is radically unlike the transitive causality we tend to imagine, where we imagine the reverse - that Thing One is in Thing Two.

In the Demonstration, Spinoza appears to move from being - "have nothing in common" and knowing - "they cannot be understood through one another" as a way of developing relationships of both objects and ideas. This is his novel way to explore connections, as he begins to develop the framework that being and knowing are basically two sides of the same coin.

Quae res nihil commune inter se habent, earum una alterius causa esse non potest.

Translated as,

Things which have nothing in common cannot be one the cause of the other.

Also translated as,

If things have nothing in common with one another, one of them cannot be the cause of the other.

Demonstratio: Si nihil commune cum se invicem habent, ergo (per axioma 5) nec per se invicem possunt intelligi adeoque (per axioma 4) una alterius causa esse non potest. Q.E.D.

Translated as,

If they have nothing in common, it follows that one cannot be apprehended by means of the other (A5), and, therefore, one cannot be the cause of the other (A4). Q.E.D.

Also translated as,

If they have nothing in common with one another, then (by A5) they cannot be understood through one another, and so (by A4) one cannot be the cause of the other, q.e.d.

Spinoza's Ethics: I.P2: Substance Commonality

If two substances have different attributes (from one another) (e.g., one has the attribute of thinking and one has the attribute of existing), then Spinoza is arguing that these two substances have nothing in common.

Clearly if two substances have different attributes (from one another), then they are essentially different. That follows from the definition of attribute (I.D4): "what the intellect perceives of substance as constituting its essence." If the perceived essences are different, then the substances are essentially different, assuming perception is reliable. So, for example, if one substance is perceived through the attribute of thinking and another substance is perceived through the attribute of existing, then the two substances are essentially different. But does that mean that the substances have "nothing in common"?

To have something in common is addressed in the definition of interactivity: (I.A5): "things that have nothing in common with one another also cannot be understood through one another." So, to have something in common is to be understood through one another. That means that either substance could be understood through the other. However, the definition of substance: (I.D3): "is in itself and is conceived through itself" denies this. Substance is only understood through itself not through another.

At this point, he does not seem to be stating there are actually two substances, but only is stating that if there were two substances did exist with different attributes (from one another), there would be an inability for such presumed substances to have anything in common with each other.

This seems to be simply a logical exercise, extending the implications of the definitions and axioms. However a key point seems to be made. If one substance were expressed with the attribute of thinking and one substance were expressed with the attribute of existing, then this two substance universe would have its thinking and existing having nothing to do with each other.   

Duae substantiae diversa attributa habentes nihil inter se commune habent.

Translated as,

Two substances, whose attributes are different, have nothing in common.

Also translated as,

Two substances having different attributes have nothing in common with one another.

Demonstratio: Patet etiam ex definitione 3. Unaquaeque enim in se debet esse et per se debet concipi sive conceptus unius conceptum alterius non involvit.

Translated as,

This also evident from D3. For each ·substance· must be in itself and be conceived through itself, which is to say that the concept of the one doesn’t involve the concept of the other.

Also translated as,

Dem.: This is also evident from I.D3. For each must be in itself and be conceived through itself, or the concept of the one does not involve the concept of the other.