Argument
In Part I, Proposition 13 of Baruch Spinoza’s Ethics, Depicted in Geometrical Order, Spinoza quotes the following argument to show that an infinite substance cannot be divisible.
Spinoza Quotes: If an infinite substance could be divided into equal parts, the parts would either retain the nature of substance or they would not.
Due to Spinoza’s presupposition that two substances of the same nature cannot coexist, it follows that an infinite substance divided into equal parts could not retain the same nature:
Spinoza Quotes: If the parts did retain the nature of substance, then there would be several substances of the same nature, which is absurd because two substances of the same nature cannot exist in the universe (Proposition 5).
If several substances of the same nature existed, they would have to be distinguished from one another, either by the difference of their attributes or by the difference of their modifications (Proposition 4). If by their attributes, then they could not share an identical attribute, as attributes constitute the nature of substance and identical attributes would render the substances indistinguishable from one another. If by modifications, then this would not truly distinguish the nature of the substances because substance is prior to its modifications (Proposition 1).
Spinoza Quotes: If the parts did not retain the nature of substance, then substance would cease to exist, which is absurd because existence belongs to the nature of substance and substance necessarily exists.
Existence belongs to the nature of substance (Proposition 7) because it cannot be produced by anything else (Proposition 6) and therefore must be self-caused. Also, it is impossible to have two substances that share an identical attribute (Proposition 5), and two substances with different attributes share nothing in common (Proposition 2), and substances that share nothing in common cannot be the cause of each other (Proposition 3, Axiom 5). Substance necessarily exists (Proposition 11) because if substance could be conceived as not existing, then its existence would no longer belong to its nature, which is absurd (Proposition 7).
Conclusion: Therefore, infinite substance cannot be divisible.
Counter-Argument
In Premise 2 of Proposition 13, it does not necessarily follow that there would be more than one substance if the divided parts retained the nature of substance. Spinoza makes the following two assertions from Definition 4 and Proposition 10:
Spinoza Quotes: An attribute constitutes the nature of substance (Definition 4).
Spinoza Quotes: The attributes of substance must be conceived as distinct (Proposition 10).
Since attributes constitute the nature of substance, and the nature of substance is in itself and conceived through itself (Definition 3), it follows that an attribute must be conceived through itself. Since substance consists of more than one attribute (Proposition 11), and an attribute cannot be conceived through anything else, the attributes must be conceived as distinct from one another.
Spinoza Quotes: If substance were to be divided into equal parts, it would be divided into attributes, because attributes constitute the nature of substance.
This brings Spinoza to the following logical assertion:
Spinoza Quotes: The conception of substance is inherently divided by attributes.
Since attributes must be conceived as distinct, the conception of substance is, by nature, divided into attributes. It follows that if the attributes did retain the nature of substance, then there would still only be one substance, because the conception of substance would be unchanged.
Conclusion: Therefore, if substance was to be divided and the parts retained the nature of substance, it would not follow that there would be more than one substance.
Counter-Defense
Regarding Premise 4, that the conception of substance is, by nature, divided by its attributes, is absurd.
Spinoza Quotes: An individual attribute can only be conceived through itself, however, attributes cannot be conceived as separate entities or substances (Proposition 10).
To conceive of attributes as separate entities would be to conceive of more than one substance, which is absurd (Proposition 5).
Spinoza Quotes: Attributes are only conceived as distinct insofar as they express a particular infinite nature of substance.
That is to say that the conception of substance can only be a conception of one substance (Proposition 14). Since substance is infinite, the intellect conceives of many attributes that each express its nature in a particular way. However, since the human intellect as a mode of thinking is finite (Definition 5, Proposition 31), it cannot conceive of all of the attributes of substance, or even conceive of the entirety of an attribute, as each attribute is also infinite (Proposition 11).
Therefore, the intellect can only conceive of finite expressions of substance and its attributes. Within these incomplete conceptions, substance could be divided by its attributes. However, the following premises will show that substance and its attributes, by their truly infinite nature, cannot be divided.
Spinoza Quotes: Neither substance nor attributes can be limited by anything else.
By definition, neither substance nor its attributes can be limited, because that which is infinite cannot be limited by anything else. This is so because a finite thing can only be conceived in relation to a larger body that necessarily limits the particular thing (Definition 2). Alternatively, an infinite thing can only be conceived through the absence of a larger, limiting body. Therefore, substance cannot be limited by its attributes, and the attributes cannot be limited by substance.
Spinoza Quotes: To divide substance by its attributes would necessarily limit the substance and the attributes.
The act of dividing a whole into parts presupposes a finite whole with finite parts. For example, a finite square can be divided into 2 finite rectangles, each comprising a finite half of the square. However, an infinite thing cannot be divided into two equal halves because the halves would necessarily be finite, by virtue of being portions of a larger whole. This would be absurd because it is impossible to conceive of a half of infinity.
Similarly, to divide substance into an infinite number of attributes would limit the attributes, as each attribute would consist of a portion of the infinite substance, which is absurd, because the attributes are infinite by nature and cannot be limited. Thus, substance could only be divided by its attributes if they were both finite things, which is absurd.
Conclusion: Therefore, the conception of substance as inherently divided by its attributes is absurd.
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