Propositions 12

12. No attribute of substance can be conceived from which it would follow that substance can be divided.

Proof: The parts into which substance as thus conceived would be divided either will retain the nature of substance, or they will not.

If the former, then (by Prop. 8) each part will necessarily be infinite, and (by Prop. 6) self—caused, and (by Prop. 5) will perforce consist of a different attribute, so that, in that case, several substances could be formed out of one substance, which (by Prop. 6) is absurd.

Moreover, the parts (by Prop. 2) would have nothing in common with their whole, and the whole (by Def. iv. and Prop. 10) could both exist and be conceived without its parts, which everyone will admit to be absurd.

If we adopt the second alternative—namely, that the parts will not retain the nature of substance—then, if the whole substance were divided into equal parts, it would lose the nature of substance, and would cease to exist, which (by Prop. 7) is absurd. –>

13. Substance absolutely infinite is indivisible.

Proof: If it could be divided, the parts into which it was divided would either retain the nature of absolutely infinite substance, or they would not.

If the former, we should have several substances of the same nature, which (by Prop. 5) is absurd.

If the latter, then (by Prop. 7) substance absolutely infinite could cease to exist, which (by Prop. 11) is also absurd.

Corollary: It follows that no substance, and consequently no extended substance, in so far as it is substance, is divisible.

Note: The indivisibility of substance may be more easily understood as follows.

The nature of substance can only be conceived as infinite, and by a part of substance, nothing else can be understood than finite substance, which (by Prop. 8) involves a manifest contradiction.

14. Besides God, no substance can be granted or conceived.

Proof: God is a being absolutely infinite. of whom no attribute that expresses the essence of substance can be denied (by Def. 6), and he necessarily exists (by Prop. 11);

If any substance besides God were granted, it would have to be explained by some attribute of God, and thus two substances with the same attribute would exist, which (by Prop. 5) is absurd; therefore, besides God no substance can be granted, or, consequently, be conceived.

If it could be conceived, it would necessarily have to be conceived as existent; but this (by the first part of this proof) is absurd.

Therefore, besides God no substance can be granted or conceived. Q.E.D.

Corollary 1: Clearly, therefore= 1. God is one, that is (by Def. 6) only one substance can be granted in the universe, and that substance is absolutely infinite, as we have already indicated (in the note to Prop. 10).

Corollary 2: It follows 2. That extension and thought are either attributes of God or (by Axiom 1) accidents (affectiones) of the attributes of God.

15. Whatsoever is, is in God, and without God nothing can be, or be conceived.

Proof: Besides God, no substance is granted or can be conceived (by Prop. 14), that is (by Def. 3) nothing which is in itself and is conceived through itself.

But modes (by Def. 5) can neither be, nor be conceived without substance; wherefore they can only be in the divine nature, and can only through it be conceived.

But substances and modes form the sum total of existence (by Axiom 1), therefore, without God nothing can be, or be conceived. Q.E.D.

Note: Some assert that God, like a man, consists of body and mind, and is susceptible of passions. How far such persons have strayed from the truth is sufficiently evident from what has been said. But these I pass over.

For all who have in anywise reflected on the divine nature deny that God has a body. Of this they find excellent proof in the fact that we understand by body a definite quantity, so long, so broad, so deep, bounded by a certain shape, and it is the height of absurdity to predicate such a thing of God, a being absolutely infinite. But meanwhile by other reasons with which they try to prove their point, they show that they think corporeal or extended substance wholly apart from the divine nature, and say it was created by God.

Wherefrom the divine nature can have been created, they are wholly ignorant; thus they clearly show, that they do not know the meaning of their own words.

I myself have proved sufficiently clearly, at any rate in my own judgment (Coroll. Prop. 6 and note 2, Prop. 8), that no substance can be produced or created by anything other than itself.

Further, I showed (in Prop. 14), that besides God no substance can be granted or conceived.

Hence we drew the conclusion that extended substance is one of the infinite attributes of God.

However, in order to explain more fully, I will refute the arguments of my adversaries, which all start from the following points:

Extended substance, in so far as it is substance, consists, as they think, in parts, wherefore they deny that it can be infinite, or consequently, that it can appertain to God.

This they illustrate with many examples, of which I will take one or two.

If extended substance, they say, is infinite, let it be conceived to be divided into two parts; each part will then be either finite or infinite.

If the former, then infinite substance is composed of two finite parts, which is absurd.

If the latter, then one infinite will be twice as large as another infinite, which is also absurd.

Further, if an infinite line be measured out in foot lengths, it will consist of an infinite number of such parts; it would equally consist of an infinite number of parts, if each part measured only an inch=

Therefore, one infinity would be twelve times as great as the other.

Lastly, if from a single point there be conceived to be drawn two diverging lines which at first are at a definite distance apart, but are produced to infinity, it is certain that the distance between the two lines will be continually increased, until at length it changes from definite to indefinable.

As these absurdities follow, it is said, from considering quantity as infinite, the conclusion is drawn, that extended substance must necessarily be finite, and, consequently, cannot appertain to the nature of God.

The second argument is also drawn from God’s supreme perfection.

It is said that God, as he is a supremely perfect being, cannot be passive.

But extended substance, as it is divisible, is passive.

It follows that extended substance does not appertain to God’s essence.

These are arguments from writers who try to prove that extended substance is:

• unworthy of the divine nature, and
• cannot possibly appertain thereto.

However, I have already answered their propositions.

All their arguments are founded on the hypothesis that extended substance is composed of parts.

Prop. 12, and Coroll. Prop. 13 showed that such a hypothesis is absurd.

Moreover, anyone who reflects will see that all these absurdities from which it is sought to extract the conclusion that extended substance is finite, do not at all follow from the notion of an infinite quantity, but merely from the notion that an infinite quantity is measurable, and composed of finite parts

Therefore, the only fair conclusion is that infinite quantity= is not measurable, and cannot be composed of finite parts.

This is exactly what we have already proved (in Prop. 12).

Thus, their weapon against us has in reality recoiled on themselves.

If, from this absurdity of theirs, they persist in drawing the conclusion that extended substance must be finite, they will in good sooth be acting like a man who asserts that circles have the properties of squares, and, finding himself thereby landed in absurdities, proceeds to deny that circles have any center, from which all lines drawn to the circumference are equal.

For, taking extended substance, which can only be conceived as infinite, one, and indivisible (Props. 8, 5, 12) they assert, in order to prove that it is finite, that it is composed of finite parts, and that it can be multiplied and divided.

So, also, others, after asserting that a line is composed of points, can produce many arguments to prove that a line cannot be infinitely divided.

Assuredly, it is not less absurd to assert that extended substance is made up of bodies or parts, than it would be to assert that a solid is made up of surfaces, a surface of lines, and a line of points.

This must be admitted by all who know clear reason to be infallible, and most of all by those who deny the possibility of a vacuum.

If an extended substance could be so divided that its parts were really separate, why should one part not admit of being destroyed, the others remaining joined together as before?

Why should all be so fitted into one another as to leave no vacuum?

Surely in the case of things, which are really distinct one from the other, one can exist without the other, and can remain in its original condition.

As, then, there does not exist a vacuum in nature (of which anon), but all parts are bound to come together to prevent it, it follows from this that the parts cannot really be distinguished, and that extended substance in so far as it is substance cannot be divided.

Regarding the question= Why are we naturally so prone to divide quantity?

I answer that quantity is conceived by us in two ways:

• in the abstract and superficially, as we imagine it; or
• as substance, as we conceive it solely by the intellect.

If, then, we regard quantity as it is represented in our imagination, which we often and more easily do, we shall find that it is finite, divisible, and compounded of parts; but if we regard it as it is represented in our intellect, and conceive it as substance, which it is very difficult to do, we shall then, as I have sufficiently proved, find that it is infinite, one, and indivisible. This will be plain enough to all who make a distinction between the intellect and the imagination, especially if it be remembered, that matter is everywhere the same, that its parts are not distinguishable, except in so far as we conceive matter as diversely modified, whence its parts are distinguished, not really, but modally. For instance, we conceive water, as it is water, to be divided. Its parts are separated from one another, but not in so far as it is extended substance. From this point of view, it is neither separated nor divisible. Further, water, as it is water, is produced and corrupted. But, as it is substance, it is neither produced nor corrupted.

I have now answered the second argument.

It is, in fact, founded on the same assumption as the first= that matter, as it is substance, is divisible and composed of parts.

Even if it were so, I do not know why it should be considered unworthy of the divine nature, inasmuch as besides God (by Prop. 14) no substance can be granted, wherefrom it could receive its modifications.

All things are in God.

All things which come to pass:

• come to pass solely through the laws of the infinite nature of God, and follow from the necessity of his essence. Wherefore it can in nowise be said, that God is passive in respect to anything other than himself, or that extended substance is unworthy of the Divine nature, even if it be supposed divisible, so long as it is granted to be infinite and eternal.