P8: Every substance is necessarily infinite.

Dem.: A substance of one attribute does not exist unless it is unique (P5), and it pertains to its nature to exist (P7). Of its nature, therefore, it will exist either as finite or as infinite. But not as finite. For then (by D2) it would have to be limited by something else of the same nature, which would also have to exist necessarily (by P7), and so there would be two substances of the same attribute, which is absurd (by P5). Therefore, it exists as infinite, q.e.d.

Schol. 1: Since being finite is really, in part, a negation, and being infinite is an absolute affirmation of the existence of some nature, it follows from P7 alone that every substance must be infinite. {NS: For if we assumed a finite substance, we would, in part, deny existence to its nature, which (by P7) is absurd.}

Schol. 2: I do not doubt that the demonstration of P7 will be difficult to conceive for all who judge things confusedly, and have not been accustomed to know things through their first causes—because they do not distinguish between the odifications of substances and the substances themselves, nor do they know how things are produced. So it happens that they fictitiously ascribe to substances the beginning which they see that natural things have; for those who do not know the true causes of things confuse everything and without any conflict of mind feign that both trees and men speak, imagine that men are formed both from stones and from seed, and that any form whatever is changed into any other. So also, those who confuse the divine nature with the human easily ascribe human affects to God, particularly so long as they are also ignorant of how those affects are produced in the mind. But if men would attend to the nature of substance, they would have no doubt at all of the truth of P7. Indeed, this proposition would be an axiom for everyone, and would be numbered among the common notions. For by substance they would understand what is in itself and is conceived through itself, i.e., that the knowledge of which does not require the knowledge of any other thing. But by modifications they would understand what is in another, those things whose concept is formed from the concept of the thing in which they are. This is how we can have true ideas of modifications which do not exist; for though they do not actually exist outside the intellect, nevertheless their essences are comprehended in another in such a way that they can be conceived through it. But the truth of substances is not outside the intellect unless it is in them themselves, because they are conceived through themselves.

Hence, if someone were to say that he had a clear and distinct, i.e., true, idea of a substance, and nevertheless doubted whether such a substance existed, that would indeed be the same as if he were to say that he had a true idea, and nevertheless doubted whether it was false (as is evident to anyone who is sufficiently attentive). Or if someone maintains that a substance is created, he maintains at the same time that a false idea has become true. Of course nothing more absurd can be conceived. So it must be confessed that the existence of a substance, like its essence, is an eternal truth. And from this we can infer in another way that there is only one {substance} of the same nature, which I have considered it worth the trouble of showing here. But to do this in order, it must be noted, I. that the true definition of each thing neither involves nor expresses anything except the nature of the thing defined. From which it follows, II. that no definition involves or expresses any certain number of individuals, since it expresses nothing other than the nature of the thing defined. E.g., the definition of the triangle expresses nothing but the simple nature of the triangle, but not any certain number of triangles. It is to be noted, III. that there must be, for each existing thing, a certain cause on account of which it exists.

Finally, it is to be noted, that this cause, on account of which a thing exists, either must be contained in the very nature and definition of the existing thing (viz. that it pertains to its nature to exist) or must be outside it. From these propositions it follows that if, in nature, a certain number of individuals exists, there must be a cause why those individuals, and why neither more nor fewer, exist. For example, if 20 men exist in nature (to make the matter clearer, I assume that they exist at the same time, and that no others previously existed in nature), it will not be enough (i.e., to give a reason why 20 men exist) to show the cause of human nature in general; but it will be necessary in addition to show the cause why not more and not fewer than 20 exist. For (by III) there must necessarily be a cause why each {NS: particular man} exists. But this cause (by II and III) cannot be contained in human nature itself, since the true definition of man does not involve the number 20. So (by IV) the cause why these 20 men exist, and consequently, why each of them exists, must necessarily be outside each of them. For that reason it is to be inferred absolutely that whatever is of such a nature that there can be many individuals {of that nature} must, to exist, have an external cause to exist. Now since it pertains to the nature of a substance to exist (by what we have already shown in this Scholium), its definition must involve necessary existence, and consequently its existence must be inferred from its definition alone. But from its definition (as we have shown from II and III) the existence of a number of substances cannot follow. Therefore it follows necessarily from this, that there exists only one of the same nature, as was proposed.