Martial Gueroult, “Spinoza’s Letter on the Infinite,” in Spinoza. A Collection of Critical Essays, ed. Marjorie Grene, trans. Kathleen McLaughlin (Garden City, New York: Anchor/Doubleday, 1973): 182-212.
I. The problem of the infinite and of the indivisibility of substance is treated in Book I of the Ethics, Proposition xii demonstrates infinity; Proposition xiii, its Corollary and its Scholium, and lastly the Scholium of Proposition xv demonstrate indivisibility. Infinity and indivisibility are two unique properties of substance which derive immediately from its fundamental property: causa sui. Indeed, whatever necessarily exists of itself cannot, without contradiction, be deprived of any part whatever of its existence; Consequently, it is necessarily infinite and excludes any partitioning. Infinity and indivisibility being two sides of the same property, there results a radical antinomy between the infinite and the divisible. If we affirm one we must deny the other: the dogmatist,·affirming divisibility, denies the infinite; Spinoza, affirming the infinite, denies divisibility. This is an irreducible conflict, as long as we ignore the nature of substance, but one that is instantly resolved as soon as we know that substance necessarily exists of itself.
Given this, however, the problem is far from exhausted. The antinomy opposing infinity and divisibility, resolved in the Ethics·on the level of substance by excluding the divisible, reappears on the level of the mode, where we must affirm infinite divisibility, that is, both the infinite and the divisible. If it is true that the solution to the second part of the problem is included in that of the first part, the Ethics did not expressly develop it. It is Letter XII to Louis Meyer, called by Spinoza and by his correspondents Letter on the lnfinite , which, embracing the problem in its entirety, answers this difficulty as well as many others.
Its character, at once succinct and exhaustive, is emphasized by the author himself: "I have," he wrote toward the end, "briefly ex-
posed to you… the causes (causas) of the errors and confusions which have arisen on the subject of this question of the Infinite, and I have explained these errors in such a way that, if I am not mistaken, there no longer remains a single question relative to the Infinite that I have not touched upon, nor one whose solution cannot be quite easily found from what I have said" .
We see by these last lines that this letter presents above all a refutative quality, and it owes to this a great part of its obscurity. The doctrine is not directly expounded, but indicated through the errors whose causes are exposed.
These causes are first of all confusions among things, and secondly the reason for such confusions, which is itself also a confusion, but among our cognitions.
II. The difficulties relating to the Infinite flow from three kinds of confusions arising from our failure to distinguish between six different cases.
These six cases are divided into three pairs of opposing terms:
1. The thing infinite by its essence or by virtue of its definition .
2. The thing without limits, not by virtue of its essence, but by virtue of its cause .
3. The thing infinite insofar as without limits ]5].
4. The thing infinite insofar as its parts, although included within a maximum and a minimum known to us, cannot be expressed by any number .
5. The things representable by understanding alone and not by imagination .
6. The things representable at once by imagination and by understanding .
The confusion between the two cases of each of these pairs has made us unable to understand: a) which Infinity cannot be divided into parts and is without parts; b) which, on the contrary, is divisible without contradiction; c) which can, without difficulty, be conceived as larger than another; d) which, on the other hand, cannot be so conceived .