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[The following summarizes Hegel's Lectures on the History of Philosophy, Section Two: Period of the Thinking Understanding, Chapter I. The Metaphysics of the Understanding, A 2. Spinoza, f.
My interpretation is not informed by a complete read of the text.]
For Spinoza, the infinite has two senses: the infinitely many and the absolutely infinite [for more on this distinction, see Spinoza's 12th Letter and Gueroult's commentary, Deleuze's Cours Vincennes: 10/03/1981, and Deleuze's commentary on Spinoza's critique of Boyle's notion of divisibility.]
Hegel then quotes from Definition 6 of the Ethics to parallel the infinitely many with infinity after its kind. [The following is the entire definition, the underlined part is what Hegel quotes.]
VI. By God, I mean a being absolutely infinite-that is, a substance consisting in infinite attributes, of which each expresses eternal and infinite essentiality.
Explanation-I say absolutely infinite, not infinite after its kind : for, of a thing infinite only after its kind, infinite attributes may be denied ; but that which is absolutely infinite, contains in its essence whatever expresses reality, and involves no negation.
Hegel then cites the 12th Letter to parallel the infinitely-many/infinite-after-its-kind with the infinite of the imagination; and he parallels the absolutely infinite with the infinite of thought.
Those who strive after the infinite – as when saying "and so on to infinity" to represent infinite cosmic distances – do not get further than the infinite of the imagination. This sort of "progressive infinite" is found in mathematics as an infinite numerical series, resulting for example when we divide 1 by 7 [see Science of Logic §558 for more on progressive infinities; see §§552-562 for an elaboration of this "spurious infinity" in his example of 2/7.] A fraction represented as a decimal fraction [such as 0.285714...] is incomplete, but 1/7 is a "true" mathematical infinity, because it represents the quantum more qualitatively in terms of relation rather than quantitatively in terms of a finite quantum. But even this infinite does not suffice, because it is still in one sense considered negatively as a determinate quantum that is not fundamentally an actual infinite. [for more on actual infinity, see Spinoza's 12th Letter and Gueroult's commentary, Deleuze's Cours Vincennes: 10/03/1981, and Deleuze's commentary on Spinoza's critique of Boyle's notion of divisibility].
Spinoza's infinite is not a sensuous infinite, that is, a "fixing of a limit and then passing beyond the limit fixed." Rather, Spinoza's infinite is the absolute infinity, which is "the positive" itself, bearing within it an unbounded absolute multiplicity [Hegel writes precisely: "But for Spinoza the infinite is not the fixing of a limit and then passing beyond the limit fixed — the sensuous infinity — but absolute infinity, the positive, which has complete and present in itself an absolute multiplicity which has no Beyond."] Hegel's "philosophical infinite" corresponds with Spinoza's actual infinite, which Spinoza characterizes as "the absolute affirmation of itself."
But Hegel wants to correct this formulation. It is more accurately rendered "the negation of negation," because it removes the limitation of delimited quantity.
To illustrate, Hegel turns to Spinoza's famous double-circle diagram [which Hegel also analyzes in his Science of Logic §566.]
Hegel quotes Spinoza as writing:
“The inequalities of the space between A B and C D exceed every number; and yet the space which lies between is not so very great.”
Hegel interprets this to mean that determining all the inequalities of distance between the limits requires that we deal with them as an infinite series [see Gueroult's commentary on the Letter, section XIII, for more on Hegel's translation of "inequalities of distance."]. But an infinite series is always "affected with negation" [Hegel explains in § 558 of Science of Logic that infinite series still represent a quantum in terms of determination, because even though there is always another determination, still it is presumed that the value is determined by means of an infinite series of these determinations. And because every determination lacks something, the whole of the series is defined by this negativity. So in this way it is "affected with negation." As Hegel writes in §558 "The consequence of this is that the amount which is expressed in the series always lacks something, so that in order to reach the required determinateness, we must always go further than the terms already posited."]
But even though this indeterminate infinity is false, it is still circumscribed between the limits of the circle. Analogously, a bounded line is composed of an infinity of points, yet the line has a limited extent.
So because there are always more smaller spaces to be found through infinite divisions of the bounded region, there is a "beyond" found within that limited area of extension; for, there is always another determination beyond any given one. But because this beyond is contained within a delimited boundary, it unifies with the finite values.
When we divide this space, our act of division is the cause that produces the divided parts; or it could also be said, that the whole space is the cause of the divided parts, which are its effects. But because these divided parts are then like wholes that are further divided, what once were effects themselves become causes. And this process repeats endlessly. But the infinite should only be found in the cause, for Spinoza. So the division effects a negation, [[either because it delimits one part against the other, or because it destroys the whole.] But when the effect becomes a cause, it negates that prior negation. So for Spinoza, the affirmative of infinity is a double negation, which is logically and grammatically equivalent to an affirmation. But this is not to be found in a progressive mathematical infinity which ever produces new negations on to infinity.
However, we see such a double negation that produces a true infinity in the case of substance, which is the "cause of itself:" It causes itself as its effect, which negates that negation to again become cause of itself. In other words, the negation that is negated is not a new negation, but that very negation doing the negation, so it is an actualized and present affirmation.
So Notion and existence [perhaps equivalent to Spinoza's essence and existence] are beyond each other, hence it would seem they cannot unify. But the existence of the self-causing substance follows necessarily from its essence. Hence even though Notion (essence) and existence have each other as a beyond, in the self-causing infinite substance these beyonds unify.
Although Spinoza did not realize it, he was here expressing absolute Notion itself, because infinite substance's "Notion is its Being, and its Being its Notion."
Hegel. Lectures on the History of Philosophy. Transl. E.S. Haldane.
See electronic text site for further publication information, and for the entirety of the text itself:
Hegel. Science of Logic. Transl. A.V. Miller. George Allen & Unwin, 1969.
Text available online at:
Spinoza. Ethics. Transl. Elwes. available online at:
http://ebooks.adelaide.edu.au/s/spinoza/benedict/ethics/index.html
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