How does one think infinity? The question seems at first to place an unreasonable demand for provisioning an answer; the structure of the human mind immediately appears finite, conditioned. Yet one soon discovers that it is reason itself which places this demand. Man is irrepressibly driven by his rational faculty to apprehend the infinitely unconditioned ground(s) upon which the finite phenomena of experience are grounded. Limitation is anathema to the most primordial desire of humanity. For nothing is more human than to reject the human — to reject finitude and become God.
The spiritual epic of man is thus guided by his cognitive romance with the Absolute, qua true infinity. In the course of its unfolding, philosophers have variously located the metaphysical domain of infinity as either belonging to the structure of the world or the mind. Classical (pre-Kantian) metaphysics naïvely sought infinity in the predicate structure of the world, a world it had imparted with universality by virtue of its deductions. In other words, infinity was for this metaphysics a mere predicate in its determinations, and bore no necessary relation to its subject apart from its copular attachment (God is infinitely powerful, infinitely knowledgeable, etc.). Rationalist ontology, pneumatology, cosmology, and theology were borne of its efforts. But cracks began to emerge in its objective edifice, and soon Hume arose to shatter the great deductive systems of philosophy. Only with Kant was universality rehabilitated, and even then only at a price. The phenomenal world was recognized for its objective finitude, but infinity was subjectively retained in the pure (a priori) faculties of the understanding. Within this categorical matrix, objectivity was granted to judgments which arranged the manifold of intuition under the twin categories of universality and necessity. Objective laws could be hoped to have infinite application to finite phenomena. But even then this infinity was strictly formal, hence empty, having been methodologically stripped of empirical (a posteriori) content. The philosophers of subjectivity (Kant, Jacobi, Fichte) had correctly diagnosed the dogmatism of the objective infinite, but the infinite they had replaced it with remained definite in its separation from the finite.
Both such conceptions of infinity (objective and subjective alike) ultimately fell short for Hegel. The objective infinity of being and the subjective infinity of thought each failed in its non-relation to finitude, i.e. its abstract isolation from infinity’s negative. The former thought the world all too gracious in its accommodation of the human mind; the latter, by contrast, “sen[t] man to feed upon husks and chaff.” Always seeking some mediating ground between two dialectical opposites, Hegel hoped to recast abstract infinity and abstract finitude into the concrete unity of the speculative infinite, or the infinitum actu of Spinoza. The notion of the “true” or “good” infinity of speculation recurs throughout Hegel’s mature works, from his early collaboration with Schelling in Faith and Knowledge (1802) to the final edition of his Encyclopedia Logic (1831), his last published work. This is hardly a coincidence. For in Hegel’s estimation, “the true infinite is [my italics] the absolute Idea” — the grounding principle of all genuinely philosophical knowledge. As such, a grasp of this immanent feature of Hegel’s thought is central to an appreciation of his philosophy. With reference to the pertinent texts that deal with this topic, its fine points might be thoroughly excogitated. As the concept takes shape, the speculative implications of its particulars will be briefly discussed, wherever appropriate.†
Bounded Infinity, or the Metaphysic of Subjectivity
Infinity introduces to thought a number of intellectual quandaries. Its æsthetic (spatio-temporal) application alone – i.e., the idea of limitless expansion and limitless duration – inevitably leads to a paralogism of the understanding. The implications of such an attempt are famously addressed in Kant’s first antinomy. Conversely, the very notion of matter as infinitely divisible also creates problems for the understanding, since any object in space could be judged to be infinite in its own (delimited) respect. This constitutes the infinitesimal quality of matter. Spinoza was careful to distinguish the infinity of his substance from this limited notion as absolutely infinite,‡ as opposed to “infinite in its kind” (in suo genere infinita). Material objects would not even qualify as infinite under this latter criterion, since they would be limited outside their measured sphere by objects sharing their essential attribute (matter, res extensa). This was meant to anticipate invocations of Zeno’s spatial paradoxes, which could easily have been raised if Spinoza had left his notion of infinity ambiguous. In his arguments against motion (the Achilles, for instance), Zeno had already thrown the idea of an infinite temporal sequence into question. The metaphysic of motion analytically presumes the existence of a spatial and temporal field. Each successive position therein would be the negation of its previous position. But if an infinite number of points exist which spatially separate one object from another, then it would logically follow that infinite time would be required in order to reach either object starting from the position of its other.
Clearly, infinity was problematic for the intellectual consistency of classical metaphysics. Hegel thus maintained that the Critical philosophy was justified in its evaluation of objectively applied metaphysics, but only insofar as the latter not sought to “go beyond the thinking of mere understanding.” The intellectual determinations of classical metaphysics were, in fact, just that. That is to say, qua determinate, they were finite (they contained a terminus or limit). The understanding was only suited to finite thinking. Hegel therefore characterized classical metaphysics as such: “The thinking of the older metaphysics was finite, because that metaphysics moved in thought-determinations whose restrictions counted for it as something fixed, that would not be negated again.” For Kant, infinity was instead supposed to be found in the subjective categories of the understanding, which were, in their formal purity, abstractly infinite.† However, this conception of infinity hardly did justice to the sense of the Latin word (infinitas) from which it was derived, a term usually translated as “unboundedness.” For if infinity belongs solely to the pure concepts of the subject’s understanding, opposed by his finite determinations of the objective world set over and against him, then the original meaning of infinity has been betrayed. The mental infinity of the subject would be effectively bounded by its failure to encompass the objective world of finitude. Nevertheless, the Critical philosophy judged that infinity only keeps its identity by virtue of its opposition to finitude, and consequently abandoned any hope for a synthesis. Jacobi, a prominent critic of Kant, and Fichte (the latter’s self-styled successor) each made gestures toward solving the problem of the infinite, but achieved no greater insight than had their illustrious predecessor.
For this reason, the subjective revisions to infinity accomplished in the empirical and Critical philosophies did not go far enough in their emendation of objective metaphysics. They had stopped shy of the final (speculative) steps which Hegel felt necessary to the realization of the true nature of infinity. Only by recourse to speculative reason could the finite objects of experience realize their truth in the infinite. The finitude of their determination would then be mediated by reference to the indeterminate “I” who thinks them, an object which always eludes determination. Finite objectivity would be recognized as belonging to the boundless (read “infinite”) speculative structure of subject who considers it. The “metaphysic of subjectivity,” as Hegel referred to it, contained within itself the tools necessary for discovering the true nature of infinity — tools which had remained obscure to its proponents. But where their efforts had been frustrated, Hegel resolved to press forward. He sensed the immediate proximity of the beyond; the Absolute seemed to lie in just this true infinity. For the distance that was to be overcome was not a distance at all: rather, the structure of the Idea was to emerge out of the (speculative) spiritual activity of man itself. The qualitative re-imagination of the nature of the infinite against its previously quantitative status (as spurious infinity) can thus be seen to be the key which freed the unhappy consciousness from the barren metaphysic of subjectivity and unlocked the gates to the metaphysic of the Absolute.
The Speculative “Good Friday”: Infinity in Faith and Knowledge
Indeed, Hegel’s initial formulation of the speculative infinite developed out of his early critique of the subjective philosophies of Kant, Jacobi, and Fichte in Faith and Knowledge. Hegel argued, echoing Jacobi, that the empty formalism of transcendental ideal-ism was symptomatic of a nihilism that had beset mankind in the waning years of the Enlightenment. Reason, in its quest for the Unconditioned, had laid bare the naïve intuitions of a personal God, eroding with it the principle of free will. Asserting that “thinking is infinity,” Hegel stressed the negative function of thinking in its infinite mediation of the mind’s representations. As such, “[i]nfinity is the pure nullification of the antithesis or of finitude; but it is at the same time also the spring of eternal movement, the spring of that finitude which is infinite, because it eternally nullifies itself.” The annihilative quality of this conception should be obvious. Contrary to Jacobi, however, he held this process to be necessary for the maturity of human consciousness. Nihilism was not to be the cause of despair; rather, it was to be wholeheartedly embraced. For “[o]ut of this nothing and pure night of infinity, as out of the secret abyss which is its birthplace, the truth lifts itself upward.” The void of pure conceptuality appears as the abysmal womb from which the truth emerges, ex nihilo, as it were. The anguish of nihilism would vanish before the “heaven of truth” which would shine forth. “[T]he pure concept or infinity as the abyss of nothingness in which all being is engulfed, must signify the infinite grief [of the finite] purely as a moment of the supreme Idea, and no more than a moment.”
Hegel’s astonishing conclusion to this piece, which deserves much more attention than it has hitherto received, lays out in explicit terms the messianic role he expected speculative philosophy to play in overcoming this grief. Recapitulating these sentiments in brief would fall utterly short of the drama and symbolism of the original. This final paragraph will instead be quoted in full:
Formerly, the infinite grief only existed historically in the formative process of culture. It existed as the feeling that ‘God Himself is dead,’ upon which the religion of more recent times rests; the same feeling that Pascal expressed in so to speak sheerly empirical terms: ‘Nature is such that it signifies everywhere a lost God both within and outside man.’† By marking this feeling as a moment of the supreme Idea, the pure concept must give philosophical existence to what used to be either the moral precept that we must sacrifice the empirical being, or the concept of formal abstraction [e.g., the categorical imperative]. Thereby it must re-establish for philosophy the Idea of absolute freedom and along with it the absolute Passion,‡ the speculative Good Friday in place of the historic Good Friday. Good Friday must be speculatively re-established in the whole truth and harshness of its Godforsakenness. Since the [more] serene, less well grounded, and more individual style of the dogmatic philosophies and of the natural religions must vanish, the highest totality can and must achieve its resurrection solely from the harsh consciousness of loss, encompassing everything, and ascending in all its earnestness and out of its deepest ground to the most serene freedom of its shape.
Speculative philosophy was thus supposed to give rise to the spiritual rebirth of mankind. Hegel understood the suffering that this project would entail. Speculative inquiry would be unrelenting in the infinity of its criticism, tearing down the finite (or false) infinites of the old objects of metaphysical faith. Skepticism and nihilism would surely appear along the way, but the spirit must be undeterred. It must move, unblinkingly, through “the whole truth and harshness of its Godforsakenness.” At no point can it revert to the faith of the old metaphysics, to the “historic Good Friday.” Nor can it turn to false idol of the Ought, into the insincere (and ultimately self-righteous) humility of finitude, consigning oneself to faithful ignorance. It must philosophically reenact the Passion itself, and experience its resurrection following the “speculative Good Friday.” This moment would herald its transfiguration — the absolution of spirit.
These remarks, which appear in the final section of Faith and Knowledge, are as programmatic as they are conclusive. They constitute Hegel’s prescription for philosophy in the modern era, an age fraught with anxiety and nihilism. This is all the more true in light of the system Hegel would eventually build upon these foundations. Within this work, however, they also served to synthetically summarize the essentials of the arguments laid out earlier in the piece. In contrast to the more methodical presentation of the Idea or speculative infinite in his later Science of Logic, Hegel’s views on the matter were formed in specific contrast to those belonging to his opponents. The style, therefore, is plainly polemical. Hegel’s overarching thesis regarding subjective philosophy’s failed notion of the infinite is provided more generally in the introduction, in such a way that it groups all three thinkers (Kant, Jacobi, Fichte) under a common objection:
The philosophies of Kant, Jacobi, and Fichte are the completion and idealization of…empirical psychology; they consist in coming to understand that the infinite concept is strictly opposed to the empirical. They understood the sphere of this antithesis, a finite and an infinite, to be absolute: but [they did not see that] if infinity is thus set up against finitude, each is as finite as the other.
His position thus described, Hegel proceeds to critique each of these systems. In the first and third sections of the article, he challenges the fixed infinities of the subjective idealism of Kant and Fichte. The longest section of the work appears between these two philosophers, and is dedicated to the metacritique of Jacobi’s various criticisms of the Spinozist, Kantian, and Fichtean systems. Hegel’s interpretation of Jacobi with regard to the first of these philosophers is especially harsh, however. His apology for Spinoza’s infinitum actu, against Jacobi’s cavil concerning empirical infinites, and time yields his most sustained account of the speculative infinite in Faith and Knowledge. Its discussion in this section bears directly upon Hegel’s final analysis, alluded to above, and lays forth the formula for the speculative infinite that would become a hallmark of his philosophy.
(α) Kant and the rejection of the genuinely infinite
In his treatment of Kant, Hegel reviews the paralogisms of the Critique of Pure Reason. In a gesture of respect, he credits Kant with the effective critique of attaching predicate infinities to objects of the mere understanding: “The polemical side of Reason, as expressed in the Paralogisms [of Pure Reason], has no other concern save that of setting aside (aufheben) the concepts of the intellect [i.e., the categories] as predicates of the Ego. The Ego is to be raised up into the intelligible realm out of the sphere of the thing, and of objective, finite determinations.” Following this, however, Hegel rebukes Kant for his evasive solution to the “mathematical antinomies [the first and third antinomies].” Instead of taking the ambitious speculative move to “suspend finitude itself,” Kant retreated from the contradiction. He “remove[d] the conflict by making it absolute.” For Hegel, resolution of the problem in such an “insipid fashion” was completely unacceptable; here lay the root of Kantian dualism. Speculative reason is left neutered, “crushed completely,” while the “[i]ntellect and finitude are quite properly exultant.” Its partial rehabilitation in the Canon of Pure Reason chapter of his first Critique, for purely practical purposes, was insufficient for the kind of scientific philosophy that Hegel envisioned. Kant, in his critical examination of our experience of the world (particularly nature, in the Critique of Judgment), always flirted with the possibility of infinite insight for which speculative reason would allow. But each time he reached this point, he would withdraw, convinced in the necessary limitation of human reason. As soon as one is confronted by the possibility of a speculative solution, “there the matter must rest; we must absolutely not go beyond finite cognition.”
Clearly, Hegel was not satisfied with Kant’s conclusions on these points. So far, we have seen Hegel’s interpretation of the Critical philosophy with reference to the specifics of Kant’s system. However, in the final paragraph of his chapter on Kant in Faith and Knowledge, Hegel frames his objection more generally:
The common ground which [the Kantian philosophy] shares with the philosophies of reflection that we are talking about is that knowledge is formal knowledge; Reason as a pure negativity is an absolute Beyond; as a Beyond and as negativity it is conditioned by this-worldliness and by positivity; infinity and finitude, each with its opposite, are all equally absolute…The highest Idea of the Kantian philosophy is the complete emptiness of subjectivity, or the purity of the infinite concept, which is also posited as what is objective in the sphere of the intellect, though there it has the dimensions of the categories; where on the practical side the infinite concept is posited as objective law. On one side there is infinity infected with finitude, on the other side, there is pure infinity, and in the middle there is the posited the identity of the finite and the infinite, though once more only in the form of the infinite, that is, as concept.
The translation of Kant’s architectonic into Hegel’s unique language of infinity and fin-itude is immediately confounding; nevertheless, reference to the more lucid exposition in his later Encyclopedia Logic will help clarify the matter. To begin with, Hegel describes Kant’s division of knowledge into two distinct forms: the understanding (or intellect) on one side, speculative reason on the other. This division is one shared by the “philosophies of reflection that we are talking about” — or subjective philosophy in general. “Kant was the first,” Hegel later wrote, “to emphasize the distinction between understanding and reason in a definite way, establishing the finite and conditioned as the subject-matter of the former, and the infinite and unconditioned as that of the latter.” Comparison with the excerpt from Faith and Knowledge sheds light upon many of its more obscure phrases. The understanding described in this second way corresponds to the knowledge “conditioned by this-worldliness and by positivity.” Reason, qua infinite and unconditioned, is the “pure negativity” of an “absolute Beyond.” The utility (or definition, one might even say) of the pure concept vacillates between these two spheres, finite in its intellectual determinations of empirical objects, but infinite in its rational speculations (see the schema in Figure 1 of Kant’s system on the following page).
The greatest fault of this distinction, according to Hegel, lies in its implicit dualism — in the incommensurability of the two sides. His dissatisfaction is obvious when he writes that, for Kant, “infinity and finitude, each with its opposite, are all equally absolute.” By positing the two sides of the antithesis as simultaneously absolute, it becomes unclear as to the way in which these opposites interact. And, as we have seen, when the two sides do come into conflict, Kant clearly sides with the finite determinations of the understanding over the infinite speculation of reason. The antithesis is thus made tragic; mankind is inexorably torn between reason’s infinite demand and the finitude of our intellects, which insuperably limits us. The transcendence of this limitation appears as an Ought, and deontologically governs our moral action by its reasonable injunctions. But this Ought is never allowed to come to pass ontologically, to be conjoined with the Is, and hence the tragic disjunction is made permanent. Kant might well be compared to the Solomon of Ecclesiastes: a lifetime of accruing wisdom and reflecting upon our epistemic capacities had revealed to him that all objective metaphysics is ultimately a “vanity and a striving after the wind” (Ecclesiastes 1:14). But the feeling of tragedy was not lost on Kant — for “God [Kant might say ‘Reason’] has put eternity into man’s mind, yet so that he cannot find out what God has done from the beginning to the end” (Ecclesiastes 3:11). This is why, as he wrote in the preface to the 1787 edition of the first Critique, Kant “had to deny knowledge in order to make room for faith.”
|THE CONCEPT||THE UNDERSTANDING||SPECULATIVE REASON|
THEORETICAL REASON– “the identity of the finite and the infinite”; finite insofar as they are determined as distinct categories of the understanding, but infinite in their speculative purity; dual aspect
– the “form of the infinite”
– the “infinite as such” – based on experience, empirical; “conditioned by this-worldliness”
– “infinity [of the pure concept] infected with finitude” — i.e., infinity of pure concepts applied to finite objects of experience
– concept as “objective,” positive, “dimensions of the categories”– abstracted from experience, purely rational; Unconditioned
– realm of the Idea; the “purity of the infinite concept”
– “the complete emptiness of subjectivity,” simply formal; hence “pure negativity”
PRACTICAL REASON– the “infinite concept” of speculative reason, simply formal– purely formal, infinite concept of speculative reason “posited as objective law,” as in determinations of the understanding; however, qua speculative, are nevertheless Unconditioned and Ideal, appearing in the form of the “Ought”
Figure 1: A schematic of Kant’s system according to Hegel.
(β) The Fichtean ego as Tantalus
Fichte’s foundationalist reworking of the Kantian philosophy likewise failed to bridge the gap between infinite formality and finite materiality. The first principle of the Fichtean system, the infinitely self-identical Ego, was (in spite of its proclaimed infinity) deemed deficient, and stood in need of an opposed non-Ego to complete it. For Fichte, Hegel explains, “[t]he starting point [the Ego] is absolute, yet finite. Its finitude makes it im-possible for the birth of cognition to be [the construction of] a genuine whole; for a genuine whole where no part exists in itself. So, a true Ideal in which the finitude of empirical reality would disappear and [subjective] affections would become Nature is strictly impossible.”
This requires some background. Following the initial reception of Kant’s philosophy, Reinhold and his successor Fichte in Jena attempted to ground the results of the Kantian system in a more stable first principle. Naturally, this principle would have to be Unconditioned. The procedure in Fichte’s Science of Logic begins from the bare tautology of A = A, which, when bourn out, is replaced with the proposition Ego = Ego, which seems confirmed in the statement “I am.” This mathematical tautology was then substituted for the sentential logic of a conditional tautology (Ego É Ego; if I am, then I am). As per the structure of this proposition, the Ego of the if clause is relationally un-conditioned, whereas the Ego which follows after then is conditioned by the former. In other words, the first Ego might be considered antecedent to the second Ego’s consequence, though their relation is rationally simultaneous. The second is posited by the first as the empirical Ego. The connection between the two is termed X – probably an allusion to the deduction to the first edition of the Critique of Pure Reason – and the determination of the one from the other is judged by Fichte to be thetic, or analytic, in Kantian terms. X thus being implicitly posited as the necessary relation of something to itself as identical, one is thus able to infer the existence of the unconditioned Ego or A as positing. They reciprocally imply one another. Both elements of this principle (A and X) are assumed to belong to a self-conscious I which posits them.† This Ego is responsible for its own positing act, or, as Fichte famously christened it, the Tathandlung (deed-action). This might be called the properly subjective side of Fichte’s philosophy.
Opposed to the infinite self-identity of the Ego, Fichte also claims that “the I, over and above positing itself, has the ability (1) posit unconditionally the Not-I, that is, it has the power, by what Fichte calls ‘an absolute act,’ to counter posit something that is exactly the opposite of, or in opposition to, the I. The I is also in the position (2) to posit unconditionally the divisibility of the I and the Not-I.” This, for Hegel, is the origin of Fichte’s dualism. The formal emptiness of the Fichtean subject (the I) is conceived as standing in need of objective content, which can only be provided by that which is Not-I. By positing the Not-I, Fichte accounted for the objective feature of experience. However, the fatal error was committed in positing both the empirical I and the Not-I unconditionally. The problem with maintaining two separate entities as unconditionally posited is the means be which they relate to one another; should the two sides interact, each could correctly be said to have conditioned the other. Hegel joked that the formal void of Fichte’s Ego was somehow supposed to give rise, by its very lack, to a plenum of objective content that might fill it: “its infinite poverty being the infinite possibility of wealth.” Nevertheless, the subjective side of Fichte’s foundational Ego, the I (or Self), was pitted absolutely against the objective side, the Not-I (or Nature). The Self, by dint of its essential predilection, seeks to rationalize or cultivate Nature, which by itself is worthless, mere potentiality. It strives to envelop Nature into its rational fold, but is destined never to overcome its division. This dynamic comprises one of the central uses of infinity in the Fichtean system. Or, as Hegel puts it: “In Fichte, th[e] subjectivity of yearning is itself turned into the infinite, it is something thought; it is an absolute requirement, and as such it is the climax of the system: the Ego ought to be equal to the non-Ego. But no point of indifference can be recognized in it.”
Locating infinity in the “infinite progress” or “infinite striving” of the Fichtean Ego toward overcoming its negation, Hegel admonished the imposition of an insurmountable limit on human knowledge. In place of knowledge, Fichte, like Kant, ultimately adopted a position of faith regarding objects of our apperception. Fichte’s popular work The Vocation of Man, published in 1801, had clearly marked “Faith” as the ultimate goal of the human enterprise. It was placed as the culmination of the progression from Doubt to Knowledge to Faith. In this final chapter, Fichte’s narrator admits the impossibility of the subjective striving. Speaking to the Absolute, he confesses that “[a]fter thousands upon thousands of spirit-lives, I shall comprehend Thee as little as I do now in this earthly house. That which I conceive becomes finite through my very conception of it; and this can never, even by endless exaltation, rise into the Infinite.” Thus the epistemic capacity of man walled off from truly comprehending the Absolute Idea, the speculative infinite. Theoretically, knowledge is again replaced by faith. This is, as before, unacceptable to Hegel’s sense of the scope of philosophy. Pure reason is handicapped at the very outset; no wonder knowledge is inevitably exchanged for faith. “Because of the absolute subjectivity of Reason and its being set against reality,” Hegel wrote, “the world [qua objective] is, then, absolutely opposed to Reason. Hence it is absolute finitude devoid of Reason, a sense-world lacking [internal] organization. It is supposed to become equal to Ego in the course of an infinite progress, or in other words it is absolute and remains so.”
In the practical realm, too, a notion of the perpetual progression of spirit (for Fichte, the Ego) toward the Absolute similarly stands in for the genuine absolution of spirit. Fichte writes:
The universe is to me no longer what it was before — the ever-recurring circle, the eternally-repeated play, the monster swallowing itself up only to bring itself forth again; it has become transfigured before me, and now bears the one stamp of spiritual life — a constant progress toward higher perfection in a line that runs out into the Infinite.
This progress is guided formally by the Ought, and as such its object (the Absolute as such) is never able to obtain ontologically. Hegel writes that in Fichte’s moral philosophy, “the infinite is posited as originally un-unified and un-unifiable with the finite, the Ideal (das Ideelle) cannot be united with the real or pure Reason with existence.” The progress quickly reveals itself as a vain advance. The infinite, or absolute Idea, is no closer for this striving. This striving is the linear infinity of mathematics, which is often presented as a serial progression of finite determinations. Fichte’s moral system is thoroughly Lutheran, in that the subject is justified by faith alone, in his intentions and not in the consequences of his actions. “In Fichte,” Hegel writes, “the moral world order that exists in the context of faith (welche im Glauben ist) is strictly outside the Ego. Only through an infinite progress does it acquire reality for the Ego, only so does the Ego enter into it or vice versa.” Earlier, Hegel points out the “perennial” nature of the Ought, a form which it can never shed, for all its so-called “progress”: “[B]ecause formal thought does not ever truly give itself up, the Ought is perennial: it is an enduring will which can only achieve a break-through to infinity and the Nothing; it remains incapable of breaking through infinity and the Nothing to positive rational cognition.”
The Fichtean Ego might easily be compared to the tormented Tantalus of Greek mythology, reaching tirelessly after its sole desire — insight into the fundamental unity of all thought, the Absolute. But no sooner is it reached for than it recedes from his grasp. The demand of reason can only be sated by the Absolute, but it is given no hope for ever eating of its fruit. For Hegel, adhering to such a philosophy is madness. One is shown the goal, the way things rationally Ought to be, but is told at the same time that it will never be obtained. The alleged “infinite progress” revealed itself to be an infinite nothing, without hope of redemption. For one can hardly consider himself closer to his goal when at one moment appears infinitely distanced from him, and when, some time later, after further “progressing,” the goal remains infinitely removed. The integration of Is and Ought, of finite and infinite, is suspended indefinitely, and mankind if condemned to an everlasting dissatisfaction.
(γ) Jacobi’s spurious critique of Spinozism
Hegel’s interpretation of Jacobi in Faith and Knowledge appears in the chapter between his chapters on Kant and Fichte. The critique of Jacobi, especially as concerns his judg-ment of Spinoza’s infinitum actu, provides the clearest exposition of Hegel’s Idea of the genuine infinite. Hegel’s speculative infinite thus bears a remarkable affinity with Spinoza’s objective formulation, and sharply contrasts with Jacobi’s subjective alternative. Though it upsets the order of argumentation as it is laid out in the original text, the more definitive nature of Hegel’s argument for the genuine infinite here lends itself best to the conclusion of our discussion. For this reason, it appears last.
Hegel charged Jacobi with gravely misreading Spinoza’s philosophy, and excoriated Jacobi’s central criticism of Spinozism as fundamentally misconceived. Quite often Hegel borrowed from Jacobi’s critical observations, but on this matter he parted with him emphatically. Jacobi had declared Spinoza’s notion of the infinite to be ulti-mately confused, especially when it came to his conception of time. According to Hegel, Jacobi had ignored the basic distinction Spinoza had drawn between the faculties of intellectus (or understanding) and imaginatio (the productive imagination). The latter of these allows us to imagine existent modes as potentially inexistent. From this, the illusion of metaphysical Possibility arises. The Substance which grounds its attributional modifications must (necessarily) be understood (via the intellectus) to be eternal, or atemporal — properly “infinite, indivisible, and single.” The imagination, by contrast, conceives the modal objects of perception as possessing only a limited durability, hence temporal, and is therefore “divisible, finite, composed of parts [composite], and mani-fold.” Owing to this misprision, Jacobi exaggerated the problematic of an infinite temporal succession of finite things within a notion of eternity, and thus took modal time to stand in contradiction to the eternal Substance which grounds them. This, Hegel claims, overlooks the thoroughly holistic approach of Spinoza’s system, and conceives of a partial negation (which Spinoza defines as the finite) as somehow incompatible with an affirmation of the whole (the infinite). Jacobi thus takes time to be something in-itself, or absolute, something which Substance cannot consume without collapsing in upon itself. He fails to grasp the relativity of “empirical infinites” (Hegel’s terminology for infinite linear sequences of finite beings) within Spinoza’s system: “Jacobi makes Spinoza responsible for th[e] empirical infinite with no more ado, although no philosopher was ever farther removed from assuming anything of the sort than Spinoza was; for since he regards finite things as nothing in themselves, this empirical infinite and time disappear at once.”
The problem hinges upon a conception of totality, which places multiplicity within unity, in an understanding which explains finitude as issuing forth from infinity. Swapping Spinoza’s term, “imagination,” for his own, “reflection,” Hegel attempts to disentangle the mess of an objective metaphysic of infinity:
[I]t is only imagination, as Spinoza calls it, or, in general, only reflection that posits and partially negates the finite; and this partially negated thing, which, when posited for itself and opposed to what is in itself not negated, to what is strictly affirmative, turns this infinite itself into something partially negated. The infinite, being thus brought into antithesis with the finite becomes an abstraction, the pure Reason, or the infinite of Kant. The eternal is to be posited as the absolute identity of both; and in the eternal the infinite on the one side, the finite on the other, are once more nullified as to the antithesis between them.
This first view belongs to Spinoza, and, as we shall see, shares many features with Hegel’s notion of the speculative infinite (replace “the eternal” with “the Absolute” or “Absolute Spirit”). Contrariwise, Jacobi’s (mis)interpretation of Spinoza’s philosophy proceeds as follows:
It is quite another matter when the finite and infinite in their abstractness are to remain what they are and each is to be taken up into the form of an opposite. Here, the one is determined as not being what the other is, and each is posited and not posited, as being this determinate being and as something else. Anything posited in this fashion runs to empirical infinity. Duration, simply as it is posited by imagination, is a time moment, a finite; fixed as such, it is something partially negated, something that is in and for itself determined as being also another moment; and this other moment which likewise receives its actual being through the imagination is yet another moment. This negation, remaining what it is, and made positive through imagination, results in the empirical infinite, that is, in an absolute, unresolved contradiction.
The lucidity of this distinction might easily be questioned, but Hegel’s reasoning is sound. In Spinoza’s philosophy, the eternity of Substance is posited as the identity of its finite modalities with their infinite temporal succession. Each side of this operation is clearly definite, determined by the mutual negativity of their relationship. As such, not even the temporally infinite is truly infinite. Only in the seamless, affirmative indeterminacy of Substance (deus sive natura) is genuine infinity to be found. Since both parts of this antithesis are mediated into a greater synthetic unity, Spinoza’s formulation is thus concrete. This exercise is identical to the one Hegel later performed in The Science of Logic. By contrast, the empirical infinite of duration is posited by Jacobi as something which is in-itself ultimate, and is thereby kept immediate. But it is only so by abstracting from its relation with the finite things that make up its succession. Jacobi, following the path taken by the subjective rationalists (Kant and Fichte), falls prey to abstract thinking.
Hegel believed that reflection upon this problem “yields the true character of thought which is infinity.” This might be thought of as analogically related to “mathematical similes,” as Spinoza had sought to provide, but ultimately must be explained qualitatively, not in terms of quanta. In the most singularly systematic exposition of infinity contained in Faith and Knowledge, Hegel resolves to
…briefly bring together the forms of infinity. The true infinite is the absolute Idea, identity of the universal and particular, or identity of the infinite and finite themselves (i.e., of the infinite as opposed to a finite). This [opposed] infinite is pure thinking. Posited as this abstraction, it is pure, absolutely formal identity, pure concept, Kant’s Reason, Fichte’s Ego. But when it is set against the finite, it is for this very reason, the absolute nothing of the finite: + A – A = 0. It is the negative side of the absolute Idea. If this nothing is posited as reality, if infinity itself is taken, not as subject or as producing — where infinity is pure identity as well as nothing — but as object or product, infinity is the + A – A, the positing of opposites. But none of these forms is empirical infinity, the infinity of imagination. The first infinity is that of absolute Reason. The infinity of pure identity or of negativity is that of formal or negative Reason. But the infinite in its reality as + A – A, where one term is itself determined as infinite and the other as finite, this finitude in general is the infinite of reflection or imagination.
The genuine infinite, insofar as it is the absolute Idea, is the vanishing point for all dist-inctions between universality and particularity, infinity and finitude. It is the composition of the simple, and the simplicity of the composite. The formally infinite (“pure concept, Kant’s Reason, Fichte’s Ego”) is such only by its definite identity, set over against and thereby bounded from the materially finite. In other words, it is not truly infinite. Rather, it is what Hegel later refers to as the “bad infinity” or “spurious infinity” of The Science of Logic. “A” stands for that which is infinite; its naked affirmation as that which is infinite (“+ A”) is set against its negation, or that which is not infinite (“– A”), of the finite. This infinity, along with its bare antinomy, the finite, are both sublated in the absolute Idea, qua genuinely infinite, pure unboundedness. The Absolute, in its subsumption (one might even say “consumption”) of the abstractly infinite and finite, is therefore everything and nothing(“+A – A = 0”) all at once. It contains within itself the determination of every particular condition which issues from it, but is, from the stand-point of the whole, completely indeterminate.
The Paschal Spirit: Transfiguration and Absolution
In The Science of Logic, Hegel famously defined the finite as that which “ceases to be; and its ceasing to be is not merely a possibility, so that it could be without ceasing to be, but the being as such of finite things is to have the germ of decease as their being-within-self: the hour of their birth is the hour of their death.” Finitude is by its limitation, its own fated death. Its immediate essence is negative. The reflexive self-reference of its qualities thus conjures plainly Christian imagery: death itself will die (Revelations 21:4). This is the promise which transcends the spiritual torment of the Good Friday, in Hegel’s speculative conception of it as well as the historical conception. The former will invert the latter: just as God became man by kenotically emptying Himself of content (Philippians 2:7), so man must become God through its speculative reclamation. The spirit of mankind is thus transfigured, and realizes the “spark of divinity” it innately contains by its absolution.
For Hegel, true science can stop at nothing short of omniscience (infinite, all encompassing knowledge). Spirit in the formal infinity of its isolation is desolate; only by means of its reunification with the world’s material finitude can it achieve consolation.
 See the diagram in footnote #51 at the bottom of the page. Georg Wilhelm Freidrich Hegel. Faith and Knowledge, or the Reflective Philosophy of Subjectivity in the complete range of its forms as Kantian, Jacobian, and Fichtean Philosophy. Translated by Walter Cerf and H.S. Harris. (State University of New York Press. Albany, NY: 1977). Pg. 111
 “…Kant calls the thought-product – and, to be precise, the universal and the necessary – ‘objective,’ and what is only sensed, he calls subjective.” Georg Wilhelm Friedrich Hegel. Encyclopedia Logic. From The Hegel Reader, translated by … and edited by Stephen Houlgate. Pg. 154. §41
 Ibid., pgs. 143-162. §26-51
 Ibid., pg. 144. §28, Addition
 Hegel, Faith and Knowledge. 113
 Immanuel Kant. The Critique of Pure Reason. Translated by Paul Guyer, with an introduction by Allen Wood. (Cambridge University Press. New York, NY: 1998). Pgs. 470-472
 Baruch Spinoza. The Ethics. From The Ethics and Selected Letters. Translated by Samuel Shirley, edited and introduced by Seymour Feldman. (Hackett Publishing Company. Indianapolis, IN: 1982). Pg. 31. Part I, Definition 6, Explication
 Aristotle. The Physics: Books V-VIII. Translated by P.H. Wicksteed and F.M. Cornford. (Loeb Classical Library, Harvard University Press. Cambridge, MA: 2006). Pg. 239b
 Hegel, Encyclopedia Logic. Pg. 144-145. §28, Addition
 Ibid., pgs. 144-145. §28, Addition
 Hegel, Faith and Knowledge. Pg. 190
 Ibid., pgs. 190-191
 Hegel earlier rejects the “holier-than-thou” self-righteousness of the Ought: “The monstrous arrogance, the conceited frenzy of this self, filled with loathing and sadness, at the thought that he is one with the universe, that eternal nature acts in him — to be filled with loathing, be horrified and sad over the resolve to subjugate oneself to the eternal laws of nature and to its hallowed and strict necessity, to be in despair because one is not free, free from the eternal laws of nature and its strict necessity, to believe that one makes oneself indescribably miserable by this obedience — all this presupposes an utterly vulgar view of nature and of the relation of the singular person to nature.” Ibid., pg. 176
 Ibid., pgs. 63-64
 Ibid., pgs. 82-83
 Ibid., pgs. 84-85
 Kant, Critique of Pure Reason. Pgs. 672-684
 Hegel, Faith and Knowledge. Pg. 91
 Ibid., pg. 96
 Hegel, Encyclopedia Logic. Pg. 157. §45, Addition
 Kant, The Critique of Pure Reason. Pg. 117
 Hegel, Faith and Knowledge. Pg. 162
 Kant, The Critique of Pure Reason. Pgs. 130-131
 Though this argument is given forcefully enough in the Fichte’s own Science of Knowledge, Horstmann’s succinct article on the matter brilliantly sums it up. Rolf-Peter Horstmann. “The Early Philosophy of Fichte and Schelling.” From The Cambridge Com-panion to German Idealism. Edited and introduced by Karl Ameriks. (Cambridge University Press. Cambridge, England: 2000). Pgs. 121-122
 Ibid., pg. 124
 “The acknowledged incompleteness of the absolute principle and the acknowledged necessity of going on to something else in consequence form the principle of the deduction of the world of sense. Because of its absolute deficiency the completely empty principle [Ego = Ego] from which [Fichte] begins has the advantage of carrying the immediate necessity of self-fulfilment [sic] immanently within itself. It must go on to something other [than itself] and from that to something else in an infinite objective world. The necessity rests upon the principle’s being nothing but a part and upon its infinite poverty being the infinite possibility of wealth…In one role it is absolute, in the other strictly finite; and in the later quality it can serve as the point of departure for the entire empirical infinity.” Hegel, Faith and Knowledge. Pg. 157
 Ibid., pg. 153
 Johann Gottlieb Fichte. The Vocation of Man. Translated, edited, and introduced by Roderick M. Chisholm. (The Bobbs-Merrill Company, Inc. New York, NY: 1956). Pg. 140
 Hegel, Faith and Knowledge. Pg. 179
 Fichte, The Vocation of Man. Pg. 152
 Hegel, Faith and Knowledge. Pg. 182
 Ibid., pg. 170
 Ibid., pg. 165
 “The controlling basic principle of this integration consists in this: the one side is absolutely not what the other one is, and no genuine identity emerges from any linkage between them. Just as for knowledge the true identity and eternity are in a Beyond that is faith, so in the practical [i.e., moral] sphere, the sphere of reality, they are in a Beyond that is the infinite progress.” Ibid., pg. 172
 Ibid., pg. 106
 Baruch Spinoza. 29th Letter.
 Hegel, Faith and Knowledge. Pgs. 106-107
 Ibid., pg. 108
 Ibid., pgs. 107-108
 Ibid., pg. 112
 Ibid., pgs. 113-114
 Georg Wilhelm Friedrich Hegel. Science of Logic. Translated by A.V. Miller, with a foreword by J.N. Findlay. (Humanity Books. Amherst, NY: 1969). Pg. 129
Johann Gottlieb Fichte. The Vocation of Man. Translated, edited, and introduced by Roderick M. Chisholm. (The Bobbs-Merrill Company, Inc. New York, NY: 1956)
Georg Wilhelm Friedrich Hegel. Encyclopedia Logic. From The Hegel Reader, translated by … and edited by Stephen Houlgate
Georg Wilhelm Freidrich Hegel. Faith and Knowledge, or the Reflective Philosophy of Subjectivity in the complete range of its forms as Kantian, Jacobian, and Fichtean Philosophy. Translated by Walter Cerf and H.S. Harris. (State University of New York Press. Albany, NY: 1977)
Georg Wilhelm Friedrich Hegel. Science of Logic. Translated by A.V. Miller, with a foreword by J.N. Findlay. (Humanity Books. Amherst, NY: 1969)
Rolf-Peter Horstmann. “The Early Philosophy of Fichte and Schelling.” From The Cambridge Companion to German Idealism. Edited and introduced by Karl Ameriks. (Cambridge University Press. Cambridge, England: 2000)
Immanuel Kant. The Critique of Pure Reason. Translated by Paul Guyer, with an introduction by Allen Wood. (Cambridge University Press. New York, NY: 1998)
Baruch Spinoza. The Ethics. From The Ethics and Selected Letters. Translated by Samuel Shirley, edited and introduced by Seymour Feldman. (Hackett Publishing Company. Indianapolis, IN: 1982)
† The passages in which Hegel addresses the subject of infinity appear in Faith and Knowledge, The Jena System, 1804-1805: Logic and Metaphysics, The Science of Logic, and the Encyclopedia Logic.
‡ “By God I mean an absolutely infinite being; that is, substance consisting of infinite attributes, each of which expresses eternal and infinite essence.” Spinoza. The Ethics. Part I, Definition 6
† Of course, Kant also postulates the infinity of the world of the noumenal things-in-themselves. But since he claims this is wholly inaccessible to the understanding, this will be excluded from the present discussion.
† Translated into English from the original in place of the French of the quoted text.
‡ As in Christ’s Passion.
† Incidentally, this is the point where Fichte introduces the Ego to replace the symbol A as the absolutely unconditioned principle of his philosophy. Of course, the Ego can be retroactively interchanged for A in the preceding formula.