Full text of one of Aadi Arya Karthik submissions for a class at Oxford.
Finitude and Infinitude
There is an obvious and immediate inconsistency in treating the finite and the infinite as
two poles standing against each other, two ends of a continuum: to do so is to strip the infinite of
its infinitude. Every something is definitionally finite, bounded in both time and space: comes to
be at its birth, and comes to be at its edges. To set the finite against the infinite as something
against something else is thereby to bound the infinite, to make it “just another finitude,” so to
speak—to render it an untruth, a “bad infinity.” And “bad infinity” is that infinity which is
proximally understood: the infinity of the ad infinitum, the interminable. For to declare that a
given process does not terminate is to note its repeating of the same (finite) algorithmic steps in a
manner that will not lead to termination, and so to represent infinity as a stack of finite blocks
whose apex is simply always further than the eye can see. The word interminable indicates a
negation in the proximal sense, but not a Hegelian negation, there being nothing affirmative
about it; not a critique, interminability prompting or resolving no crisis in terminability. This
discrete infinity presents itself as and in a swindle of sorts: what is discretely infinite is never
something that one can actually come to know in any sense; one is simply told to take on faith
that what one sees will go on forever. Any concept of discrete infinity, then, covertly imports
with it a concept (a vulgar concept, that is, not a Grenzbegriff) of the thing-in-itself, or at least a
concept with the same intolerable flaw as the concept of the thing-in-itself: its artificial positing
of some fraction of nature which we cannot and do not know.
A more sustainable infinitude is needed; and Hegel’s solution is what one might think of
as continuous infinitude: that is, a rendering of infinitude immanent and therefore a making of it
philosophically proper in some sense—not a swindle. This new concept is quite easily
exemplified mathematically (whence my motivation for using the descriptors discrete and
continuous): the infinity of the set of all natural numbers is a “bad” infinity; the infinity of the set
of all real numbers between zero and one is proper, being the infinity that constitutes an interval
“finite enough” to present itself to us at once. (It is notable, of course, that even mathematically
these are two different infinities, the latter being “bigger” than the former.) Hegel’s own example
is perhaps more philosophically profound: true infinity is evidenced in the “relation [of an entity]
to itself in its transition and in the other”;1 true infinity is the finite entity’s “attain[ing] its being-
in-itself,” “rejoin[ing] itself,” in its termination, its death.2 After all, what characterizes the finite
entity in its finitude is its coming-to-an-end (equivalently, its coming-to-be); a finite entity is not
finite unless and until it is resolved that it will have died. But a purely finite entity also cannot
undergo death in any real sense, since at the moment of death it is no longer itself but something
else: that is, it is correct for me to insist that I will never die, in the sense that at any point there is
an I to speak of it has not yet died; death is always yet to come. Hegel gives the spatial
equivalent of this argument: “something can be known, even felt to be a barrier, a lack only
insofar as one has at the same time gone beyond it”;3 the finite entity must at once remain within
its spatial limits and extend past them. The declared finitude of the entity conceals an infinitude
immanent to the entity, an infinitude which is the very act of reaching into the other, whether this
reaching is conceived of as the entity’s reaching past its limits—its being related to that which is
other than it—or the unit interval’s being in bijection with the set of all reals, this bijection
perhaps metaphorically constituting its own relating to the other, its own reaching-past.4 True
infinitude is not the infinitude to which the finite is opposed, but the infinitude which all
finitudes betray upon inspection.
However, true infinitude is not the finitude which all “bad infinitudes” betray upon
inspection; true infinitude is intended as a replacement for “bad infinitude,” not as a sublation of
it.5 Elsewhere, Hegel in fact assigns the axiom which identifies the “finitized infinite” with the
“infinitized finite” to the fallen understanding, names this identification “scandalous.”6 At very
least it is made clear that there is no structural, topological, methodological equivalence between
the dialectic consisting of being, nothing, and becoming and that consisting of the bad-infinite,
finite, and true-infinite: the revealings, vanishings which take place in the former seem not at all
paralleled by those taking place in the latter. Being vanishes into nothing and nothing into being
the moment they are articulated (whether in the intellect or in the world): the concepts are
volatile. Likewise, it is not difficult to conceive of becoming as “spontaneously generated” by
this vanishing; that is, the vanishing of being into nothing, and of nothing into being, is not to be
explained by becoming: becoming is simply and plainly exhibited there. In other words, the
being, nothing, becoming triad is exemplary in that it presents a phenomenon that may be clearly
phenomenologically described as taking place in the same way “in thought” and “in reality”: its
exposition parallels its conceptual form, which in turn parallels the affinities of the concepts
themselves, their action independent of the observer. This exemplarity seems not to be present in
the second triad, as we will see below.
It is worth first discussing exactly what the first two moments of this triad—whose third
member must be true infinitude—are to be. The triad bad-infinite, finite, true-infinite seems most
consonant with the facts of Hegel’s exposition, as well as his broader aims, given that he argues
from the inadequacy of the bad-infinite to the adequacy of the true-infinite, and that Hegelian
dialectic is to be a method of arguing from inadequate concepts to adequate ones. But Hegel is
careful to specify that the bad-infinite is not sublated in the true-infinite, is nothing more than a
failed attempt at sublating the finite;7 to take as axiomatic the self-consistency, the intelligence of
a great thinker like Hegel—the notion, more precisely, that he knew what he was doing, that any
glaring or critical error one finds is more likely an error on the part of the reader—this being
really the only axiom I am comfortable regularly taking—is to discount the idea that Hegel is
arriving at the truly infinite via this particular dialectic. Hegel also hints at a possible alternative
dialectic triad, that of something, Other, and true-infinite as the reaching of the (finite) something
into this Other. However, this dialectic movement presents its own difficulties. For one, both
being and nothingness are perfectly fine starting-points for their respective dialectic motion,
given that the vanishing of one into the other is the vanishing of the other into the one, thus
suggesting a symmetry between the first two terms of the dialectic. But one cannot posit an
Other before one posits a something; that is, a positing of a (finite) something—for Hegel, at
least, it must be said; I do not think the following obtains at all—immediately mandates and
indeed brings about the positing of an Other into which this something may reach, but a positing
of an Other cannot even take place without the prior positing of a something for the Other to
stake its Otherness upon. In addition, it is not entirely clear why the positing of a particular
something and its Other ought to bring about true infinity as concept and not a true-infinity, nor
is it clear that the above dialectic works when something is replaced by something-ness and
Other is replaced by Otherness. (One objection: Hegel does, I think, hint at this approach in
identifying determinate being, i.e. existence, with being-so, being something—reality8—but the
difficulty lies in articulating Otherness as a universal rather than a particular, which Hegel does
not even seem to attempt.) Finally, and most vitally, Hegel at no point indicates that he is using
this dialectic, making this particular maneuver; given that the rational structure of Hegel’s
exposition is actually to be the rational structure of the Umwelt, I think that claims of the type
Hegel meant to say [something] but did not do so do not hold water: the system is the system as
it is presented.
In either case, basic failures to parallel the architecture of the being, nothing, becoming
dialectic are exhibited; I think that it is only fair to mark these failures as intended. We are then
left with the problem: what grounds does Hegel have to use two different “dialectical”
procedures in the course of the exposition of his system? One might attempt to take dialectic to
indicate critique of critique—as I suggested it might be in my last essay—rather than the more
peculiar threefold process which is typically said to be Hegel’s; it does not seem, however, that
the derivation of true infinity might even be said to have the architecture of a critique of critique.
I believe this problem to be identical to, or at least answered by the same realization as, another
pressing one I have come to in my reading of Hegel: how does Hegel know where to stop his
derivation? The being, nothing, becoming dialectic is in itself a closed loop, a perfectly self-
similar system seemingly adhering to all of Hegel’s aesthetic commitments (I do not mean
aesthetic as a pejorative; all good commitments are aesthetic commitments); the only reason to
move beyond it is because one brings into the derivation some notion of what philosophy ought
to be and do—a notion which is a facticity, a product of one’s thrownness, perhaps—and finds
this dialectic insufficient. It is accepted enough that what is remarkable in a sentence is not the
words that constitute it but their arrangement and context both inside and outside the sentence,
signs being interchangeable; it is not too different to assert that what is remarkable in a thought is
the arrangement of “thought-content” within it, rather than the thought-content itself. A
meaningful portion of what is remarkable in the derivation of a system, then, is where the
derivation ends, where the system declares itself entirely systematic—which seems in Hegel to
evince an infiltration of the empirical into the system itself. I am sure Hegel will prove to be able
to evade this line of argument; I am simply at the moment not quite sure how.
—-
1 Hegel, The Encyclopedia Logic, s. 95.
2 Hegel, The Science of Logic, 21.123.
3 Hegel, The Encyclopedia Logic, s. 60; quoted in Moore, The Evolution of Modern Metaphysics, 165.
4 I am not particularly happy with this metaphor, but it’s orthogonal enough to the main direction of argument and
does its job well enough that I’ll leave it in for now.
5 See Hegel, The Encyclopedia Logic, s. 94.
6 Hegel, The Science of Logic, 21:132.
7 Hegel, The Encyclopedia Logic, s. 94.
8 See Hegel, The Science of Logic, 21:102.
Aadi Arya Karthik 