What is Radical Recursion?

Steven M. Rosen

Departments of Psychology and Philosophy (Emeritus)

College of Staten Island/City

©This paper is not for reproduction without permission of the author.

ABSTRACT

Recursion or self-reference is a key
feature of contemporary research and writing in semiotics. The paper commences
by focusing on the role of recursion in poststructuralism. It is suggested that
much of what passes for recursion in this field is in fact not recursive all
the way down. After the paradoxical meaning of *radical* recursion is adumbrated, topology is employed to provide
some examples. The properties of the Moebius strip prove helpful in bringing
out the dialectical nature of radical recursion. The Moebius is employed to
explore the recursive interplay of terms that are classically regarded as
binary opposites: identity and difference, object and subject, continuity and
discontinuity, etc. To realize radical recursion in an even more concrete
manner, a higher-dimensional counterpart of the Moebius strip is utilized,
namely, the Klein bottle. The presentation concludes by enlisting
phenomenological philosopher Maurice Merleau-Ponty’s concept of *depth* to interpret the Klein
bottle’s extra dimension.

1. SEMIOTICS, POSTSTRUCTURALISM, AND RECURSION

In classical signification, the stability of the
relationship between the signifier and what it signifies is maintained by
preserving the anonymity of the former. Attention is fixed solely on the
meanings that are signified, not on the act of signification itself. With the
advent of semiotics this changes. Semiotics is the discipline that *studies* the process of signification.
Here the sign becomes recursive; instead of focusing exclusively on signified
meanings, it comes to focus on itself. The signifier, which had played a
predominantly tacit role in classical semiosis, is now itself explicitly
signified.

Despite this role reversal inherent in the very
existence of the discipline of semiotics, structuralist semioticians like
Saussure still sought to preserve the invariance of the link between the given
signifier and what it signifies. The problem is that, once classical signification
is surpassed by signifying the signifier, the door is opened to an infinite
regress. For now, it seems that no signifier is exempted from mutation into
that which is signified. A new signifier is presumably needed to signify what *had* been the signifier, but this new signifier
is subject to signification by a still newer signifier, and so on *ad infinitum*. And each time the tacit
operation of the signifier is undermined by being explicitly signified, the
functioning of what had been signified by that signifier is also affected.
Ultimately then, we have in this “hall of mirrors” neither
signifier nor signified in any stable, abidingly meaningful form.

Poststructuralist writing exemplifies the recursive
“sliding” or “slippage” of the signifier. The approach
of psychoanalyst Jacques Lacan is a prime illustration. For Lacan, language
“is constituted by a set of signifiers” that involves what “I
call the Other” (1966/1970: 193). The “otherness” of language
results from the fact that, in its “chain of signifiers” (194),
every act of self-reference, rather than affirming the identity of the self or
subject that is referred to, always slips away into what is *other*, into a new and anonymous
signifier. As Lacan puts it:

All that is language is lent from this otherness and this is why the subject is always a fading thing that runs under the chain of signifiers. For the definition of a signifier is that it represents a subject not for another subject but for another signifier. This is the only definition possible of the signifier as different from the sign. The sign is something that represents something for somebody, but the signifier is something that represents a subject for another signifier. The consequence is that the subject disappears….” (1966/1970: 194)

In this way, the sign—which had constituted for earlier semioticians a fixed relationship between a signifier and its signified meaning, with the subject operating stably behind the scenes (the “somebody” to which Lacan alludes)—now dissolves into an evanescent flux of differences wherein the subject loses its substance, becoming a “nobody,” a ghost-like quasi-presence.

Much the same process of dissolution is reflected in
the deconstructionist writings of Jacques Derrida. In the “primary
writing” (1976: 7) of which he speaks, “[s]ign
will always lead to sign, one substituting the other...as signifier and
signified in turn” (Spivak 1976: xix). In Derrida’s own words,
language must be understood as a field “of *freeplay*, that is to say, a field
of infinite substitutions” (cited by Spivak 1976: xix) in which identity
fragments into sheer difference (*différance*).
The specific way this takes place is by the process of self-referential
mirroring in which, time and again, the signifier is displaced by being made
into what is signified by a newly implicit signifier.

It seems clear that the slippage of the signifier results from the indefinite repetition of recursion. This is the nature of semiosis, we are told; it is a “mirror game” in which self-identity is perpetually subverted and we continually slide into otherness and difference. While I certainly agree with the general consensus among semioticians that symbolic operations are inherently recursive, I propose that the infinite regress to which poststructuralism is prone actually derives from its failure fully to achieve recursion.

Again, when the signifier “slips,” it
becomes something that is now itself signified. Yet does this really constitute
a concrete instance of self-signification, or does it merely entail the close
juxtaposition of two semiotic acts neither of which are recursive in
themselves? Initially, X signifies Y. Then there is the reversal of this in
which X itself becomes signified by a new signifier, Z. This is obviously not
to say that X signifies X. Poststructuralist “self-signification”
then does not truly involve a signifier’s reference to itself within the
same *actual occasion*, to use
Whitehead’s (1978) term for a fundamental concrete event; it entails only
a switching of roles between signifier and signified from one occasion to
another. (No doubt occasions may follow each other in close succession and may
be broadly construed as belonging to the “same” occurrence; but, on
a more concrete level, the occasions of poststructuralism constitute distinct
semiotic acts.) Therefore, poststructuralist signification is not radically recursive,
not recursive all the way down into the roots of semiosis, for the signifier of
occasion 2 does not signify itself but only that which was the signifier on
occasion 1.

If poststructuralism fails to meet the challenge of radical self-reference because its signification within the actual occasion is strictly a reference to what is other, does radical recursion involve reference to the self in the sense of simple self-identity (X≡X)? It surely cannot. For without the aspect of the other, of difference, meaning is trivialized and collapses. Radical recursion therefore entails neither external reference nor self-identity. What it constitutes, I suggest, is the dialectical interplay of these.

I propose that, in radical recursion, though the
self that is signified is not simply the same self that does the signifying;
though the very act of reflecting upon the self turns it into what is other;
this other flows right back into the source from which it arises, rather than
appearing *merely *as an other cast
before a new self. The semiotic act I am intimating thus would give us neither
self nor other, in the categorically opposed sense of these terms. We would
realize instead their paradoxical interpenetration. I suggest that this
dialectic is what we require to supersede the supremacy of linear signification
in a meaningful way. Signifier and signified would be more than reciprocally
interdependent in such a self-signification. They would be identical, utterly
one. Yet they also would be two. By virtue of the latter aspect, meaningful
signification would continue; by virtue of the former, recursion would go all
the way down; it would be realized concretely in the heart of the actual
occasion. To be sure, this construal of radical recursion requires further
explication.

2. RADICAL RECURSION IN TOPOLOGY

Psychoanalyst Jacques Lacan had turned to the science of linguistics in order to clarify the language of the psyche. What we see precisely in the slippage of the signifier through which the subject is “a fading thing” (1966/1970: 194) is the functioning of the unconscious. But Lacan was not content to stop with a merely linguistic clarification of psychic process. In an effort to achieve an even higher level of precision, he appealed to mathematics, and, in particular, to topology, the qualitative sub-discipline that deals with the properties of surfaces. By way of elucidating the signifying activity that constitutes the unconscious discourse of the human subject, Lacan presented a diagram of a Moebius strip:

This diagram can be considered the basis of a sort of essential inscription at the origin, in the knot which constitutes the subject. This goes much further than you may think at first, because you can search for the sort of surface able to receive such inscriptions. You can perhaps see that the sphere, that old symbol for totality, is unsuitable. A torus, a Klein bottle, a cross-cut surface, are able to receive such a cut. And this diversity is very important as it explains many things about the structure of mental disease. If one can symbolize the subject by this fundamental cut, in the same way one can show that a cut on a torus corresponds to the neurotic subject, and on a cross-cut surface to another sort of mental disease. (Lacan 1966/1970: 192–193)

Comparing the sphere and the Moebius strip, we can say that both are recursive, insofar as they both turn back upon themselves. But unlike “that old symbol for totality,” the Moebius possesses a “fundamental cut”; a knot, twist or gap. In Lacan’s view, this cut represents the division inherent in the subject that prevents it from realizing the self-identity symbolized by the sphere. The cut functions like a crack in a mirror, leading the Moebius to signify itself in such a way that it distorts or displaces itself. Or, speaking diachronically, we can say that Moebius recursion is interrupted by the cut and we cut to a new occasion. In moving into the Moebius’s twist, the signifier we started with is twisted into that which is signified by a newly implicit signifier that is a “mirror image” of the original yet out of step with it. On Lacan’s reading then, the Moebius strip embodies the self-alienating kind of recursion that falls short of what I have called radical recursion. I suggest, however, that there is a different way of reading the Moebius. To see how Moebius recursion can be grasped in the radical sense adumbrated above, let us look more closely at this curious topological structure.

We may bring out most effectively the dialectical character of the Moebius strip by comparing it to a non-dialectical structure more similar to it than is the sphere: the cylindrical ring (see Rosen 1994, 2004a).

**
Figure 1. Cylindrical ring (a) and Moebius strip (b)**

A cylindrical ring (Fig. 1a) is constructed by cutting out a narrow strip of paper and joining the ends. The surface of Moebius (Fig. 1b) is produced by giving one end of such a strip a half twist (through an angle of 180°) before linking it with the other. The cylindrical ring possesses the familiar property of two-sidedness: at any point along its surface, two distinct sides can be identified. Commencing on either side, rotation about the ring traces out a circle of simple self-return like that found on the sphere. The two-sidedness of the cylinder of course precludes continuous passage from one side to the other. Such a transition is inevitably cut short at the surface’s edge; the singularity we encounter there tells us that we cannot reach the far side without a break in contact, a cut to a new occasion. We are therefore able to say that, whereas rotation about a single side of the two-sided ring signifies the simply continuous affirmation of self-identity, passage between sides expresses a simply discontinuous cut to what is other.

Now, in the case of the Moebius strip, it is true
that if you place your index finger anywhere on the surface, you will be able
to put your thumb on a corresponding point on the opposite side. The Moebius
strip does have two sides, like the cylinder. But this only holds for the local
cross-section of the strip defined by thumb and forefinger. Taking the full
length of the strip into account, we discover that points on opposite sides are
intimately connected—they can be thought of as twisting or dissolving
into each other continuously, as being bound up internally. Accordingly,
mathematicians define such pairs of points as *single* points, and the two sides of the Moebius strip as but *one* side.

I want to emphasize that the Moebius surface is not
one-sided in the homogeneous sense of a single side of the cylindrical ring. It
is one-sided in the paradoxical sense, one-sided and also two-sided, for the
local distinction between sides is not just negated with expansion to the
Moebius as a whole. In coming to interpenetrate each other, the sides do not
merely lose their distinct identities. And yet, though the sides remain
different, they also become one and the same. Thus, if the cylindrical ring
embodies the dualism of identity and difference, of continuity and
discontinuity, the Moebius strip signifies their dialectical entwinement. We
can say as well that while the cylinder dualistically expresses both trivial
recursion (through movement on a single side) and non-recursion (through
passage to the other side), the Moebius models *radical* recursion.

Let us focus on the unique recursive action of the
Moebius. With 360° of rotation about this surface, we appear to return to
our point of origin. But this return is in fact also a departure, since,
instead of remaining on the same side of the strip as in the case of cylindrical
rotation, we are carried to the opposite side. So Moebius recursion
incorporates an element of discontinuity not evident in its cylindrical
counterpart. It is this distinctive feature that Lacan picked up on in
contrasting the Moebius with the sphere. What Lacan apparently missed is that
the discontinuity of the Moebius, its twist or cut—unlike the cut
required in passing from one side of the cylinder to the other—is also *continuous*. It is in glossing over the paradox of
Moebius recursion that Lacan stopped short of radical recursion. Lacan was
apparently unable to recognize that the Moebius signifier does not merely
short-circuit its reference to itself by prematurely cutting away from itself
to an alter-self operative on a new occasion. Rather, the Moebius signification
of self as other (and other as self) transpires within the same concrete occasion
thereby surpassing the dualism of self and other.

The application of Moebius topology has been taken up by thinkers with diverse orientations and disciplinary backgrounds. From a feminist perspective emphasizing embodiment, theorist Elizabeth Grosz expands on Lacan’s use of the Moebius by portraying it as expressing “the inflection of mind into body and body into mind” (1994: xii). Anthropologist Peter Harries-Jones (2002) suggests that the paradoxical link between culture and environment as understood by Bateson is best depicted in the form of a Moebius strip. Communications philosopher Brian Massumi (carrying forward Deleuze and Guattari’s call for a “topology of multiplicities” [1987: 483]), demonstrates the need to reconceive human transactions via a “strange one-sided topology” that recursively surmounts the old dichotomies by working at a “paradoxically creative edge” (1998). Semiotician Floyd Merrell (1998) uses the paradox of the Moebius to model C. S. Peirce’s concept of abduction. And philosopher Yair Neuman—in this issue of SEED—applies the Moebius to the structure of boundary events in semiotic systems. (My own work with the Moebius dates back to the 1970s; see Rosen 1994.) In these writings, sustained emphasis on paradox allows the authors to surmount Lacanian “slippage” and employ Moebius topology to question effectively “the binary oppositions…[of] mind/body, nature/culture, subject/object and interior/exterior” (Grosz 1994: 164).

It is all too easy, however, to lose one’s
paradoxical edge. This is evidenced in Grosz and Massumi when—after using
topology to successfully challenge binary opposition on one level of analysis,
they appear to fall prey to it on another. Thus, in the case of certain root
philosophical oppositions that implicitly structure their thinking—such
as the one and the many, identity and difference, being and becoming—they
wind up privileging “the fields of difference, the trajectories of
becoming” (Grosz 1994: 210). The “one-sidedness” of such a
reaction to the totalizing propensities of structuralism (and
classico-modernism in general) is certainly not of the Moebius kind. Instead of
genuinely questioning the categorial purity of the old approach by consistently
applying topological paradox to the most basic philosophical dichotomies, there
is a slippage into a sort of “reverse purism” (Rosen, 2004b). Pure
identity (totality, unity, being, continuity, etc.) is supplanted by a mode of
difference every bit as pure: Derridean *différance*.
It is in the process of unambiguously affirming one member of the philosophical
binary over the other that the Moebius edge is lost. So what I am proposing is
that the application of topological paradox needs to be implemented in a
consistent and thoroughgoing manner all the way down. From my own experience, I
know how difficult this is to achieve and it would not surprise me to learn
that I myself lose my edge in places in this very text. As a dweller in a
“glass house,” I must be careful then about the
“stones” I hurl. We are all challenged to avoid limiting our
applications of topological paradox to the surface of our discourse while
allowing our deepest assumptions and forms of expression to remain tacitly governed
by the system of binary logic that has controlled our thinking for so many
centuries. To keep our topological edge all the way down and thus achieve
radical recursion, we must consistently exceed mere “logics of presence
or position” and employ “qualitative topologics,”
as Massumi (1998) so well puts it.

3. THE KLEIN BOTTLE

I must now acknowledge a limitation in the Moebius
expression of radical recursion. The Moebius does effectively signify the
dialectic of continuity and discontinuity. In traversing the twist, we depart
from the circle of self-identity associated with continuous rotation about a
single side of the cylindrical ring, and we make the transition to the other
side of the surface, which, when enacted on the cylinder, brings simple
discontinuity. Yet, in the Moebius case, the departure from cylindrical
continuity happens *continuously*.
However, the discontinuous aspect of the Moebius dialectic is in fact something
of an abstraction. While the *effect*
of discontinuity is surely created in passing through the twist to the far side,
there is never any true cut or break, as occurs when actually crossing an edge.
The Moebius therefore signifies the continuity-discontinuity dialectic in a
continuous way; the discontinuous element is symbolized but not concretely
embodied. I suggest that, for a full-fledged realization of the radically
recursive dialectic, we require a topological structure in which continuity and
discontinuity are interwoven not merely in effect but in actual fact.

There exists a higher-dimensional counterpart of the
Moebius surface. By way of introduction, consider an interesting attribute of
the Moebius: its asymmetry. Unlike the cylindrical ring, the Moebius has a
definite orientation in space; it can be produced either in a left- or
right-handed form (depending on the direction in which it is twisted). If both
a left- and right-oriented Moebius surface were constructed and then
"glued together," superimposed on one another point for point, a* Klein bottle* would result
(Lacan’s passing allusion to this topological structure is cited above).

**Figure 2.
The Klein bottle**

The Klein bottle (Fig. 2) has the same property of asymmetric
one-sidedness as the two-dimensional Moebius surface but incorporates an added
dimension (Rosen 1994). Note, however, that we cannot actually produce a
continuous model of this curious container, for left- and right-facing Moebius
bands cannot be superimposed on each other in three-dimensional space without* *tearing the surfaces. Therefore, while
each Moebius enantiomorph is continuous within itself, joining these mirror
twins to form a Klein bottle brings discontinuity.

The feature of Kleinian discontinuity can be illustrated by means of a different but mathematically equivalent way of making the bottle. Once again a comparison is called for.

**Figure 3. Construction of torus (upper row) and Klein bottle
(lower row)**

Both rows of Figure 3 depict the progressive closing of a
tubular surface that initially is open. In the upper row, the end circles of
the tube are joined in the conventional way, brought together through the
three-dimensional space outside the body of the tube to produce a
doughnut-shaped form technically known as a torus (a higher-order analogue of
the cylindrical ring). By contrast, the end circles in the lower row are
superimposed from *inside* the body of
the tube, an operation requiring the tube to pass* through* itself. This results in the formation of the Klein bottle.
Indeed, if the structure so produced were cut in half, the halves would be
Moebius bands of opposite handedness. But in three-dimensional space, no
structure can penetrate itself without cutting a hole in its surface. So, from
a second standpoint, we see that the continuous construction of a Klein bottle
cannot be carried out in the three dimensions available to us. The Klein bottle
thus seems to possess the element of concrete discontinuity missing from its
lower-dimensional Moebius counterpart. Whereas the twist in the Moebius
mediates the transition from one side of the surface to the other in a
continuous fashion, the Kleinian passage from inside to outside requires a
hole. Of course, a *simply*
discontinuous structure will serve us no better than a simply continuous one if
we are seeking to express the dialectic of continuity and discontinuity. What
is needed is a structure that embodies the *paradoxical
interweaving *of continuity and discontinuity. And, in fact, the Klein
bottle does just that, provided that we approach it in a truly dialectical way.

How does modernist mathematics approach the Klein
bottle? Mathematicians certainly do not just accept the discontinuity of the
Klein bottle. Instead they rely on the idea that a form that penetrates itself
in a given number of dimensions can be produced *without* cutting a hole by invoking an added dimension. The point is
nicely illustrated by the mathematician Rudolph Rucker (1977). He asks us to
imagine a species of "flatlanders" attempting to assemble a Moebius
strip. Rucker shows that, since the space inhabited by these creatures would be
limited to* two* dimensions, when they
would try to make an actual model of the Moebius, they would be forced to cut a
hole in it. Of course, no such problem arises for us human beings, who have
full access to three dimensions. It is the continuous construction of the Klein
bottle that seems problematic for us, since this would appear to require a *fourth* dimension, but, try as we might,
we find no fourth dimension in which to execute the operation. For modernist
mathematics, however, there is actually no problem. Although dimensions higher
than the third may be unavailable to concrete experience, mathematicians feel
free to proceed abstractly, calling forth as many extra dimensions as they
wish. Added dimensions are summoned into being by extrapolation from the known
three-dimensionality of the physical world. This theoretical procedure of
dimensional proliferation presupposes that the nature of dimensionality itself
is left unchanged. In the case of the Klein bottle, the "fourth
dimension" required to complete its formation remains an extensive
continuum as is three-dimensional space, though the "higher” space
is taken as "imaginary"; the Klein bottle, for its part, is regarded
as an "imaginary object" embedded in this space. Enclosed as it is in
the hypothesized four-dimensional continuum, the imaginary Klein bottle itself
is presumed simply continuous. Like the Moebius strip of three-dimensional
space, it is thought to possess nary a hole.

Now, in his phenomenological study of topology, the
mathematician Stephen Barr advised that we should not be intimidated by the
“higher mathematician....We must not be put off because he is interested
only in the higher abstractions: we have an equal right to be interested in the
tangible” (1964: 20). The tangible fact about the Klein bottle that is
glossed over in the higher abstractions of modernist mathematics is its *hole*. Because the standard approach has
always presupposed extensive continuity, it cannot come to terms with the
inherent *dis*continuity of the Klein
bottle created by its self-intersection. Therefore, all too quickly,
“higher” mathematics circumvents this hole by an act of abstraction
in which the Klein bottle is treated as a closed object embedded in a
hyper-dimensional continuum. To be sure, an “added dimension” is
needed if the Klein bottle is not to be regarded as *merely* discontinuous. When limited to the three dimensions of
ordinary space, the Klein bottle cannot give expression to the dialectic of
continuity and discontinuity. But the “added dimension,” rather
than being a continuum, must itself blend continuity and discontinuity.

In a continuum, all interactions occur between fixed
terms that are externally related. The point elements of which the continuum is
composed are themselves related in this manner. As philosopher Milič Čapek put it in
his reflection on classical space, “no matter how minute a spatial
interval may be, it must always be an *interval*
separating two points, each of which is *external*
to the other” (1961: 19). In the words of Martin Heidegger, the continuum
is essentially constituted by the “‘outside-of-one-another’
of the multiplicity of points” (1927/1962: 481). Given the fundamental
exteriority of classical space, relations among objects and events contained
within it must also be external. In the continuum, systems “interact
through forces that do not bring about any changes in their essential
natures…[they interact] only through some kind of external contact”
(Bohm 1980: 173). Generally speaking then, the notion of the continuum implies
that all boundaries are external in nature. This includes the point elements
that bound space; the boundaries between and among interacting objects,
systems, and events in space; and the figure-ground boundary that distinguishes
an entity from its spatial context.

One other kind of exterior boundary is implicit in
the classico-modernist approach: the one that separates the object being
observed from the subject that observes or analyzes it. Whereas objects are
embedded in the extensive continuum, the subject entails discontinuity. In the
language of Descartes, the object is *res
extensa* and the subject *res cogitans*,
thus unextended, not manifested in space. However, this distinction is
complicated by the subtlety of the continuum idea. The continuum actually
possesses its own aspect of discontinuity. Even though the points composing
space are packed densely together, because these points are related to one
another externally, the continuum is infinitely divisible; it can be
indefinitely partitioned into ever smaller segments (the mathematician Charles
Muses was thus prompted to describe the continuum as actually constituting an
“infinite *dis*continuum”
[1968: 37]). It naturally follows that the objects embedded in this medium are
themselves partible; they can be rendered discontinuous. The discontinuity
associated with the subject, on the other hand, signifies its transcendence of
the continuum. So, whereas the breach one may produce in an object in fact
reflects a property of the continuum, the subject constitutes a break with that
continuum. Although it may rightly be said that classico-modernism favors
continuity over discontinuity, what we are seeing is that there is indeed a
place for discontinuity in the conventional paradigm, albeit a tacit or
negative one. At the deepest level, it is the *division* of continuity and discontinuity that classico-modernism
upholds.

4.
RADICAL RECURSION AND THE DIMENSION OF *DEPTH*

The classical concept of dimension has prevailed from the
time of Descartes and Kant to the physics and mathematics of today. In
mainstream science and philosophy, the exteriority of relations among objects
in space, and between object and subject, has not been questioned in a
fundamental way. Yet countercurrents do exist. We find evidence of these in the
works of process-oriented thinkers such as Heidegger (1962/1972), Gendlin and
Lemke (1983), and Bateson (see Neuman’s
topological interpretation of Bateson in this issue of SEED, and Harries-Jones’s
[1995] account of Bateson’s recursive vision); in each case, internal
dynamics are given precedence over the static externality of the spatial
continuum (see also Rosen 1994, 2004a). One of the most explicit formulations
of process dimensionality is found in the notion of *depth* advanced by the phenomenological philosopher Maurice
Merleau-Ponty (1964). This idea provides us with an insight into dimension that
permits us to surpass the limits of classico-modernism and arrive at a
radically recursive understanding of space that is well suited for expressing
the Kleinian dialectic of continuity and discontinuity.

By way of introducing Merleau-Ponty’s *depth* dimension, let us consider in
greater detail the traditional dichotomy between the objects contained in space
and their spatial container, or, as Plato put it, between “that which
becomes [and] that in which it becomes” (1965: 69). A visible form
“becomes,” whereas that “*in*
which it becomes” is “invisible and formless” (1965: 70).
Whatever changes may transpire in the objects that “become,”
however they may be transformed, the containing space itself does not change.
Indeed, for there to be change, there must be difference, contrast, dialectical
opposition of some kind. But the point-elements that make up the classical
continuum, rather than entailing opposition, involve mere *juxta*position. Unextended and thus devoid of inner structure, the
elements of space possess no gradations of depth; no shading, texture, or
nuance; no contrasts or distinctions of any sort. Instead of expressing the
dialectical interplay of shadow and light, space itself is *all light*, as it were. A condition of “total exposure”
prevails for the point-elements of the continuum, since these elements, having
no interior recesses, must be said to exist solely “on the
outside.” All that can be said of the relations among such eviscerated
beings is what Heidegger said: the points of classical space are
“‘outside-of-one-another’” (1927/1962: 481). So, rather
than actively engaging each other as the beings that are contained in space
seem to do, the densely packed elements of the classical container sit inertly
side by side, like identical beads on a string.

In fact, even though the beings that dwell in such a space can be described as “actively engaged,” we have seen that the quality of their interaction is affected by the context in which they are embedded: since the continuum is constituted by sheer externality, the relations among its inhabitants must also be external. Classical dynamics are essentially mechanistic; instead of involving a full-fledged dialectic of opposition and identity wherein beings influence each other from core to core, influence is exerted in a more superficial fashion, “only through some kind of external contact” (Bohm 1980: 173). We may say then that classical space contains dialectical process in such a way that it externalizes it, divesting it of its depth and vitality.

It is the classico-modernist view of space that
Merleau-Ponty calls into question. What he demonstrates is that the spatial
continuum appearing to contain dialectical process actually *originates* from it.

In his essay “Eye and Mind,”
Merleau-Ponty emphasizes the “absolute positivity” of traditional
Cartesian space (1964: 173). For Descartes, space simply is *there*; possessing no folds or nuances,
it is the utterly explicit openness, the sheer positive extension that constitutes
the field of strictly external relations wherein unambiguous measurements can
be made. Merleau-Ponty speaks of

this space without hiding places which in
each of its points is only what it is....Space is in-itself; rather, it is the
in-itself *par excellence. *Its
definition is *to be* in itself. Every
point of space is and is thought to be right where it is—one here,
another there; space is the evidence of the “where.” Orientation,
polarity, envelopment are, in space, derived phenomena inextricably bound to my
presence [thus “merely subjective”]. *Space *remains absolutely in itself, everywhere equal to itself,
homogeneous; its dimensions, for example, are interchangeable. (1964: 173)

Merleau-Ponty concludes that, for Descartes, space is a purely “positive being, outside all points of view, beyond all latency and all depth, having no true thickness” (1964: 174).

Challenging the Cartesian view, Merleau-Ponty insists
that the dialectical features of perceptual experience (“[o]rientation, polarity, [and] envelopment”) are *not* merely secondary to a space that
itself is devoid of such features. He begins his own account of spatiality by exploring
the paradoxical interplay of the visible and invisible, of identity and
difference, that is characteristic of true depth:

The enigma consists in the fact that I
see things, each one in its place, precisely because they eclipse one another,
and that they are rivals before my sight precisely because each one is in its
own place. Their exteriority is known in their envelopment and their mutual
dependence in their autonomy. Once depth is understood in this way, we can no
longer call it a third dimension. In the first place, if it were a dimension,
it would be the *first *one; there are
forms and definite planes only if it is stipulated how far from me their
different parts are. But a *first*
dimension that contains all the others is no longer a dimension, at least in
the ordinary sense of a *certain
relationship* according to which we make measurements. Depth thus understood
is, rather, the experience of the reversibility of dimensions, of a global
“locality”—everything in the same place at the same time, a
locality from which height, width, and depth [the classical dimensions] are
abstracted. (1964: 180)

Speaking in the same vein, Merleau-Ponty characterizes depth as “a single dimensionality, a polymorphous Being,” from which the Cartesian dimensions of linear extension derive, and “which justifies all [Cartesian dimensions] without being fully expressed by any” (1964: 174). The dimension of depth is “both natal space and matrix of every other existing space” (1964: 176).

Merleau-Ponty goes on to
observe that primal dimensionality must be understood as *self-containing*. This is illustrated through a discussion of
contemporary art, and, in particular, the work of Paul Cézanne:
“Cézanne knows already what cubism will repeat: that the external
form, the envelope, is secondary and derived, that it is not that which causes
a thing to take form, that this shell of space must be shattered, this fruit
bowl broken” (1964: 180). In breaking the “shell,” one
disrupts the classical representation of objects in space. Merleau-Ponty asks:

[W]hat is there to paint, then? Cubes,
spheres, and cones...? Pure forms which have the solidity of what could be
defined by an internal law of construction...? Cézanne made an
experiment of this kind in his middle period. He opted for the solid, for
space—and came to find that inside this space, a box or container too
large for them, the things began to move, color against color; they began to
modulate in instability. Thus we must seek space and its content *as *together. (1964: 180)

The work of Cézanne is Merleau-Ponty’s
primary example of the exploration of depth as originary dimension. The
foregoing passage describes Cézanne’s discovery that primal dimensionality
is not space taken in *abstraction*
from its content, but is the *unbroken
flow* from container to content. It is in this sense of the internal
mediation of container and content that Cézanne’s depth dimension
is self-containing.

Merleau-Ponty also makes
it clear that the primal dimension engages embodied subjectivity: the dimension
of depth “goes toward things from, as starting point, this body to which
I myself am fastened” (1964: 173). In commenting that, “there are
forms and definite planes only if it is stipulated how far from *me* their different parts are”
(180; italics mine), Merleau-Ponty is conveying the same idea. A little later,
Merleau-Ponty goes further:

The painter’s vision is not a view
upon the *outside*, a merely
“physical-optical” relation with the world. The world no longer
stands before him through representation; rather, it is the painter to whom the
things of the world give birth by a sort of concentration or coming-to-itself
of the visible. Ultimately the painting relates to nothing at all among
experienced things unless it is first of all
“autofigurative.”....The spectacle is first of all a spectacle of
itself before it is a spectacle of something outside of it. (1964: 181)

In this passage, the painting of which Merleau-Ponty
speaks, in drawing upon the originary dimension of depth, recursively draws in
upon itself. Painting of this kind is not merely a signification of what is
other, but a concrete *self*-signification
that undercuts the external boundary between signifier and signified.

In sum, the phenomenological dimension of depth as described by Merleau-Ponty is (1) the “first” dimension, inasmuch as it is the source of the Cartesian dimensions, which are idealizations of it; it is (2) a self-containing dimension, not merely a container for contents that are taken as separate from it; and it is (3) a dimension that blends subject and object concretely, rather than serving as a static staging platform for the objectifications of a detached subject. In realizing depth, we surpass the concept of space as but an inert container and come to understand it as an aspect of an indivisible cycle of action in which container, contained, and “uncontained”—space, object, and subject—are integrally incorporated.

The work of Merleau-Ponty provides us with an
insight into the nature of the “added dimension” that is required
for the Kleinian signification of radical recursion. It would not be enough to
say that the Klein bottle *makes* *use* of the dimension of depth to realize
its dialectic of continuity and discontinuity—not if “makes
use” connotes the operation of a model employing a containing medium to
signify a meaning external to itself. It is perhaps more accurate to say that
the Klein bottle *is* the depth
dimension. For, rather than being a model contained as object-in-space, the
Klein bottle—grasped in terms of depth—is the inseparability of
object, space, and subject, the unbroken circulation of these intimated by
Merleau-Ponty.

It is the unique hole in the Klein bottle that plays
the pivotal role. This loss in continuity is *necessary*. One certainly could make a hole in the torus, or in any
other object in three-dimensional space, but such discontinuities would not be
necessary inasmuch as these objects could be fully assembled in space *without* rupturing them. It is clear that
whether an object like the torus is cut open or left intact, the closure of the
space containing that object will not be brought into question; in rendering
such an object discontinuous, we do not affect the assumption that the space in
which it is embedded is a continuum. Indeed, we have seen that the divisibility
of an ordinary object *derives* from
the infinite divisibility of the continuum itself. With the Klein bottle it is
different. Its discontinuity does challenge the continuity of three-dimensional
space as such, for the *necessity *of
the hole in the bottle indicates that space is unable to contain the bottle the
way ordinary objects appear containable. We know that for the Kleinian
“object” to be brought to completion, assembled without* *a hole, an “added dimension”
is required, and I am proposing that the dimension to be engaged is that of
Merleau-Pontean depth (assuming we do not wish merely to skip over the hole by
a continuity-maintaining act of abstraction, as in the standard mathematical
stratagem for dealing with the Klein bottle).

In the depth dimension, while the hole in the Klein
bottle is no mere breach in an object in space, neither is it simply a rupture
in space per se that corresponds to the subject. Rather, the Kleinian
“hole” is in fact a *dialectical*
*(w)hole* resulting from an act of
self-intersection wherein the purported object does the
“impossible”: it passes unbrokenly through itself, and, in so
doing, flows backward into its own subjective ground (in Merleau-Ponty’s
terms, it is “autofigurative”). Elsewhere, I noted the resemblance
of the Klein bottle to the hermetic vessel of old alchemy (Rosen 1995). The design of the enigmatic vessel is
essentially that of the* uroboros*, the
serpent that consumes itself by swallowing its own tail. To contain itself, the serpent must
intersect itself, an operation requiring a hole (corresponding to the opening
that is its mouth). The hole in the
Klein bottle is of this sort. It is
neither solely a hole in a container, nor a hole in that which it contains, but
the hole produced by the recursive act of *self*-containment
that integrates the container with its contents in this way giving (w)holeness.

The Kleinian process of
self-containment enacted through the dimension of depth is surely no *trivial* recursion, no regression of
meaning to simple self-identity. The Klein bottle refers to itself, but it also
makes reference to what is other and a boundary is crossed. Of course, the
boundary in question is not of the exterior sort so familiar to us; instead it
is paradoxical, a boundary that is *not*
a boundary (see Neuman, this issue of SEED, and Rosen 1997, 2004a). In passing
through Kleinian depths from self to other (subject to object, the
discontinuous to the continuum), we cross over the boundary to the “far
side,” yet at once remain on the “near side.” In this way,
while the self-other distinction is not just abrogated, the *supremacy* of
this distinction is overcome and we realize a harmony of self and other so intimate that the prior meanings
of these terms are transmuted. The erstwhile categorical purity of self and
other is supplanted by an odd uroboric hybrid, a “hermaphroditic”
fusion wherein self and other, though assuredly different, are one and the
same. This profoundly paradoxical manner of self-reference is what I mean by *radical recursion*.

5. SEMIOTIC POSTSCRIPT

The subtlety of the notion of radical recursion has not been exhausted by what I have written above. I will note another layer of meaning before I conclude.

Consider the words “Klein bottle.” Although these signifiers point to the depth-dimensional structure that embodies the paradox of radical self-signification, the signifiers themselves—“K-l-e-i-n” and “b-o-t-t-l-e”—are but arbitrarily devised, conventionally agreed upon tokens that refer to their content in a merely external manner. These one-dimensional typographic marks appearing on the two-dimensional surface of this page obviously fall short of tangibly delivering the three-dimensional Kleinian depth they signify. To the extent that “unmotivated” conventional marks constitute the primary mode of signification for this text, the old division between signifier and signified will be upheld and the meaning of the Klein bottle will remain an abstraction.

In seeking to close the gap between signifier and
signified, it might be feasible to place greater emphasis on our two-dimensional
images of the Klein bottle (Figs. 2 and 3), or, better still, to work with a
full three-dimensional model of this paradoxical structure (a model can be
constructed with a flexible length of tubing such as that illustrated in Fig.
3). However, it should be clear by now that for the Klein bottle to be fully
dimensioned, our model cannot be limited to an object in three-dimensional
space. The Klein bottle must be realized in Merleau-Pontian depth. To this end,
rather than regarding the Klein bottle as but an object appearing before us,
something “out there” in space that we see and can handle, we must
resist this compulsion of naïve realism and take the bottle as something
that we *read*. What I am suggesting is
that, while the Klein bottle cannot be actualized in the abstract universal
medium of words alone, neither can it be brought to fruition as but a
particular concrete thing; to be realized in depth, it must be realized as a *hybrid* of word and thing (a
“general thing”; Merleau-Ponty 1968: 139), as a “tangible
word” or iconic text. In the capacity of iconic sign, the Klein bottle
can serve to “motivate” the process of signification by raising it
from one-dimensional arbitrariness to a fully committed three-dimensional
intercourse of signifier and signified.

In reading our iconic Kleinian text, we must of
course read its *hole*. Instead of
interpreting the hole as a gap in an ordinary object contained in space, we are
to read it as an opening to a “higher dimension,” and read that
dimension “autofiguratively,” grasping it as the prereflective
source of our very own reading. The hole then becomes a *(w)hole* that we actualize in depth as we pass unbrokenly from text
back to subtext in a uroboric act of radical recursion. Here the Klein bottle
signifies radical recursion by signifying itself. Or in Peircian terms, we may
say with semiotician Paul Ryan that the Klein bottle is a “sign of
itself” (1993: 345–347).

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