Topology
La Topologie Perdue de la Psychanalyse
Independently of the propositions on culture and nature that thematize
the field of psychoanalysis, Lacan has constructed its discipline
in a topology (Greek topos [place] + logos [discourse/reason/logic])
which has become a basis on which to achieve Freudian psychoanalysis
in its theory and practice.
What is called the structuralism
of Lacan is inseparable from a psychoanalysis where place
— topos — is more
primary than the contents that fill it. The crucial problem of
contemporary psychoanalysis,then, is neither the multiplication
of it contents through applications to cultural or clinical fields,
nor that of borrowing ideas from the sciences and philosophy to renovate
its theory, but of establishing a practice of structure that Freud
had left at the level of a mythology of the drives and the Oedipus complex.
This is very well to set forth the proposition that psychoanalysis,
since Lacan, operates an involution in theory and practice that marks
a passage from psychoanalysis in intension to psychoanalysis in
extension; said otherwise, that marks the passage from a place
of a psychoanalytic practice to a practice of a psychoanalytic place.
Yet, with the marketing of bookcovers and paraphrase on Lacan and
topology so far advanced that one can no longer ignore
it, we must raise the unavoidable question whether this commerce
is compatible with a construction of a psychoanalytic place -
or clinic -beyond an illustrative use. Without prohibiting anyone from
the use of a fantasy, we should ask if there is not a more fundamental
un-doing, un-reading, or indeed traversing
of such fantasies in the production of an effective topological
and psychoanalytic work. In this short web essay, we only begin to respond
to such a question by bringing out a geneology of the problem of a topological
structure in the tradition of Freud and Lacan. We will begin, therefore,
by a 'soft' historical introduction to the problem and will only follow
in the next chapters to determine our structural monstrations. No doubt,
the current opinion on Lacanian topology — including those
of many psychoanalysts and topologists — is that such a topology would
be, at best, an illustrative metaphor, or at worst, an imposture.
Here,then, a fitting name for our project is La Topologie Perdue de la Psychanalyse
- The Lost Topology of Psychoanalysis -; a title which appears to
be so simple and regular that it renders account of the vertigo of
the savant and the ignorant alike.
Psychoanalytic Topology: Metaphor, Analogy, or just Structure
?
If as Jacques Derrida would have us believe, "stricto sensu, the notion of structure only
refers to space, morphological or geometrical space […]" and
"it is only by metaphor that this literality topographic can be
displaced towards its aristotelian and topic signification (the
theory of places in language and the manipulation of motifs or arguments)",
then Derrida's conception of structure — and thus psychoanalysis
— is, no doubt, decidedly different than that of Lacan who stated
that his topology was "not metaphorical" and has as its goal
"to articulate the space of the
speaking being".
If Sokal and Bricmont's critique of Lacan: "Everything is based — at best —
on analogies between topology and psychoanalysis that are unsupported
by any argument" is founded, then it becomes difficult
to explain Lacan's response to a question posed by Harry Woolf
in an American seminar:
Woolf: May I ask if this fundamental arithmetic
and topology are not in themselves a myth or merely at best
an analogy for an explanation of the life of the mind?
Lacan: Analogy to what? […] It is not an analogy.
It is really in some part of reality, this sort of torus. This
torus really exists and it is exactly the structure of the neurotic.
To attempt to either justify such a response
or contradict it without marking the difficulty of situating the
concept of structure in a theory itself, is to take for granted
what requires explaining in the first instance: the relation of
structure to topology.
If structure is presupposed given as a concept
and defined in a mathematical theory, then topology is one structure
among others: group structures, vector space structure, manifold
structure, etc. — and the conjunction of topology and psychoanalysis
is a metaphor. But if the definition of the concept of structure cannot
be taken for granted in a mathematical text — but is always more or
less assimilated or trivialized to the particular theory of the mathematician1
— then the term topology can only be taken as an adjective in the
phrase topological structure
if one considers that the question of structure has been regulated
beforehand. Consequently, the question of whether the conjunction
of topology and psychoanalysis is a metaphor or not, is secondary
to the question of how to situate precisely the relation between topology,
structure, and theory. Indeed, in following the great
strides that have been achieved by the logicians and mathematicians
in determining a concept of structure, nothing prohibits
the psychoanalyst from proceeding to give him or herself the same litteral
methods of transliteration between theories. For,
in fact, this method of translilteration is comparable to both Champollion's
method of going between two languages in the deciphering of the rosetta
stone and Freud's method of going between two languages in the interpretation
of a dream.
For instance, just as it is possible to introduce a structural
interpretation of a site by posing a Functor F from
the category (or theory) of mathematics to music:
Category: Mathematics
-----------> Music
12
(relations/elements)
12 Tones
Cyclic
Well-tempered
Group
Scale
it is possible to introduce a structural interpretation by posing
a Functor (or Signifier) from the theory
of mathematics to psychoanalysis:
Mathematics -------------->
Psychoanalysis
Torus
Se Faire Tore
Fundamental
Neurosis
Group
(theory of the Drives)
No doubt, the arrow between the two theories requires a concept
of structure that is not a simple isomorphism, but a more modern Functorial
conception corresponding to what in analysis is called a
Signifier. It remains to be shown in what way a functorial
doctrine of structure would suffice to encompass a psychoanalytic gesture
in its work with tori; or further still, whether the arrow itself
carries with it the semantics of a topology (as an initial probe one
could present this double category as the homology
- or pasting together - of two triangles:see footnote 1/below).
In the end, it is neither a question of applying mathematics to
psychoanalysis, nor of doing a psychoanalysis of mathematics. Just
as it is not a question of whether someone must become a mathematician in
order to do psychoanalysis, but rather one of asking what type of ignorance
does a psychoanalyst participate in by avoiding the use of such a rigorous
writing ? Or inversely, it is not a question of asking someone to become
a psychoanalyst in order to do mathematics, but rather of asking what type
of ignorance would a topologist participate in by avoiding an analysis
of the space of the dream, illusion, and hallucination ?
Psychoanalytic Topology: The Nature of its Lack
We aim to situate how a notion of structure emerges in terms of a
topology which has always been missing from mathematics, or at least has
always posed an obstacle to its mathematization. In this regard,
such a topology is not merely in default, but as the celebrated
biologist Buffon remarked "absolutely missing" or as Husserl would
claim, that "Whatever the development of an exact science, that
is to say operating on ideal infrastructures, it can not resolve
the authorized and original tasks of a pure description". Without
stopping here to argue for or against what is being presumed in such
statements, it suffices to note that the notion of structure has always
emerged, whether in physics, biology, Gestalt psychology, or phenomenology
precisely at the point of formulating the desire for something like a
topology or analysis situs
which has been thought missing.
A Short Geneology
In all the domains — physics, biology, psychology, anthropology,
linguistics, economy, philosophy, etc. — where a structure
posseses an empirical or cultural value and is revealed as necessary
to the explication of it's experience, its theoretical function
has always been to go beyond one of the most ancient obstacles (in the sense of Bachelard)
posed to scientific rationality: that between mechanism and vitalism.
Briefly, this opposition comes to the fore once the former is determined
as an entity in which the whole is reducible to the sum of its parts
(a machine), while the latter can be determined as an entity of which
the whole is greater than the sum of its parts (an organ). It is by
no means established that a mechanism and organism are incompatible.
What is established is that their compatibility, or lack of it, is discovered
when the totality of the organism contains a supplement, an 'error
of the vital', that creates an obstacle in the representation of an
organ in terms of a machine.
For example, in its origin and development psychology
has been the province of philosophers of the soul/organism (a
Gestalt Psychology in the
sense of Kohler or Descriptive
Psychology in the sense of Brentano)and the mind/machine
(Atomistic Psychology based on
the association of ideas in the sense of Locke and Helmholtz).
Yet, neither explication has succeeded to unite a mechanistic
or organistic explication of the human being in one single synthetic
judgement.
What ought to be recognized, but is rarely,
is that this obstacle
of representing the human being, is found in a more significant
way the moment the human being attempts to represent the act of
representation itself. From a purely logical viewpoint, it is under
the aegis of a paradox of representation (more properly spoken of
here as 'mimesis') that the problem of structure should be interrogated.
For instance, it would involve recognizing a conflict between a classical
conception of the function of art as composition with a romantic conception
of the work of art (the
oeuvre) as symbolic construction
(see Wofflin's distinction between graphic and the painterly styles). For
the attention to the aporia
that exists between a mechanist attempt to explain art as a composition
of its parts - a representation - is not the same as a holistic attempt
to explain the work of art
as a contruction of a whole, an oeuvre. The latter, requires
a reference to an 'essence of Life'
- a style, plasticity, vital error, subjectivity, genius, spirituality,
etc. - which is a stranger to the composition of its parts.
Finally, all those psychologies that have taken
Life as their domain of
analysis have ended up replacing an analysis of representation
with that of expression and giving it a content — in a Psychology of the Mind, or Philosophy of the Personality —
irreducible to an atomist psychology. Heidegger reformulates this holistic
reading of Life as a question
of Dasein. For as long as one opposes an objectal domain of
man as mere machine and automaton to a subject of analysis which escapes
such representations, then the place of the subject is reduced to a
science of the whole: genres of being, intentionality, spirit, psyche,
mind, etc. annotated in an anthropology or a science of the personality.
For Heidegger, such common readings of the totality of beings, impedes
posing the fundamental question of Dasein and the place of Being itself. Indeed, it is well known
that it is this ontic-ontological
difference that Heidegger aims to establish as the basis of
not only aporias of representation, but of all philosophical discourse.
In this way, his critique of psychology and phenomenology occurs
in the name of existential structures and a Topology of Being defined as the "locality
of all localities". A 'dwelling place' or 'locality' which Heidegger
was to underline was "lacking a place", or rather constantly being
assimilated to the places of beings and facticity.
Towards a Topology of Repression
Without going further here into the problem, it suffices here to
restate it: Why, then, the constant lack of place in the formulation
of problems of topology and structure?
It appears that Lacan's return to Freud by way
of topology — whether it occurs casually or in line with highly
self-conscious, philosophical mutations —upsets the taken for
granted assumptions with which the profession of psychoanalysis
has been operating for some time now. As a result, the attribution
of a reliable or even exemplary knowledgeable and ethical function
to psychoanalysis becomes much more difficult. But this is a recurrent
psychoanalytic quandary that has never been resolved. The latest version
of the question, which still determines many analysts' present-day convictions
about the aims of psychoanalysis, goes back to the rise of neurophysiology
as an independent discipline in the 19th century. The link between representation,
knowledge, and truth is the burden of Freud's failed attempt to explain
the psyche in a purely mechanical and quantitative modelization. Or
rather, the construction of such a neuronal model is lacking, because
it fails to render account of qualitative elements: the observing ego
and the occurrence of consciousness. It is because it is common to
teach psychoanalysis as a form of psychology, at least in the anglo-saxon
inspired countries, that such 'qualitative elements' are classified
as psychological properties, and not more precisely as invariants of
a topological theory. This echoes a generally admitted position among
professors of the 'psy' and is the reason one can nowadays pass so easily
from neurophysiology to its apparent prolongations in the field of psychology.
Whether a reading of Freud's Scientific
Project (Entwurf) and Beyond
the Pleasure Principle would confirm this passage between
mechanism and organism certainly stands in need of careful examination,
which we can only touch on in such an essay.
In its origins and development, Freud proposes
a vitalist hypothesis of the finality of an organism in order to
explain repression
(verdrangung) and attention.
Otherwise said, he supplements the psychic machine determined by
an analysis partes extra partes
with a reflex circuit — an inertial system voiding quantity according
to a principle of pleasure/unpleasure — in the attempt to explain
the functioning of the whole in terms of an organism. From this ideal
principle of 'least action', Freud progressively recognizes a spatio-temporal deployment of an internal
organizing force and gives it the name of the libido and the
drive. But one of the central results of Freud is the discovery
of a drive irremediably disjoint from the spatio-temporality of
Life or the auto-conservation
of the organism: the Death Drive.
It is in this respect, in the recognition of a non-biological relation of the sexed
individual, that Freud passes from a neurophysiology
seeking to model adaptation as 'least action' and defense (movement from
an exterior stimuli), to a neuropathology
seeking to model nonadaptation
as 'excessive or missed acts' and pathological defense ( repression or an impossible flight).
At this point, Beyond the Pleasure Principle, Freud's reading of in-forms
and discontinuity become more important than a continuous hypothesis of
an internal organizing force of Life.
Psychoanalytic theory raises here the unavoidable
question whether neurophysiological and psychological themes
of Life can be compatible
with the topological structures that make up the reality of the
entities from which these themes are derived: their Sexuality
and Death. What is established is that the insufficiency
of such themes has to remain an open question and that the manner
in which the teaching of psychoanalysis, at least since the middle
of the 21st century, has foreclosed the question is unsound, even
if motivated by the best of intentions. What also ought to be established,
but rarely is, is that the clinic of psychoanalysis ought to take
place under the aegis of this question.
From a pedagogical point of view, this is not
difficult to achieve. It involves a change by which psychoanalysis,
instead of being taught only as a historical and a humanistic subject,
should be taught as a topology prior to any psychology or hermeneutics.
The resistance to such a Lacanian move in France has been predictable,
but can today be used to show what is being avoided by culturalist
and humanist programs. For by re-theorizing the qualitative element
of Freud topologically — as we will see in terms of the petite a — Lacan removed an
obstacle to its theory
and practice that continues to haunt many contemporary analysts.
Attention to the formal
differentiation of the individual and its milieu - or we will indicate
later one signifier, then another - is prior to any theory of contents which would
try to understand the individual as adapting to its environment or
as developing according to an innate program of heredity. Counter the
dominant continuist interpretations of the human subject emerging in the
20th century - primarily focussed on the individual's adaptation to the
environment and an innate genetic formation – Freudian psychoanalysis began
to recognize a discontinuous relation of the individual to the formation
of its milieu: one that determines the very differentiation of the individual
in a theory of narcissism. Lacanian psychoanalysis disengages this natural
observation in the formation of a structural concept: the obstacle between a theory of the
individual and the milieu (the group, society, culture, etc.), between
the part and the whole, is resolved by a seemingly innocuous move founded
on a definition of structure
exemplified in Cantor's set theory. Here, the whole is equal to its
part (even numbers are equal to the natural numbers for instance) which
is further complicated by a problem of the invariancy of dimension (the
first dimension seemed, to Cantor, to be equal to the second). Psychoanalytically,
these problems can be used to landmark the emergence of structure in the
topological isolation of the individual: the milieu (whole) is equivalent
to the individual (part) as an oddly interior exteriority, thus making
the problem of the individuation and the relation to an object particularly
delicate. In a parallel manner, the obstacle opposing a vitalist to a
mechanist explication of the formation of the individual becomes obsolete
the moment one begins to theorize the organism as a transfinite machine or 'cyborg' (see George Canguillem's
Machine and Organisme and John
Conway's Game of Life).
Later, Lacan would state that language itself
is a biological organ (as Chomsky had), and as such it is in
a discontinuous relation to its milieu that the problem of the
signifier is first raised. Lacan states "The speaking subject has the privilege
of revealing the mortifying sense of this organ, and by this his
or her relation to sexuality. This is because the signifier as such
has, in barring the subject in its first intention, made the sense
of death enter into it (the letter kills but we learn this by the
letter itself). This is why every pulsion is virtually a pulsion of death"
[E.848].
This place of language is now the milieu proper
to the subject: it is no longer a question of situating the milieu
of the biological individual or explaining its paradoxical defenses,
but of situating the place of the speaking being in a topology of the subject.
To conclude, the early elaboration of structure
by the biologists is in accord with the observations of the psychoanalysis
in recognizing that a topology sufficient to the formalization
of Life has always been radically
in default — not because science has not yet discovered its secret
— but because it is repressing. One sees that the whole problem
is to determine a purely structural reformulation of the vitalist's
'part maudite' (in the sense
of Bataille); or in an algebraic 'petite a' (in the sense of Lacan);
or the alpha and beta functions (in the sense of
Bion) permitting psychoanalysis to escape from a purely speculative
biology and the philosophies of Life.
The Missing Clinic of Psychoanalysis and the Current Lack of Psychoanalytic
Literature on the Subject
Lacan drew attention to structure in a clinical theory by showing
how the reading of the relation between what is seen and what is heard is able to transform the
methodology of psychoanalysis in a manner that would only be
subversive to those who think of teaching of psychoanalysis as
a substitute for teaching philosophy, psychology, theology, zen,
or literature. Clinical psychoanalysis becomes a topology almost
in spite of itself because it cannot fail to respond to the problem
of structure which it has been more or less the implicit aim of analytic
teaching to avoid.
Since Lacan, students, as they began to analyze
a case were instructed not to consider anything that was not
derived from the case at hand. They were asked not to make
statements that they could not support by a specific construction:
instead of merely trying to understand the place of Man discursively
in terms of philosophy, Lacan asked himself and his students to construct
a thing — graphs, surfaces, or knots — that would exhibit the
properties of holding together or connectedness and disconnectedness.
They were asked, in other words, to begin by not taking for granted
the materiality of language and its place in the presentations of
their own practice, while not to remain at the level of representing
others in the context of human history or experience. Much more humbly,
they were to start out from the misunderstandings that such a practice
was bound to produce in clinicians and practicioners perceptive enough
to notice them and courageous enough not to hide their ignorance behind
the screen of received ideas that often passes in psychoanalytic instruction
for humanistic research.
These very simple guidelines have had far reaching
consequences in the training analysis. Yet, some never saw the
point of focusing their attention on the matters at hand, and of
presenting a structure as a way to construct experience, rather than
commenting on experience itself. Indeed, the norm of the current literature
in Lacanian psychoanalysis and its clinic was created by those who
never found the time for such lessons. For what they see as the great
power of Lacanian theory to critique and generalize just about anything
— from politics to science — is in fact, nothing but a loss of precision
and the foreclosure of any possible discourse on structure and the clinic.
Others, no doubt, took the time, so much so that psychoanalysis, at
least in France, would never be the same. Yet, a didactic in topology
did not make their writing in psychoanalysis any easier for they no longer
felt free to 'free associate' and fantasize without at least having
a manner to construct the results of their practice. Today, in spite
of the facticious paraphrases of Lacan, a book on clinical Lacanian
analysis has yet to be written — as sound practicianers are slow writers,
and in the present state of psychoanalysis, this can only be a sign of
progress.
Robert Groome
Autumn 2004
Santa Monica, CA
NOTES: 1 For instance, the notion of structure
as modelled by sets in the early theory of Bourbaki is insufficient
as it still pressuposes an ontology without ever clarifying the functional
status of its object. Yet the latter Bourbaki rectifies this misreading,
beginning with the use of echelons (or species) in its
last chapters with Charles Ehresmann, then with the theory of faisceaux
and esquisses, a concept of structure is de-ontologized from sets
resulting in the birth of the notion of a category. In the United
States this revolution of method began with Sanders Maclane, Eilenberg,
and Lawvere. Needless to say, this algebraization of topology,
whether in the notion of an esquisse or in the modern topos
theory still carries with it the semantics of its algebraization to the
extent a topology remains bound to an algebra. It remains to
be shown if there is, then, an opening here in what has been called the
'old' polish school which proceeds in the opposite way: in the topologization
of algebra (see Tarski's: Topological
Algebra and forthcoming article).
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