On the History of Consciousness

On the Origin of Species (1859)

Capital, Volume 1 (1867)

The Interpretation of Dreams (1900)

The General Theory of Relativity (1915)

Course in General Linguistics (1916) - Lectures 1906-1911

Quantum Uncertainty (Heisenberg) (1927)

Lacanian Mathematics

Lacan asserts that language is a formulation only of "the correlations between signifier and signifier" with no recourse to any referent, either in the real world or at the core of our self, both of which are cut off from any true communicability (The Norton Anthology of Literary Criticism, 2nd Ed., p. 1174). If we take him at his word, language is no more than an internally consistent system. This at first seems supremely ironic: the very system we employ to relate our ideas about our selves and the outside world is limited precisely in its ability to reference these two 'objects.' Language itself defines a 'no-man's-land' of physical existence between Self and Other. However, it seems to make sense insofar as language is a social phenomenon, and so must have meanings which are in constant flux according to their current status in the social construct.

Lacan's formulation effectively reduces almost every possible argument to a question of semantics. If we can make no reference to the real in our communication with other people, what we must essentially be arguing for is a specific set of relationships between words. That is to say, if I argue that a Jellyfish Sandwich from the Corner Cafe in Carrollton, GA is the best sandwich in the southeast, what I am really arguing for (as far as Lacan is concerned) is a certain relationship between the word (and conceptual utterances surrounding the word) "Jellyfish" and the word "best" and "sandwich" and "southeast" and "Corner Cafe" (along with their associated utterances--what Saussure might call the paradigmatic connections).

This fact might shed some light on Lacan's interest in mathematical formulas. After all, what theoretical construct illustrates relationships more specifically than math? Instead of attempting to justify, in any more conventional (and more traditionally rhetorical--read by Lacan as 'roundabout') way his ideas concerning desire, Lacan simply constructs the following mathematical relationship:

“...desire is neither the appetite for satisfaction, nor the demand for love, but the difference that results from the subtraction of the first from the second, the phenomenon of their splitting” (1186).

Reduced in Lacanian fashion to mathematical symbols, this amounts to

D = d - a

Thus desire is by its nature unsatisfiable, if for no other reason than it is represented by the removal of our real appetite for its satisfaction. By this reduction, Lacan sidesteps the necessity to "argue" for his formulation, as well as all the difficulties which he perceives might necessarily result from trying to get us, the audience, to understand exactly what relationship between terms he is attempting to achieve. These mathematical formulas tend to come off as somewhat humorous, embedded as they are within pages of text, convoluted in its system of representation, ostensibly explaining the very complexities of language that they work ironically to undercut. However, what Lacan sacrifices in the these formulations in the way of 'straightforward' complexity, he more than gains back in 'straightforward' ambiguity.

Another benefit to Lacan's recourse to mathematical formulas in his discussion of language is to highlight language's status as an abstract system. From the time we experience complex algebra or Euclidean geometry, we grow comfortable with the idea that mathematics works only insofar as it depends on the rules of a system. Euclidean geometry does not need to correspond to the real world in order to be usefully applied. Part of its strength, in fact, stems from the regularizing effect that systematization has on the 'real' world of experience. By foregrounding mathematical formulations, Lacan is making the same argument for our ideas about language, which we are perhaps less inclined to think of as an abstract system if only because it is the system by which we must necessarily navigate our everyday lives.

That said, I think it might be fun to play around a little with Lacanian mathematics. By the application of the transitive property to the above formulation, Lacan also seems to set up a whole range of relationships which are achievable through simple mathematic maneuvering. Each term in the equation can be isolated:

d = D + a

a = d - D

What does it mean to suggest that our demand for love is simply our Desire added to our appetite for satisfaction? Perhaps in love we find our unquenchable Desire at least partially satiated? As interesting is the idea that our appetite for satisfaction is the difference between our demand for love and our unquenchable Desire to know directly, which suggests a familiar dichotomy between 'love' and 'lust', between 'need' and 'want'. However, rather than opposing them to one another (as the more traditional cliche might be inclined to do) Lacan suggests that they are intricately related through the same conduit of Desire which shapes our relationship to language, and thus both to our inner selves and the outside world.

Lacan also represents the relationship between signifier and signified in terms of the formulation S/s. While it seems that a primary reason for this visualization is that Lacan wishes us to see the Signifier as Over the Signified, in prominence and importance as well as in the role it plays in language (the signified 'reigns', so to speak), Lacan's obvious penchant for mathematics suggests that this might equally be read as "the sign = the Signifier divided by the signified," or perhaps better: "the sign = the signified divided into the Signifier." And Lacan's use of big and small letters is also employed with regards to the Other, which is opposed to an inner self that Lacan terms "objet petit autre", or the object of the little o. So we are left with a conception of the individual as represented by O/o, where O represents our outer self, the Other, which Lacan defines as “the very locus evoked by the recourse to speech” (1185). Once again we confront the bar, at which position Lacan locates the all-but-uncrossable gap between truth and symbol. It is all to tempting then, to set up an equivalency:

S/s ~ O/o

And a whole range of suggestions is evident from this formulation. First of all, the relationship of the Other to the Signifier is made clear, and the signified is consigned in the formula to a position equivalent to that of the inner self (o). This all seems to fit Lacan's own pronouncements very well. However, what happens when we play our little mathematics games with this formula and come up with:

S(o) ~ O(s)

or further

S ~ O(s) / o

Perhaps we might read the first formulation as "the Signifier is multiplied through the objet petit a as the Other is multiplied through the signified." If we take the two sides of the equation to be stating the same thing in different ways (in the same sense that 2+2=4 does not necessarily indicate an operation so much as a simple equivalency), we might achieve a clearer understanding of the relation of the signified as Saussurian "concept" to the multiplicity of things which each concept must reference. The multiple and fragmented Other from which all signifiers ultimately must spring is bound up by the seemingly singular signified concept. The inverse of this relationship is represented by the first half of the equation, where the signifying linguistic element becomes multiple and fractured as it attempts to enter the inner self.

The second equation manages to isolate the Signifier, which we can now usefully take to equal "the inner self divided into the Other as multiplied through the signified."

Lacan's Mirror, Letter

Through his synthesis of Saussurian linguistics and Freudian psychology, Lacan attempts to give us a more complete and nuanced picture of both systems of thought as well as of the nature of our own existence.

I think it is important to note, from the outset, that by applying (and to a great extent, also extending) the Saussurian model of language into the core of the human psyche (as it relates to our subconscious), Lacan necessarily pushes any theory of existence into a realm of uncertainty and relativity. In his essay, "The Agency of the Letter in the Unconscious," (excerpted in The Norton Anthology of Theory and Criticism, 2nd ed.), Lacan quickly does away with the possibility of discussing reality at all when he states that

“...this primordial distinction [between signifier (S) and signified (s)] goes well beyond...the bi-univocal correspondence between the word and the thing....[N]o signification can be sustained other than by reference to another signification..." (1171).

The reason for this impossibility, Lacan posits, is the fact that there necessarily remains within us sections of ourselves that are untranslatable in any language. This quality of the incompleteness of language Lacan develops into a complete explanation of human existence as it manifests itself in human psychology.

In "The Mirror Stage as Formative..." (also excerpted in the Norton), Lacan sets about explaining what he conceives to be the moment of an initial separation of self from self-image. He locates this moment at the time that a baby first identifies itself in a mirror. (Thus, "Mirror Stage"). Before this moment, Lacan suggests, the baby is a bundle of "turbulent movements" that he can feel animating him (1165). We might find an analogy for this state in Nietzsche's Dionysiac, associated as it is with immediate sensory experience and fleeting sensation. Lacan's pre-Mirror baby has no sense of itself beyond the fragmented pieces of sensory data that it compiles from the world around it. It does not conceive of itself as a unified being. During this time, it might be inclined to consider its mother's breast as much a part of its existence as the feeling of hunger which the milk helps to subside. The sensation of warmth provided by blankets likewise is not conceived as separate from the self.

All of this changes, Lacan says, when the baby makes the "primary identification" of its self in the mirror (1164). In this identification, the baby conceives of itself as a unified whole for the first time. This has the effect of drawing forth the child's ego, as he conceives of a certain power and agency associated with a unified self. Lacan says that the idea of a unified self is primarily an aesthetic realization, based as it is on the form of unity presented in the mirror. However, the "agency of the ego," in order to establish it's self-image, must necessarily subject that pre-Mirror sensory data to a degree of repression, and so, Lacan says, the ego in this way is from the beginning situated "in a fictional direction" (1165). The mirror image, Lacan continues, "symbolizes the mental permanence of the I, at the same time as it prefigures its alienating destination" (1165). This alienation occurs because at the same time that the child conceives of himself as a unified being, he must also recognize all of those things which are evidently not a part of that self, including (not least of all) his mother's breast.

We can see Hegel's Master-Slave Dialectic at play in the interaction which Lacan conceives of between the child and his mirror image. Here, the child understands the image in the mirror to be Other than himself. It is not actually part of himself, but the child incorporates this "specular I" as Other into himself when he begins to interact with others. Thus Lacan's concept of "paranoiac alienation" occurs as the true self (the inner, inexpressible, and fragmented self that the child first knew) is subdued, enslaved and repressed by this "specular I" in its first attempts to interact socially.

From a scientific standpoint (one to which I, despite [because of?] my deep-seated romanticism, am ever inclined to resort), the question arises whether the mirror in this stage must be a literal mirror (in the sense that the identification taking place must be an actual literal reflection of the child), or if it might as easily be seen as a figurative mirror, in which the child’s individual mirror image comes together finally in his mind as he unifies the sensed anatomy of his own body and the perceived unity of other humans. It seems that if we are to consider the mirror to be literal (there is some suggestion that we are), Lacan’s theory has implications for our understanding of pre-industrial societies. For thousands of years the only mirrors available might have been the calm surfaces of bodies of water. Prehistoric babies would not likely encounter such a reflection until well into their lives. How much earlier today’s young children are surely exposed to any literal reflection of themselves. (The little musical swing that we bought for my son Will when he was born had a little mirror positioned just above his newborn head, for easy viewing.) Does this earlier exposure suggest that the mirror stage might be reached earlier in life for modern man? If, in order to avoid such a difficulty, we consider the latter to be true, and the mirror to be figurative, we are put in the position of asking what the immediate stimulus might be for the child to unify his self-image. Truthfully, this same issue arises in the literal-mirror model, since I think we can safely assume that Will did not unify his self-image the first time we laid him in the musical swing. In this sense, Lacan's theory lacks clarity as so many other developmental models seem to (my wife, in this connection, suggests Piaget's cognitive development model). Lacan goes farther than Freud in explaining the roots of the consciousness we experience, but he shirks the question of why it should be this moment, this stimulus, rather than any other which causes the necessary fracture.

In "The Agency of the Letter in the Unconscious," Lacan attempts to trace out the implications of the fracture of self-consciousness which occurs in the Mirror Stage. He begins by positioning the individual, from the very start, in the context of society and culture through our relationship with language, to which he attributes the structure of the unconscious. Lacan observes:

“...language and its structure exist prior to the moment at which each subject at a certain point in his mental development makes his entry into it....[T]he ethnographic duality of nature and culture is giving way to a ternary conception of the human condition--nature, society, and culture--the last term of which would well be reduced to language...” (1169-70)

He perceptively downplays the traditional dichotomy between individual and society, between nature and nurture. Lacan sees these two elements as so intimately interwoven as to be ultimately indivisible. However, the real revolution is Lacan's assertion that the unconscious itself contains "the whole structure of language" (1169). Saussure was astute in his recognition that language is something complete which we enter into as we learn it. We do not have to create language anew because it already exists in the world separate us. Lacan goes a step further by stating that language is the structure by which our subconscious is organized. Language, in other words, at least in its structure is an inborn trait. By identifying himself with an image, the child has already undergone a shift in his consciousness from the 'real' (that state of immediate sensual interaction with the world) to the 'imaginary' (literally, associated with his image, his fractured ego self). When the child enters into language, he undergoes a second transition, into the realm of the 'symbolic.' Interactions at the level of society and culture, undergone through our employment of language, are therefore removed from reality by two degrees.

In explaining the structure of this Language itself, Lacan reproduces the formula put forward by Saussure, translating it in Lacanian fashion into an algorithm: S/s, where S represents the signifier and s the signified. Lacan here places a great deal of stress on the bar separating the two, a barrier which he says "initially...resist[s] signification" (1171). This bar takes on greater significance later, as he explore the signification of the phallus. For now, it is enough to say that this bar is largely uncrossable, and that any suggestion of a direct relationship between S and s is misleading. To demostrate this, Lacan reproduces the diagram employed by Saussure to explain the relationship between signifier and concept:

And couples it with his own, corrected and refined image:

It should be noted that, in reproducing Saussure's image, Lacan has flipped it, in order to make it match with his formulation S/s. (in Saussure's original, the concept is on top, the sound-image on bottom, which accords with his view that the sound-image flows from an attempt to represent, to signify, the concept). This change also has the effect of placing the primacy on the signifier, rather than the signified, which is key to Lacan's understanding of the final inaccessibility of the signified in any attempt at signification.

The difference between these two images lies in Lacan's conception of language as a power in identity formation. The second image presents two signifiers with seemingly identical signifieds. The speaker then is left with a choice which requires that he enter into one or the other, and in so doing, fit himself (or herself) to a notion of identity which is defined by language (as a system external to the self). While Lacan's example ostensibly deals with sexual identity (which he seems to consider as primary in identity formation), the same diagram might well serve in any number of cases, as far as it relates to Saussure's pardigmatic axis of signification. The choices we make with regards to language, what word we choose as signifier for any given signified, help to define our identity in relation to society.

Lacan here introduces what he terms "the signifying chain," which is the chain of signification by which we attempt communication, and which he characterizes as "rings of a necklace that is a ring in another necklace made of rings" (1174). The image is appropriately abstruse, and suggests in language the quality of a fractal, ever diverting from any specific connection to reality. The signifying chain is indicative in grammar of "the level of the unit immediately superior to the sentence," and in verbal utterance, "the level of the verbal locution" (1174). In other words, the signifying chain is active at the moment that a complete meaning is attempted in language. Lacan allows that this meaning is developed either by recourse to metonymy (the substitution of words based on partial signification) or by metaphor (the substitution of signifiers which cross the barrier of signification).

Mytonymy is basically a repetitive act, and does not require any creation, as it simply substitutes signifier for signifier based on similarity or relationship in meaning. This seems similar to Saussure's paradigmatic axis, which finds attached to any signifier other signifiers which have a related meaning. Indeed, Lacan echoes Saussure when he states that "[t]here is in effect no signifying chain that does not have, as if attached to the punctuation of each of its units, a whole articulation of relevant contexts suspended 'vertically', as it were, from that point" (1175). This suggests that any utterance implies a substitution, some other chain which would work equally well to perform the intended meaning. It also suggests that any utterance might well imply its opposite, and that the implication of opposing meaning is equally as important in our attempts to understand the meanings of opposite. (This seems again to suggest language as a system of negative relations: we define any single concept not so much by what it is as by what it isn't).

Metaphor is more complicated, and I'm not quite sure what to make of it. Lacan allows that metaphoric construction is "creative or poetic" in that it represents a "crossing of the bar" of signification which allows for the "passage of the signifier into the signified" (1179). This formulation is confusing, considering Lacan's insistence up until now (and later, in his discussion of the Phallus) of the inviolability of the bar, the essential inaccessibility of the signified in any attempt at signification. Perhaps we must understand this simply as stating that metaphor is creative in that it sets up new linguistic relationships between signifiers, new arrows pointing to new signifieds (which perhaps still remain essentially out of reach). If at one time 'love' and 'rose' were metonymically isolated, metaphor has since crossed the bar, so that the two words now are bound by associative meaning.

Lacan finishes his essay on "The Agency of the Letter" with a deconstruction of Descartes assertion that "I think, therefore I am." Lacan identifies the error in this assertion as the assumption that human thought represents the epitome of existence. Thought, Lacan has shown, is a determinate of language, which is necessarily symbolic and repressive of reality. To hold to Descartes' formula is to "deny oneself access to what might be called the Freudian universe," all that part of existence which eludes consciousness (1180). The reformulation of human consciousness around linguistic construction means that, henceforth, it is no longer "a question of knowing whether I speak of myself in a way that conforms to what I am, but rather of knowing whether I am the same as that of which I speak" (1180). Because the true inner self, the 'real', is eternally cut off by our recourse to language, we must forever wonder whether what we are 'really' bares any relation to how we are represented in our speech. Lacan thus reformulates the saying, first as, "I think where I am not, therefore I am where I do not think," and finally as, "I am not wherever I am the playing of my thought; I think of what I am where I do not think to think" (1180). While these reformulations can be seen partially as a humorous Lacanian subversion, they suggest what is really at the core of his philosophy: that human 'reality' must necessarily lie forever beyond the scope of explanation.