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.”