Lacan’s formula for the paternal metaphor is famous:
What is less famous is Lacan’s more general formula for the mechanism of metaphor inspired by Freud’s forgetting of “Signorelli” (from Seminar 5 page 51):
In both cases, the intermediate term (mother’s desire, Signor) cancels itself out, so that the top term (Name-of-the-Father, X) and the bottom term (signified to the subject, Herr) remain operative. These two examples are also fundamentally different in that the top term is present in the paternal metaphor whereas it’s absent in “Signorelli”. That’s why the mother’s desire facilitates the subject’s social link as a familiarity whereas “Signor” blocks Freud’s memory as a foreign word. Nonetheless, these two formulas of metaphor are structurally the same.
It is quite surprising that nobody seems to have officially stated the pretty obvious fact that the chain rule in mathematics looks exactly the same with Lacan’s metaphor formula:
This is chain rule in Leibniz’s notation in terms of dependent variables. The same rule can also be written in Lagrange’s notation in terms of mathematical functions:
If f(x) = p(m(x)), then f'(x) = p'(m(x)) m'(x).
In both metaphor and chain rule, a basic mediation is established on the basis of cancelling further mediation. In the case of the paternal metaphor, mother’s desire assumes the self-cancelling role; in the case of “Signorelli” the foreign word “Signor” assumes the self-cancelling role. This prior role of the (m)Other is a crucial condition for the symbolic castration to take place. Only if the prior self-cancelling succeeds is the subject (or Herr Freud) subjected to the symbolic castration by the father (or the question about the enigmatic X). When the self-cancelling that conditions the symbolic castration is undone, the furious superego is unleashed (which explains male violence).
In the last equation, take p = papa, m = mama, f = family. As usual the papa embraces the mama in the family: f(x) = p(m(x)). This can be protective and/or suffocating or even abusive. Remember that the mama is the self-cancelling element.
But what really matters is the structural consequence of this paternal embrace in the derivative equation that we must interpret: f'(x) = p'(m(x)) m'(x). To reliably tell anything about this family, we have to talk to the papa about the mama but this is not enough! We also have to talk to the mama separately.
This is a dynamistically significant consequence : Precisely because she is the self-cancelling element, she has an independent term in the derivative equation that the man can never bypass. The self-cancelling element prevents other terms from cancelling it, so in a sense it remains uncastrated. This is like the famous tic of Sygne de Coûfontaine. Remember that the same role was assumed by the metonymic “Signor” that blocked Freud’s memory.
To conclude, Lacan’s version of the chain rule shows the singular maternal strength of metonymy against the particular force of the paternal metaphor. And this singularity can be universalized through language!