Toward a Dynamics of Being: From the Barred Subject to the Borromean Knot

The Lacanian conception of the subject is essentially non-substantial. The subject is not a stable ego, but an effect of the signifier, as formulated in Seminar III (Lacan, 1955–1956). Each time one speaks, the subject appears divided between what is said and what is lost. Lacan writes it as barred, $, to indicate this structural division (Lacan, 1964). Every act of speech leaves an irreducible remainder, the object a, cause of desire, formalized in “Subversion of the Subject and the Dialectic of Desire” (Lacan, 1966). The classical matheme of this relation is:

$ ◊ a

Yet this formula is static. It does not explain how the subject transforms over time nor how the knot of the Real (R), Imaginary (I) and Symbolic (S) tightens or reorganizes across a lifetime. This post proposes a dynamic formalization that extends Lacan’s mathematical gesture, especially his late work on the Borromean knot (Lacan, 1972–1973).

1. SIR as temporal functions of consistency

The three Lacanian registers (R, I and S) are not measurable quantities but forms of psychic consistency (Milner, 2002). However, their relative intensity changes through time. Each register can activate in pulses, sometimes brief and sometimes prolonged, depending on memories, affects, or structural tensions that trigger it. They are not continuous vibrations but pulsatile modulations that mark the dynamics of the subject. To formalize these variations, we use Greek coefficients that indicate the momentary pressure of each register. These coefficients do not represent fixed numbers but dynamic modulations, consistent with readings of the late teaching (Miller, 1981).

  • ρR(t) names the pressure of the Real, that which erupts without being symbolized, following the definition of the impossible in Seminar XI (Lacan, 1964). The coefficient ρ marks how much force the Real exerts at that instant.
  • ιI(t) expresses imaginary consistency, the stability of the ego, the solidity or fragility of identifications, following the “mirror stage” (Lacan, 1949). The coefficient ι indicates its fluctuating intensity.
  • σS(t) represents symbolic efficacy, the capacity of Language and Law to sustain meaning (Fink, 1995). The coefficient σ marks its momentary degree of functioning.

The aim is not to sum these functions but to explore how the tensions between them reconfigure over time. The subjective structure depends not on how much R, I or S is present, but on how they knot, compete or compensate one another in each moment of life.

2. The Borromean operator N and the consistency function C(t)

The operator N does not compute numerical values; it configures a knotting. This extends the Borromean formalization developed in Seminar XX (Lacan, 1972–1973). We define:

C(t) = N(ρR(t), ιI(t), σS(t))

Where:

  • C(t) = 1 stable knot
  • C(t) = 0 un-knotting
  • 0 < C(t) < 1 tensions or symptomatic compensations

The operator N gives form. If one register slips, the entire knot disarms.

3. BEING as the temporal integral of consistency

The subject is a knot maintained over time. We define:

BEING = ∫ C(t) dt

This conception is inspired by the idea of temporal insistence of the signifier (Lacan, 1957) and by structural readings of consistency (Milner, 2002). The BEING is the historical trajectory of the SIR knotting. Whereas $ designates the punctual appearance of the subject at each signifying cut, the BEING describes its continuity through time.

4. The object a as structural derivative

The object a does not arise from a register but from the variation of the knot. This follows the formulation of object a as remainder of the signifying cut (Lacan, 1960; Lacan, 1966). We formalize it as:

a(t) = d/dt [ C(t) ]

The object a is the remainder that appears when the knot tightens or shifts. It is the cause of desire, the trace of lack created by the variation itself (Lacan, 1959–1960).

5. Dynamics of affects

Each Lacanian affect can be defined as a variation between registers. We use natural logarithms to capture relative changes rather than absolute magnitudes (Fink, 1995). We define:

  • Anxiety ≈ Δ ln(R/I)
  • Inhibition ≈ Δ ln(S/I)
  • Symptom ≈ Δ ln(R/S)

Anxiety arises from a mismatch between the Real and the Imaginary, when the Real erupts or when the image sustaining the ego collapses (Lacan, 1962–1963; Lacan, 1964). Inhibition appears when the Symbolic imposes itself over imaginary consistency, blocking action under the pressure of Law or Ideal (Lacan, 1926/1950; Miller, 1981). The symptom emerges when an excess of Real overwhelms the capacity of symbolization, fixing itself as repetition to sustain the knot’s consistency (Freud, 1926; Lacan, 1959–1960).

An increase in ρR(t) forces a contraction of σS(t) and instability of ιI(t). Consistency C(t) is preserved only if the registers adjust to avoid surpassing their structural limit. In that variation appears the remainder, a(t), as the non-symbolizable trace of the knot’s change.

6. The Symbolic Man or Perfect Being

We can imagine a limiting case where the RSI knot is perfectly stable over time, that is, where consistencies do not vary under external or internal perturbations:

ρR(t) = 1, ιI(t) = 1, σS(t) = 1, C(t) = 1

Then the derivative is zero:

a(t) = 0

Without remainder there is no lack, without lack no desire, and without desire no barred subject. This describes an immutable consistency that does not belong to human experience. In the Lacanian field, such a limit can only be thought as divine structure or as outside the speaking subject (Lacan, 1972–1973).

Conclusion

This dynamic model articulates a precise vision:

  • The BEING is the integral of the SIR knot
  • The object a(t) is the derivative of that variation in consistency
  • The subject $ appears at every signifying cut
  • Desire is the movement driven by that remainder

The subject is not a substance but a rhythm. A consistency that holds, a lack that persists, and a variation that propels. Lacan designed the mathemes. This model seeks to give them a dynamic.

Bibliography

Brief note on editions: Lacan’s pagination varies by edition and language. Below I list commonly used versions in Spanish and French. If you use others, replace page numbers accordingly.

  • Freud, S. (1926). Inhibition, Symptom and Anxiety. In Complete Works, vol. XX. Buenos Aires: Amorrortu, 2003, pp. 73–161.
  • Lacan, J. (1949). “The Mirror Stage as Formative of the I Function”. In Écrits. Buenos Aires: Siglo XXI, 2008, pp. 93–100.
  • Lacan, J. (1955–1956). Seminar III: The Psychoses. Ed. J A Miller. Buenos Aires: Paidós, 1984.
  • Lacan, J. (1957). “The Agency of the Letter in the Unconscious”. In Écrits. Buenos Aires: Siglo XXI, 2008, pp. 493–528.
  • Lacan, J. (1959–1960). Seminar VII: The Ethics of Psychoanalysis. Buenos Aires: Paidós, 1988.
  • Lacan, J. (1960). “Position of the Unconscious”. In Écrits. Buenos Aires: Siglo XXI, 2008, pp. 795–811.
  • Lacan, J. (1964). Seminar XI: The Four Fundamental Concepts of Psychoanalysis. Buenos Aires: Paidós, 1987.
  • Lacan, J. (1966). Écrits. Buenos Aires: Siglo XXI, 2008.
  • Lacan, J. (1972–1973). Seminar XX: Encore. Buenos Aires: Paidós, 1982.
  • Miller, J-A. (1981). Introduction to the Reading of Lacan. Barcelona: Paidós, 1999.
  • Milner, J-C. (2002). The Structural Odyssey. Buenos Aires: Manantial, 2003.
  • Fink, B. (1995). The Lacanian Subject: Between Language and Jouissance. Princeton University Press, 1995.

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