As the inversion-induction thesis develops, this page will index contributions. Titles and short summaries will link to full papers and YouTube videos. The ultimate aim is to create an alternative to the mathematical and quantum physics approaches advocated by Slavoj Žižek, mathematician Alain Connes and psychoanalyst Patrick Gauthier-Lafaye, and the mathematician and psychoanalyst Daniel Sibony. These excellent introductions require considerable mathematical knowledge, however. The approaches below are not exclusively “non-mathematical” (they sometimes involve the non-numerical calculus of George Spencer-Brown) but they argue that Lacan’s topological-diagrammatic approach links directly to ethnology and popular culture, something lacking in the quantum-physics approaches. Space and time are cultural constructs, integrated into language, rituals, customs, and artifacts. They are also mathematical constructs, just without the tie-ins to the operations of culture.

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Don Kunze, The Psyche Is Extended — but HOW? [YouTube; 25 minutes]

This video elaboration of how the psyche is extended in time as well as space is really not for beginners. It presumes a reasonable familiarity with Lacan’s L-schema, Freud’s pithy note about how “Psyche is extended; knows nothing of it”; Lacan’s formula for metaphor; and Freud’s “parapraxis” (the forgetting of the proper name, “Signorelli”). On top of this, the argument carries into the idea of travel as involving three kinds of circularities; then it claims that circular triplicity relates to the dilation and cathesis of the Imaginary as evidenced in Lacan’s “butterfly diagram” in Seminar XI, “The Four Fundamental Concepts of Psychoanalysis.” Really! It’s not just “not for beginners,” it’s not even for puzzle-loving Lacanians with years of experience. Yet, there are benefits by comparing all of these models of spatial and temporal extension. The hope is that they may offer a polymorphic-perverse methodology for combining topology with ethnology and, in the process, aligning Lacan’s theory of metaphor with that of the 18c. Neapolitan philosopher of culture, Giambattista Vico. If you’re up for it, give it a spin but be patient and persistent. This is one of those projects that requires as much work from the viewer as the producer, maybe more! A script of this video is available here.

Don Kunze, The Butterfly and the Circuit: Shibboleth for the Imaginary [YouTube]

This video is not for beginners! It as an attempt to convert some of Lacan’s most difficult materials into a new protocol of diagram relations involving the classic “butterfly” diagram of the look and gaze, the L-schema, and topological figures. This dense series of diagrams argues for an anti-historicist view of Lacan’s teachings. His first topological thinking begins with his early essay on Logical Time, with the use of inversion and induction in the production of cathesis as paralysis. None of this will make sense without considering how the line between the Analyst and Analysand in the clinical session amounts to a sagittal Imaginary, permeable and resistant. Each of these panels amount to a week or more of study, so don’t be discouraged. With time, some new themes, such as triplicity, cathesis, and dilation, will become clearer. Read the script here.

Don Kunze, Žižek’s Favorite Crime Plot. Lacanians are in debt to Slavoj Žižek for his lifetime dedication to maintaining and restoring the legacy of psychoanalysis and adopting it to new critical, academic, and political challenges. There is literally no one who has contributed as much to the Freudian-Lacanian “cause.” In a recent short essay in Crisis and Critique, Žižek cites his favorite kind of plot, where a character, usually a woman, is victimized because of what she knows but doesn’t know she knows. It is typical for Žižek to articulate, correctly and imaginatively, an idea that has been around for 2500 years without citing its pedegree or giving it its proper and customary name. (In this case, not knowing what you know is called kenosis). More important is that Žižek forgets or does not know the name of the plot-form that he not only likes the most but identifies with, as his own intellectual predicament: the induction puzzle.

Don Kunze, Ethnotopology. A New Lacanian Topology will not be produced by competent mathematicians. It will be hobbled together by amateurs who carry out ersatz conjectures based on the speculative wager, that topology begins with ethnology. This is an intentionally erratic theory about the metaphoric structure of “mythic mentality,” initiated by the subject-as-signifier in relation to the Other, a hysterical sublimation of a traumatic Real that resurfaces in signifying chains. This ethnological beginning reverberates through successive periods of development of thought, from mythic to representational (“heroic”) thought, to the conceptual thinking of modernity.

Don Kunze, Lacan’s Inversion Circle and the Injunction of Popilius. In “La troisième” Lacan presents a Brunnnian variation of the Borromean circles representing the Real and Symbolic as crossed lines and the Imaginary as an interlacing circle. This modest example is compounded by references to “the Injunction of Popilius,” the story of the Roman consul who deflected the armies of Antiochus by drawing a circle in the sand around the invader king. If at least three people have known what this circle meant (Antiochus, Popilius, and Jacques Lacan), countless others may be added if we take into account folk practices involving encirclement and consider the works of art, narratives, and rituals, both formal and unconscious. Inversion circles are “everywhere if one knows where and how to look.” The foundation of Rome, for example, if read correctly, carefully establishes the efficacy of a renewable pomœrium that was legally the only actual ground constituting the city: the areas to either side of it were regarded as territories. A psychoanalytical understanding of the inversion circle, however, is limited to Lacan. This collection of texts establishes Lacan primacy in theorizing the inversion circle, leaving no doubt that the ethnological study of this geometric form must begin with Lacan’s topology, but this topology in turn must begin with the inversion circle.

Frances Conrad, “The Second Parallax of the Heroic Traveller.” In this early attempt to relate Lacan’s middle seminars to independent examples of inversive geometry, the (fictional) Conrad jumps into the idea of travel as a methodology for aligning the toroidal thesis inverting Descartes’ je pense, donc je suis into a self-intersection, non-orientable structure of instrumental convergence.

Iraj Ismailpoor Ghoochani, Simonides’ Tale and Freud’s Case of “Emma. The tale of Simonides and Freud’s case of “Emma” both revolve around the complex structures of memory, trauma, and the chiastic interplay of past and present events. In analyzing these stories structurally, we can see how concepts like chiasmus and inversion are close to each other. They both rely on a deferred action (Nachträglichkeit). The goal here is to show how these structures resonate with Lacanian and Lévi-Straussian approaches, which empahsize the deep, often hidden structural elements of human experience as a narrative.

Iraj Ismailpoor Ghoochani, The Inversion Circle: The X-Axis Revelation. The inversion circle isn’t just an abstract construct or a distant geometric idea; it’s rather an intrinsic part of the x-axis itself. By rethinking zero as an infinitesimal inversion circle, we reveal that the x-axis itself is a kind of infinite inversion circle—every point on the axis is a reflection of this underlying geometric relationship. The inversion circle, in its limit, becomes indistinguishable from the number line, embedding itself into the structure of the x-axis.

Iraj Ismailpoor Ghoochani, Perpendicularity as the Holy Grail: A Manifesto for Revisiting (Post)Structuralism through the Mirror of Orthogonality. Ghoochani: “I encourage you in this essay to explore the concept of Orthogonality through a system deeply concerned with acts rather than facts, where attitudes like rotations take precedence over magnitudes. This article proposes a refinement of the binary model central to structuralism by reinterpreting traditional dichotomies—such as Word and Flesh, symbol and symbolized, positive and negative, men and women—as orthogonal dimensions rather than opposites. Building on the structural analysis of Lévi-Strauss, we argue that these dichotomies should be reconceptualized as two orthogonal spaces, akin to the complex plane in mathematics. This perspective allows for a more nuanced understanding of cultural structures, recognizing the coexistence and interaction of distinct yet interrelated domains, and paving the way for a richer exploration of the complexities within human experience. It also enables us to integrate humanistic materials, in the form of  mythemes, with insights from cultural studies into a cohesive set of mathemes, without succumbing to the pitfalls of “physics envy.“

Iraj Ismailpoor Ghoochani, Imaginary Axis as a Helix. In the complex plane, examining these mathematical structures reveals a toroidal form, where the imaginary axis loops back onto itself. This insight is crucial for grasping Lacan’s ideas about the cyclical nature of demand and desire. To engage with Lacan’s theoretical constructs effectively, one must be acquainted with vector mathematics and topological principles, as they provide a foundational perspective on his psychoanalytic concepts. This essay serves as an initial guide to these mathematical principles, introducing the imaginary axis in the complex plane as the cut.

Iraj Ismailpoor Ghoochani, The Orthogonality of the Will: From Apple to Appeal—Curvature and the Inversion Circle. Spinoza discusses the illusion of human freedom by comparing it to a stone that, if it were conscious, would believe its motion was a result of its own will rather than external forces. This analogy highlights Spinoza’s broader argument that humans, like everything else in nature, are driven by necessity, and our sense of free will is simply a result of our ignorance of the true causes of our actions. This concept paves the way for exploring the inversion circle in psychoanalysis, where we can examine how the illusion of autonomy— similar to Spinoza’s stone—might be understood through the lens of object petit a and the “acting at a distance” dynamics of desire.

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Dan Collins, “Lacan with Frost,” Canadian Review of American Studies 51, Number 1, (Spring 2021): 32–43. [Collins:] “In this article, the author argues that the unlikely pair of Jacques Lacan and Robert Frost have more in common than one might think. Several examples from Frost’s poetry and prose are adduced to show that he anticipated Lacan’s psychoanalytic insights in several respects. The similarities cannot be attributed to influence as the two were unaware of each other’s work. Frost was certainly aware of Freud but largely took issue with Freud’s view of human nature. It is argued that the basis of the similarities between the two thinkers is structural.” Collins uses “For Once, Then, Something,” Frost’s poem about looking down a well, where the viewer assumes a position with Lacan’s famous diagram of the look and the gaze. The rim (Frost: “curb”) of the well functions as an inversion circle, condensing the upward view into a modified reflection including the viewer with the viewed.  The relation between Lacan’s work in Seminar XI, The Four Fundamental Concepts of Psychoanalysis, perhaps requires Xavier Audouard’s presentation in Seminar XIII, The Object of Psychoanalysis. This essay is available through Project Muse.