The non-numerical calculus of George Spencer-Brown, concepts from physics and mathematics, and literature techniques such as chiasmus, lipograms, acrostics, magic squares, palindromes, quantum physics, illusions (e.g. the Ames Window), magicians techniques, the “body loading” of the pickpocket,  confidence tricks, and so on provide inversive geometry study with a broad array of contexts and technical expertise from which theory may move fluidly across media, historical periods, and cultural contexts. This page indexes topics with brief descriptions linked to in-depth studies as they develop. Read the proposal for “ethnotopology” as the first response to the call for an ersatz hysteria.

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The Hysteria of Physics Envy

Lacanian psychoanalysts and scholars are rarely trained in mathematics or physics, yet they aspire to not a little surplus jouissance by using sexy terms such as “inversion circles,” “non-orientation,” and “Euler circles.” It is commonplace now for academics who flunked calculus to use “quantum” without warning the reader that they are experiencing a metaphoric if not a delusional moment. Lack of understanding gives rise to a specific form of envy, in this case “physics envy,” which leads to pseudo-mathematizing one’s research to the point of absurdity. Is this not precisely what is going on with this project?*

There are two tactical realities, imposed those without any proper background in higher-level mathematics, that turn out to have an upside:

(1) The first is the Hysteric’s Discourse, where the scholar is in the position of the hysteric who questions the (defective) Other, Ⱥ, on behalf of its suppression of the essential S’…S’ (signifying chains constituting the fundament of knowledge in a field), while all the time concealing the special, “scholarly,” jouissance that comes with living above one’s station. The upside comes from the occasional unexpected find that checks out and happens to reveal what no one else has noticed.

(2) The second reality is a forced choice. The non-mathematician has no option but to employ an “ersatz/ansatz methodology” — the creation  of a lot of wrong answers in which is sometimes embedded a lucky guess. The upside is two-fold. First, this method has been used by mathematicians themselves who, facing an intractable problem, invent an “ersatz” theorem and allow it to fail. Naturally, error data will be produced, but this restructures the problem and the ersatz investigator constructs a second, better-informed approach. With luck, the error data will have patterns and structures that overcome the preponderance of ignorance. What IS discovered by this method turns out to be more original and earth-shaking than what would have been discovered by more conservative techniques, constructed by presuppositions and old paradigms. Even for the seasoned expert, the ersatz/ansatz method had much to recommend it.

The hybrid approach, Hysterical Ersatz Conjecture (HEC), can defend itself on the grounds that standard approaches are (1) not very exciting and (2) rarely advance beyond their initial expectations. Ersatz Conjecture, when it regards hysteria as a discipline and not an excuse, develops a nose for high-payoff situations, where the improbability of connections is balanced by the productivity of new relations overlooked by conventional approaches.

The hysterical speculator values his/her partnership with the orthodox theorist, whose effort and results are roughly at a 1:1 ratio. The orthodox theorist, in return, benefits from HEC without having to run the risks of ersatz conjecture. Although the original inputs of HEC require wide reading, high levels of uncertainty and numerous false leads, the benefits are proportionately better. As in the case of Ghoochani’s discovery of Lacan’s use of the inversion circle in “La troisième,” the implications of knowing “what Lacan knew and when he knew it” are multiplied by the fact that, until this point, no other Lacanian had noticed this; and the few who had noticed it mistook it for precisely the opposite of what it was.

Similar payoffs await future ersatz theorists willing to investigate Lacan’s middle seminars, the remaining resistant core of Lacan’s topologizing. Little understood topics await clarification: the Injunction of Popilius; Xavier Audouard’s parallelogram argument in Seminar XIII; Lacan’s “slide-rule analogy,” begin in XIV, The Logic of Phantasy and taken up again in XVI, From the Other to the other.

The question of mathematics should be put thus: it is not about finding the objectively correct answer but to find the answer that Lacan himself found useful, which became critical to his work. A competent mathematician perhaps would not be willing to cut Lacan a break, but an amateur whose ultimate concern is for Lacan’s project of “the signifier that represents the subject for another signifier” will not forsake this for any abstract notion of correctness. When Lacan uses Richard Feynman’s idea of the conservation of energy, we must not ask “Is it correct?” but “How will he use it to structure his next move?” Lacan’s thinking, not mathematics, is the point of HEC.

The following issues are critical and must be investigated:

1. What is methodology for a hysteric, and what advantages are their over the discourses of the University and Analysis (the standard discursive forms assigned to psychoanalytic investigation)?
2. What are the obligations of the ersatz theorist to make full disclosure, when nothing of the sort is required of traditional investigations?
3. Is there not a benefit of allying the HEC with the principles of the fou littéraire — the free mixing of fictional and factual elements, given that the ersatz investigation is obliged to disclose all of its investigatory procedures, and at a higher standard than conventional research?
4. When Karl Popper joined conjecture and refutation in stating the standards of science, should not conjecture be held to as high a standard as refutation? Should not conjecture return to key historical resources, such as Gongórismo, Agutezza, and other critical initiatives that generated numerous works of genius? Conjecture should be more exacting and accountable than Refutation, which after all is too easily simplified into “just saying no.”
5. If Lacanian theory is not to devolve into the drudge task of understanding what Lacan meant, it needs to play out the implications of Lacan’s more heroic gestures, including his recognition of “fictional partners” such as Marguerite Duras’ novel, The Ravishment of Lol Stein. Like Freud’s interest in Wilhelm Jensen’s Gradiva, Lacan’s interest was that of the seasoned professional in the ersatz amateur, who had, against all odds, discovered precisely the same truths without any theoretical efforts. When the theorist and the artist arrive at the same point at the same time, but the former is exhausted and the latter seems not to have broken a sweat, is it not time to re-evaluate the role of the ersatz?
6. If Lacanians are, finally, to accept Lacan’s mathematical commitments fully; and few if any will be fully informed of the mathematics behind this dedication; and if hysteric/ersatz speculation is proven to be productive, effective, and testable, should not the next step be a “study program” that can be objectively adapted to diverse projects? This is the context of the current study of inversive geometry. It is not to complicate the already bizarre variety of Lacanian studies but, rather, offer a means to “cut to the chase”; to consolidate what is known based on what can be confirmed through accepted mathematical authority, to allow conjecture, under the conditions of full disclosure, the means and benefits of experiment.

hysteria + ersatz: inversion circle topology

For the non-mathematical Lacanian (presumably the majority of Lacanians) hysteria (speculation) requiring an ersatz methodology is not optional but necessary. The advantage of this is seen from the other side. What mathematician (besides George Spencer-Brown) has produced a psychology out of pure mathematics? Lacan’s coupling of psychoanalysis with topology (which we restate in relation to the central functions of the inversion circle) is the unavoidable critical aporia which Lacanians have mainly bungled. Were the Lacanian the Analysand in this situation, there would be an admission of a symptom, in relation to the Symbolic, and a sinthome in relation to the Real.

The middle term, the Imaginary, operates silently, enthymemically. The enthymeme is the principle of the analytic session. The Analysand must say it, or, rather, find that “he/she is being said.” With his/her back to the wall, theorists must, in their hysteria, understand their division, the signifier that is not just divided but is the division, in relation to other signifiers, as primordial, as a truth of their own making. Admission (of authorship) is commission. The enthymeme, silent transfer, pure re-(non-)orientation, negates position: paralysis. This is the basis of the hysteric’s complaint, which exposes the Big Other in the guise of the master signifier (Ⱥ), but only the part that faces and is, thus, a profile behind which there is a “perfect” (collinear) shadow: S2 as the consecutive  S′… S′ “seen from above” (orthogonality) so that its curvature, as in Eratosthenes’ experiment at noon on the solstice, reveals a new center, a new signified: x → s′′. The gaps in the signifying chain give rise to the Real, which becomes generative (s′′→ x→ S′… S′). Metaphor, in other words, is implicit in any S′… S′.

The law of parapraxis is that of a “stranger in a strange land” (the post-Babel condition of the speaker amongst other speakers), the production of orthogonality (metaphor) within linearity (metonymy) according to the same graphical relation that indicates independence as ⏊. Beneath the horizontal bar we might write x → s′′. But, we should note that the | emerges from the signifying chain, S′… S′. The x is the face that produces the perfect shadow, s′′. This collinearity is the signified of the unary trait: 1/s′′. This is the teaching of the hysteric. The product of the ersatz is the ersatz itself. Verum ipsum factum.

the alternative: ethnotopology

Between Lacan’s “pre-topological” period and his formal declaration of commitment to topology in 1961, there was a decisive event that shifted his conception of topology, from a primarily mathematical phenomenon to an ethnological one. This was Seminar VII, The Ethics of Psychoanalysis (1959–1960). In a decisive session focused on architecture, Lacan described the Baroque as a “surface of pain” and gave, as a mythical correlate, the story of Apollo’s pursuit of the nymph Daphne. Unable to flee the amorous god’s advances, the nymph froze in place; her compensation was the eternal life of the ever-green laurel. Although Lacan did not present the back-story, his geometry was projective and correct. Daphne was trapped “as soon as she thought to flee.” Ethnotopology takes this as a paradigm of the way the thinking of the first humans embeds a latent signifier of structure, which shapes it inversively and re-enacts this inversion with every telling.

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*Henry Krips has raised precisely this point, LACK Conference, Burlington, VT, Spring 2023. Krips, who actually has a Ph.D. in mathematical physics, observed that the broader interpretation of the mathematical concept of the “Hamiltonian” in a presentation gave a false impression of mastery without any clear benefit to theory. This was an accurate observation, but one that did not think forward to the predicament of most in the audience who, without the advantage of a tested mastery of mathematics or physics, nonetheless wish to understand Lacan’s forays into the difficult territory of projective topology, knot theory, and inversive geometry. In such cases, where disadvantage of (non-)education cannot be avoided or dismissed, a different strategy must evolve in order to obey Lacan’s maxim: Real>Structure>Topology. This Che vuoi? conditions a methodology which is best faced directly as a Hysteric’s Discourse, supported through an ersatz/ansatz (inebriated idea to generate, then sober analysis of error) that has long served mathematicians facing seemingly intractable problems.