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Alain Connes brought out a book containing a dialogue with psychoanalyst Patrick Gauthier-Lafaye called À l’ombre de Grothendieck et de Lacan. Developing Lacan’s ’The unconscious is structured like a language’, the thesis of the book is
The unconscious is structured like a topos.
I found the book very light on details. Attention is given to the idea of a classifying topos for a geometric theory encoding what Lacan calls the fundamental fantasy.
Surprisingly, there’s nothing on the dynamics of therapy, which in this story ought to correspond to modifying the topos/refining the theory. That got me wondering outside of anything psychoanalytic how to get the classifying topos concept to relate to Aufhebung, level of a topos, etc. Whenever we have some resolution of an opposition, should we be able to describe it via theory extension? This from Aufhebung suggests so
We can think of the inclusion of the sheaf category of a lower level into the higher sheaf category as an analytic relation between the concepts involved: when viewed as a relation between the geometric theories classified by the respective subtoposes an inclusion relation corresponds indeed to an unpacking of the richer theory of the smaller subtopos e.g. the subtopos corresponding to the theory of local rings is included in the topos corresponding to the theory of rings which on the conceptual side is spelled out as a local ring is a ring, or, the concept ’local ring’ implies the concept ’ring’. So the passage from subtopos to including supratopos corresponds to an unfolding of the concepts implied in the subtopos concept.
So if $\empty\dashv \ast$ corresponds to the inconsistent geometric theory. what should we say for $\flat\dashv \sharp$?
I guess to answer my final question, it’s the passage from the inconsistent geometric theory to the empty geometric theory.
Surprisingly, there’s nothing on the dynamics of therapy,
Are you being serious in your surprise here? It sounds to me like “He built a castle in the sky — surprisingly he forgot to add a stable for the horses.” :-)
Well, even someone sketching out an imaginary castle should mention essential components, walls, towers, drawbridges, etc., so you might expect a conversation on the unconscious lasting over 160 pages to touch on therapy.
But anyway, the mathematical question (which I could pose elsewhere) concerns the relationship between related classifying toposes and levels of a topos/Aufhebung. I hadn’t considered thinking of the latter in terms of the former.
So I gather you are not being ironic; you believe it makes sense to say that psychotherapy is about modifying a topos?
I feel that statements like this do little good unless accompanied by a substantial argument that makes plausible why what superficially sounds inane may have something noteworthy to it.
Compare for instance Conway & Kochen’s “Free will theorem”: The verbiage is comparably whimsical, but there it serves as advertisement for an observation that is noteworthy in itself.
Another similar case is the canonical formula of myth, which (for reasons that escape me, admittedly) has inspired mathematicians to associate contentful statements with the colorful verbiage.
Did you know I spent about 18 months in Paris studying Lacanian psychoanalysis, before returning to London to work on philosophy and category theory? Naturally, I was interested to see what someone like Connes made of the task of relating these worlds. I think very little. It seemed to me an obvious question he needed to pose to himself concerned change if he was to even begin to “make plausible why what superficially sounds inane may have something noteworthy to it.” But he didn’t.
That sparked off the purely mathematical issue raised at the end of #4, which is why I started the discussion.
As to whether I think there’s any hope of a mathematical formulation of ideas from psychoanalysis, I sense some loose connections, but I agree that substantial work is required to reach something as developed as Science of Logic. There are strains of Jung that resemble Hegel.
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