Post cartesian rationality - Presentation abstract

THE SUBJECT AS MATTER OF THE SPEECH

- Resumption of an old project by Descartes and Leibnitz: to regulate one's thinking with a method.

- ENTROPOLOGY is taken from Lévi-Strauss in "Tristes Tropiques": how to consider an exchange between two interlocutors as an upgrade of their information systems, in the sens that one would have a thermodynamic balance (cold water+hot water=warm water) (cf.: Huyguens pendulums).

LESSON 1 : Imaginary Levels

A Sujet is a being of language (Lacan among others)

• His Imagination is bordered by the Real and the Symbolic (triptych of Lévi-Strauss taken up by Lacan)
• His Imagination is articulated like a language (Lacan)
• A language is composed of : (Saussure)
  • "synchronic" levels or language/ metalanguage levels etc.;
  • the gap between a language and a metalanguage is a "diachronic gap";
• The Imaninary of the Subject is strutured in "leaflets" which, shaking each other, transmit the stimuli received frome de Teal to the Symbolic level like a pendulum of Newton. Idea of Freud in 1896 in a letter to Wilhem Fliess. (see Newton's pedulum).

I started from this base to explore the language of category theory in mathmatics.

After this first exploration, I revisited what I had learned from physics to arrive at the idea that our Imaginary can be brought back to 5 different levels, bounded by the Real and the Symbolic, what I represent thus (my own place as the "bearer of the discourse" being spotted by the symbol 𓂀 to the extreme right, closing he statement) :

☯[∃] [⚤] [#] [♲] [∅]☯𓂀

Definition of glyphs used :

• ☯ the Real in the sense of Lacan : what traumatizes or takes out teh Subject of is Imagination, what questions him;
• [∃] this level    has only one element : the final objet (*) of category theory;
• [⚤] characterized by the dichotomy of concepts. It contains the diecriminating object of category theory (for Set : {0,1}). It is the level of sets, discrete, N, logic etc.;
• [#] characterized by orthogonality of concepts. It is the transition from discrete to continuous, from N to R, then C and H, from algebra to topology etc.;
• [♲] after the explosion of concepts in [⚤] and [#], it is the level of equivalences and relativity (principle of least action, conservation, triptych of Emmy Noether);
• [∅] this level is only the empty initial object of the category theory;
• ☯ the Symbolisum in the sense of Lacan and Lévi-Strauss.

Both levels [∃] and [∅] are the boundary of an Imaginary, which is essentially articulated between [⚤], [#] and [♲].

LESSON 2 :  The postures of the Subject

1. The Subject has two basic attitudes:
   • Ex post or global𓁜: turned towards the Real ☯𓁜
   • Ex ante or local 𓁝: turns towards the Symbolic 𓁝☯
2. We will follow the neurologist J.P. Changeux according which the "awareness" of the Subject consists in a connexion between a "percept" coming from the organes of the senses, and a "concept", already in the cortex, saying that the subject :
   • "becomes aware of himself" :  𓁝I𓁜
   • "becomes aware of an object" : 𓁝[α]𓁜

Exemple with these pendulums.

The Subject is always in motion in is Imagination, and thought is movement.

We find Spinoza, and his 3 types of understanding :

• 1^st understanding or immanent : ☯↑☯ 
  • The Subject rises from floor to floor (diachronic "jump" or "Imaginary progression"):
    • ☯ [∃]𓁜[⚤] [#] [♲] [∅] ☯
    • ☯ [∃] [⚤]𓁜[#] [♲] [∅] ☯
    • ☯ [∃] [⚤] [#]𓁜[♲] [∅] ☯
    • ☯ [∃] [⚤] [#] [♲]𓁜[∅] ☯
• 2^nd understanding or transcendent : ☯↓☯
  • The Subject descends from floor to floor (or "Imaginary régression")
    • ☯ [∃] [⚤] [#] [♲]𓁝[∅] ☯
    • ☯ [∃] [⚤] [#]𓁝[♲] [∅] ☯
    • ☯ [∃] [⚤]𓁝[#] [♲] [∅] ☯
    • ☯ [∃]𓁝[⚤] [#] [♲] [∅] ☯
• 3^rd understanding which corresponds to the "awareness" of J.P. Changeux. Il is the meeting of the both previous, which I note generically: 𓁝I𓁜.
  • In this sense, the "awareness" can occure at each level of the Imaginary of the Subject :
    • ☯ [∃]𓁝I𓁜[⚤] [#] [♲] [∅] ☯ <=> awareness of the cogito
    • ☯ [∃] [⚤]𓁝I𓁜[#] [♲] [∅] ☯ <=> the mirror stage
    • ☯ [∃] [⚤] [#]𓁝I𓁜[♲] [∅] ☯ <=> principle of conservation
    • ☯ [∃] [⚤] [#] [♲]𓁝I𓁜[∅] ☯ <=> limit of rational thinking

From this point of view, fixed attitudes are seen as dysfunctions in relation to a "normal" functioning of the subject:

• Ex post blocking ☯𓁜: psychotic attitude (he always knows)
• Ex ante blocking 𓁝☯: neurotic attitude (he still doubts)

In short, the postures of the Subject in his own Imaginary are as follows :

☯ [∃]𓁝I𓁜[⚤]𓁝I𓁜[#]𓁝I𓁜[♲]𓁝I𓁜[∅] ☯

Some writting conventions, in order to situate our concepts before discussing them:

• The Imaginary level [α] of a concept is indicated as an index : concept_α ;
• The concept_α considerated ex ante is noted by exposing : ^𓁝concept_α
• Without an exponent, a concept is implicitly seen ex post.

LESSON 3 : Repetition automation

The fundamental point is that each level [⚤], [#] & [♲] defined a specific diachronic principle : 

1/ In [⚤], the concepts of successor_⚤ ou _⇅ , or order_⚤, or causality_⚤ are used to identify the jumps [∃] [⚤]𓁜.

Freud's repetition automatism is expressed temporally.

2/ In [#] the concepts of alterity_# or orthogonality_# or ⊥ are used to characterize the jumps [⚤] [#]𓁜. 

The repetition automatism destroys the previous notion of order_⚤, to give representations of the object by its surface_#, in an increasingly complex space: from R (with still de notion of order) repetition_# leads to R² (where it disappears), R³,...R^∞.

3/ In [♲] is expressed the concept of equivalence_♲ or ⇆ allowing to compare what at the lower levels is dissimilar. The object is no longer represented by a surface but by a volume_♲ or quantity_♲ that is measured_♲. Both equivalence and conservation relate to comparisons of these measures. (Example : a Picasso table worth 20.000 tons of green lemons). We come to the following diagram :

☯[∃]𓁝⇅𓁜[⚤]𓁝⇅𓁜[#]𓁝⊥𓁜[♲]𓁝⇆𓁜[∅]☯𓂀

THE AUTHOR OF THE SPEECH

By hypothesis :

1. The general schema makes it possible to represent the Imaginary of any Subject, in particular that of the author 𓂀, who expresses himself by some discourse: (...)𓂀;
2. If the content of the synchronic levels can be different, for the Subject and  for the author (i.e.: Lévi-Strauss can speak oa Subject Jivaro and vice versa) the diachronic concepts used are those of the author ⇅, ⊥ and ⇆.

This can be done by indicating as an index of 𓂀 the diachronic concept used by the author in his speach:

• [⚤]𓂀 <=> (...)_⇅𓂀
• [#]𓂀 <=> (...)_⊥𓂀
• [♲]𓂀 <=> (...)_⇆𓂀

The same author being able to express several speeches about the same object (in this case a Subject as "Other" of himself), one can draw up a table where each line would correspond to a particlar level of the speech.

An indication of the depth of the discours, would then be the line level of the table.

Example : the description of the difference of positions of the Subejct around the Imaginary level  [#] (mirror stage) :

• The discourse (a) of the form ((𓁝_⇅[#]𓁜)⇅(𓁝[#]_⊥𓁜))_⇅𓂀 indicates a succession of two states or "temporal evolution" from 𓁝_⇅ to _⊥𓁜; that we can mark more simply by (𓁝⏩𓁜)_⇅𓂀, it is the narrative (historical) of the shock felt by the child understanding suddenly that his image in the mirror represents him. 
• The discourse (b) ((𓁝_⇅[#]𓁜)⊥(𓁝[#]_⊥𓁜))_⊥𓂀 indicates a double otherness:      
  • "I 𓁜 am not my image 𓁝" in the mirror :  𓁝⊥𓁜;
  • "I 𓂀 am not the Other 𓁝I𓁜" : (𓁝I𓁜)_⊥𓂀 
• The discourse (c) marks at a higher level the equivalence between the two previous points of view: ((𓁝_⇅[#]𓁜)⇆(𓁝[#]_⊥𓁜))_⇆𓂀, or (𓁝⇆ 𓁜)_⇆𓂀. It is the guardian word confirming to the child, showing his reflexion in the mirror: "yes, it is you", and correspondingly "I am another".

The fundamental difference between the author 𓂀 and the Subject 𓁝I𓁜 is that :

• The Subject, as an object of discourse can take dual positions 𓁝I𓁜
• The author 𓂀 is always in a global or ex post position in relation to is discourse (...)𓂀.

1/ Closing of the discourse in [♲]⇆𓂀:

1.1/ Once the author identifies with the Subject on the highest degree of rationality of his Imaginary, in other words, after having expressed: (𓁝⇆𓁜)⇆𓂀, he can no longer add anything and by hypothesis  ((𓁝⇆𓁜)⇆𓂀⇆((𓁝⇆𓁜)⇆𓂀)⇆𓂀)⇆𓂀, so there is no discourse in the form of "Russian dolls" developping to infinity.

1.2/ We can represent the speach space in [#]_⊥𓂀 by two orthogonal axes :

Table 1

The discours in [♲]_⇆𓂀 is a tantanum to noting a redoubled equivalence :

• The autor 𓂀 seen as the Subject 𓁝I𓁜;
• The Real ☯ closing on the Symbolic ☯.

Which is as folding our space delimited by these 4 points : 

like closing an hancherchief by tying its 4 corners. In topology it is called "projective space", with here our author/ Subject in place of the projection point.

1.3 Emmy Noether Triptych :

• The preservation_♲ of 𓂀, specific to [♲]𓂀 level, is the volume_♲ bonded by this surface_# (note: strictly inexpressible, since as soon as he speaks, the autor 𓂀 poses himself at the previous "projection point", which pierces the surface_# of the volume_♲ in question and erease any difference between interior/ exterior);
• The element of minimal symetry_♲ is the neutral element_♲, corresponding to the equivalence_♲ between the dual points of view after the evolution  𓁝I𓁜⏩𓂀;
• The uncertainty_♲ is the author 𓂀's freedom to choice between 𓁝 or 𓁜 postures.

2/ Spatial representation in [#]_⊥𓂀 : 

The author 𓂀 and the Subject 𓁝I𓁜 can share a common understanding of the concept of orthogonality_#, which makes it possible to conceive for each of them a different geometry respecting the same concept_#, and in particular to:

• Define an hyperbolic space for my Subject 𓁝I𓁜 of experience;
• Represent this space of the Subject on a sheet of paper, in other words my Euclidean space of author 𓂀.

The only concept we share is the idea of orthogonality_# !

It's a good thing, since this concept of orthogonality_# can be deduced, as we have seen, with Bachmann'geometry, to a principle of symmetry_♲ of level [♲]!

3/ Temporal Representation in [⚤]_⇅𓂀: 

By definition, the Subject in ex ante position 𓁝 has no temporal reference (since precisely he is not in [∃] [⚤]𓁜 position), so there are only two times :

• That of the author 𓂀, holding the discourse and making measurements, in relation to the fixed benchmarks of his discourse [∃] [⚤]𓂀;
• That of the Subject ex post 𓁜 within the discourse, mobile in it, and counting the time in relation to the markers attached to him : [∃] [⚤]𓁜.
• Both movements are equivalents_♲ for 𓂀.

Example : the time measured by a station chief on the plateform and a passenger in the moving car :

• In a narrative, 𓂀 takes the place of the station chief 𓁜 and looks at the traveller 𓁝;
• In a second narrative, 𓂀 takes the traveler's place 𓁜 and looks at the station chief 𓁝;
• At level [#] both disourses are orthogonal_#;
• At level [♲] both discourses are equivalent_♲.

In summary  

1/ Having in mind that "I am another", we form the hypothesis that the Imaginary structure of the author 𓂀 of the discourse is similar_♲ to that of the Subject 𓁝I𓁜 of which he speaks.

☯[∃] [⚤] [#] [♲] [∅]☯

2/ When the concepts of similarity_♲, orthogonality_# and successor_⚤ are common at the respective levels [♲], [#] and [⚤] of the Subject 𓁝I𓁜 (in the speech) and the author 𓂀, closing the speech (note: think of the Brouwer fixed point theorem which explains the need for [∅]), then 

• The common concept of similarity_♲ allows the author 𓂀 to express locally in 𓁝 and globally in 𓁜 the laws of physics (relativity, quantum physics included).
• The common concept of orthogonality_# allows to articulate an Euclidean expression (for 𓂀) of a non-Euclidean space (for 𓁝I𓁜).
• The common concept of successor_⇅ allows the author 𓂀 to describe the evolutions of the Subject 𓁝I𓁜 in a "narrative time" (i.e.: symbolized by ⏩), decoupled from the time proper to the Subject (i.e.: symbolized by ⇅).

Return to the identity of the Subject:

When the Author speaks about himself, he identifies each of the levels of his Imagination, which can be represented by identity Id.

I am another (𓁝_⇆𓁜)_⇆𓂀

Our Subject becoming self-aware at the mirror stage is represented by a symmetry_♲ around this diagonal, like the fold of the Subject folding back upon itself.

THE 3 DISCOURSES OF LACAN, LÉVI-STRAUSS AND DESCARTES

Lacan L-diagram:

• The Subject 𓁝 is in a ex ante posture regarding his Symbolic 𓁝☯ ;
• The author 𓂀 (Lacan) defined 
  • This Symbolic level ☯ by ((S⇆A)_☯)_{​​​​​​⇆}𓂀;
  • The awareness of the Subject as not to be the "Other", in the mirror stage such as : (𓁝(S⇆A)_☯⏩𓁝(s⊥a)_#𓁜)𓂀;

Lacan indicates a transcendent approach, which must be completed by an immanent process of  symetrization by steps, implemented by the Subject to structure his Imaginary in synchronic levels, stopping at the awareness of his own consistency, in terms of volume retention_♲. 

The transcendent process corresponding to this permanence_♲ of the Subject is seen by Lacan as a decoherence from the Other. This process not being closed in the Imaginary of the Subject (i.e.: 𓁝☯) remains for him, irrational ! 

Canonical form of the myths of Lévi-Strauss:

When an object cannot be reduced to a dichotomous criterion (the object is (white or black)_⚤ it must be concluded that the criteria chosen to describe it are foreign to each other (pottery⊥jealousy)_# and that the object must be spotted as a product_# of these two concepts_# : the woman_# is (potter; jealous)_#.

What Lévi-Strauss points out in this process is the way to progress from a level [⚤] to a level [#] through a process developped in mythical thought, and which he defines by the canonical form of myths.

Example : Among the Jivagos, the woman has for totem the nighthawk, symbol of jealousy and chaos, but she is also a potter and shows meticulousness in the choice of pottery ground and the care that she puts to follow the rites of manufacture. The woman is at the same time nighthawk and no-nighthawk, which shocks the Cartesian logic of our good Jivaros !

The mythical process thus consists in sacrificing the disturbing concept (here the nighthawk) ceasing as well as being in contradiction with the observation (the woman is potter), to become an element of her définition as cartesian product : woman=(jealous; potter).

Nothing has changed untill today, and we find the same process to define "a" as a²=2 or "i" as i²=-1.

The myth does not solve the contradiction that appears in a dialectic way (nighthawk/ oven-bird) by staying at the level [⚤], but indeed by a "diachronic jump" at level [#] regulated by a principle in [♲]𓁝☯.

Complementarity of the two processes : 

• Lacan tells us about the constitution of the Imaginary self of the Subject from the fall of the Other into his Imaginary (or his Imaginary sacrifice).
• Lévi-Strauss tells us about the structuring of the Imagination of the Subject from the sacrifice of a symbol represented here by the nighthaw. The Subject is already there, and it is an Imaginary rise, caused by the shock of the Real (i.e.: the woman is a potter, in opposition to the Imaginary representation of the woman by her totem the nighthawk).

Now, we have just seen this last immanent process as the reverse of the mirror stage described by Lacan! We are just going through the same sequence in two different directions : 

Immanent ([⚤]𓁝⇅𓁜⏩[⚤](𓁝⇅𓁜[#]⏩[#]𓁝⊥𓁜)⏩[#]𓁝⊥𓁜[♲])_⇅𓂀

transcendent (𓁝⊥𓁜[♲]⏩([#]𓁝⊥𓁜⏩[⚤]𓁝⇅𓁜[#])⏩[⚤]𓁝⇅𓁜)_⇅𓂀

When the storyteller 𓂀 expresses himself in the form of a process, (𓁝[#]⏩[#]𓁜)_⇅𓂀, he remains outside the audience ([#]𓁜⏩[#]𓁝[♲])𓂀 and thus, marks his own permanence (𓁝⇆𓁜)_⇆𓂀, as a teacher giving a lesson, and it is here that he becomes the Other of the Subject to which he is intriged, at the Symbolic level (𓁝☯)𓂀.

Descartes' cogito :

Descartes becomes aware in [⚤]𓂀 of two postures:

• "I think" <=> ([⚤]𓁜)_⇅𓂀
• "I do exist" <=> ([∃]𓁜)_⇅𓂀

Which cover two movements, namely :

• "I think so I am" :  (𓁝[⚤]_⇅𓁜⏩[∃]𓁝_⇅𓁜[⚤]⏩[∃]𓁝_⇅𓁜)_⇅𓂀
• "I am so I think" : ([∃]𓁝_⇅𓁜⏩[∃]𓁝_⇅𓁜[⚤]⏩𓁝[⚤]_⇅𓁜)_⇅𓂀
• The awaereness of Descartes as a Subject holding to the equivalence of the two disourses  (𓁝_⇅𓁜)_⇆𓂀.

In this scheme, the coming and going of through [∃]⇅[⚤], allows two observations, according to the point which one takes at origin :

• "I exist": from there, all thought confirms my existence, and identifies me it is the meaning of 1_A in [∃].
• "I think": all my thoughts reflect me in one way or another, since at least I thought once that I am, even if I express my no-existence otherwise or if I lie (i.e.: (*)↑{0}), this is the meaning of the idempotence of my thoughts in [⚤].

Discussion: The sequence of the 2 disocourses recalls a fundamental distinction between identity / Idempotence, specific to topology_# and is expressed particularly well in category theory.

• s.r=e <=> "I thought_⚤"↓"I am_∃" ↑"I thought_⚤" : idempotence
• r.s=1_A <=> "I am_∃"↑"I thought_⚤"↓"I am_∃" : identity

The thesis that can be argued is that Descartes brings back in [∃][⚤] an immanent experience of the mirror lived by the child in [⚤][#].

His hope to base his method just on an immanent process would be doomed from the outset to failure on several levels :

1. His awareness  (𓁝_⇅𓁜)_⇆𓂀 is at level [♲]𓂀;
2. The process used is already acquired by the Subject in is childhood, in [#]𓂀;
3. The process of killig God to put the Subject in his place is a mythical process aiming at an "ascent" Imaginary, but here, it is a "regression";
4. The very project of a closing of the discourse on the Subject, runs up against the theorem of Gödel (i.e.: in a closed language, one can have contradictory propositions), which invalidades his hope of absolute rationality.

The 3 discourses : ☯([∃]𓁝⇅𓁜([⚤](𓁝⇅𓁜)[#]𓁝⊥𓁜)[♲]𓁝I⇆𓁜[∅]☯)𓂀

• ([∃]𓁜⇅[⚤]𓁜) Descartes' discourse ;
• ([⚤]𓁝⇅𓁜⏩[⚤]𓁝⇅𓁜[#]⏩[#]𓁝⊥𓁜) Lévi-Strauss' discourse (immanent); 
• (𓁝[#]☯⏩[#]𓁝⊥𓁜) Lacan's discourse (transcendent).

- What about Noether's one ?

- It covers the entire Imaginary field of the author 𓂀 as of the Subject 𓁝|𓁜 and is expessed by [♲]𓂀, this is what we checked at each jump [∃]/[⚤], [⚤]/[#] and [#]/[♲], until it desappears in the last jump [♲]/[∅]!

I hope that I responded to an attack launched by Derrida to the structuralists, at a congress in Cerisy : "how do you structure the genesis of your structures?"

CONCLUSION

We can distinguish three stages of our European evolution :

1. Imaginary "unipolar mythical" = [⚤] [#] [♲]☯𓂀 
2. Imaginary "unipolar cartesian" =  ☯[∃] [⚤] [#]𓂀
3. Imaginary "bipolar" or "post-cartesian" = ☯[∃] [⚤] [#] [♲] [∅]☯𓂀 

I would like to find the three modes of understanding of Spinoza :

• Mythical thought = trancendent or second-kind thought;
• Cartesian thought = immanent or first-kind thought;
• "Modern thought" = mixed thought or third-kind thought