created 23 avril 2008



Est enim de ratione infiniti, ut a me, qui sum finitus, non comprhendatur.

Descartes. Meditationes de prima philosophia III


This is a consultation page, linked to other pages on the site where the words “categorical”, “categoricity”, “complete” or “incomplete” are used. I will use it to briefly outline and declare my epistemic opinions so it can be used by readers to understand the other pages. If you like, it could be considered the “manifesto” of the site which presents a possible scientific approach to psychoanalysis as a science of incomplete knowledge (See the page “forme del sapere- forms of knowledge).
I will explain the meaning of the word “categorical”, from a technical point of view, disregarding its common use or philosophical meaning, which roughly see it as being synonymous with “conceptual”, “unequivocal”, and even “compulsory”. I will also provide a brief outline of the relationship between “completeness” and “categoricity”.
“Requiring that a theory be categorical (with the meaning proposed by J. Dewey, but used for the first time with a mathematical connotation by O. Veblen in a paper published in 1905) means requiring that all its models be isomorphic; that is, they must have the same structure (C. Mangione and S. Bozzi, Storia della logica. Da Boole ai nostri giorni - History of logic. From Boole to the present - Garzanti, Milan 1993, p. 371).
The ultimate example of the categorical structure is Euclidean geometry. However it is presented, it fundamentally remains true to itself. There are no two different Euclidean geometries. At the most, there are non-Euclidean geometries, which however were theorized at a much later date, because of the orthodoxy imposed by Euclid’s Elements and the dogmatism which, from the heights of their formal perfection, they have exerted on mathematical intelligence over the centuries, to the point of  frustrating the development of the notion of number, suppressing it with the notion of quantity. (On this point see Carl Benjamin Boyer, The History of the Calculus and Its Conceptual Development (1949)). Euclidean geometry is fundamentally Aristotlean, materializing the classical metaphysics of the One, ideal of perfect knowledge free from ambiguities and deficiencies. This is luckily only an ideal. (Geometry as an ideal knowledge is a topos of phenomenology, considered as a science of the essences. Refer to E. Husserl, Crisis of European Sciences §§ 8,9).
If all knowledge were Euclidean, there would be no place for science in general or for psychoanalysis in particular. As Euclidean geometry clearly shows, a categorical theory completes itself and, in a certain sense, is sterile. It can produce no new theories because the full theoretical potential is addressed inwards. Thus, it cannot accommodate a maxim of truth, deemed to be a breeding ground of new truths, adopted herein and purpose of psychoanalysis.
Obviously, if at least two different models exist, the theory is non-categorical. Reference is made on the website to the theory of the actual infinite (or proper infinite, according to Cantor), opposed to the potential infinite (or improper infinite, again according to Cantor). Cantor’s theory is non-categorical. Indeed, two non-isomorphic models of the infinite can be evinced: the numerable infinite model (series of integers) and the continuous infinite model (series of real numbers or segments of the straight line). The diagonal method, invented by Cantor in 1890, can be used to demonstrate that the set of integers is a different infinite, that is, “inferior” to the infinite of real numbers. This is a generalisation of the proof by contradiction presented by Mangione and Bozzi in the text mentioned above (ibidem, pp. 312-313) and which I will now recapitulate in a way which even a psychoanalyst with a humanistic background should be able to understand.

Cantor’s Diagonal method

As regards the epistemic considerations made on this site, non-categoricity is more important than categoricity for two reasons. The first is negative: non-categoricity banishes considerations on essences and their conceptual determination from the realm of science. If a theory has several non-equivalent models, the essence cannot be determined in an unequivocal way. The second is positive: non-categoricity allows us to begin imagining a specific aspect of modern science found in all its variants (or models): incompleteness and fertility. “Indeed, it is clear that, assuming reasonable hypotheses, categoricity implies completeness; but it is not obvious, and indeed, after having defined terms, it can be established that the opposite is not true” (Ibidem, p.377). In turn, fecundity is the offspring of incompleteness. A complete theory can produce nothing new and has nothing new to propose.
Speaking of this, allow me to remind readers that semantic completeness (or sufficiency) is the property whereby all valid expressions are theorems of the theory, i.e. inferable from the axioms of the theory by applying the rules set for the inference. (The reciprocal, i.e. that all theorems are valid expressions, is the meta-theorem of coherence). Completeness establishes the equivalence of the syntactic (or inference) and semantic (or interpretation) planes of a theory.
We owe two enlightening theorems on the relationship between these two planes to Gödel. In fact, they are equivalent in the first-order predicate logic (completeness theorem of the first-order predicative logic, 1930), but they are not equivalent in ordinary arithmetic, if this is coherent (arithmetic incompleteness theorem, 1931). According to the incompleteness theorem, there are true statements that cannot be proved. Characteristically, arithmetic coherence is “hypothetically” true but it cannot be proved within aritmetic. Nowadays we know several true arithmetical statements which cannot be proved in finitary arithmetic; that is, that doesn’t resort to a strong axiom of the infinite (See, for example, E’ necessario l’infinito?Is the infinite necessary?)
This is the logical (not mythological or aetiological) rationale for the existence of Freudian “primary repression” (Urverdrängung) and for which true expressions, but not yet available to analysis, exist and will continue to exist. Primary displacement is indispensable in understanding Freud’s notion of the unconscious mind; it founds the unconscious on an underlying impossibility that cannot be eliminated even with infinite analysis. In technical terms, we say that the unconscious mind is “essentially incomplete”. Even if we add the displaced material, a new unconscious mind is obtained which maintains an unattainable earlier displacement. In a certain sense, it is the earlier displacement, and not the displacement, that represents the “essence” of the unconscious. Yet it is an essence that cannot be defined in a categorical way.
Non-categoricity and incompleteness are the elementary epistemic ingredients of many notions used by human sciences, including psychoanalysis. Notions such as culture, female, paternal, unconscious, madness, language, other, object, infinite and many others are non-categorical and incomplete, that is, not referable to a unifying essence.
Generally speaking, proper classes as defined by Von Neumann and Gödel, that is, classes that do not belong to metaclasses, are non-categorical and incomplete. Attempts to conceptualize these notions within higher metanotions all share the fundamental flaw that condemns individual human sciences to ideology or, worse still, to fetishism. A clear example of this is historical materialism which has fetishized the very notion of class. Seyla Benhabib sustains that many misunderstandings of multiculturalism derive from the essentialistic approach to culture. (For arguments against the essentialistic sociology of culture refer to Seyla Benhabib, Reclaiming cultural identity. Equality and diversity in the global era, 2002).
In the same way, we understand nothing about madness – Lacan would say that it has been precluded from the discourse – if we analyse it using essentialistic schematisms, whether psychopathologic, sociologic, or even psychoanalytical (see the preclusion of the Name of the Father). In his last work, Lacan himself acknowledges that the aetiological schematism he proposed does not work in Hölderlin’s madness. (Refer to L’Etourdit, “Scilicet”, 4, Seuil, Paris 1973, p. 22. See page La cura della scienza – The cure of science - for further information).
Essentialism does not explain the madness just as it does not explain the scientific procedure. Husserl’s eidetic science is not a Cartesian science but a revitalisation of Aristotle’s tode ti and of prime substances at the service of the master.
So, the theory of incompleteness marks the end of pre-scientific logocentrism, based on deterministic considerations surrounding essences and sufficient reason. (Causes and essences are two sides of the same structural determinism: essences are determined in theory as in practice causes determine effects).The psychoanalyst must understand that these forms of logocentrism, even although they have been superseded by Cartesian thought, reappear either in some contemporary philosophies, above all phenomenological, or in the forms of techno-sciences that make up the cognitivist galaxy, an interdisciplinary (buzz word!) mix of cybernetics, computer science and neurosciences. This resurgence of the old, dressed as rigour in philosophy, and/or as ultramodern in knowledge, bears witness to the widespread and undying nostalgia for the One which completely encompasses the whole without leaving clamorous gaps. In short, the psychoanalyst must realize that, even though religions are flourishing, throughout the world,


And he must also be aware that this epoch-making event of modernity, before being an empirical and documentable event, is an epistemic one, impressed in collective knowledge; that is, neither empirical nor rational but on the borderline – which in turn is not conceptually determinable - between reason and irrationality.
For a more in-depth study of the relationship between completeness and incompleteness see the page on Wittgenstein.
It is worthwhile mentioning a topos. How does contemporary philosophy deal with the question of non-categoricity and non-conceptualizibilty? By means of diverse theories of the metaphor. Every philosopher builds his own. They are the last effects of logocentrism at the moment in which it eclipses along with the categoricity of the concept. On the subject of the function of the metaphor I attach the final chapter of René Scheu’s doctoral dissertation on the weak subject, Das schwache Subjekt, published recently in Vienna by Turia und Kant. The Italian version is entitled La luce scura della metafora -The dark light of the metaphor.

A scientific approach has no need to take into consideration these theories of rhetoric, considering the lack of significance given to the logos. As regards the non-categorical scientific discourse, attempts have been made to create a “sciences of ignorance” field of study based on self-deception, centred around the Freudian institution of psychoanalysis. The metaphors of rhetoric are mere instruments in sustaining self and hetero deception. Do you need counter-evidence? When a metaphor appears in scientific debate – for example the model of the solar system for Bohr’s quantum atom – more confusion than clarity is created. Indeed, the metaphor has cognitive but not scientific value. It tries to illuminate the unknown with the glimmer of light of the known. The classical Aristotlean metaphor “old age is the evening of life” tries to explain something we know nothing about –– old age–  with something we know – evening. But science doesn’t have cognitive problems. When faced with the unknown it simply invents new models.

 A warning to all Lacanian surfers. On this site the words “non-categorical” and “incomplete”, are used in the place of Lacan’s improbable concept of “not everything”. See. Lacan intuizionista. As Alan Sokal points out in his Intellectual Impostures -  in the seminar Encore of 13 March 1973, Lacan makes a banal spelling mistake when he “defines” the female as “not everything”, using the overbar to indicate a negation. In the formula for female sexuation he overbars the universal “quantor” (foolish play on words between universal “quantifier” and “Kant”) instead of the entire formula (showing that the phenomonologist has little mastery of algebraic writing). It is on this spelling mistake that Lacan builds his theory of “not everything”, that Wolfgang Pauli wrote off as “not even wrong”. Das ist nicht einmal falsch!
In practice, Lacan’s theory of “not all” is not completely wrong, it is theological. It distinguishes two universes: male and female. The male universe is defined by the exception which ex-sists everything. The Oedipal formula of the male universe is: “Everyone is castrated except the father”. The father is the transcendent god who represents the essence of the immanent universe under him (A similar theory, based on the generality/exception theory was developed by Giorgio Agamben in his formulation of biopolitics staring from Soeren Kierkegaard and Carl Schmitt in the series Homo sacer.) On the contrary, the female universe is “defined” by the absence of the paternal exceptions and is consequently more “open”, in Heidegger’s (Rilke’s) meaning of the word. In practice, the female universe is what should be exorcised and distanced from culture because it is without father and therefore cannot be traced back to a pre-codified culture; it is potentially mad, without saying even savage. In the best case scenario, we can speak of the female as a source of paradoxes . (My essay Contro i paradossi, per le topologieAgainst paradoxes, for topologies - may be useful here).

To be indulgent, we could see in Lacan’s lucubration a faint parallelism with the scientific conception. The male universe would be a set, which belongs to the set of its sub-sets, while this does not belong to that. The set of sub-sets would be the exception to the initial set because its power is greater. (If the initial set possesses two elements, the set of subsets possesses two to the power of two, that is four elements). The female universe would be so all-encompassing as to include its own set of the parts which paradoxically, however, does not include it. I have developed these considerations in the afterword to my translation of Isterico sublime – Sublime Hysteric - by Slavoj Zizek, entitled A proposito del Tutto e dell’Eccezione  - On Everything and the exception - (Mimesis, Milan 2003, p. 205).

The main reason why I cannot accept the doctrine of the general founded on the exceptional is not only because it is a theology but because it prevents me from thinking non-categorical generalities. The exceptional represents the essence of the general in the sense of the cliché “the exception proves the rule”. But if this is the case then the general always has its own essence, that is, the exception. This means that the general is always categorical. To sum up, I cannot accept this doctrine because it prevents me from thinking non-categorical generalities. In short, it prevents me from beginning to think of the infinite... in all its incompleteness. So it cannot help me to theorize psychoanalysis where the infinite acts as the object of desire.

Further material can be found in my essay: L’“unfinito” ovvero l’uno, gli uno e l’infinito, “aut aut”, 283-284, 1998, pp. 81-106.

From an aesthetic point of view see as well

Una struttura, molti modelli (Eine Struktur, mehrere Modelle).

















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