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Erotetics
tags:eroteticslogiccschallenging
Transcendental syntax
Transcendental Syntax is the name of a proposal which means to re-think the fundamental aspects of formal logic, of syntax/semantics. According to Girard, linear logic and Geometry of interaction are but exercises in transcendent syntax.
While Girard’s prose is notoriously demanding, exegesis may be found in Abrusci-Pistone 12 for the philosophical side and Rouleau 13 which was written before modern developments of transcendental syntax by means of stars and constellations (Girard 13b)
the use (usage): it corresponds to Ludwig Wittgenstein meaning-as-use. The computational behavior of epistates and their potential interaction with other ones induces a classification into dichologies.
the factory (usine): we want our computational objects to pass a specific set of tests in order to accept them in the corresponding type.
This is related to the distinction between existentialism and essentialism in philosophy. The logical certainty corresponds to the adequacy between usine and usage: we would like the usine to be finite and sufficient to ensure what we consider a right use of epistates. According to Girard Girard 16, the separation between constat and performance comes from undecidability (as in the halting problem) because the potential of programs cannot always be reduced to their result (which is not always defined). As for the separation between usine and usage, it would come from Gödel’s incompleteness theorem which intuitively means that the possible uses of a logical object may go beyond what we expect from our definition (our formatting of the concept of logic).
Introduction
Modern logic remains basically impregnated with the 1900 spirit, this sort of pretension at simplifying everything since one can solve all problems.
When after 1930 incompleteness shook this haughtiness, one hardly observed more than a complexification of discourse: instead of explaining from simpler, one started to explain from the meta. There began a time of counterfeit coinage. Since that time, logic, unable to effect its own reformation, severed its links with mathematics, physics etc.
This reminds me of my daughter Isabelle - then very young — “why not call the door “spoon” and the spoon “door” to which I answered, When one says “make for the door” it should not be taken as an invitation to supper.” Among the magisterial mistakes of logic, one will first mention quantum logic, whose ridiculousness can only be ascribed to a feeling of superiority of the language - and ideas, even bad, as soon as they take a written form - over the physical world.
At the beginning of the last century, Einstein’s relativity and, in a more radical way, quantum physics, called in question our “fundamental intuitions.” Logic, because of its excesses decided to catapult itself into emptiness; the non-structure, the non-significant “everything can be coded in everything” and also into the sea of idea of translating images into sound, or rather gurgling.
The domain, as it is ossified during the XXth century is indeed everything but crazy: a cemetary of ideas. In other words, the only excuse in the XXIth century for indulging in “foundations” is a “grain de folie” i.e. a slight madness.
Existence vs essence
Essentialists - those who think that everything is already there, that one can but repeat archetypes; they believe in flying saucers, especially above Aztec pyramids. In logic, they believe in inverted foundations: a system can be explained by a deeper meta system; which in turn can be explained by a meta-meta system… and this never ends. What could be taken as a faulty construction appears as the fascination with the irreducible.
Existentialists - this term would rather qualify those who find the ideas of Kubrick infantile and do not believe in civilizing saucers. Those who do not find “deep” the fact of defining truth as supposedly done by Tarski.
The opposition essence/existence is at the very heart of typing
Projects
- set theory - simplified contradiction by Russell, {set of sets not belong to themselves}. This was not terrible, since naive set theory was still an experimental system, and it only took a couple of years until Zermelo formulated restrictions essential those of ZF set-theoretic systems.
- hillbertian - mathematics is pure symbol pushing, with no more significance than the game of chess; all that matters is the formal consistency of the rule of the game.
- brouwer
Among slightly decaying ideas, let us mention (for the sake of good laugh) iterations of theories. Starting with a system of arithmetic, say one founds it over it a meta-system . Since it doesn’t prove it’s own meta-consistency we end up with matrioshka-turtles each one sustaining the previous one. Transfinite progressions of meta-theories were eventually produced.
Observe that a bad idea (matrioshka-turtles) does not improve through transfinite iteration: it simply becomes a bad transfinite idea.
My hypothesis is the absolute, complete, inadequacy of classical logic and - from the foundational viewpoint - classic mathematics. My hypothesis is that classical, logic, classical truth, are only self-justifying essentialist illusions.
Perfection corresponds to unique, well-defined actions, while imperfection is the mode of repetition.
Incompleteness
On the theorem:
- the result, like Monet painting is easy to perceive from a distance. A close look reveals only details one perhaps does not want to know.
- neither is there a need to know, since it is a scientific cul-de-sac, exposing a way in but no way out.
Incompleteness is not a deficiency. It is not a disease that can be healed and it is ridiculous to look for the missing piece of the puzzle.
Remember consistency is the fact that we cannot prove the absurdity ⊥ but this relies on the fact that ⊥ implies no matter what the formula. A system is consistent when there is something it cannot prove.
Technically: completeness of predicate calculus is stated “a closed formula B true in all models is provable”
Digression - Artificial Intelligence
If one means the possibility of mechanizing the activity of certain zones of the brain.. of course yes. But should one call this intelligence or rather instinct. Artificial instinct is unproblematic. If one seeks real intelligence or creativity, this is more complex and doomed to failure.
What constitutes intelligence in the creative sense, is that ones does not expect it, it places itself in a position of unrest: (it affords itself) the right to error. Intelligence follows in deviant ways, unexpected, including those of prejudice , ambition, wrath, the seven deadly sins and worse. The remains tolerable in society, individual power being limited, this internal whirlpool only has weakly dramatic effects. When a smart individual attains supreme power, it is unmistakably the intent that the world will benefit from the wanderings of individual’s thought.
… the most genuine ambition dresses in the most modest clothes.