Exhibition In Copenhagen

I have the chance to have three paintings on the walls of the Lighthouse cultural center in Copenhagen for few weeks. To visit, look at the calendar if the space is open, the paitings are available only when the space is open for activities.

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The paintings relate to my researches in narratology that can be found in favious articles of the blog. The main useful articles are:

Narrative Sculptures: Graph Theory, Topology and New Perspectives in Narratology.

and for french readers:

Narration et mathématiques: L’utilisation des graphes au cinéma et dans la bande dessinée (1, 2, 3, 4)

The four pieces presented there are the following:

Infinite Walls

Infinite Walls
Infinite Walls by Felix Lambert. Two Circles on a torus. Copyright: Felix Lambert, 2017

Handcuffs

Handcuffs
Handcuffs by Felix Lambert. Two circles on the plane. Copyright: Felix Lambert, 2017.

Lost in Days and Nights

Lost in Days and Nights
Lost in Days and Nights by Felix Lambert. Two circles on the torus. Copyright: Felix Lambert, 2017.

To complete stories represented by the paintings are presented by the side of the paintings at The Lighthouse. Visit the place, Pasteursvenj 8, Copenhagen, Denmark, to read the stories.

To visit the rest of my work, please visit my website.

http://metonym.io/

Felix Lambert

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Arrival: On the other side of a narrative language

Today is a great day for narratology. It is so for the simple reason that we agreed to make things more complicated, beautifully so. It doesn’t matter how many Oscars end up in the hands of Denis Villeneuve for his sci-fi movie Arrival, what matter is its inescapable presence.

Storytelling, in all forms, has always been a way to shape our minds. Stories need to be entertaining in order to stand out from all the available ways to occupy ourselves, but also need to be challenging so we can step out of our habits, and learn to think something new.

Many movies have been able to satisfy both sides of the balance. Some by the order the story is been told, as for movies from Pulp Fiction (Tarantino, 1994) to Memento (Nolan, 2000) and some reached similar effect by the inner story structure like Primer (Carruth, 2004), Looper (Johnson, 2012) or Triangle (Smith, 2009).

What makes Arrival particularly interesting is the prominent place given to a language itself, a language that allows for more intricate patterns and storytelling process.

arrival-1-476x250

Indeed, being presented on the other side of the mirror, within the diegetic world itself, this language permits time traveling by itself, understanding both future and past events at the same time and acting coherently with all of them.

Sadly, there is no proof such a language exist in our world. What we do have though is languages that help us understanding stories as groups of logical interactions of groups of events, just like in the classic publicity against drug abuse where one works more, to make more money, to make more drugs, to work more.

Of such languages are mathematics and the way we can use them to represent and understand stories, our own stories, and our own patterns. Such simple examples can easily be drawn and analyzed with graph theory and topology.

A conclusion we can drawn from Arrival, both in its content and its worldwide popularity, is that we might need to accept the fact that we need to learn something new to be able to solve, as humans, the the various pressing worldwide problems that could lead to our extermination, or at least mass decimation.

The fact we all seemed to touch so many of us could simply be the fact that it addresses the conclusion we have all already made, somewhere in our self-regulating surviving minds: we need to find solutions, solutions to problems deep enough that it could involve restructuring the way we tell ourselves, as a species, our own story, through mass media, through education, through thinking.

The fact that Arrival is there tonight might be a very indirect way of admitting it. Movies that create intricate story structures are a strong first step, since they are also a language. Arrival stands clearly as its own pertinent example. It’s entertaining enough that masses want to watch it, and complex enough so that we need to make links ourselves, conclude ourselves, think a step further.

The real arrival that is needed is not the aliens’ one, it’s the arrival of new languages, new paradigms.

Félix Lambert.

Sharing Paradoxes: Impossible Spaces, Impossible Times and Impossible Facts. The Function of Self-Contradictory Structures in Arts, Sciences and Philosophy. (Part 4)

A first step in dealing with paradoxes is then to accept their double existence as being true and false. Outside academic studies, it is a natural process implied in the appreciation of any narrative art. As it has been described about cinema, there is a point where we accept the false to be true, as if trying to find a proof reducto ad absurdum. This process is called the suspension of disbelief (Walton, 7). Proposed first by the poet Samuel Taylor Coleridge, it has extended in the study of literature, cinema and videogames to name a few. This refers to any action where the false is taken as possibly true in order to appreciate a narration and facilitate the immersion process. Youssef Ishaghpour describes the duality implied as the reality of the image and the image of reality (Ishaghpour, 8-11). The suspension of disbelief is therefore a way to conciliate this duality in order to appreciate the fiction.

The difference when working in a fictional environment rather than in a scientific one is that every time a contradiction or unearthly events appears, it is usually taken as an extension of the perceived diegetic world: when the staircase optical illusion appears in Inception, it is accepted as part of the fictional world. Instead of defying physical rules, it is simply accepted as a new information contained within the fiction. Again, as in Kierkegaard or Bohr’s vision, it is an extension of the paradigm. The same happens for multiple worlds’ diegetic construction such as previously mentioned in movies like ExistenZ or Avalon. The multiple ontological world, very similar in their nature to the Russell’s paradox are accepted as such. The suspension of disbelief catalyses the conceptual acceptation of such constructions and even changes them into interesting and pleasing artistic visions.

Cat'.s Paradox

Figure 7: The cat’s Paradox. Source: Wikipedia

The use of paradoxical constructions taken from science does not end here. Another case comes from quantic physic. Erwin Schrödinger described the nature of some quantic events by the metaphor of a cat in a box. Let say there is a cat and some poison inside a box. There is as well a 50% chances for the poison to be relieved and therefore for the cat to die. The way quantic physics works is that as long as the information about the cat has not been extract from the situation, the cat is in both states: alive and dead. Both states excludes each other and therefore it leads to a paradox that of ‘’ the living and the dead cat mixed or smeared out in equal parts.’’ (Schrödinger) What modern physics proposes as a solution the acceptation of both state for a certain period of time. This process is known as quantic bifurcation. Even if this is very difficult to accept as being true for neophytes, when transferred to fiction it leads to acceptable and interesting narrative constructions. An example of a movie using this type of multi-linear time frame is Source Code by Duncan Jones. In this movie, a soldier is sent multiple times in the past to prevent a terrorist attack. After failing multiple times, he achieves his goal and life continue normally in this new independent timeline. The use of quantic bifurcation appears in multiple science-fiction movies and communities of fans are sketching schemas to understand the structure behind the film. Movies like Primer by Shane Carruth and Looper by Rian Johnson have generate numbers of complicated charts using quantic bifurcation in order to explain these narrations. (Figure 8)

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Figure 8: Looper movie chart by Rick Slusher. Source: Film.com

Another paradoxical time construction that has caused many problems is the time loop. Circular construction of time was accepted by many cultures around the world: Egyptians had circular time named Neheh (Assman, 137), the tzolkin, the Mayan calendar based on cycles of 260 days (Falcón, 19-21) and Hinduism has constructions of multiple intricate circles (Eliade, 134-136). This vision does not conciliate with the European linear construction of time, but it still easily apply to fiction.

Time being both in the future and the past appears in various fictional cases. First of all, whenever there is a time loop a cyclic time has to be accepted. Movies like Terminator (Cameron, 1984), Before the Rain (Manchevsky, 1994) or Chin Chin el Teporocho (Retes, 1976) all present this cyclic time construction. Indeed, time loops can be multiple and quantic bifurcation might again apply.

In the quantic bifurcation case as in the circular time construction, the paradoxical construction induces multivalued time states, discrete moment can be different but at a same time distance from a specific moment, the bifurcation point or those previous, or they can stand both in the future and in the past of a referential moment. It can also be seen as a specific case of multiple ontological states, as previously described, but with the specificity that the ontological state is defined by a time value.

It can be presented in a more mystic way as in the movie Voyage in Time by Tarkovsky and Tonino Guerra. In this case, the movie shows the directors talking about the film they will make about a trip they once had. The anecdotes supposedly in the past appears as well in front of the camera and therefore the time of the movie is triple, it stands for the past, the present and the future as in Three Sundays in a week, but without the logical explanation.

Perhaps the most well-known results about paradoxes is Gödel’s incompleteness theorem. After Cantor and Russell discoveries, logicians have tried to build a perfect and complete system for logic. The project happened to be more problematic than expected and new set of axioms surfaced. The outstanding result obtained by the mathematician Kurt Gödel changed radically the conception of logic and left the community in crisis. The incompleteness theorem states that no matter how many axioms we add to a logic system, there will always appear some statements that will be undecidable, meaning it will be impossible to prove them right or wrong (Nagel, 19-20). This is a perfect example of Paz’s perspective of grasping dual objects as such instead of considering them as problematic undefined concepts to reach a better understanding of it. In this case, the conclusion obtained by paradox is that paradoxes are inherent part of complex logical systems.

Paz’s consideration encompasses a big range of logical instance and, as seen previously, they apply to a wide variety of paradoxical objects: from optical illusions to narrative charts passing through quantum physics. It still does not hold for a type of undecidable statements. Some facts are not necessarily true or false; they stand somewhere in between as a result of incomplete definitions. They work as ambigrams but instead of offering mainly a finite amount of elements, they offer a continuous range of possible information. Such problems are common in everyday life since more situations are not clearly defined. For instance, we can pretend the sky is blue but it can’t be proven without adding precisions to the statement offered; at night the statement does not hold for instance. A relatively new branch of mathematics dedicates itself to such logical system. The idea behind this fuzzy logic, as it is coined, is to attribute truth values that varies continuously between the usual zero and one (Kandell). Therefore allowing any probability of truth ranging from zero to 100%. Such logical system coincides with perspective of quantum physics allowing diverse states with various probabilities. It is the case for instance for electrons in the atomic model were they navigate through a probabilistic area instead of following a precise trajectory.

Finally, paradoxes can appear within humoristic or philosophical functions. The twist are often used in usually called intellectual humour such as Woody Allen’s work. In Allen’s quote from Annie Hall ‘’ The food here is terrible and the portions are too small’’, the double statement stands in the contradiction that, in fact, if the food is terrible there is no reason to ask for more, but complaining about small portions implies asking for more food. This kind of construction can be found as well in Annie Hall: ‘’ Life is full of mystery, loneliness, and suffering –and it’s all over much too soon’’. The role of the paradox is then, in this case, to release a tension constructed around the paradoxical statement. In this situation, the contradiction, or double truth value, stands as a sign that the joke has reached its climax. The contradictory aspect of the logic involves is to be read as a sign to character does not make sense anymore, therefore the humoristic relief. The humour can follow as a comment on a paradox: way before Russell, Lewis Carroll underlines that no one can contain himself because of excitement because nothing can contain itself (Benayoun, 84). These considerations follow the seriousness of Ambrose Bierce’s definition of logic as the art of thinking within human capacities (Benayoun, 113), the presence of this limit is in itself both humoristic and a relief.

Paradox Humour

Figure 9: Paradox humour

In a broader perspective, the same applies to koans, small stories or statement present in the zen tradition. The sentences serve to increase doubt and questioning. The simple logic behind the koan ‘’What is the sound of one hand clapping’’ is similar; it states the possibility and impossibility of the referred sound. Possible since it is stated there is a clap sound and the impossibility by the uniqueness of the hand producing it. In this case, the paradoxical information serves again to release tension. The same holds for the koan: If you have a stick, I’ll give one to you, if you don’t I’ll steal it from you. The tension is released with acceptation to work outside a strict logical frame, to accept our humanity as proposed by Bierce.

This work outside logic may serve as well, paradoxically, for theological arguments. To understand we have to go back to the unliftable rock paradox. God, being almighty, should be able to create an unliftable rock, but then if he can lift the rock he is not almighty. An easy solution to this problem is to state that God’s work beyond human based logic.

As we have shown, the contradictory dialectic raising from paradoxes only cause problem within its own paradigm of binary logical values as being true or false. When grasped as specific concepts gathering both values, or, even infinitely many values ranging from absolutely true to absolutely false, many applications can be found. Accepting such condition standing in between these poles is what allows us to enjoy narratives in different ways; first to enhance the emotive effect of a diegetic world by accepting the ontological quality of fiction as being an image of reality that is itself included in and presented as a simulacra of reality, secondly as to define science fiction or fantastic narratives as legitimate by extending the accepted diegetic world. Logical statements sharing both truth and false value are integral parts of human scientific and cultural knowledge[1]. It is hoped that more research to consider paradoxes in their social appearances will be provided as to understand better their functions as a fundamental part of human thinking rather than solely as odd mythological thoughts gravitating in the abstract spheres of philosophy and logic.

Felix Lambert

First version September 2015

[1] As it is the case in Dialetheism. For a good review of this as a philosophe, the reader is invited to read the entry in the Stanford Encyclopedia of Philosophy by Francesco Berto.

 

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Sharing Paradoxes: Impossible Spaces, Impossible Times and Impossible Facts. The Function of Self-Contradictory Structures in Arts, Sciences and Philosophy. (Part 1)

By Félix Lambert

Relativity-escher

Relativity by M.C. Escher

Logical twists and games have always seemed to intrigued thinkers from all times and civilizations. As being mostly curiosities, they appeared in an unorganised fashion in many disciplines, arts and games. One such logical twists is the paradox. From Antiquity’s philosophy to modern mathematics, paradoxes have brought various questions and, in some cases, answers about human knowledge. The main goal of this paper is to demonstrate how paradoxes have proven useful in various cases in sciences and narrative arts, therefore justifying them as proper object of study instead of being considered simply as odd singularities. Different definitions of paradoxes will first be discussed and a series of paradoxes will be presented. We will then use Octavio Paz’s discussion over dual concepts to approach paradoxes. Finally, we will come back to some examples where a dualistic study of some paradoxical structures has been useful. This will show how paradoxes now constitute a significant part of contemporary knowledge, art and to a certain extent, mythology.

The first common point found in various definitions of paradoxes is the self-contradictory aspect of its claim. One of the most common paradoxes that clearly exemplifies this fact is the liar’s paradox. It seems to first have been proposed by Epimenide when saying that all Cretans lie. The problem appears when we realise Epimenide is himself native from Crete and therefore two options are possible: first, if he lies then his statement is right and therefore we cannot trust his saying, secondly if he tells the truth we because he is Cretan then he tells a lie. A more condensed version of this paradox was expressed by Eubulibe of Millet (Vidal-Rosset, 31): ‘’I’m lying’’ of which the literary equivalent is ‘’This sentence is false’’. The common point of all these statements is that they all appear as contradicting themselves, leaving us incapable of deciding the rightness of their claim.

Many authors have proposed classifications for paradoxes. One interesting proposition has been Willard Van Orman Quine’s tripartite classification. Quines distinguishes falsifical paradoxes, veridical paradoxes and antinomy (Vidal-Rosset, 105). The first category includes paradoxes that are finally proved to be false. Such an example is given during the Renaissance by Guido Ubaldus finding that 0=1, which was interpreted at the time by implying that matters can be creating out of nothing. The claim seems to contradict itself by giving two different values to an integer, one being equal to zero and of course to itself. The proof uses infinitely many addition of ones and zeros by reorganising them in such a way to obtain the result (Stewart, 578). The paradox is false because it does not use the allowed operations between infinite series (Labelle, 262-264). Another common example is 1=2 obtained by a division by zero, which is of course prohibited. The second type of paradoxes, the veridical ones, contains paradoxes that seem false but end up being true. The most common example is the Monty Hall paradox. A player is offered three choices of doors behind which one of them a price is hidden. The player picks a door. After the choice, one of the two remaining doors is opened and shows no price. The player is then asked to choose again. Although it is commonly believed that the chances on the last pick are even, it is in fact false: it seems that there is a 50% chance of winning when there is actually a higher chance to win if the player switches their choice. We can compute all possible options and the results shows the player stands better chances, in fact 2 out of 3, if he doesn’t change his mind. This paradox is a veridical paradox because we can prove it to be true. (Figure 1)

Monty Hall Paradox

Figure 1: The Monty Hall paradox

Another veridical paradox has been used by Edgar Allan Poe in one of his short story “Three Sundays in a Week’’. In this story, two young lovers want to get married but the uncle in charge decides it will only happen when a week will have three Sundays. A year later on a Sunday afternoon, the couple meet the uncle with two captains that traveled the globe in opposite directions at such speed that they respectively lost and gained a day, therefore thinking that Sunday was the day before or the day after and fulfilling the uncle’s condition (Poe, 225-232). Yet again, the statement seems contradictory but an analysis on the matter shows in fact in it is a veridical paradox; Poe constructs the story around the fact that the referential for the day was not specified.

Finally, there are paradoxes that can be both, true or false. Grelling’s paradox falls into this category: let us divide all adjectives in two sets, the autological and heterological ones. The first ones are those that describes themselves, for instance short is a short word. Heterological are those that does not describes themselves: long is a short word. The paradox arises when we try to classify the adjective heterological: if its autological then it describes itself, it is then heterological which contradicts the statement. On the other hand, if it is heterological, then it cannot be in its own category, it must then be autological, but we already showed that it can’t be (Vidal-Rosset, 26). We will see later that another paradox by Bertrand Russell is similar to this case.

Trying to classify paradoxes into one of these categories is the first way to turn paradoxes into useful abstract objects enlarging the scope of knowledge. The categorisation implies gathering enough knowledge and understanding of the paradigm in which the paradox is stated to be able to classify it into the proper box, but this procedure might redefined the paradigm or lead to new theories. Although, it is quite natural to take some assumption as true -as axiomatic- in part of building knowledge, but the interrogation about these keystones of knowledge really comes unavoidable when paradoxes are found. This thought is well express by many great thinkers in sciences and philosophy. As Niels Bohr said when working on theoretical model for the atom ‘’How wonderful that we have met with a paradox. Now we have some hope of making progress.’’ (Moore, 196). The idea of meditating on the meaning of paradoxes have even brought to an almost mystical perspective in Kierkegaard’s’ writings: ‘’ The paradox’s really the pathos of intellectual life and just as only great souls are exposed to passions it is only the great thinker who is exposed to what I call paradoxes which are nothing else than grandiose thoughts in embryo’’ (Slaatté, 64). In these cases, paradoxes technically mean self-contradiction, but it is seen as well as a possible extension of knowledge, and wider possible range for the concerned paradigm. An interesting and humoristic quote sometimes attributed to Mark Twain expresses this state of mind about knowledge: ‘’All generalisations are false, even this one’’.